From 5623cfee42b971e4156f85344f52a2ccf54c8f15 Mon Sep 17 00:00:00 2001
From: "Documenter.jl" <documenter@juliadocs.github.io>
Date: Mon, 25 Nov 2024 10:52:04 +0000
Subject: [PATCH] build based on f2c8823

---
 docs/.documenter-siteinfo.json              |    1 -
 docs/assets/documenter.js                   | 1012 +--
 docs/assets/search.js                       |  251 +
 docs/assets/themes/catppuccin-frappe.css    |    1 -
 docs/assets/themes/catppuccin-latte.css     |    1 -
 docs/assets/themes/catppuccin-macchiato.css |    1 -
 docs/assets/themes/catppuccin-mocha.css     |    1 -
 docs/assets/themes/documenter-dark.css      | 7647 ++++++++++++++++++-
 docs/assets/themes/documenter-light.css     | 7639 +++++++++++++++++-
 docs/assets/themeswap.js                    |   82 +-
 docs/assets/warner.js                       |   52 -
 docs/convergence/index.html                 |    2 +
 docs/dev/index.html                         |    2 +-
 docs/examples/anderson-model/index.html     |   42 +-
 docs/examples/bath-observables/index.html   |   26 +-
 docs/examples/protontransfer/index.html     |  235 -
 docs/examples/puredephasing/index.html      |   20 +-
 docs/examples/sbm/index.html                |   16 +-
 docs/examples/timedep/index.html            |   16 +-
 docs/index.html                             |   20 +-
 docs/methods/index.html                     |   75 +-
 docs/nutshell/index.html                    |    2 +-
 docs/objects.inv                            |  Bin 3180 -> 0 bytes
 docs/search/index.html                      |    2 +
 docs/search_index.js                        |    2 +-
 docs/theory/index.html                      |   10 +-
 docs/user-guide/index.html                  |   16 +-
 versions.js                                 |    2 -
 28 files changed, 15789 insertions(+), 1387 deletions(-)
 delete mode 100644 docs/.documenter-siteinfo.json
 create mode 100644 docs/assets/search.js
 delete mode 100644 docs/assets/themes/catppuccin-frappe.css
 delete mode 100644 docs/assets/themes/catppuccin-latte.css
 delete mode 100644 docs/assets/themes/catppuccin-macchiato.css
 delete mode 100644 docs/assets/themes/catppuccin-mocha.css
 delete mode 100644 docs/assets/warner.js
 create mode 100644 docs/convergence/index.html
 delete mode 100644 docs/examples/protontransfer/index.html
 delete mode 100644 docs/objects.inv
 create mode 100644 docs/search/index.html

diff --git a/docs/.documenter-siteinfo.json b/docs/.documenter-siteinfo.json
deleted file mode 100644
index 1a06912..0000000
--- a/docs/.documenter-siteinfo.json
+++ /dev/null
@@ -1 +0,0 @@
-{"documenter":{"julia_version":"1.7.3","generation_timestamp":"2024-11-22T16:28:17","documenter_version":"1.8.0"}}
\ No newline at end of file
diff --git a/docs/assets/documenter.js b/docs/assets/documenter.js
index 7d68cd8..15dc682 100644
--- a/docs/assets/documenter.js
+++ b/docs/assets/documenter.js
@@ -1,15 +1,15 @@
 // Generated by Documenter.jl
 requirejs.config({
   paths: {
-    'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min',
-    'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min',
-    'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min',
-    'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min',
-    'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min',
-    'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min',
-    'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min',
-    'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min',
-    'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min',
+    'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.15.10/languages/julia.min',
+    'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.10.3/headroom.min',
+    'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min',
+    'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/contrib/auto-render.min',
+    'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min',
+    'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.10.3/jQuery.headroom.min',
+    'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min',
+    'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.15.10/highlight.min',
+    'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.15.10/languages/julia-repl.min',
   },
   shim: {
   "highlight-julia": {
@@ -65,155 +65,11 @@ $(document).ready(function() {
 
 })
 ////////////////////////////////////////////////////////////////////////////////
-require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) {
+require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($, hljs) {
 $(document).ready(function() {
-    hljs.highlightAll();
+    hljs.initHighlighting();
 })
 
-})
-////////////////////////////////////////////////////////////////////////////////
-require(['jquery'], function($) {
-
-let timer = 0;
-var isExpanded = true;
-
-$(document).on(
-  "click",
-  ".docstring .docstring-article-toggle-button",
-  function () {
-    let articleToggleTitle = "Expand docstring";
-    const parent = $(this).parent();
-
-    debounce(() => {
-      if (parent.siblings("section").is(":visible")) {
-        parent
-          .find("a.docstring-article-toggle-button")
-          .removeClass("fa-chevron-down")
-          .addClass("fa-chevron-right");
-      } else {
-        parent
-          .find("a.docstring-article-toggle-button")
-          .removeClass("fa-chevron-right")
-          .addClass("fa-chevron-down");
-
-        articleToggleTitle = "Collapse docstring";
-      }
-
-      parent
-        .children(".docstring-article-toggle-button")
-        .prop("title", articleToggleTitle);
-      parent.siblings("section").slideToggle();
-    });
-  }
-);
-
-$(document).on("click", ".docs-article-toggle-button", function (event) {
-  let articleToggleTitle = "Expand docstring";
-  let navArticleToggleTitle = "Expand all docstrings";
-  let animationSpeed = event.noToggleAnimation ? 0 : 400;
-
-  debounce(() => {
-    if (isExpanded) {
-      $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down");
-      $("a.docstring-article-toggle-button")
-        .removeClass("fa-chevron-down")
-        .addClass("fa-chevron-right");
-
-      isExpanded = false;
-
-      $(".docstring section").slideUp(animationSpeed);
-    } else {
-      $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up");
-      $("a.docstring-article-toggle-button")
-        .removeClass("fa-chevron-right")
-        .addClass("fa-chevron-down");
-
-      isExpanded = true;
-      articleToggleTitle = "Collapse docstring";
-      navArticleToggleTitle = "Collapse all docstrings";
-
-      $(".docstring section").slideDown(animationSpeed);
-    }
-
-    $(this).prop("title", navArticleToggleTitle);
-    $(".docstring-article-toggle-button").prop("title", articleToggleTitle);
-  });
-});
-
-function debounce(callback, timeout = 300) {
-  if (Date.now() - timer > timeout) {
-    callback();
-  }
-
-  clearTimeout(timer);
-
-  timer = Date.now();
-}
-
-})
-////////////////////////////////////////////////////////////////////////////////
-require([], function() {
-function addCopyButtonCallbacks() {
-  for (const el of document.getElementsByTagName("pre")) {
-    const button = document.createElement("button");
-    button.classList.add("copy-button", "fa-solid", "fa-copy");
-    button.setAttribute("aria-label", "Copy this code block");
-    button.setAttribute("title", "Copy");
-
-    el.appendChild(button);
-
-    const success = function () {
-      button.classList.add("success", "fa-check");
-      button.classList.remove("fa-copy");
-    };
-
-    const failure = function () {
-      button.classList.add("error", "fa-xmark");
-      button.classList.remove("fa-copy");
-    };
-
-    button.addEventListener("click", function () {
-      copyToClipboard(el.innerText).then(success, failure);
-
-      setTimeout(function () {
-        button.classList.add("fa-copy");
-        button.classList.remove("success", "fa-check", "fa-xmark");
-      }, 5000);
-    });
-  }
-}
-
-function copyToClipboard(text) {
-  // clipboard API is only available in secure contexts
-  if (window.navigator && window.navigator.clipboard) {
-    return window.navigator.clipboard.writeText(text);
-  } else {
-    return new Promise(function (resolve, reject) {
-      try {
-        const el = document.createElement("textarea");
-        el.textContent = text;
-        el.style.position = "fixed";
-        el.style.opacity = 0;
-        document.body.appendChild(el);
-        el.select();
-        document.execCommand("copy");
-
-        resolve();
-      } catch (err) {
-        reject(err);
-      } finally {
-        document.body.removeChild(el);
-      }
-    });
-  }
-}
-
-if (document.readyState === "loading") {
-  document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks);
-} else {
-  addCopyButtonCallbacks();
-}
-
 })
 ////////////////////////////////////////////////////////////////////////////////
 require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) {
@@ -221,602 +77,29 @@ require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) {
 // Manages the top navigation bar (hides it when the user starts scrolling down on the
 // mobile).
 window.Headroom = Headroom; // work around buggy module loading?
-$(document).ready(function () {
-  $("#documenter .docs-navbar").headroom({
-    tolerance: { up: 10, down: 10 },
+$(document).ready(function() {
+  $('#documenter .docs-navbar').headroom({
+    "tolerance": {"up": 10, "down": 10},
   });
-});
-
 })
-////////////////////////////////////////////////////////////////////////////////
-require(['jquery'], function($) {
-
-$(document).ready(function () {
-  let meta = $("div[data-docstringscollapsed]").data();
-
-  if (meta?.docstringscollapsed) {
-    $("#documenter-article-toggle-button").trigger({
-      type: "click",
-      noToggleAnimation: true,
-    });
-  }
-});
-
-})
-////////////////////////////////////////////////////////////////////////////////
-require(['jquery'], function($) {
-
-/*
-To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc
-
-PSEUDOCODE:
-
-Searching happens automatically as the user types or adjusts the selected filters.
-To preserve responsiveness, as much as possible of the slow parts of the search are done
-in a web worker. Searching and result generation are done in the worker, and filtering and
-DOM updates are done in the main thread. The filters are in the main thread as they should
-be very quick to apply. This lets filters be changed without re-searching with minisearch
-(which is possible even if filtering is on the worker thread) and also lets filters be
-changed _while_ the worker is searching and without message passing (neither of which are
-possible if filtering is on the worker thread)
-
-SEARCH WORKER:
-
-Import minisearch
-
-Build index
-
-On message from main thread
-  run search
-  find the first 200 unique results from each category, and compute their divs for display
-    note that this is necessary and sufficient information for the main thread to find the
-    first 200 unique results from any given filter set
-  post results to main thread
-
-MAIN:
-
-Launch worker
-
-Declare nonconstant globals (worker_is_running,  last_search_text, unfiltered_results)
-
-On text update
-  if worker is not running, launch_search()
-
-launch_search
-  set worker_is_running to true, set last_search_text to the search text
-  post the search query to worker
-
-on message from worker
-  if last_search_text is not the same as the text in the search field,
-    the latest search result is not reflective of the latest search query, so update again
-    launch_search()
-  otherwise
-    set worker_is_running to false
-
-  regardless, display the new search results to the user
-  save the unfiltered_results as a global
-  update_search()
-
-on filter click
-  adjust the filter selection
-  update_search()
-
-update_search
-  apply search filters by looping through the unfiltered_results and finding the first 200
-    unique results that match the filters
-
-  Update the DOM
-*/
-
-/////// SEARCH WORKER ///////
-
-function worker_function(documenterSearchIndex, documenterBaseURL, filters) {
-  importScripts(
-    "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js"
-  );
-
-  let data = documenterSearchIndex.map((x, key) => {
-    x["id"] = key; // minisearch requires a unique for each object
-    return x;
-  });
-
-  // list below is the lunr 2.1.3 list minus the intersect with names(Base)
-  // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with)
-  // ideally we'd just filter the original list but it's not available as a variable
-  const stopWords = new Set([
-    "a",
-    "able",
-    "about",
-    "across",
-    "after",
-    "almost",
-    "also",
-    "am",
-    "among",
-    "an",
-    "and",
-    "are",
-    "as",
-    "at",
-    "be",
-    "because",
-    "been",
-    "but",
-    "by",
-    "can",
-    "cannot",
-    "could",
-    "dear",
-    "did",
-    "does",
-    "either",
-    "ever",
-    "every",
-    "from",
-    "got",
-    "had",
-    "has",
-    "have",
-    "he",
-    "her",
-    "hers",
-    "him",
-    "his",
-    "how",
-    "however",
-    "i",
-    "if",
-    "into",
-    "it",
-    "its",
-    "just",
-    "least",
-    "like",
-    "likely",
-    "may",
-    "me",
-    "might",
-    "most",
-    "must",
-    "my",
-    "neither",
-    "no",
-    "nor",
-    "not",
-    "of",
-    "off",
-    "often",
-    "on",
-    "or",
-    "other",
-    "our",
-    "own",
-    "rather",
-    "said",
-    "say",
-    "says",
-    "she",
-    "should",
-    "since",
-    "so",
-    "some",
-    "than",
-    "that",
-    "the",
-    "their",
-    "them",
-    "then",
-    "there",
-    "these",
-    "they",
-    "this",
-    "tis",
-    "to",
-    "too",
-    "twas",
-    "us",
-    "wants",
-    "was",
-    "we",
-    "were",
-    "what",
-    "when",
-    "who",
-    "whom",
-    "why",
-    "will",
-    "would",
-    "yet",
-    "you",
-    "your",
-  ]);
-
-  let index = new MiniSearch({
-    fields: ["title", "text"], // fields to index for full-text search
-    storeFields: ["location", "title", "text", "category", "page"], // fields to return with results
-    processTerm: (term) => {
-      let word = stopWords.has(term) ? null : term;
-      if (word) {
-        // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names
-        word = word
-          .replace(/^[^a-zA-Z0-9@!]+/, "")
-          .replace(/[^a-zA-Z0-9@!]+$/, "");
-
-        word = word.toLowerCase();
-      }
-
-      return word ?? null;
-    },
-    // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not
-    // find anything if searching for "add!", only for the entire qualification
-    tokenize: (string) => string.split(/[\s\-\.]+/),
-    // options which will be applied during the search
-    searchOptions: {
-      prefix: true,
-      boost: { title: 100 },
-      fuzzy: 2,
-    },
-  });
-
-  index.addAll(data);
-
-  /**
-   *  Used to map characters to HTML entities.
-   * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts
-   */
-  const htmlEscapes = {
-    "&": "&amp;",
-    "<": "&lt;",
-    ">": "&gt;",
-    '"': "&quot;",
-    "'": "&#39;",
-  };
-
-  /**
-   * Used to match HTML entities and HTML characters.
-   * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts
-   */
-  const reUnescapedHtml = /[&<>"']/g;
-  const reHasUnescapedHtml = RegExp(reUnescapedHtml.source);
-
-  /**
-   * Escape function from lodash
-   * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts
-   */
-  function escape(string) {
-    return string && reHasUnescapedHtml.test(string)
-      ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr])
-      : string || "";
-  }
-
-  /**
-   * RegX escape function from MDN
-   * Refer: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping
-   */
-  function escapeRegExp(string) {
-    return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string
-  }
-
-  /**
-   * Make the result component given a minisearch result data object and the value
-   * of the search input as queryString. To view the result object structure, refer:
-   * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult
-   *
-   * @param {object} result
-   * @param {string} querystring
-   * @returns string
-   */
-  function make_search_result(result, querystring) {
-    let search_divider = `<div class="search-divider w-100"></div>`;
-    let display_link =
-      result.location.slice(Math.max(0), Math.min(50, result.location.length)) +
-      (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div
-
-    if (result.page !== "") {
-      display_link += ` (${result.page})`;
-    }
-    searchstring = escapeRegExp(querystring);
-    let textindex = new RegExp(`${searchstring}`, "i").exec(result.text);
-    let text =
-      textindex !== null
-        ? result.text.slice(
-            Math.max(textindex.index - 100, 0),
-            Math.min(
-              textindex.index + querystring.length + 100,
-              result.text.length
-            )
-          )
-        : ""; // cut-off text before and after from the match
-
-    text = text.length ? escape(text) : "";
-
-    let display_result = text.length
-      ? "..." +
-        text.replace(
-          new RegExp(`${escape(searchstring)}`, "i"), // For first occurrence
-          '<span class="search-result-highlight py-1">$&</span>'
-        ) +
-        "..."
-      : ""; // highlights the match
-
-    let in_code = false;
-    if (!["page", "section"].includes(result.category.toLowerCase())) {
-      in_code = true;
-    }
-
-    // We encode the full url to escape some special characters which can lead to broken links
-    let result_div = `
-        <a href="${encodeURI(
-          documenterBaseURL + "/" + result.location
-        )}" class="search-result-link w-100 is-flex is-flex-direction-column gap-2 px-4 py-2">
-          <div class="w-100 is-flex is-flex-wrap-wrap is-justify-content-space-between is-align-items-flex-start">
-            <div class="search-result-title has-text-weight-bold ${
-              in_code ? "search-result-code-title" : ""
-            }">${escape(result.title)}</div>
-            <div class="property-search-result-badge">${result.category}</div>
-          </div>
-          <p>
-            ${display_result}
-          </p>
-          <div
-            class="has-text-left"
-            style="font-size: smaller;"
-            title="${result.location}"
-          >
-            <i class="fas fa-link"></i> ${display_link}
-          </div>
-        </a>
-        ${search_divider}
-      `;
-
-    return result_div;
-  }
-
-  self.onmessage = function (e) {
-    let query = e.data;
-    let results = index.search(query, {
-      filter: (result) => {
-        // Only return relevant results
-        return result.score >= 1;
-      },
-      combineWith: "AND",
-    });
-
-    // Pre-filter to deduplicate and limit to 200 per category to the extent
-    // possible without knowing what the filters are.
-    let filtered_results = [];
-    let counts = {};
-    for (let filter of filters) {
-      counts[filter] = 0;
-    }
-    let present = {};
-
-    for (let result of results) {
-      cat = result.category;
-      cnt = counts[cat];
-      if (cnt < 200) {
-        id = cat + "---" + result.location;
-        if (present[id]) {
-          continue;
-        }
-        present[id] = true;
-        filtered_results.push({
-          location: result.location,
-          category: cat,
-          div: make_search_result(result, query),
-        });
-      }
-    }
-
-    postMessage(filtered_results);
-  };
-}
-
-/////// SEARCH MAIN ///////
-
-function runSearchMainCode() {
-  // `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript!
-  const filters = [
-    ...new Set(documenterSearchIndex["docs"].map((x) => x.category)),
-  ];
-  const worker_str =
-    "(" +
-    worker_function.toString() +
-    ")(" +
-    JSON.stringify(documenterSearchIndex["docs"]) +
-    "," +
-    JSON.stringify(documenterBaseURL) +
-    "," +
-    JSON.stringify(filters) +
-    ")";
-  const worker_blob = new Blob([worker_str], { type: "text/javascript" });
-  const worker = new Worker(URL.createObjectURL(worker_blob));
-
-  // Whether the worker is currently handling a search. This is a boolean
-  // as the worker only ever handles 1 or 0 searches at a time.
-  var worker_is_running = false;
-
-  // The last search text that was sent to the worker. This is used to determine
-  // if the worker should be launched again when it reports back results.
-  var last_search_text = "";
-
-  // The results of the last search. This, in combination with the state of the filters
-  // in the DOM, is used compute the results to display on calls to update_search.
-  var unfiltered_results = [];
-
-  // Which filter is currently selected
-  var selected_filter = "";
-
-  $(document).on("input", ".documenter-search-input", function (event) {
-    if (!worker_is_running) {
-      launch_search();
-    }
-  });
-
-  function launch_search() {
-    worker_is_running = true;
-    last_search_text = $(".documenter-search-input").val();
-    worker.postMessage(last_search_text);
-  }
-
-  worker.onmessage = function (e) {
-    if (last_search_text !== $(".documenter-search-input").val()) {
-      launch_search();
-    } else {
-      worker_is_running = false;
-    }
-
-    unfiltered_results = e.data;
-    update_search();
-  };
-
-  $(document).on("click", ".search-filter", function () {
-    if ($(this).hasClass("search-filter-selected")) {
-      selected_filter = "";
-    } else {
-      selected_filter = $(this).text().toLowerCase();
-    }
-
-    // This updates search results and toggles classes for UI:
-    update_search();
-  });
-
-  /**
-   * Make/Update the search component
-   */
-  function update_search() {
-    let querystring = $(".documenter-search-input").val();
-
-    if (querystring.trim()) {
-      if (selected_filter == "") {
-        results = unfiltered_results;
-      } else {
-        results = unfiltered_results.filter((result) => {
-          return selected_filter == result.category.toLowerCase();
-        });
-      }
-
-      let search_result_container = ``;
-      let modal_filters = make_modal_body_filters();
-      let search_divider = `<div class="search-divider w-100"></div>`;
-
-      if (results.length) {
-        let links = [];
-        let count = 0;
-        let search_results = "";
-
-        for (var i = 0, n = results.length; i < n && count < 200; ++i) {
-          let result = results[i];
-          if (result.location && !links.includes(result.location)) {
-            search_results += result.div;
-            count++;
-            links.push(result.location);
-          }
-        }
-
-        if (count == 1) {
-          count_str = "1 result";
-        } else if (count == 200) {
-          count_str = "200+ results";
-        } else {
-          count_str = count + " results";
-        }
-        let result_count = `<div class="is-size-6">${count_str}</div>`;
-
-        search_result_container = `
-              <div class="is-flex is-flex-direction-column gap-2 is-align-items-flex-start">
-                  ${modal_filters}
-                  ${search_divider}
-                  ${result_count}
-                  <div class="is-clipped w-100 is-flex is-flex-direction-column gap-2 is-align-items-flex-start has-text-justified mt-1">
-                    ${search_results}
-                  </div>
-              </div>
-          `;
-      } else {
-        search_result_container = `
-            <div class="is-flex is-flex-direction-column gap-2 is-align-items-flex-start">
-                ${modal_filters}
-                ${search_divider}
-                <div class="is-size-6">0 result(s)</div>
-              </div>
-              <div class="has-text-centered my-5 py-5">No result found!</div>
-        `;
-      }
-
-      if ($(".search-modal-card-body").hasClass("is-justify-content-center")) {
-        $(".search-modal-card-body").removeClass("is-justify-content-center");
-      }
-
-      $(".search-modal-card-body").html(search_result_container);
-    } else {
-      if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) {
-        $(".search-modal-card-body").addClass("is-justify-content-center");
-      }
-
-      $(".search-modal-card-body").html(`
-        <div class="has-text-centered my-5 py-5">Type something to get started!</div>
-      `);
-    }
-  }
-
-  /**
-   * Make the modal filter html
-   *
-   * @returns string
-   */
-  function make_modal_body_filters() {
-    let str = filters
-      .map((val) => {
-        if (selected_filter == val.toLowerCase()) {
-          return `<a href="javascript:;" class="search-filter search-filter-selected"><span>${val}</span></a>`;
-        } else {
-          return `<a href="javascript:;" class="search-filter"><span>${val}</span></a>`;
-        }
-      })
-      .join("");
-
-    return `
-          <div class="is-flex gap-2 is-flex-wrap-wrap is-justify-content-flex-start is-align-items-center search-filters">
-              <span class="is-size-6">Filters:</span>
-              ${str}
-          </div>`;
-  }
-}
-
-function waitUntilSearchIndexAvailable() {
-  // It is possible that the documenter.js script runs before the page
-  // has finished loading and documenterSearchIndex gets defined.
-  // So we need to wait until the search index actually loads before setting
-  // up all the search-related stuff.
-  if (typeof documenterSearchIndex !== "undefined") {
-    runSearchMainCode();
-  } else {
-    console.warn("Search Index not available, waiting");
-    setTimeout(waitUntilSearchIndexAvailable, 1000);
-  }
-}
-
-// The actual entry point to the search code
-waitUntilSearchIndexAvailable();
 
 })
 ////////////////////////////////////////////////////////////////////////////////
 require(['jquery'], function($) {
 
 // Modal settings dialog
-$(document).ready(function () {
-  var settings = $("#documenter-settings");
-  $("#documenter-settings-button").click(function () {
-    settings.toggleClass("is-active");
+$(document).ready(function() {
+  var settings = $('#documenter-settings');
+  $('#documenter-settings-button').click(function(){
+    settings.toggleClass('is-active');
   });
   // Close the dialog if X is clicked
-  $("#documenter-settings button.delete").click(function () {
-    settings.removeClass("is-active");
+  $('#documenter-settings button.delete').click(function(){
+    settings.removeClass('is-active');
   });
   // Close dialog if ESC is pressed
-  $(document).keyup(function (e) {
-    if (e.keyCode == 27) settings.removeClass("is-active");
+  $(document).keyup(function(e) {
+    if (e.keyCode == 27) settings.removeClass('is-active');
   });
 });
 
@@ -824,246 +107,145 @@ $(document).ready(function () {
 ////////////////////////////////////////////////////////////////////////////////
 require(['jquery'], function($) {
 
-$(document).ready(function () {
-  let search_modal_header = `
-    <header class="modal-card-head gap-2 is-align-items-center is-justify-content-space-between w-100 px-3">
-      <div class="field mb-0 w-100">
-        <p class="control has-icons-right">
-          <input class="input documenter-search-input" type="text" placeholder="Search" />
-          <span class="icon is-small is-right has-text-primary-dark">
-            <i class="fas fa-magnifying-glass"></i>
-          </span>
-        </p>
-      </div>
-      <div class="icon is-size-4 is-clickable close-search-modal">
-        <i class="fas fa-times"></i>
-      </div>
-    </header>
-  `;
-
-  let initial_search_body = `
-    <div class="has-text-centered my-5 py-5">Type something to get started!</div>
-  `;
-
-  let search_modal_footer = `
-    <footer class="modal-card-foot">
-      <span>
-        <kbd class="search-modal-key-hints">Ctrl</kbd> +
-        <kbd class="search-modal-key-hints">/</kbd> to search
-      </span>
-      <span class="ml-3"> <kbd class="search-modal-key-hints">esc</kbd> to close </span>
-    </footer>
-  `;
-
-  $(document.body).append(
-    `
-      <div class="modal" id="search-modal">
-        <div class="modal-background"></div>
-        <div class="modal-card search-min-width-50 search-min-height-100 is-justify-content-center">
-          ${search_modal_header}
-          <section class="modal-card-body is-flex is-flex-direction-column is-justify-content-center gap-4 search-modal-card-body">
-            ${initial_search_body}
-          </section>
-          ${search_modal_footer}
-        </div>
-      </div>
-    `
-  );
-
-  document.querySelector(".docs-search-query").addEventListener("click", () => {
-    openModal();
-  });
-
-  document
-    .querySelector(".close-search-modal")
-    .addEventListener("click", () => {
-      closeModal();
-    });
-
-  $(document).on("click", ".search-result-link", function () {
-    closeModal();
-  });
-
-  document.addEventListener("keydown", (event) => {
-    if ((event.ctrlKey || event.metaKey) && event.key === "/") {
-      openModal();
-    } else if (event.key === "Escape") {
-      closeModal();
-    }
-
-    return false;
-  });
-
-  // Functions to open and close a modal
-  function openModal() {
-    let searchModal = document.querySelector("#search-modal");
-
-    searchModal.classList.add("is-active");
-    document.querySelector(".documenter-search-input").focus();
-  }
-
-  function closeModal() {
-    let searchModal = document.querySelector("#search-modal");
-    let initial_search_body = `
-      <div class="has-text-centered my-5 py-5">Type something to get started!</div>
-    `;
-
-    searchModal.classList.remove("is-active");
-    document.querySelector(".documenter-search-input").blur();
-
-    if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) {
-      $(".search-modal-card-body").addClass("is-justify-content-center");
-    }
-
-    $(".documenter-search-input").val("");
-    $(".search-modal-card-body").html(initial_search_body);
-  }
-
-  document
-    .querySelector("#search-modal .modal-background")
-    .addEventListener("click", () => {
-      closeModal();
-    });
-});
-
-})
-////////////////////////////////////////////////////////////////////////////////
-require(['jquery'], function($) {
-
 // Manages the showing and hiding of the sidebar.
-$(document).ready(function () {
+$(document).ready(function() {
   var sidebar = $("#documenter > .docs-sidebar");
-  var sidebar_button = $("#documenter-sidebar-button");
-  sidebar_button.click(function (ev) {
+  var sidebar_button = $("#documenter-sidebar-button")
+  sidebar_button.click(function(ev) {
     ev.preventDefault();
-    sidebar.toggleClass("visible");
-    if (sidebar.hasClass("visible")) {
+    sidebar.toggleClass('visible');
+    if (sidebar.hasClass('visible')) {
       // Makes sure that the current menu item is visible in the sidebar.
       $("#documenter .docs-menu a.is-active").focus();
     }
   });
-  $("#documenter > .docs-main").bind("click", function (ev) {
+  $("#documenter > .docs-main").bind('click', function(ev) {
     if ($(ev.target).is(sidebar_button)) {
       return;
     }
-    if (sidebar.hasClass("visible")) {
-      sidebar.removeClass("visible");
+    if (sidebar.hasClass('visible')) {
+      sidebar.removeClass('visible');
     }
   });
-});
+})
 
 // Resizes the package name / sitename in the sidebar if it is too wide.
 // Inspired by: https://github.com/davatron5000/FitText.js
-$(document).ready(function () {
+$(document).ready(function() {
   e = $("#documenter .docs-autofit");
   function resize() {
-    var L = parseInt(e.css("max-width"), 10);
+    var L = parseInt(e.css('max-width'), 10);
     var L0 = e.width();
-    if (L0 > L) {
-      var h0 = parseInt(e.css("font-size"), 10);
-      e.css("font-size", (L * h0) / L0);
+    if(L0 > L) {
+      var h0 = parseInt(e.css('font-size'), 10);
+      e.css('font-size', L * h0 / L0);
       // TODO: make sure it survives resizes?
     }
   }
   // call once and then register events
   resize();
   $(window).resize(resize);
-  $(window).on("orientationchange", resize);
+  $(window).on('orientationchange', resize);
 });
 
 // Scroll the navigation bar to the currently selected menu item
-$(document).ready(function () {
+$(document).ready(function() {
   var sidebar = $("#documenter .docs-menu").get(0);
   var active = $("#documenter .docs-menu .is-active").get(0);
-  if (typeof active !== "undefined") {
+  if(typeof active !== 'undefined') {
     sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15;
   }
-});
+})
 
 })
 ////////////////////////////////////////////////////////////////////////////////
 require(['jquery'], function($) {
 
-// Theme picker setup
-$(document).ready(function () {
-  // onchange callback
-  $("#documenter-themepicker").change(function themepick_callback(ev) {
-    var themename = $("#documenter-themepicker option:selected").attr("value");
-    if (themename === "auto") {
-      // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light');
-      window.localStorage.removeItem("documenter-theme");
+function set_theme(theme) {
+  var active = null;
+  var disabled = [];
+  for (var i = 0; i < document.styleSheets.length; i++) {
+    var ss = document.styleSheets[i];
+    var themename = ss.ownerNode.getAttribute("data-theme-name");
+    if(themename === null) continue; // ignore non-theme stylesheets
+    // Find the active theme
+    if(themename === theme) active = ss;
+    else disabled.push(ss);
+  }
+  if(active !== null) {
+    active.disabled = false;
+    if(active.ownerNode.getAttribute("data-theme-primary") === null) {
+      document.getElementsByTagName('html')[0].className = "theme--" + theme;
     } else {
-      // set_theme(themename);
-      window.localStorage.setItem("documenter-theme", themename);
+      document.getElementsByTagName('html')[0].className = "";
     }
-    // We re-use the global function from themeswap.js to actually do the swapping.
-    set_theme_from_local_storage();
+    disabled.forEach(function(ss){
+      ss.disabled = true;
+    });
+  }
+
+  // Store the theme in localStorage
+  if(typeof(window.localStorage) !== "undefined") {
+    window.localStorage.setItem("documenter-theme", theme);
+  } else {
+    console.error("Browser does not support window.localStorage");
+  }
+}
+
+// Theme picker setup
+$(document).ready(function() {
+  // onchange callback
+  $('#documenter-themepicker').change(function themepick_callback(ev){
+    var themename = $('#documenter-themepicker option:selected').attr('value');
+    set_theme(themename);
   });
 
   // Make sure that the themepicker displays the correct theme when the theme is retrieved
   // from localStorage
-  if (typeof window.localStorage !== "undefined") {
-    var theme = window.localStorage.getItem("documenter-theme");
-    if (theme !== null) {
-      $("#documenter-themepicker option").each(function (i, e) {
-        e.selected = e.value === theme;
-      });
+  if(typeof(window.localStorage) !== "undefined") {
+    var theme =  window.localStorage.getItem("documenter-theme");
+    if(theme !== null) {
+      $('#documenter-themepicker option').each(function(i,e) {
+        e.selected = (e.value === theme);
+      })
+    } else {
+      $('#documenter-themepicker option').each(function(i,e) {
+        e.selected = $("html").hasClass(`theme--${e.value}`);
+      })
     }
   }
-});
+})
 
 })
 ////////////////////////////////////////////////////////////////////////////////
 require(['jquery'], function($) {
 
 // update the version selector with info from the siteinfo.js and ../versions.js files
-$(document).ready(function () {
-  // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the
-  // siteinfo.js file, we just return immediately and not display the version selector.
-  if (
-    typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" &&
-    DOCUMENTER_VERSION_SELECTOR_DISABLED
-  ) {
-    return;
-  }
-
+$(document).ready(function() {
   var version_selector = $("#documenter .docs-version-selector");
   var version_selector_select = $("#documenter .docs-version-selector select");
 
-  version_selector_select.change(function (x) {
-    target_href = version_selector_select
-      .children("option:selected")
-      .get(0).value;
+  version_selector_select.change(function(x) {
+    target_href = version_selector_select.children("option:selected").get(0).value;
     window.location.href = target_href;
   });
 
   // add the current version to the selector based on siteinfo.js, but only if the selector is empty
-  if (
-    typeof DOCUMENTER_CURRENT_VERSION !== "undefined" &&
-    $("#version-selector > option").length == 0
-  ) {
-    var option = $(
-      "<option value='#' selected='selected'>" +
-        DOCUMENTER_CURRENT_VERSION +
-        "</option>"
-    );
+  if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) {
+    var option = $("<option value='#' selected='selected'>" + DOCUMENTER_CURRENT_VERSION + "</option>");
     version_selector_select.append(option);
   }
 
-  if (typeof DOC_VERSIONS !== "undefined") {
+  if (typeof DOC_VERSIONS !== 'undefined') {
     var existing_versions = version_selector_select.children("option");
-    var existing_versions_texts = existing_versions.map(function (i, x) {
-      return x.text;
-    });
-    DOC_VERSIONS.forEach(function (each) {
-      var version_url = documenterBaseURL + "/../" + each + "/";
+    var existing_versions_texts = existing_versions.map(function(i,x){return x.text});
+    DOC_VERSIONS.forEach(function(each) {
+      var version_url = documenterBaseURL + "/../" + each;
       var existing_id = $.inArray(each, existing_versions_texts);
       // if not already in the version selector, add it as a new option,
       // otherwise update the old option with the URL and enable it
       if (existing_id == -1) {
-        var option = $(
-          "<option value='" + version_url + "'>" + each + "</option>"
-        );
+        var option = $("<option value='" + version_url + "'>" + each + "</option>");
         version_selector_select.append(option);
       } else {
         var option = existing_versions[existing_id];
@@ -1077,6 +259,6 @@ $(document).ready(function () {
   if (version_selector_select.children("option").length > 0) {
     version_selector.toggleClass("visible");
   }
-});
+})
 
 })
diff --git a/docs/assets/search.js b/docs/assets/search.js
new file mode 100644
index 0000000..71ebd87
--- /dev/null
+++ b/docs/assets/search.js
@@ -0,0 +1,251 @@
+// Generated by Documenter.jl
+requirejs.config({
+  paths: {
+    'lunr': 'https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.6/lunr.min',
+    'lodash': 'https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.15/lodash.min',
+    'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min',
+  }
+});
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'lunr', 'lodash'], function($, lunr, _) {
+
+$(document).ready(function() {
+  // parseUri 1.2.2
+  // (c) Steven Levithan <stevenlevithan.com>
+  // MIT License
+  function parseUri (str) {
+    var  o   = parseUri.options,
+      m   = o.parser[o.strictMode ? "strict" : "loose"].exec(str),
+      uri = {},
+      i   = 14;
+
+    while (i--) uri[o.key[i]] = m[i] || "";
+
+    uri[o.q.name] = {};
+    uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) {
+      if ($1) uri[o.q.name][$1] = $2;
+    });
+
+    return uri;
+  };
+  parseUri.options = {
+    strictMode: false,
+    key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"],
+    q:   {
+      name:   "queryKey",
+      parser: /(?:^|&)([^&=]*)=?([^&]*)/g
+    },
+    parser: {
+      strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/,
+      loose:  /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/
+    }
+  };
+
+  $("#search-form").submit(function(e) {
+    e.preventDefault()
+  })
+
+  // list below is the lunr 2.1.3 list minus the intersect with names(Base)
+  // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with)
+  // ideally we'd just filter the original list but it's not available as a variable
+  lunr.stopWordFilter = lunr.generateStopWordFilter([
+    'a',
+    'able',
+    'about',
+    'across',
+    'after',
+    'almost',
+    'also',
+    'am',
+    'among',
+    'an',
+    'and',
+    'are',
+    'as',
+    'at',
+    'be',
+    'because',
+    'been',
+    'but',
+    'by',
+    'can',
+    'cannot',
+    'could',
+    'dear',
+    'did',
+    'does',
+    'either',
+    'ever',
+    'every',
+    'from',
+    'got',
+    'had',
+    'has',
+    'have',
+    'he',
+    'her',
+    'hers',
+    'him',
+    'his',
+    'how',
+    'however',
+    'i',
+    'if',
+    'into',
+    'it',
+    'its',
+    'just',
+    'least',
+    'like',
+    'likely',
+    'may',
+    'me',
+    'might',
+    'most',
+    'must',
+    'my',
+    'neither',
+    'no',
+    'nor',
+    'not',
+    'of',
+    'off',
+    'often',
+    'on',
+    'or',
+    'other',
+    'our',
+    'own',
+    'rather',
+    'said',
+    'say',
+    'says',
+    'she',
+    'should',
+    'since',
+    'so',
+    'some',
+    'than',
+    'that',
+    'the',
+    'their',
+    'them',
+    'then',
+    'there',
+    'these',
+    'they',
+    'this',
+    'tis',
+    'to',
+    'too',
+    'twas',
+    'us',
+    'wants',
+    'was',
+    'we',
+    'were',
+    'what',
+    'when',
+    'who',
+    'whom',
+    'why',
+    'will',
+    'would',
+    'yet',
+    'you',
+    'your'
+  ])
+
+  // add . as a separator, because otherwise "title": "Documenter.Anchors.add!"
+  // would not find anything if searching for "add!", only for the entire qualification
+  lunr.tokenizer.separator = /[\s\-\.]+/
+
+  // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names
+  lunr.trimmer = function (token) {
+    return token.update(function (s) {
+      return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '')
+    })
+  }
+
+  lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter')
+  lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer')
+
+  var index = lunr(function () {
+    this.ref('location')
+    this.field('title',{boost: 100})
+    this.field('text')
+    documenterSearchIndex['docs'].forEach(function(e) {
+        this.add(e)
+    }, this)
+  })
+  var store = {}
+
+  documenterSearchIndex['docs'].forEach(function(e) {
+      store[e.location] = {title: e.title, category: e.category, page: e.page}
+  })
+
+  $(function(){
+    searchresults = $('#documenter-search-results');
+    searchinfo = $('#documenter-search-info');
+    searchbox = $('#documenter-search-query');
+    function update_search(querystring) {
+      tokens = lunr.tokenizer(querystring)
+      results = index.query(function (q) {
+        tokens.forEach(function (t) {
+          q.term(t.toString(), {
+            fields: ["title"],
+            boost: 100,
+            usePipeline: true,
+            editDistance: 0,
+            wildcard: lunr.Query.wildcard.NONE
+          })
+          q.term(t.toString(), {
+            fields: ["title"],
+            boost: 10,
+            usePipeline: true,
+            editDistance: 2,
+            wildcard: lunr.Query.wildcard.NONE
+          })
+          q.term(t.toString(), {
+            fields: ["text"],
+            boost: 1,
+            usePipeline: true,
+            editDistance: 0,
+            wildcard: lunr.Query.wildcard.NONE
+          })
+        })
+      })
+      searchinfo.text("Number of results: " + results.length)
+      searchresults.empty()
+      results.forEach(function(result) {
+        data = store[result.ref]
+        link = $('<a class="docs-label">'+data.title+'</a>')
+        link.attr('href', documenterBaseURL+'/'+result.ref)
+        if (data.category != "page"){
+          cat = $('<span class="docs-category">('+data.category+', '+data.page+')</span>')
+        } else {
+          cat = $('<span class="docs-category">('+data.category+')</span>')
+        }
+        li = $('<li>').append(link).append(" ").append(cat)
+        searchresults.append(li)
+      })
+    }
+
+    function update_search_box() {
+      querystring = searchbox.val()
+      update_search(querystring)
+    }
+
+    searchbox.keyup(_.debounce(update_search_box, 250))
+    searchbox.change(update_search_box)
+
+    search_query_uri = parseUri(window.location).queryKey["q"]
+    if(search_query_uri !== undefined) {
+      search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20'))
+      searchbox.val(search_query)
+    }
+    update_search_box();
+  })
+})
+
+})
diff --git a/docs/assets/themes/catppuccin-frappe.css b/docs/assets/themes/catppuccin-frappe.css
deleted file mode 100644
index 32e3f00..0000000
--- a/docs/assets/themes/catppuccin-frappe.css
+++ /dev/null
@@ -1 +0,0 @@
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html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content 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.button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-outlined:focus,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-focused{background-color:#8caaee;border-color:#8caaee;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe 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.button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-light{background-color:#edf2fc;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:hover,html.theme--catppuccin-frappe .button.is-link.is-light.is-hovered{background-color:#e2eafb;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:active,html.theme--catppuccin-frappe .button.is-link.is-light.is-active{background-color:#d7e1f9;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:hover,html.theme--catppuccin-frappe .button.is-info.is-hovered{background-color:#78c4b9;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus,html.theme--catppuccin-frappe .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus:not(:active),html.theme--catppuccin-frappe .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(129,200,190,0.25)}html.theme--catppuccin-frappe .button.is-info:active,html.theme--catppuccin-frappe .button.is-info.is-active{background-color:#6fc0b5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:#81c8be;box-shadow:none}html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-focused{background-color:#81c8be;border-color:#81c8be;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe 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.button.is-success{background-color:#a6d189;border-color:#a6d189;box-shadow:none}html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-focused{background-color:#a6d189;border-color:#a6d189;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe 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.button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-frappe .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-frappe .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-frappe .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-frappe .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-frappe .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-frappe .content li+li{margin-top:0.25em}html.theme--catppuccin-frappe .content p:not(:last-child),html.theme--catppuccin-frappe .content dl:not(:last-child),html.theme--catppuccin-frappe .content ol:not(:last-child),html.theme--catppuccin-frappe .content ul:not(:last-child),html.theme--catppuccin-frappe .content blockquote:not(:last-child),html.theme--catppuccin-frappe .content pre:not(:last-child),html.theme--catppuccin-frappe .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-frappe .content h1,html.theme--catppuccin-frappe .content h2,html.theme--catppuccin-frappe .content h3,html.theme--catppuccin-frappe .content h4,html.theme--catppuccin-frappe .content h5,html.theme--catppuccin-frappe .content h6{color:#c6d0f5;font-weight:600;line-height:1.125}html.theme--catppuccin-frappe .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-frappe .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-frappe .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-frappe .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-frappe .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-frappe .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-frappe .content ul ul ul{list-style-type:square}html.theme--catppuccin-frappe .content dd{margin-left:2em}html.theme--catppuccin-frappe .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-frappe .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-frappe .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-frappe .content figure img{display:inline-block}html.theme--catppuccin-frappe .content figure figcaption{font-style:italic}html.theme--catppuccin-frappe .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-frappe .content sup,html.theme--catppuccin-frappe .content sub{font-size:75%}html.theme--catppuccin-frappe .content table{width:100%}html.theme--catppuccin-frappe .content table td,html.theme--catppuccin-frappe .content table th{border:1px solid #626880;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-frappe .content table th{color:#b0bef1}html.theme--catppuccin-frappe .content table th:not([align]){text-align:inherit}html.theme--catppuccin-frappe .content table thead td,html.theme--catppuccin-frappe .content table thead th{border-width:0 0 2px;color:#b0bef1}html.theme--catppuccin-frappe .content table tfoot td,html.theme--catppuccin-frappe .content table tfoot th{border-width:2px 0 0;color:#b0bef1}html.theme--catppuccin-frappe .content table tbody tr:last-child td,html.theme--catppuccin-frappe .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-frappe .content .tabs li+li{margin-top:0}html.theme--catppuccin-frappe .content.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-frappe .content.is-normal{font-size:1rem}html.theme--catppuccin-frappe .content.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .content.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-frappe .icon.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-frappe .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-frappe .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-frappe .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-frappe .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-frappe .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-frappe .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-frappe div.icon-text{display:flex}html.theme--catppuccin-frappe .image,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-frappe .image img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-frappe .image img.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-frappe .image.is-fullwidth,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-frappe .image.is-square img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-frappe .image.is-square .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-frappe .image.is-1by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-frappe .image.is-1by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-frappe .image.is-5by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-frappe .image.is-5by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-frappe .image.is-4by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-frappe .image.is-4by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-frappe .image.is-3by2 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-frappe .image.is-5by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-frappe .image.is-5by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-frappe .image.is-16by9 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-frappe .image.is-16by9 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-frappe .image.is-2by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-frappe .image.is-2by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-frappe .image.is-3by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-frappe .image.is-3by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--catppuccin-frappe .image.is-4by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--catppuccin-frappe .image.is-4by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--catppuccin-frappe .image.is-3by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--catppuccin-frappe .image.is-3by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--catppuccin-frappe .image.is-2by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--catppuccin-frappe .image.is-2by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-frappe .image.is-3by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-frappe .image.is-9by16 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-frappe .image.is-9by16 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-frappe .image.is-1by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-frappe .image.is-1by2 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-frappe .image.is-1by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-frappe .image.is-1by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-frappe .image.is-square,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square,html.theme--catppuccin-frappe .image.is-1by1,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1{padding-top:100%}html.theme--catppuccin-frappe .image.is-5by4,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4{padding-top:80%}html.theme--catppuccin-frappe .image.is-4by3,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--catppuccin-frappe 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.modal-card{display:flex;flex-direction:column;max-height:calc(100vh - 40px);overflow:hidden;-ms-overflow-y:visible}html.theme--catppuccin-frappe .modal-card-head,html.theme--catppuccin-frappe .modal-card-foot{align-items:center;background-color:#292c3c;display:flex;flex-shrink:0;justify-content:flex-start;padding:20px;position:relative}html.theme--catppuccin-frappe .modal-card-head{border-bottom:1px solid #626880;border-top-left-radius:8px;border-top-right-radius:8px}html.theme--catppuccin-frappe .modal-card-title{color:#c6d0f5;flex-grow:1;flex-shrink:0;font-size:1.5rem;line-height:1}html.theme--catppuccin-frappe .modal-card-foot{border-bottom-left-radius:8px;border-bottom-right-radius:8px;border-top:1px solid #626880}html.theme--catppuccin-frappe .modal-card-foot .button:not(:last-child){margin-right:.5em}html.theme--catppuccin-frappe 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diff --git a/docs/assets/themes/catppuccin-latte.css b/docs/assets/themes/catppuccin-latte.css
deleted file mode 100644
index 63160d3..0000000
--- a/docs/assets/themes/catppuccin-latte.css
+++ /dev/null
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kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte 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.docstring>section>a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-primary.is-light,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-light.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-hovered.docs-sourcelink{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:active,html.theme--catppuccin-latte .button.is-primary.is-light.is-active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:hover,html.theme--catppuccin-latte .button.is-link.is-hovered{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus,html.theme--catppuccin-latte .button.is-link.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus:not(:active),html.theme--catppuccin-latte .button.is-link.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(30,102,245,0.25)}html.theme--catppuccin-latte .button.is-link:active,html.theme--catppuccin-latte .button.is-link.is-active{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:#1e66f5;box-shadow:none}html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-link.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-outlined.is-focused{background-color:#1e66f5;border-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-link.is-light{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:hover,html.theme--catppuccin-latte .button.is-link.is-light.is-hovered{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:active,html.theme--catppuccin-latte .button.is-link.is-light.is-active{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:hover,html.theme--catppuccin-latte .button.is-info.is-hovered{background-color:#15878e;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus,html.theme--catppuccin-latte .button.is-info.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus:not(:active),html.theme--catppuccin-latte .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(23,146,153,0.25)}html.theme--catppuccin-latte .button.is-info:active,html.theme--catppuccin-latte .button.is-info.is-active{background-color:#147d83;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:#179299;box-shadow:none}html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;color:#179299}html.theme--catppuccin-latte .button.is-info.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-outlined.is-focused{background-color:#179299;border-color:#179299;color:#fff}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-info.is-light{background-color:#edfcfc;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:hover,html.theme--catppuccin-latte .button.is-info.is-light.is-hovered{background-color:#e2f9fb;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:active,html.theme--catppuccin-latte .button.is-info.is-light.is-active{background-color:#d7f7f9;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:hover,html.theme--catppuccin-latte .button.is-success.is-hovered{background-color:#3c9628;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus,html.theme--catppuccin-latte .button.is-success.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus:not(:active),html.theme--catppuccin-latte .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(64,160,43,0.25)}html.theme--catppuccin-latte .button.is-success:active,html.theme--catppuccin-latte .button.is-success.is-active{background-color:#388c26;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:#40a02b;box-shadow:none}html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-outlined.is-focused{background-color:#40a02b;border-color:#40a02b;color:#fff}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-latte .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-latte .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-latte .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-latte .content li+li{margin-top:0.25em}html.theme--catppuccin-latte .content p:not(:last-child),html.theme--catppuccin-latte .content dl:not(:last-child),html.theme--catppuccin-latte .content ol:not(:last-child),html.theme--catppuccin-latte .content ul:not(:last-child),html.theme--catppuccin-latte .content blockquote:not(:last-child),html.theme--catppuccin-latte .content pre:not(:last-child),html.theme--catppuccin-latte .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-latte .content h1,html.theme--catppuccin-latte .content h2,html.theme--catppuccin-latte .content h3,html.theme--catppuccin-latte .content h4,html.theme--catppuccin-latte .content h5,html.theme--catppuccin-latte .content h6{color:#4c4f69;font-weight:600;line-height:1.125}html.theme--catppuccin-latte .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-latte .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-latte .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-latte .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-latte .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-latte .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-latte .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-latte .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-latte .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-latte .content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-latte .content ul ul ul{list-style-type:square}html.theme--catppuccin-latte .content dd{margin-left:2em}html.theme--catppuccin-latte .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-latte .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-latte .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-latte .content figure img{display:inline-block}html.theme--catppuccin-latte .content figure figcaption{font-style:italic}html.theme--catppuccin-latte .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-latte .content sup,html.theme--catppuccin-latte .content sub{font-size:75%}html.theme--catppuccin-latte .content table{width:100%}html.theme--catppuccin-latte .content table td,html.theme--catppuccin-latte .content table th{border:1px solid #acb0be;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-latte .content table th{color:#41445a}html.theme--catppuccin-latte .content table th:not([align]){text-align:inherit}html.theme--catppuccin-latte .content table thead td,html.theme--catppuccin-latte .content table thead th{border-width:0 0 2px;color:#41445a}html.theme--catppuccin-latte .content table tfoot td,html.theme--catppuccin-latte .content table tfoot th{border-width:2px 0 0;color:#41445a}html.theme--catppuccin-latte .content table tbody tr:last-child td,html.theme--catppuccin-latte .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-latte .content .tabs li+li{margin-top:0}html.theme--catppuccin-latte .content.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-latte .content.is-normal{font-size:1rem}html.theme--catppuccin-latte .content.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .content.is-large{font-size:1.5rem}html.theme--catppuccin-latte .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-latte .icon.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-latte .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-latte .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-latte .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-latte .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-latte .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-latte .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-latte div.icon-text{display:flex}html.theme--catppuccin-latte .image,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-latte .image img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-latte .image img.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-latte .image.is-fullwidth,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-latte .image.is-square img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-latte .image.is-square .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-latte .image.is-1by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-latte .image.is-1by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-latte .image.is-5by4 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-latte .image.is-5by4 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-latte .image.is-4by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-latte .image.is-4by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-latte .image.is-3by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-latte .image.is-3by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-latte .image.is-5by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-latte .image.is-5by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-latte .image.is-16by9 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-latte .image.is-16by9 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-latte .image.is-2by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-latte .image.is-2by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-latte .image.is-3by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-latte 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img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-latte .image.is-3by5 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-latte .image.is-9by16 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-latte .image.is-9by16 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-latte .image.is-1by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-latte .image.is-1by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-latte .image.is-1by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-latte .image.is-1by3 .has-ratio,html.theme--catppuccin-latte 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diff --git a/docs/assets/themes/catppuccin-macchiato.css b/docs/assets/themes/catppuccin-macchiato.css
deleted file mode 100644
index a9cf9c5..0000000
--- a/docs/assets/themes/catppuccin-macchiato.css
+++ /dev/null
@@ -1 +0,0 @@
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.button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:hover,html.theme--catppuccin-macchiato .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus,html.theme--catppuccin-macchiato .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus:not(:active),html.theme--catppuccin-macchiato .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-macchiato .button.is-black:active,html.theme--catppuccin-macchiato .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:hover,html.theme--catppuccin-macchiato .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus,html.theme--catppuccin-macchiato .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus:not(:active),html.theme--catppuccin-macchiato .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-macchiato .button.is-light:active,html.theme--catppuccin-macchiato .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-light.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-macchiato 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.content kbd.button{background-color:#363a4f;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:hover,html.theme--catppuccin-macchiato .content kbd.button:hover,html.theme--catppuccin-macchiato .button.is-dark.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-hovered{background-color:#313447;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus,html.theme--catppuccin-macchiato .content kbd.button:focus,html.theme--catppuccin-macchiato .button.is-dark.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus:not(:active),html.theme--catppuccin-macchiato .content kbd.button:focus:not(:active),html.theme--catppuccin-macchiato .button.is-dark.is-focused:not(:active),html.theme--catppuccin-macchiato .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(54,58,79,0.25)}html.theme--catppuccin-macchiato .button.is-dark:active,html.theme--catppuccin-macchiato .content kbd.button:active,html.theme--catppuccin-macchiato .button.is-dark.is-active,html.theme--catppuccin-macchiato .content kbd.button.is-active{background-color:#2c2f40;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark[disabled],html.theme--catppuccin-macchiato .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button{background-color:#363a4f;border-color:#363a4f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-dark.is-inverted,html.theme--catppuccin-macchiato .content kbd.button.is-inverted{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted:hover,html.theme--catppuccin-macchiato 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.button.is-dark.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-focused{background-color:#363a4f;border-color:#363a4f;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content 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.button.is-info.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-focused{background-color:#8bd5ca;border-color:#8bd5ca;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-outlined{background-color:transparent;border-color:#8bd5ca;box-shadow:none;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-light{background-color:#f0faf8;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:hover,html.theme--catppuccin-macchiato .button.is-info.is-light.is-hovered{background-color:#e7f6f4;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:active,html.theme--catppuccin-macchiato .button.is-info.is-light.is-active{background-color:#ddf3f0;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:hover,html.theme--catppuccin-macchiato .button.is-success.is-hovered{background-color:#9ed78c;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus,html.theme--catppuccin-macchiato .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus:not(:active),html.theme--catppuccin-macchiato .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,218,149,0.25)}html.theme--catppuccin-macchiato .button.is-success:active,html.theme--catppuccin-macchiato .button.is-success.is-active{background-color:#96d382;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:#a6da95;box-shadow:none}html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-macchiato .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-macchiato .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused,html.theme--catppuccin-macchiato .buttons.has-addons .button:active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button:active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-macchiato .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-macchiato .buttons.is-centered{justify-content:center}html.theme--catppuccin-macchiato .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-macchiato 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diff --git a/docs/assets/themes/catppuccin-mocha.css b/docs/assets/themes/catppuccin-mocha.css
deleted file mode 100644
index 8b82652..0000000
--- a/docs/assets/themes/catppuccin-mocha.css
+++ /dev/null
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kbd.button.is-inverted{background-color:#fff;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted:hover,html.theme--catppuccin-mocha .content kbd.button.is-inverted:hover,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-dark.is-inverted[disabled],html.theme--catppuccin-mocha .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-loading::after,html.theme--catppuccin-mocha .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-dark.is-outlined,html.theme--catppuccin-mocha 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.button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-light{background-color:#effbf9;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:hover,html.theme--catppuccin-mocha .button.is-info.is-light.is-hovered{background-color:#e5f8f5;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:active,html.theme--catppuccin-mocha .button.is-info.is-light.is-active{background-color:#dbf5f1;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:hover,html.theme--catppuccin-mocha .button.is-success.is-hovered{background-color:#9de097;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus,html.theme--catppuccin-mocha .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus:not(:active),html.theme--catppuccin-mocha .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,227,161,0.25)}html.theme--catppuccin-mocha .button.is-success:active,html.theme--catppuccin-mocha .button.is-success.is-active{background-color:#93dd8d;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:#a6e3a1;box-shadow:none}html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons .button:focus:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-mocha .buttons.has-addons .button:active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-mocha .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-mocha .buttons.is-centered{justify-content:center}html.theme--catppuccin-mocha .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-mocha .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-mocha .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-mocha .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-mocha 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diff --git a/docs/assets/themes/documenter-dark.css b/docs/assets/themes/documenter-dark.css
index c41c82f..f866234 100644
--- a/docs/assets/themes/documenter-dark.css
+++ b/docs/assets/themes/documenter-dark.css
@@ -1,7 +1,7642 @@
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-  Theme: a11y-dark
-  Author: @ericwbailey
-  Maintainer: @ericwbailey
+@charset "UTF-8";
+/* Font Awesome 5 mixin. Can be included in any rule that should render Font Awesome icons. */
+@keyframes spinAround {
+  from {
+    transform: rotate(0deg); }
+  to {
+    transform: rotate(359deg); } }
 
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.button.is-info.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#3c5dcd}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #3c5dcd #3c5dcd !important}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark 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.button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#259a12;box-shadow:none;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #259a12 #259a12 !important}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-success.is-light{background-color:#effded;color:#2ec016}html.theme--documenter-dark .button.is-success.is-light:hover,html.theme--documenter-dark 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.button.is-warning:active,html.theme--documenter-dark .button.is-warning.is-active{background-color:#f3c017;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning{background-color:#f4c72f;border-color:#f4c72f;box-shadow:none}html.theme--documenter-dark .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#f4c72f;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-outlined.is-focused{background-color:#f4c72f;border-color:#f4c72f;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f4c72f #f4c72f !important}html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#f4c72f;box-shadow:none;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f4c72f #f4c72f !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fefaec;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fdf7e0;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fdf3d3;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:hover,html.theme--documenter-dark .button.is-danger.is-hovered{background-color:#c13930;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus,html.theme--documenter-dark .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus:not(:active),html.theme--documenter-dark .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(203,60,51,0.25)}html.theme--documenter-dark .button.is-danger:active,html.theme--documenter-dark .button.is-danger.is-active{background-color:#b7362e;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:#cb3c33;box-shadow:none}html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#cb3c33;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-outlined.is-focused{background-color:#cb3c33;border-color:#cb3c33;color:#fff}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #cb3c33 #cb3c33 !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#cb3c33;box-shadow:none;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #cb3c33 #cb3c33 !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark 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.gap-8{gap:2rem}html.theme--documenter-dark{background-color:#1f2424;font-size:16px;min-width:300px;overflow-x:auto;overflow-y:scroll;text-rendering:optimizeLegibility;text-size-adjust:100%}html.theme--documenter-dark .ansi span.sgr1{font-weight:bolder}html.theme--documenter-dark .ansi span.sgr2{font-weight:lighter}html.theme--documenter-dark .ansi span.sgr3{font-style:italic}html.theme--documenter-dark .ansi span.sgr4{text-decoration:underline}html.theme--documenter-dark .ansi span.sgr7{color:#1f2424;background-color:#fff}html.theme--documenter-dark .ansi span.sgr8{color:transparent}html.theme--documenter-dark .ansi span.sgr8 span{color:transparent}html.theme--documenter-dark .ansi span.sgr9{text-decoration:line-through}html.theme--documenter-dark .ansi span.sgr30{color:#242424}html.theme--documenter-dark .ansi span.sgr31{color:#f6705f}html.theme--documenter-dark .ansi span.sgr32{color:#4fb43a}html.theme--documenter-dark .ansi span.sgr33{color:#f4c72f}html.theme--documenter-dark .ansi span.sgr34{color:#7587f0}html.theme--documenter-dark .ansi span.sgr35{color:#bc89d3}html.theme--documenter-dark .ansi span.sgr36{color:#49b6ca}html.theme--documenter-dark .ansi span.sgr37{color:#b3bdbe}html.theme--documenter-dark .ansi span.sgr40{background-color:#242424}html.theme--documenter-dark .ansi span.sgr41{background-color:#f6705f}html.theme--documenter-dark .ansi span.sgr42{background-color:#4fb43a}html.theme--documenter-dark .ansi span.sgr43{background-color:#f4c72f}html.theme--documenter-dark .ansi span.sgr44{background-color:#7587f0}html.theme--documenter-dark .ansi span.sgr45{background-color:#bc89d3}html.theme--documenter-dark .ansi span.sgr46{background-color:#49b6ca}html.theme--documenter-dark .ansi span.sgr47{background-color:#b3bdbe}html.theme--documenter-dark .ansi span.sgr90{color:#92a0a2}html.theme--documenter-dark .ansi span.sgr91{color:#ff8674}html.theme--documenter-dark .ansi span.sgr92{color:#79d462}html.theme--documenter-dark .ansi span.sgr93{color:#ffe76b}html.theme--documenter-dark .ansi span.sgr94{color:#8a98ff}html.theme--documenter-dark .ansi span.sgr95{color:#d2a4e6}html.theme--documenter-dark .ansi span.sgr96{color:#6bc8db}html.theme--documenter-dark .ansi span.sgr97{color:#ecf0f1}html.theme--documenter-dark .ansi span.sgr100{background-color:#92a0a2}html.theme--documenter-dark .ansi span.sgr101{background-color:#ff8674}html.theme--documenter-dark .ansi span.sgr102{background-color:#79d462}html.theme--documenter-dark .ansi span.sgr103{background-color:#ffe76b}html.theme--documenter-dark .ansi span.sgr104{background-color:#8a98ff}html.theme--documenter-dark .ansi span.sgr105{background-color:#d2a4e6}html.theme--documenter-dark .ansi span.sgr106{background-color:#6bc8db}html.theme--documenter-dark .ansi span.sgr107{background-color:#ecf0f1}html.theme--documenter-dark code.language-julia-repl>span.hljs-meta{color:#4fb43a;font-weight:bolder}html.theme--documenter-dark .hljs{background:#2b2b2b;color:#f8f8f2}html.theme--documenter-dark .hljs-comment,html.theme--documenter-dark .hljs-quote{color:#d4d0ab}html.theme--documenter-dark .hljs-variable,html.theme--documenter-dark .hljs-template-variable,html.theme--documenter-dark .hljs-tag,html.theme--documenter-dark .hljs-name,html.theme--documenter-dark .hljs-selector-id,html.theme--documenter-dark .hljs-selector-class,html.theme--documenter-dark .hljs-regexp,html.theme--documenter-dark .hljs-deletion{color:#ffa07a}html.theme--documenter-dark .hljs-number,html.theme--documenter-dark .hljs-built_in,html.theme--documenter-dark .hljs-literal,html.theme--documenter-dark .hljs-type,html.theme--documenter-dark .hljs-params,html.theme--documenter-dark .hljs-meta,html.theme--documenter-dark .hljs-link{color:#f5ab35}html.theme--documenter-dark .hljs-attribute{color:#ffd700}html.theme--documenter-dark .hljs-string,html.theme--documenter-dark .hljs-symbol,html.theme--documenter-dark .hljs-bullet,html.theme--documenter-dark .hljs-addition{color:#abe338}html.theme--documenter-dark .hljs-title,html.theme--documenter-dark .hljs-section{color:#00e0e0}html.theme--documenter-dark .hljs-keyword,html.theme--documenter-dark .hljs-selector-tag{color:#dcc6e0}html.theme--documenter-dark .hljs-emphasis{font-style:italic}html.theme--documenter-dark .hljs-strong{font-weight:bold}@media screen and (-ms-high-contrast: active){html.theme--documenter-dark .hljs-addition,html.theme--documenter-dark .hljs-attribute,html.theme--documenter-dark .hljs-built_in,html.theme--documenter-dark .hljs-bullet,html.theme--documenter-dark .hljs-comment,html.theme--documenter-dark .hljs-link,html.theme--documenter-dark .hljs-literal,html.theme--documenter-dark .hljs-meta,html.theme--documenter-dark .hljs-number,html.theme--documenter-dark .hljs-params,html.theme--documenter-dark .hljs-string,html.theme--documenter-dark .hljs-symbol,html.theme--documenter-dark .hljs-type,html.theme--documenter-dark .hljs-quote{color:highlight}html.theme--documenter-dark .hljs-keyword,html.theme--documenter-dark .hljs-selector-tag{font-weight:bold}}html.theme--documenter-dark .hljs-subst{color:#f8f8f2}html.theme--documenter-dark .search-result-link{border-radius:0.7em;transition:all 300ms}html.theme--documenter-dark .search-result-link:hover,html.theme--documenter-dark .search-result-link:focus{background-color:rgba(0,128,128,0.1)}html.theme--documenter-dark .search-result-link .property-search-result-badge,html.theme--documenter-dark .search-result-link .search-filter{transition:all 300ms}html.theme--documenter-dark .search-result-link:hover .property-search-result-badge,html.theme--documenter-dark .search-result-link:hover .search-filter,html.theme--documenter-dark .search-result-link:focus .property-search-result-badge,html.theme--documenter-dark .search-result-link:focus .search-filter{color:#333 !important;background-color:#f1f5f9 !important}html.theme--documenter-dark .search-result-title{color:whitesmoke}html.theme--documenter-dark .search-result-highlight{background-color:greenyellow;color:black}html.theme--documenter-dark .search-divider{border-bottom:1px solid #5e6d6f50}html.theme--documenter-dark .w-100{width:100%}html.theme--documenter-dark .gap-2{gap:0.5rem}html.theme--documenter-dark .gap-4{gap:1rem}
+html.theme--documenter-dark .delete, html.theme--documenter-dark .modal-close, .is-unselectable, html.theme--documenter-dark .button, html.theme--documenter-dark .file, html.theme--documenter-dark .breadcrumb, html.theme--documenter-dark .pagination-previous,
+html.theme--documenter-dark .pagination-next,
+html.theme--documenter-dark .pagination-link,
+html.theme--documenter-dark .pagination-ellipsis, html.theme--documenter-dark .tabs {
+  -webkit-touch-callout: none;
+  -webkit-user-select: none;
+  -moz-user-select: none;
+  -ms-user-select: none;
+  user-select: none; }
+
+html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after, html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after {
+  border: 3px solid transparent;
+  border-radius: 2px;
+  border-right: 0;
+  border-top: 0;
+  content: " ";
+  display: block;
+  height: 0.625em;
+  margin-top: -0.4375em;
+  pointer-events: none;
+  position: absolute;
+  top: 50%;
+  transform: rotate(-45deg);
+  transform-origin: center;
+  width: 0.625em; }
+
+html.theme--documenter-dark .box:not(:last-child), html.theme--documenter-dark .content:not(:last-child), html.theme--documenter-dark .notification:not(:last-child), html.theme--documenter-dark .progress:not(:last-child), html.theme--documenter-dark .table:not(:last-child), html.theme--documenter-dark .table-container:not(:last-child), html.theme--documenter-dark .title:not(:last-child),
+html.theme--documenter-dark .subtitle:not(:last-child), html.theme--documenter-dark .block:not(:last-child), html.theme--documenter-dark .highlight:not(:last-child), html.theme--documenter-dark .breadcrumb:not(:last-child), html.theme--documenter-dark .level:not(:last-child), html.theme--documenter-dark .list:not(:last-child), html.theme--documenter-dark .message:not(:last-child), html.theme--documenter-dark .tabs:not(:last-child), html.theme--documenter-dark .admonition:not(:last-child) {
+  margin-bottom: 1.5rem; }
+
+html.theme--documenter-dark .delete, html.theme--documenter-dark .modal-close {
+  -moz-appearance: none;
+  -webkit-appearance: none;
+  background-color: rgba(10, 10, 10, 0.2);
+  border: none;
+  border-radius: 290486px;
+  cursor: pointer;
+  pointer-events: auto;
+  display: inline-block;
+  flex-grow: 0;
+  flex-shrink: 0;
+  font-size: 0;
+  height: 20px;
+  max-height: 20px;
+  max-width: 20px;
+  min-height: 20px;
+  min-width: 20px;
+  outline: none;
+  position: relative;
+  vertical-align: top;
+  width: 20px; }
+  html.theme--documenter-dark .delete::before, html.theme--documenter-dark .modal-close::before, html.theme--documenter-dark .delete::after, html.theme--documenter-dark .modal-close::after {
+    background-color: white;
+    content: "";
+    display: block;
+    left: 50%;
+    position: absolute;
+    top: 50%;
+    transform: translateX(-50%) translateY(-50%) rotate(45deg);
+    transform-origin: center center; }
+  html.theme--documenter-dark .delete::before, html.theme--documenter-dark .modal-close::before {
+    height: 2px;
+    width: 50%; }
+  html.theme--documenter-dark .delete::after, html.theme--documenter-dark .modal-close::after {
+    height: 50%;
+    width: 2px; }
+  html.theme--documenter-dark .delete:hover, html.theme--documenter-dark .modal-close:hover, html.theme--documenter-dark .delete:focus, html.theme--documenter-dark .modal-close:focus {
+    background-color: rgba(10, 10, 10, 0.3); }
+  html.theme--documenter-dark .delete:active, html.theme--documenter-dark .modal-close:active {
+    background-color: rgba(10, 10, 10, 0.4); }
+  html.theme--documenter-dark .is-small.delete, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.delete, html.theme--documenter-dark .is-small.modal-close, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.modal-close {
+    height: 16px;
+    max-height: 16px;
+    max-width: 16px;
+    min-height: 16px;
+    min-width: 16px;
+    width: 16px; }
+  html.theme--documenter-dark .is-medium.delete, html.theme--documenter-dark .is-medium.modal-close {
+    height: 24px;
+    max-height: 24px;
+    max-width: 24px;
+    min-height: 24px;
+    min-width: 24px;
+    width: 24px; }
+  html.theme--documenter-dark .is-large.delete, html.theme--documenter-dark .is-large.modal-close {
+    height: 32px;
+    max-height: 32px;
+    max-width: 32px;
+    min-height: 32px;
+    min-width: 32px;
+    width: 32px; }
+
+html.theme--documenter-dark .button.is-loading::after, html.theme--documenter-dark .loader, html.theme--documenter-dark .select.is-loading::after, html.theme--documenter-dark .control.is-loading::after {
+  animation: spinAround 500ms infinite linear;
+  border: 2px solid #dbdee0;
+  border-radius: 290486px;
+  border-right-color: transparent;
+  border-top-color: transparent;
+  content: "";
+  display: block;
+  height: 1em;
+  position: relative;
+  width: 1em; }
+
+.is-overlay, html.theme--documenter-dark .image.is-square img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square img,
+html.theme--documenter-dark .image.is-square .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, html.theme--documenter-dark .image.is-1by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 img,
+html.theme--documenter-dark .image.is-1by1 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, html.theme--documenter-dark .image.is-5by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 img,
+html.theme--documenter-dark .image.is-5by4 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, html.theme--documenter-dark .image.is-4by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 img,
+html.theme--documenter-dark .image.is-4by3 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, html.theme--documenter-dark .image.is-3by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 img,
+html.theme--documenter-dark .image.is-3by2 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, html.theme--documenter-dark .image.is-5by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 img,
+html.theme--documenter-dark .image.is-5by3 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, html.theme--documenter-dark .image.is-16by9 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 img,
+html.theme--documenter-dark .image.is-16by9 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, html.theme--documenter-dark .image.is-2by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 img,
+html.theme--documenter-dark .image.is-2by1 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, html.theme--documenter-dark .image.is-3by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 img,
+html.theme--documenter-dark .image.is-3by1 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, html.theme--documenter-dark .image.is-4by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 img,
+html.theme--documenter-dark .image.is-4by5 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, html.theme--documenter-dark .image.is-3by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 img,
+html.theme--documenter-dark .image.is-3by4 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, html.theme--documenter-dark .image.is-2by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 img,
+html.theme--documenter-dark .image.is-2by3 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, html.theme--documenter-dark .image.is-3by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 img,
+html.theme--documenter-dark .image.is-3by5 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, html.theme--documenter-dark .image.is-9by16 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 img,
+html.theme--documenter-dark .image.is-9by16 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, html.theme--documenter-dark .image.is-1by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 img,
+html.theme--documenter-dark .image.is-1by2 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, html.theme--documenter-dark .image.is-1by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 img,
+html.theme--documenter-dark .image.is-1by3 .has-ratio,
+html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio, html.theme--documenter-dark .modal, html.theme--documenter-dark .modal-background, html.theme--documenter-dark .hero-video {
+  bottom: 0;
+  left: 0;
+  position: absolute;
+  right: 0;
+  top: 0; }
+
+html.theme--documenter-dark .button, html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark .textarea, html.theme--documenter-dark .select select, html.theme--documenter-dark .file-cta,
+html.theme--documenter-dark .file-name, html.theme--documenter-dark .pagination-previous,
+html.theme--documenter-dark .pagination-next,
+html.theme--documenter-dark .pagination-link,
+html.theme--documenter-dark .pagination-ellipsis {
+  -moz-appearance: none;
+  -webkit-appearance: none;
+  align-items: center;
+  border: 1px solid transparent;
+  border-radius: 0.4em;
+  box-shadow: none;
+  display: inline-flex;
+  font-size: 15px;
+  height: 2.25em;
+  justify-content: flex-start;
+  line-height: 1.5;
+  padding-bottom: calc(0.375em - 1px);
+  padding-left: calc(0.625em - 1px);
+  padding-right: calc(0.625em - 1px);
+  padding-top: calc(0.375em - 1px);
+  position: relative;
+  vertical-align: top; }
+  html.theme--documenter-dark .button:focus, html.theme--documenter-dark .input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:focus, html.theme--documenter-dark .textarea:focus, html.theme--documenter-dark .select select:focus, html.theme--documenter-dark .file-cta:focus,
+  html.theme--documenter-dark .file-name:focus, html.theme--documenter-dark .pagination-previous:focus,
+  html.theme--documenter-dark .pagination-next:focus,
+  html.theme--documenter-dark .pagination-link:focus,
+  html.theme--documenter-dark .pagination-ellipsis:focus, html.theme--documenter-dark .is-focused.button, html.theme--documenter-dark .is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-focused.textarea, html.theme--documenter-dark .select select.is-focused, html.theme--documenter-dark .is-focused.file-cta,
+  html.theme--documenter-dark .is-focused.file-name, html.theme--documenter-dark .is-focused.pagination-previous,
+  html.theme--documenter-dark .is-focused.pagination-next,
+  html.theme--documenter-dark .is-focused.pagination-link,
+  html.theme--documenter-dark .is-focused.pagination-ellipsis, html.theme--documenter-dark .button:active, html.theme--documenter-dark .input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:active, html.theme--documenter-dark .textarea:active, html.theme--documenter-dark .select select:active, html.theme--documenter-dark .file-cta:active,
+  html.theme--documenter-dark .file-name:active, html.theme--documenter-dark .pagination-previous:active,
+  html.theme--documenter-dark .pagination-next:active,
+  html.theme--documenter-dark .pagination-link:active,
+  html.theme--documenter-dark .pagination-ellipsis:active, html.theme--documenter-dark .is-active.button, html.theme--documenter-dark .is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active, html.theme--documenter-dark .is-active.textarea, html.theme--documenter-dark .select select.is-active, html.theme--documenter-dark .is-active.file-cta,
+  html.theme--documenter-dark .is-active.file-name, html.theme--documenter-dark .is-active.pagination-previous,
+  html.theme--documenter-dark .is-active.pagination-next,
+  html.theme--documenter-dark .is-active.pagination-link,
+  html.theme--documenter-dark .is-active.pagination-ellipsis {
+    outline: none; }
+  html.theme--documenter-dark .button[disabled], html.theme--documenter-dark .input[disabled], html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled], html.theme--documenter-dark .textarea[disabled], html.theme--documenter-dark .select select[disabled], html.theme--documenter-dark .file-cta[disabled],
+  html.theme--documenter-dark .file-name[disabled], html.theme--documenter-dark .pagination-previous[disabled],
+  html.theme--documenter-dark .pagination-next[disabled],
+  html.theme--documenter-dark .pagination-link[disabled],
+  html.theme--documenter-dark .pagination-ellipsis[disabled],
+  fieldset[disabled] html.theme--documenter-dark .button,
+  html.theme--documenter-dark fieldset[disabled] .button,
+  fieldset[disabled] html.theme--documenter-dark .input,
+  html.theme--documenter-dark fieldset[disabled] .input,
+  fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input,
+  html.theme--documenter-dark #documenter .docs-sidebar fieldset[disabled] form.docs-search > input,
+  fieldset[disabled] html.theme--documenter-dark .textarea,
+  html.theme--documenter-dark fieldset[disabled] .textarea,
+  fieldset[disabled] html.theme--documenter-dark .select select,
+  html.theme--documenter-dark .select fieldset[disabled] select,
+  fieldset[disabled] html.theme--documenter-dark .file-cta,
+  html.theme--documenter-dark fieldset[disabled] .file-cta,
+  fieldset[disabled] html.theme--documenter-dark .file-name,
+  html.theme--documenter-dark fieldset[disabled] .file-name,
+  fieldset[disabled] html.theme--documenter-dark .pagination-previous,
+  html.theme--documenter-dark fieldset[disabled] .pagination-previous,
+  fieldset[disabled] html.theme--documenter-dark .pagination-next,
+  html.theme--documenter-dark fieldset[disabled] .pagination-next,
+  fieldset[disabled] html.theme--documenter-dark .pagination-link,
+  html.theme--documenter-dark fieldset[disabled] .pagination-link,
+  fieldset[disabled] html.theme--documenter-dark .pagination-ellipsis,
+  html.theme--documenter-dark fieldset[disabled] .pagination-ellipsis {
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+
+/*! minireset.css v0.0.4 | MIT License | github.com/jgthms/minireset.css */
+html,
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+pre,
+iframe,
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+h2,
+h3,
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+  padding: 0; }
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+ul {
+  list-style: none; }
+
+button,
+input,
+select,
+textarea {
+  margin: 0; }
+
+html {
+  box-sizing: border-box; }
+
+*, *::before, *::after {
+  box-sizing: inherit; }
+
+img,
+embed,
+iframe,
+object,
+video {
+  height: auto;
+  max-width: 100%; }
+
+audio {
+  max-width: 100%; }
+
+iframe {
+  border: 0; }
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+table {
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+td,
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+  font-size: 0.85em !important; }
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+@media screen and (max-width: 768px) {
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+  .is-size-7-mobile {
+    font-size: 0.85em !important; } }
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+@media screen and (min-width: 769px), print {
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+    font-size: 0.85em !important; } }
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+@media screen and (min-width: 1056px) {
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+    font-size: 1.25rem !important; }
+  .is-size-6-widescreen {
+    font-size: 15px !important; }
+  .is-size-7-widescreen {
+    font-size: 0.85em !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-size-1-fullhd {
+    font-size: 3rem !important; }
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+    font-size: 1.25rem !important; }
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+    text-align: center !important; } }
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+  .has-text-centered-tablet {
+    text-align: center !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .has-text-centered-tablet-only {
+    text-align: center !important; } }
+
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+  .has-text-centered-touch {
+    text-align: center !important; } }
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+    text-align: center !important; } }
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+    text-align: center !important; } }
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+  .has-text-centered-widescreen {
+    text-align: center !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .has-text-centered-widescreen-only {
+    text-align: center !important; } }
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+    text-align: center !important; } }
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+@media screen and (max-width: 768px) {
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+    text-align: justify !important; } }
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+@media screen and (min-width: 769px), print {
+  .has-text-justified-tablet {
+    text-align: justify !important; } }
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+@media screen and (min-width: 769px) and (max-width: 1055px) {
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+    text-align: justify !important; } }
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+    text-align: justify !important; } }
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+@media screen and (min-width: 1056px) {
+  .has-text-justified-desktop {
+    text-align: justify !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .has-text-justified-desktop-only {
+    text-align: justify !important; } }
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+@media screen and (min-width: 1216px) {
+  .has-text-justified-widescreen {
+    text-align: justify !important; } }
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+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .has-text-justified-widescreen-only {
+    text-align: justify !important; } }
+
+@media screen and (min-width: 1408px) {
+  .has-text-justified-fullhd {
+    text-align: justify !important; } }
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+@media screen and (max-width: 768px) {
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+    text-align: left !important; } }
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+@media screen and (min-width: 769px), print {
+  .has-text-left-tablet {
+    text-align: left !important; } }
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+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .has-text-left-tablet-only {
+    text-align: left !important; } }
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+    text-align: left !important; } }
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+    text-align: left !important; } }
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+  .has-text-left-desktop-only {
+    text-align: left !important; } }
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+@media screen and (min-width: 1216px) {
+  .has-text-left-widescreen {
+    text-align: left !important; } }
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+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .has-text-left-widescreen-only {
+    text-align: left !important; } }
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+@media screen and (min-width: 1408px) {
+  .has-text-left-fullhd {
+    text-align: left !important; } }
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+@media screen and (max-width: 768px) {
+  .has-text-right-mobile {
+    text-align: right !important; } }
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+@media screen and (min-width: 769px), print {
+  .has-text-right-tablet {
+    text-align: right !important; } }
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+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .has-text-right-tablet-only {
+    text-align: right !important; } }
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+    text-align: right !important; } }
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+@media screen and (min-width: 1056px) {
+  .has-text-right-desktop {
+    text-align: right !important; } }
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+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .has-text-right-desktop-only {
+    text-align: right !important; } }
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+  .has-text-right-widescreen {
+    text-align: right !important; } }
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+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .has-text-right-widescreen-only {
+    text-align: right !important; } }
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+  .has-text-right-fullhd {
+    text-align: right !important; } }
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+.is-capitalized {
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+.is-lowercase {
+  text-transform: lowercase !important; }
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+.is-uppercase {
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+  font-style: italic !important; }
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+.has-text-white {
+  color: white !important; }
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+a.has-text-white:hover, a.has-text-white:focus {
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+.has-text-black-bis {
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+.has-text-white-ter {
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+.has-background-white-ter {
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+.has-text-white-bis {
+  color: #fafafa !important; }
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+.has-background-white-bis {
+  background-color: #fafafa !important; }
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+.has-text-weight-light {
+  font-weight: 300 !important; }
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+.has-text-weight-normal {
+  font-weight: 400 !important; }
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+.has-text-weight-medium {
+  font-weight: 500 !important; }
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+.has-text-weight-semibold {
+  font-weight: 600 !important; }
+
+.has-text-weight-bold {
+  font-weight: 700 !important; }
+
+.is-family-primary {
+  font-family: "Lato", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif !important; }
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+.is-family-secondary {
+  font-family: "Lato", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif !important; }
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+.is-family-sans-serif {
+  font-family: "Lato", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif !important; }
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+.is-family-monospace {
+  font-family: "Roboto Mono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace !important; }
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+  font-family: "Roboto Mono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace !important; }
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+.is-block {
+  display: block !important; }
+
+@media screen and (max-width: 768px) {
+  .is-block-mobile {
+    display: block !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-block-tablet {
+    display: block !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-block-tablet-only {
+    display: block !important; } }
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+@media screen and (max-width: 1055px) {
+  .is-block-touch {
+    display: block !important; } }
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+@media screen and (min-width: 1056px) {
+  .is-block-desktop {
+    display: block !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-block-desktop-only {
+    display: block !important; } }
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+@media screen and (min-width: 1216px) {
+  .is-block-widescreen {
+    display: block !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-block-widescreen-only {
+    display: block !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-block-fullhd {
+    display: block !important; } }
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+.is-flex {
+  display: flex !important; }
+
+@media screen and (max-width: 768px) {
+  .is-flex-mobile {
+    display: flex !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-flex-tablet {
+    display: flex !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-flex-tablet-only {
+    display: flex !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-flex-touch {
+    display: flex !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-flex-desktop {
+    display: flex !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-flex-desktop-only {
+    display: flex !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-flex-widescreen {
+    display: flex !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-flex-widescreen-only {
+    display: flex !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-flex-fullhd {
+    display: flex !important; } }
+
+.is-inline {
+  display: inline !important; }
+
+@media screen and (max-width: 768px) {
+  .is-inline-mobile {
+    display: inline !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-inline-tablet {
+    display: inline !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-inline-tablet-only {
+    display: inline !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-inline-touch {
+    display: inline !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-inline-desktop {
+    display: inline !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-inline-desktop-only {
+    display: inline !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-inline-widescreen {
+    display: inline !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-inline-widescreen-only {
+    display: inline !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-inline-fullhd {
+    display: inline !important; } }
+
+.is-inline-block {
+  display: inline-block !important; }
+
+@media screen and (max-width: 768px) {
+  .is-inline-block-mobile {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-inline-block-tablet {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-inline-block-tablet-only {
+    display: inline-block !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-inline-block-touch {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-inline-block-desktop {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-inline-block-desktop-only {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-inline-block-widescreen {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-inline-block-widescreen-only {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-inline-block-fullhd {
+    display: inline-block !important; } }
+
+.is-inline-flex {
+  display: inline-flex !important; }
+
+@media screen and (max-width: 768px) {
+  .is-inline-flex-mobile {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-inline-flex-tablet {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-inline-flex-tablet-only {
+    display: inline-flex !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-inline-flex-touch {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-inline-flex-desktop {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-inline-flex-desktop-only {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-inline-flex-widescreen {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-inline-flex-widescreen-only {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-inline-flex-fullhd {
+    display: inline-flex !important; } }
+
+.is-hidden {
+  display: none !important; }
+
+.is-sr-only {
+  border: none !important;
+  clip: rect(0, 0, 0, 0) !important;
+  height: 0.01em !important;
+  overflow: hidden !important;
+  padding: 0 !important;
+  position: absolute !important;
+  white-space: nowrap !important;
+  width: 0.01em !important; }
+
+@media screen and (max-width: 768px) {
+  .is-hidden-mobile {
+    display: none !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-hidden-tablet {
+    display: none !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-hidden-tablet-only {
+    display: none !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-hidden-touch {
+    display: none !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-hidden-desktop {
+    display: none !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-hidden-desktop-only {
+    display: none !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-hidden-widescreen {
+    display: none !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-hidden-widescreen-only {
+    display: none !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-hidden-fullhd {
+    display: none !important; } }
+
+.is-invisible {
+  visibility: hidden !important; }
+
+@media screen and (max-width: 768px) {
+  .is-invisible-mobile {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-invisible-tablet {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-invisible-tablet-only {
+    visibility: hidden !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-invisible-touch {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-invisible-desktop {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-invisible-desktop-only {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-invisible-widescreen {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-invisible-widescreen-only {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-invisible-fullhd {
+    visibility: hidden !important; } }
+
+.is-marginless {
+  margin: 0 !important; }
+
+.is-paddingless {
+  padding: 0 !important; }
+
+.is-radiusless {
+  border-radius: 0 !important; }
+
+.is-shadowless {
+  box-shadow: none !important; }
+
+.is-relative {
+  position: relative !important; }
+
+html.theme--documenter-dark {
+  /* This file contain the overall layout.
+ *
+ * The main container is <div> that is identified by id #documenter.
+ */
+  /* a11y-dark theme */
+  /* Based on the Tomorrow Night Eighties theme: https://github.com/isagalaev/highlight.js/blob/master/src/styles/tomorrow-night-eighties.css */
+  /* @author: ericwbailey */
+  /* Comment */
+  /* Red */
+  /* Orange */
+  /* Yellow */
+  /* Green */
+  /* Blue */
+  /* Purple */ }
+  html.theme--documenter-dark html {
+    background-color: #1f2424;
+    font-size: 16px;
+    -moz-osx-font-smoothing: grayscale;
+    -webkit-font-smoothing: antialiased;
+    min-width: 300px;
+    overflow-x: auto;
+    overflow-y: scroll;
+    text-rendering: optimizeLegibility;
+    text-size-adjust: 100%; }
+  html.theme--documenter-dark article,
+  html.theme--documenter-dark aside,
+  html.theme--documenter-dark figure,
+  html.theme--documenter-dark footer,
+  html.theme--documenter-dark header,
+  html.theme--documenter-dark hgroup,
+  html.theme--documenter-dark section {
+    display: block; }
+  html.theme--documenter-dark body,
+  html.theme--documenter-dark button,
+  html.theme--documenter-dark input,
+  html.theme--documenter-dark select,
+  html.theme--documenter-dark textarea {
+    font-family: "Lato", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif; }
+  html.theme--documenter-dark code,
+  html.theme--documenter-dark pre {
+    -moz-osx-font-smoothing: auto;
+    -webkit-font-smoothing: auto;
+    font-family: "Roboto Mono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace; }
+  html.theme--documenter-dark body {
+    color: #fff;
+    font-size: 1em;
+    font-weight: 400;
+    line-height: 1.5; }
+  html.theme--documenter-dark a {
+    color: #1abc9c;
+    cursor: pointer;
+    text-decoration: none; }
+    html.theme--documenter-dark a strong {
+      color: currentColor; }
+    html.theme--documenter-dark a:hover {
+      color: #1dd2af; }
+  html.theme--documenter-dark code {
+    background-color: rgba(255, 255, 255, 0.05);
+    color: #e74c3c;
+    font-size: 0.875em;
+    font-weight: normal;
+    padding: 0.1em; }
+  html.theme--documenter-dark hr {
+    background-color: #282f2f;
+    border: none;
+    display: block;
+    height: 2px;
+    margin: 1.5rem 0; }
+  html.theme--documenter-dark img {
+    height: auto;
+    max-width: 100%; }
+  html.theme--documenter-dark input[type="checkbox"],
+  html.theme--documenter-dark input[type="radio"] {
+    vertical-align: baseline; }
+  html.theme--documenter-dark small {
+    font-size: 0.875em; }
+  html.theme--documenter-dark span {
+    font-style: inherit;
+    font-weight: inherit; }
+  html.theme--documenter-dark strong {
+    color: #f2f2f2;
+    font-weight: 700; }
+  html.theme--documenter-dark fieldset {
+    border: none; }
+  html.theme--documenter-dark pre {
+    -webkit-overflow-scrolling: touch;
+    background-color: #282f2f;
+    color: #fff;
+    font-size: 0.875em;
+    overflow-x: auto;
+    padding: 1.25rem 1.5rem;
+    white-space: pre;
+    word-wrap: normal; }
+    html.theme--documenter-dark pre code {
+      background-color: transparent;
+      color: currentColor;
+      font-size: 1em;
+      padding: 0; }
+  html.theme--documenter-dark table td,
+  html.theme--documenter-dark table th {
+    vertical-align: top; }
+    html.theme--documenter-dark table td:not([align]),
+    html.theme--documenter-dark table th:not([align]) {
+      text-align: left; }
+  html.theme--documenter-dark table th {
+    color: #f2f2f2; }
+  html.theme--documenter-dark .box {
+    background-color: #343c3d;
+    border-radius: 8px;
+    box-shadow: none;
+    color: #fff;
+    display: block;
+    padding: 1.25rem; }
+  html.theme--documenter-dark a.box:hover, html.theme--documenter-dark a.box:focus {
+    box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px #1abc9c; }
+  html.theme--documenter-dark a.box:active {
+    box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2), 0 0 0 1px #1abc9c; }
+  html.theme--documenter-dark .button {
+    background-color: #282f2f;
+    border-color: #4c5759;
+    border-width: 1px;
+    color: #375a7f;
+    cursor: pointer;
+    justify-content: center;
+    padding-bottom: calc(0.375em - 1px);
+    padding-left: 0.75em;
+    padding-right: 0.75em;
+    padding-top: calc(0.375em - 1px);
+    text-align: center;
+    white-space: nowrap; }
+    html.theme--documenter-dark .button strong {
+      color: inherit; }
+    html.theme--documenter-dark .button .icon, html.theme--documenter-dark .button .icon.is-small, html.theme--documenter-dark .button #documenter .docs-sidebar form.docs-search > input.icon, html.theme--documenter-dark #documenter .docs-sidebar .button form.docs-search > input.icon, html.theme--documenter-dark .button .icon.is-medium, html.theme--documenter-dark .button .icon.is-large {
+      height: 1.5em;
+      width: 1.5em; }
+    html.theme--documenter-dark .button .icon:first-child:not(:last-child) {
+      margin-left: calc(-0.375em - 1px);
+      margin-right: 0.1875em; }
+    html.theme--documenter-dark .button .icon:last-child:not(:first-child) {
+      margin-left: 0.1875em;
+      margin-right: calc(-0.375em - 1px); }
+    html.theme--documenter-dark .button .icon:first-child:last-child {
+      margin-left: calc(-0.375em - 1px);
+      margin-right: calc(-0.375em - 1px); }
+    html.theme--documenter-dark .button:hover, html.theme--documenter-dark .button.is-hovered {
+      border-color: #8c9b9d;
+      color: #f2f2f2; }
+    html.theme--documenter-dark .button:focus, html.theme--documenter-dark .button.is-focused {
+      border-color: #8c9b9d;
+      color: #17a689; }
+      html.theme--documenter-dark .button:focus:not(:active), html.theme--documenter-dark .button.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); }
+    html.theme--documenter-dark .button:active, html.theme--documenter-dark .button.is-active {
+      border-color: #343c3d;
+      color: #f2f2f2; }
+    html.theme--documenter-dark .button.is-text {
+      background-color: transparent;
+      border-color: transparent;
+      color: #fff;
+      text-decoration: underline; }
+      html.theme--documenter-dark .button.is-text:hover, html.theme--documenter-dark .button.is-text.is-hovered, html.theme--documenter-dark .button.is-text:focus, html.theme--documenter-dark .button.is-text.is-focused {
+        background-color: #282f2f;
+        color: #f2f2f2; }
+      html.theme--documenter-dark .button.is-text:active, html.theme--documenter-dark .button.is-text.is-active {
+        background-color: #1d2122;
+        color: #f2f2f2; }
+      html.theme--documenter-dark .button.is-text[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-text {
+        background-color: transparent;
+        border-color: transparent;
+        box-shadow: none; }
+    html.theme--documenter-dark .button.is-white {
+      background-color: white;
+      border-color: transparent;
+      color: #0a0a0a; }
+      html.theme--documenter-dark .button.is-white:hover, html.theme--documenter-dark .button.is-white.is-hovered {
+        background-color: #f9f9f9;
+        border-color: transparent;
+        color: #0a0a0a; }
+      html.theme--documenter-dark .button.is-white:focus, html.theme--documenter-dark .button.is-white.is-focused {
+        border-color: transparent;
+        color: #0a0a0a; }
+        html.theme--documenter-dark .button.is-white:focus:not(:active), html.theme--documenter-dark .button.is-white.is-focused:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); }
+      html.theme--documenter-dark .button.is-white:active, html.theme--documenter-dark .button.is-white.is-active {
+        background-color: #f2f2f2;
+        border-color: transparent;
+        color: #0a0a0a; }
+      html.theme--documenter-dark .button.is-white[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-white {
+        background-color: white;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-white.is-inverted {
+        background-color: #0a0a0a;
+        color: white; }
+        html.theme--documenter-dark .button.is-white.is-inverted:hover, html.theme--documenter-dark .button.is-white.is-inverted.is-hovered {
+          background-color: black; }
+        html.theme--documenter-dark .button.is-white.is-inverted[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted {
+          background-color: #0a0a0a;
+          border-color: transparent;
+          box-shadow: none;
+          color: white; }
+      html.theme--documenter-dark .button.is-white.is-loading::after {
+        border-color: transparent transparent #0a0a0a #0a0a0a !important; }
+      html.theme--documenter-dark .button.is-white.is-outlined {
+        background-color: transparent;
+        border-color: white;
+        color: white; }
+        html.theme--documenter-dark .button.is-white.is-outlined:hover, html.theme--documenter-dark .button.is-white.is-outlined.is-hovered, html.theme--documenter-dark .button.is-white.is-outlined:focus, html.theme--documenter-dark .button.is-white.is-outlined.is-focused {
+          background-color: white;
+          border-color: white;
+          color: #0a0a0a; }
+        html.theme--documenter-dark .button.is-white.is-outlined.is-loading::after {
+          border-color: transparent transparent white white !important; }
+        html.theme--documenter-dark .button.is-white.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-white.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #0a0a0a #0a0a0a !important; }
+        html.theme--documenter-dark .button.is-white.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined {
+          background-color: transparent;
+          border-color: white;
+          box-shadow: none;
+          color: white; }
+      html.theme--documenter-dark .button.is-white.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #0a0a0a;
+        color: #0a0a0a; }
+        html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-focused {
+          background-color: #0a0a0a;
+          color: white; }
+        html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent white white !important; }
+        html.theme--documenter-dark .button.is-white.is-inverted.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted.is-outlined {
+          background-color: transparent;
+          border-color: #0a0a0a;
+          box-shadow: none;
+          color: #0a0a0a; }
+    html.theme--documenter-dark .button.is-black {
+      background-color: #0a0a0a;
+      border-color: transparent;
+      color: white; }
+      html.theme--documenter-dark .button.is-black:hover, html.theme--documenter-dark .button.is-black.is-hovered {
+        background-color: #040404;
+        border-color: transparent;
+        color: white; }
+      html.theme--documenter-dark .button.is-black:focus, html.theme--documenter-dark .button.is-black.is-focused {
+        border-color: transparent;
+        color: white; }
+        html.theme--documenter-dark .button.is-black:focus:not(:active), html.theme--documenter-dark .button.is-black.is-focused:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); }
+      html.theme--documenter-dark .button.is-black:active, html.theme--documenter-dark .button.is-black.is-active {
+        background-color: black;
+        border-color: transparent;
+        color: white; }
+      html.theme--documenter-dark .button.is-black[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-black {
+        background-color: #0a0a0a;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-black.is-inverted {
+        background-color: white;
+        color: #0a0a0a; }
+        html.theme--documenter-dark .button.is-black.is-inverted:hover, html.theme--documenter-dark .button.is-black.is-inverted.is-hovered {
+          background-color: #f2f2f2; }
+        html.theme--documenter-dark .button.is-black.is-inverted[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted {
+          background-color: white;
+          border-color: transparent;
+          box-shadow: none;
+          color: #0a0a0a; }
+      html.theme--documenter-dark .button.is-black.is-loading::after {
+        border-color: transparent transparent white white !important; }
+      html.theme--documenter-dark .button.is-black.is-outlined {
+        background-color: transparent;
+        border-color: #0a0a0a;
+        color: #0a0a0a; }
+        html.theme--documenter-dark .button.is-black.is-outlined:hover, html.theme--documenter-dark .button.is-black.is-outlined.is-hovered, html.theme--documenter-dark .button.is-black.is-outlined:focus, html.theme--documenter-dark .button.is-black.is-outlined.is-focused {
+          background-color: #0a0a0a;
+          border-color: #0a0a0a;
+          color: white; }
+        html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after {
+          border-color: transparent transparent #0a0a0a #0a0a0a !important; }
+        html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent white white !important; }
+        html.theme--documenter-dark .button.is-black.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined {
+          background-color: transparent;
+          border-color: #0a0a0a;
+          box-shadow: none;
+          color: #0a0a0a; }
+      html.theme--documenter-dark .button.is-black.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: white;
+        color: white; }
+        html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused {
+          background-color: white;
+          color: #0a0a0a; }
+        html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #0a0a0a #0a0a0a !important; }
+        html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined {
+          background-color: transparent;
+          border-color: white;
+          box-shadow: none;
+          color: white; }
+    html.theme--documenter-dark .button.is-light {
+      background-color: #ecf0f1;
+      border-color: transparent;
+      color: #282f2f; }
+      html.theme--documenter-dark .button.is-light:hover, html.theme--documenter-dark .button.is-light.is-hovered {
+        background-color: #e5eaec;
+        border-color: transparent;
+        color: #282f2f; }
+      html.theme--documenter-dark .button.is-light:focus, html.theme--documenter-dark .button.is-light.is-focused {
+        border-color: transparent;
+        color: #282f2f; }
+        html.theme--documenter-dark .button.is-light:focus:not(:active), html.theme--documenter-dark .button.is-light.is-focused:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); }
+      html.theme--documenter-dark .button.is-light:active, html.theme--documenter-dark .button.is-light.is-active {
+        background-color: #dde4e6;
+        border-color: transparent;
+        color: #282f2f; }
+      html.theme--documenter-dark .button.is-light[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-light {
+        background-color: #ecf0f1;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-light.is-inverted {
+        background-color: #282f2f;
+        color: #ecf0f1; }
+        html.theme--documenter-dark .button.is-light.is-inverted:hover, html.theme--documenter-dark .button.is-light.is-inverted.is-hovered {
+          background-color: #1d2122; }
+        html.theme--documenter-dark .button.is-light.is-inverted[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted {
+          background-color: #282f2f;
+          border-color: transparent;
+          box-shadow: none;
+          color: #ecf0f1; }
+      html.theme--documenter-dark .button.is-light.is-loading::after {
+        border-color: transparent transparent #282f2f #282f2f !important; }
+      html.theme--documenter-dark .button.is-light.is-outlined {
+        background-color: transparent;
+        border-color: #ecf0f1;
+        color: #ecf0f1; }
+        html.theme--documenter-dark .button.is-light.is-outlined:hover, html.theme--documenter-dark .button.is-light.is-outlined.is-hovered, html.theme--documenter-dark .button.is-light.is-outlined:focus, html.theme--documenter-dark .button.is-light.is-outlined.is-focused {
+          background-color: #ecf0f1;
+          border-color: #ecf0f1;
+          color: #282f2f; }
+        html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after {
+          border-color: transparent transparent #ecf0f1 #ecf0f1 !important; }
+        html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #282f2f #282f2f !important; }
+        html.theme--documenter-dark .button.is-light.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined {
+          background-color: transparent;
+          border-color: #ecf0f1;
+          box-shadow: none;
+          color: #ecf0f1; }
+      html.theme--documenter-dark .button.is-light.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #282f2f;
+        color: #282f2f; }
+        html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused {
+          background-color: #282f2f;
+          color: #ecf0f1; }
+        html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #ecf0f1 #ecf0f1 !important; }
+        html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined {
+          background-color: transparent;
+          border-color: #282f2f;
+          box-shadow: none;
+          color: #282f2f; }
+    html.theme--documenter-dark .button.is-dark, html.theme--documenter-dark .content kbd.button {
+      background-color: #282f2f;
+      border-color: transparent;
+      color: #ecf0f1; }
+      html.theme--documenter-dark .button.is-dark:hover, html.theme--documenter-dark .content kbd.button:hover, html.theme--documenter-dark .button.is-dark.is-hovered, html.theme--documenter-dark .content kbd.button.is-hovered {
+        background-color: #232829;
+        border-color: transparent;
+        color: #ecf0f1; }
+      html.theme--documenter-dark .button.is-dark:focus, html.theme--documenter-dark .content kbd.button:focus, html.theme--documenter-dark .button.is-dark.is-focused, html.theme--documenter-dark .content kbd.button.is-focused {
+        border-color: transparent;
+        color: #ecf0f1; }
+        html.theme--documenter-dark .button.is-dark:focus:not(:active), html.theme--documenter-dark .content kbd.button:focus:not(:active), html.theme--documenter-dark .button.is-dark.is-focused:not(:active), html.theme--documenter-dark .content kbd.button.is-focused:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); }
+      html.theme--documenter-dark .button.is-dark:active, html.theme--documenter-dark .content kbd.button:active, html.theme--documenter-dark .button.is-dark.is-active, html.theme--documenter-dark .content kbd.button.is-active {
+        background-color: #1d2122;
+        border-color: transparent;
+        color: #ecf0f1; }
+      html.theme--documenter-dark .button.is-dark[disabled], html.theme--documenter-dark .content kbd.button[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-dark,
+      fieldset[disabled] html.theme--documenter-dark .content kbd.button {
+        background-color: #282f2f;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-dark.is-inverted, html.theme--documenter-dark .content kbd.button.is-inverted {
+        background-color: #ecf0f1;
+        color: #282f2f; }
+        html.theme--documenter-dark .button.is-dark.is-inverted:hover, html.theme--documenter-dark .content kbd.button.is-inverted:hover, html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered, html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered {
+          background-color: #dde4e6; }
+        html.theme--documenter-dark .button.is-dark.is-inverted[disabled], html.theme--documenter-dark .content kbd.button.is-inverted[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,
+        fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted {
+          background-color: #ecf0f1;
+          border-color: transparent;
+          box-shadow: none;
+          color: #282f2f; }
+      html.theme--documenter-dark .button.is-dark.is-loading::after, html.theme--documenter-dark .content kbd.button.is-loading::after {
+        border-color: transparent transparent #ecf0f1 #ecf0f1 !important; }
+      html.theme--documenter-dark .button.is-dark.is-outlined, html.theme--documenter-dark .content kbd.button.is-outlined {
+        background-color: transparent;
+        border-color: #282f2f;
+        color: #282f2f; }
+        html.theme--documenter-dark .button.is-dark.is-outlined:hover, html.theme--documenter-dark .content kbd.button.is-outlined:hover, html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered, html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered, html.theme--documenter-dark .button.is-dark.is-outlined:focus, html.theme--documenter-dark .content kbd.button.is-outlined:focus, html.theme--documenter-dark .button.is-dark.is-outlined.is-focused, html.theme--documenter-dark .content kbd.button.is-outlined.is-focused {
+          background-color: #282f2f;
+          border-color: #282f2f;
+          color: #ecf0f1; }
+        html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after {
+          border-color: transparent transparent #282f2f #282f2f !important; }
+        html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #ecf0f1 #ecf0f1 !important; }
+        html.theme--documenter-dark .button.is-dark.is-outlined[disabled], html.theme--documenter-dark .content kbd.button.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,
+        fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined {
+          background-color: transparent;
+          border-color: #282f2f;
+          box-shadow: none;
+          color: #282f2f; }
+      html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #ecf0f1;
+        color: #ecf0f1; }
+        html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused {
+          background-color: #ecf0f1;
+          color: #282f2f; }
+        html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #282f2f #282f2f !important; }
+        html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled], html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,
+        fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined {
+          background-color: transparent;
+          border-color: #ecf0f1;
+          box-shadow: none;
+          color: #ecf0f1; }
+    html.theme--documenter-dark .button.is-primary, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink {
+      background-color: #375a7f;
+      border-color: transparent;
+      color: #fff; }
+      html.theme--documenter-dark .button.is-primary:hover, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-hovered.docs-sourcelink {
+        background-color: #335476;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-primary:focus, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink {
+        border-color: transparent;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-primary:focus:not(:active), html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus:not(:active), html.theme--documenter-dark .button.is-primary.is-focused:not(:active), html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); }
+      html.theme--documenter-dark .button.is-primary:active, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:active, html.theme--documenter-dark .button.is-primary.is-active, html.theme--documenter-dark .docstring > section > a.button.is-active.docs-sourcelink {
+        background-color: #2f4d6d;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-primary[disabled], html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-primary,
+      fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink {
+        background-color: #375a7f;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-primary.is-inverted, html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink {
+        background-color: #fff;
+        color: #375a7f; }
+        html.theme--documenter-dark .button.is-primary.is-inverted:hover, html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-inverted.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-hovered.docs-sourcelink {
+          background-color: #f2f2f2; }
+        html.theme--documenter-dark .button.is-primary.is-inverted[disabled], html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted,
+        fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink {
+          background-color: #fff;
+          border-color: transparent;
+          box-shadow: none;
+          color: #375a7f; }
+      html.theme--documenter-dark .button.is-primary.is-loading::after, html.theme--documenter-dark .docstring > section > a.button.is-loading.docs-sourcelink::after {
+        border-color: transparent transparent #fff #fff !important; }
+      html.theme--documenter-dark .button.is-primary.is-outlined, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink {
+        background-color: transparent;
+        border-color: #375a7f;
+        color: #375a7f; }
+        html.theme--documenter-dark .button.is-primary.is-outlined:hover, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-outlined.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-outlined:focus, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-outlined.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-focused.docs-sourcelink {
+          background-color: #375a7f;
+          border-color: #375a7f;
+          color: #fff; }
+        html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink::after {
+          border-color: transparent transparent #375a7f #375a7f !important; }
+        html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:hover::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:focus::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:focus::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after {
+          border-color: transparent transparent #fff #fff !important; }
+        html.theme--documenter-dark .button.is-primary.is-outlined[disabled], html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-outlined,
+        fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink {
+          background-color: transparent;
+          border-color: #375a7f;
+          box-shadow: none;
+          color: #375a7f; }
+      html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink {
+        background-color: transparent;
+        border-color: #fff;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined:hover, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined:focus, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-focused.docs-sourcelink {
+          background-color: #fff;
+          color: #375a7f; }
+        html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after {
+          border-color: transparent transparent #375a7f #375a7f !important; }
+        html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined[disabled], html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined,
+        fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink {
+          background-color: transparent;
+          border-color: #fff;
+          box-shadow: none;
+          color: #fff; }
+    html.theme--documenter-dark .button.is-link {
+      background-color: #1abc9c;
+      border-color: transparent;
+      color: #fff; }
+      html.theme--documenter-dark .button.is-link:hover, html.theme--documenter-dark .button.is-link.is-hovered {
+        background-color: #18b193;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-link:focus, html.theme--documenter-dark .button.is-link.is-focused {
+        border-color: transparent;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-link:focus:not(:active), html.theme--documenter-dark .button.is-link.is-focused:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); }
+      html.theme--documenter-dark .button.is-link:active, html.theme--documenter-dark .button.is-link.is-active {
+        background-color: #17a689;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-link[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-link {
+        background-color: #1abc9c;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-link.is-inverted {
+        background-color: #fff;
+        color: #1abc9c; }
+        html.theme--documenter-dark .button.is-link.is-inverted:hover, html.theme--documenter-dark .button.is-link.is-inverted.is-hovered {
+          background-color: #f2f2f2; }
+        html.theme--documenter-dark .button.is-link.is-inverted[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted {
+          background-color: #fff;
+          border-color: transparent;
+          box-shadow: none;
+          color: #1abc9c; }
+      html.theme--documenter-dark .button.is-link.is-loading::after {
+        border-color: transparent transparent #fff #fff !important; }
+      html.theme--documenter-dark .button.is-link.is-outlined {
+        background-color: transparent;
+        border-color: #1abc9c;
+        color: #1abc9c; }
+        html.theme--documenter-dark .button.is-link.is-outlined:hover, html.theme--documenter-dark .button.is-link.is-outlined.is-hovered, html.theme--documenter-dark .button.is-link.is-outlined:focus, html.theme--documenter-dark .button.is-link.is-outlined.is-focused {
+          background-color: #1abc9c;
+          border-color: #1abc9c;
+          color: #fff; }
+        html.theme--documenter-dark .button.is-link.is-outlined.is-loading::after {
+          border-color: transparent transparent #1abc9c #1abc9c !important; }
+        html.theme--documenter-dark .button.is-link.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #fff #fff !important; }
+        html.theme--documenter-dark .button.is-link.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-link.is-outlined {
+          background-color: transparent;
+          border-color: #1abc9c;
+          box-shadow: none;
+          color: #1abc9c; }
+      html.theme--documenter-dark .button.is-link.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #fff;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-focused {
+          background-color: #fff;
+          color: #1abc9c; }
+        html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #1abc9c #1abc9c !important; }
+        html.theme--documenter-dark .button.is-link.is-inverted.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted.is-outlined {
+          background-color: transparent;
+          border-color: #fff;
+          box-shadow: none;
+          color: #fff; }
+    html.theme--documenter-dark .button.is-info {
+      background-color: #024c7d;
+      border-color: transparent;
+      color: #fff; }
+      html.theme--documenter-dark .button.is-info:hover, html.theme--documenter-dark .button.is-info.is-hovered {
+        background-color: #024470;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-info:focus, html.theme--documenter-dark .button.is-info.is-focused {
+        border-color: transparent;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-info:focus:not(:active), html.theme--documenter-dark .button.is-info.is-focused:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); }
+      html.theme--documenter-dark .button.is-info:active, html.theme--documenter-dark .button.is-info.is-active {
+        background-color: #023d64;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-info[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-info {
+        background-color: #024c7d;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-info.is-inverted {
+        background-color: #fff;
+        color: #024c7d; }
+        html.theme--documenter-dark .button.is-info.is-inverted:hover, html.theme--documenter-dark .button.is-info.is-inverted.is-hovered {
+          background-color: #f2f2f2; }
+        html.theme--documenter-dark .button.is-info.is-inverted[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted {
+          background-color: #fff;
+          border-color: transparent;
+          box-shadow: none;
+          color: #024c7d; }
+      html.theme--documenter-dark .button.is-info.is-loading::after {
+        border-color: transparent transparent #fff #fff !important; }
+      html.theme--documenter-dark .button.is-info.is-outlined {
+        background-color: transparent;
+        border-color: #024c7d;
+        color: #024c7d; }
+        html.theme--documenter-dark .button.is-info.is-outlined:hover, html.theme--documenter-dark .button.is-info.is-outlined.is-hovered, html.theme--documenter-dark .button.is-info.is-outlined:focus, html.theme--documenter-dark .button.is-info.is-outlined.is-focused {
+          background-color: #024c7d;
+          border-color: #024c7d;
+          color: #fff; }
+        html.theme--documenter-dark .button.is-info.is-outlined.is-loading::after {
+          border-color: transparent transparent #024c7d #024c7d !important; }
+        html.theme--documenter-dark .button.is-info.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #fff #fff !important; }
+        html.theme--documenter-dark .button.is-info.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-info.is-outlined {
+          background-color: transparent;
+          border-color: #024c7d;
+          box-shadow: none;
+          color: #024c7d; }
+      html.theme--documenter-dark .button.is-info.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #fff;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-focused {
+          background-color: #fff;
+          color: #024c7d; }
+        html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #024c7d #024c7d !important; }
+        html.theme--documenter-dark .button.is-info.is-inverted.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted.is-outlined {
+          background-color: transparent;
+          border-color: #fff;
+          box-shadow: none;
+          color: #fff; }
+    html.theme--documenter-dark .button.is-success {
+      background-color: #008438;
+      border-color: transparent;
+      color: #fff; }
+      html.theme--documenter-dark .button.is-success:hover, html.theme--documenter-dark .button.is-success.is-hovered {
+        background-color: #007733;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-success:focus, html.theme--documenter-dark .button.is-success.is-focused {
+        border-color: transparent;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-success:focus:not(:active), html.theme--documenter-dark .button.is-success.is-focused:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); }
+      html.theme--documenter-dark .button.is-success:active, html.theme--documenter-dark .button.is-success.is-active {
+        background-color: #006b2d;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-success[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-success {
+        background-color: #008438;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-success.is-inverted {
+        background-color: #fff;
+        color: #008438; }
+        html.theme--documenter-dark .button.is-success.is-inverted:hover, html.theme--documenter-dark .button.is-success.is-inverted.is-hovered {
+          background-color: #f2f2f2; }
+        html.theme--documenter-dark .button.is-success.is-inverted[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted {
+          background-color: #fff;
+          border-color: transparent;
+          box-shadow: none;
+          color: #008438; }
+      html.theme--documenter-dark .button.is-success.is-loading::after {
+        border-color: transparent transparent #fff #fff !important; }
+      html.theme--documenter-dark .button.is-success.is-outlined {
+        background-color: transparent;
+        border-color: #008438;
+        color: #008438; }
+        html.theme--documenter-dark .button.is-success.is-outlined:hover, html.theme--documenter-dark .button.is-success.is-outlined.is-hovered, html.theme--documenter-dark .button.is-success.is-outlined:focus, html.theme--documenter-dark .button.is-success.is-outlined.is-focused {
+          background-color: #008438;
+          border-color: #008438;
+          color: #fff; }
+        html.theme--documenter-dark .button.is-success.is-outlined.is-loading::after {
+          border-color: transparent transparent #008438 #008438 !important; }
+        html.theme--documenter-dark .button.is-success.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #fff #fff !important; }
+        html.theme--documenter-dark .button.is-success.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined {
+          background-color: transparent;
+          border-color: #008438;
+          box-shadow: none;
+          color: #008438; }
+      html.theme--documenter-dark .button.is-success.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #fff;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused {
+          background-color: #fff;
+          color: #008438; }
+        html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #008438 #008438 !important; }
+        html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined {
+          background-color: transparent;
+          border-color: #fff;
+          box-shadow: none;
+          color: #fff; }
+    html.theme--documenter-dark .button.is-warning {
+      background-color: #ad8100;
+      border-color: transparent;
+      color: #fff; }
+      html.theme--documenter-dark .button.is-warning:hover, html.theme--documenter-dark .button.is-warning.is-hovered {
+        background-color: #a07700;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-warning:focus, html.theme--documenter-dark .button.is-warning.is-focused {
+        border-color: transparent;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-warning:focus:not(:active), html.theme--documenter-dark .button.is-warning.is-focused:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); }
+      html.theme--documenter-dark .button.is-warning:active, html.theme--documenter-dark .button.is-warning.is-active {
+        background-color: #946e00;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-warning[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-warning {
+        background-color: #ad8100;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-warning.is-inverted {
+        background-color: #fff;
+        color: #ad8100; }
+        html.theme--documenter-dark .button.is-warning.is-inverted:hover, html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered {
+          background-color: #f2f2f2; }
+        html.theme--documenter-dark .button.is-warning.is-inverted[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted {
+          background-color: #fff;
+          border-color: transparent;
+          box-shadow: none;
+          color: #ad8100; }
+      html.theme--documenter-dark .button.is-warning.is-loading::after {
+        border-color: transparent transparent #fff #fff !important; }
+      html.theme--documenter-dark .button.is-warning.is-outlined {
+        background-color: transparent;
+        border-color: #ad8100;
+        color: #ad8100; }
+        html.theme--documenter-dark .button.is-warning.is-outlined:hover, html.theme--documenter-dark .button.is-warning.is-outlined.is-hovered, html.theme--documenter-dark .button.is-warning.is-outlined:focus, html.theme--documenter-dark .button.is-warning.is-outlined.is-focused {
+          background-color: #ad8100;
+          border-color: #ad8100;
+          color: #fff; }
+        html.theme--documenter-dark .button.is-warning.is-outlined.is-loading::after {
+          border-color: transparent transparent #ad8100 #ad8100 !important; }
+        html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #fff #fff !important; }
+        html.theme--documenter-dark .button.is-warning.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined {
+          background-color: transparent;
+          border-color: #ad8100;
+          box-shadow: none;
+          color: #ad8100; }
+      html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #fff;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused {
+          background-color: #fff;
+          color: #ad8100; }
+        html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #ad8100 #ad8100 !important; }
+        html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined {
+          background-color: transparent;
+          border-color: #fff;
+          box-shadow: none;
+          color: #fff; }
+    html.theme--documenter-dark .button.is-danger {
+      background-color: #9e1b0d;
+      border-color: transparent;
+      color: #fff; }
+      html.theme--documenter-dark .button.is-danger:hover, html.theme--documenter-dark .button.is-danger.is-hovered {
+        background-color: #92190c;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-danger:focus, html.theme--documenter-dark .button.is-danger.is-focused {
+        border-color: transparent;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-danger:focus:not(:active), html.theme--documenter-dark .button.is-danger.is-focused:not(:active) {
+          box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); }
+      html.theme--documenter-dark .button.is-danger:active, html.theme--documenter-dark .button.is-danger.is-active {
+        background-color: #86170b;
+        border-color: transparent;
+        color: #fff; }
+      html.theme--documenter-dark .button.is-danger[disabled],
+      fieldset[disabled] html.theme--documenter-dark .button.is-danger {
+        background-color: #9e1b0d;
+        border-color: transparent;
+        box-shadow: none; }
+      html.theme--documenter-dark .button.is-danger.is-inverted {
+        background-color: #fff;
+        color: #9e1b0d; }
+        html.theme--documenter-dark .button.is-danger.is-inverted:hover, html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered {
+          background-color: #f2f2f2; }
+        html.theme--documenter-dark .button.is-danger.is-inverted[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted {
+          background-color: #fff;
+          border-color: transparent;
+          box-shadow: none;
+          color: #9e1b0d; }
+      html.theme--documenter-dark .button.is-danger.is-loading::after {
+        border-color: transparent transparent #fff #fff !important; }
+      html.theme--documenter-dark .button.is-danger.is-outlined {
+        background-color: transparent;
+        border-color: #9e1b0d;
+        color: #9e1b0d; }
+        html.theme--documenter-dark .button.is-danger.is-outlined:hover, html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered, html.theme--documenter-dark .button.is-danger.is-outlined:focus, html.theme--documenter-dark .button.is-danger.is-outlined.is-focused {
+          background-color: #9e1b0d;
+          border-color: #9e1b0d;
+          color: #fff; }
+        html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after {
+          border-color: transparent transparent #9e1b0d #9e1b0d !important; }
+        html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #fff #fff !important; }
+        html.theme--documenter-dark .button.is-danger.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined {
+          background-color: transparent;
+          border-color: #9e1b0d;
+          box-shadow: none;
+          color: #9e1b0d; }
+      html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #fff;
+        color: #fff; }
+        html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused {
+          background-color: #fff;
+          color: #9e1b0d; }
+        html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after {
+          border-color: transparent transparent #9e1b0d #9e1b0d !important; }
+        html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],
+        fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined {
+          background-color: transparent;
+          border-color: #fff;
+          box-shadow: none;
+          color: #fff; }
+    html.theme--documenter-dark .button.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.button {
+      border-radius: 3px;
+      font-size: 0.85em; }
+    html.theme--documenter-dark .button.is-normal {
+      font-size: 15px; }
+    html.theme--documenter-dark .button.is-medium {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .button.is-large {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .button[disabled],
+    fieldset[disabled] html.theme--documenter-dark .button {
+      background-color: #8c9b9d;
+      border-color: #dbdee0;
+      box-shadow: none;
+      opacity: 0.5; }
+    html.theme--documenter-dark .button.is-fullwidth {
+      display: flex;
+      width: 100%; }
+    html.theme--documenter-dark .button.is-loading {
+      color: transparent !important;
+      pointer-events: none; }
+      html.theme--documenter-dark .button.is-loading::after {
+        position: absolute;
+        left: calc(50% - (1em / 2));
+        top: calc(50% - (1em / 2));
+        position: absolute !important; }
+    html.theme--documenter-dark .button.is-static {
+      background-color: #282f2f;
+      border-color: #5e6d6f;
+      color: #dbdee0;
+      box-shadow: none;
+      pointer-events: none; }
+    html.theme--documenter-dark .button.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.button {
+      border-radius: 290486px;
+      padding-left: 1em;
+      padding-right: 1em; }
+  html.theme--documenter-dark .buttons {
+    align-items: center;
+    display: flex;
+    flex-wrap: wrap;
+    justify-content: flex-start; }
+    html.theme--documenter-dark .buttons .button {
+      margin-bottom: 0.5rem; }
+      html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth) {
+        margin-right: 0.5rem; }
+    html.theme--documenter-dark .buttons:last-child {
+      margin-bottom: -0.5rem; }
+    html.theme--documenter-dark .buttons:not(:last-child) {
+      margin-bottom: 1rem; }
+    html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large) {
+      border-radius: 3px;
+      font-size: 0.85em; }
+    html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large) {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium) {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .buttons.has-addons .button:not(:first-child) {
+      border-bottom-left-radius: 0;
+      border-top-left-radius: 0; }
+    html.theme--documenter-dark .buttons.has-addons .button:not(:last-child) {
+      border-bottom-right-radius: 0;
+      border-top-right-radius: 0;
+      margin-right: -1px; }
+    html.theme--documenter-dark .buttons.has-addons .button:last-child {
+      margin-right: 0; }
+    html.theme--documenter-dark .buttons.has-addons .button:hover, html.theme--documenter-dark .buttons.has-addons .button.is-hovered {
+      z-index: 2; }
+    html.theme--documenter-dark .buttons.has-addons .button:focus, html.theme--documenter-dark .buttons.has-addons .button.is-focused, html.theme--documenter-dark .buttons.has-addons .button:active, html.theme--documenter-dark .buttons.has-addons .button.is-active, html.theme--documenter-dark .buttons.has-addons .button.is-selected {
+      z-index: 3; }
+      html.theme--documenter-dark .buttons.has-addons .button:focus:hover, html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover, html.theme--documenter-dark .buttons.has-addons .button:active:hover, html.theme--documenter-dark .buttons.has-addons .button.is-active:hover, html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover {
+        z-index: 4; }
+    html.theme--documenter-dark .buttons.has-addons .button.is-expanded {
+      flex-grow: 1;
+      flex-shrink: 1; }
+    html.theme--documenter-dark .buttons.is-centered {
+      justify-content: center; }
+      html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth) {
+        margin-left: 0.25rem;
+        margin-right: 0.25rem; }
+    html.theme--documenter-dark .buttons.is-right {
+      justify-content: flex-end; }
+      html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth) {
+        margin-left: 0.25rem;
+        margin-right: 0.25rem; }
+  html.theme--documenter-dark .container {
+    flex-grow: 1;
+    margin: 0 auto;
+    position: relative;
+    width: auto; }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .container {
+        max-width: 992px; }
+        html.theme--documenter-dark .container.is-fluid {
+          margin-left: 32px;
+          margin-right: 32px;
+          max-width: none; } }
+    @media screen and (max-width: 1215px) {
+      html.theme--documenter-dark .container.is-widescreen {
+        max-width: 1152px; } }
+    @media screen and (max-width: 1407px) {
+      html.theme--documenter-dark .container.is-fullhd {
+        max-width: 1344px; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .container {
+        max-width: 1152px; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .container {
+        max-width: 1344px; } }
+  html.theme--documenter-dark .content li + li {
+    margin-top: 0.25em; }
+  html.theme--documenter-dark .content p:not(:last-child),
+  html.theme--documenter-dark .content dl:not(:last-child),
+  html.theme--documenter-dark .content ol:not(:last-child),
+  html.theme--documenter-dark .content ul:not(:last-child),
+  html.theme--documenter-dark .content blockquote:not(:last-child),
+  html.theme--documenter-dark .content pre:not(:last-child),
+  html.theme--documenter-dark .content table:not(:last-child) {
+    margin-bottom: 1em; }
+  html.theme--documenter-dark .content h1,
+  html.theme--documenter-dark .content h2,
+  html.theme--documenter-dark .content h3,
+  html.theme--documenter-dark .content h4,
+  html.theme--documenter-dark .content h5,
+  html.theme--documenter-dark .content h6 {
+    color: #f2f2f2;
+    font-weight: 600;
+    line-height: 1.125; }
+  html.theme--documenter-dark .content h1 {
+    font-size: 2em;
+    margin-bottom: 0.5em; }
+    html.theme--documenter-dark .content h1:not(:first-child) {
+      margin-top: 1em; }
+  html.theme--documenter-dark .content h2 {
+    font-size: 1.75em;
+    margin-bottom: 0.5714em; }
+    html.theme--documenter-dark .content h2:not(:first-child) {
+      margin-top: 1.1428em; }
+  html.theme--documenter-dark .content h3 {
+    font-size: 1.5em;
+    margin-bottom: 0.6666em; }
+    html.theme--documenter-dark .content h3:not(:first-child) {
+      margin-top: 1.3333em; }
+  html.theme--documenter-dark .content h4 {
+    font-size: 1.25em;
+    margin-bottom: 0.8em; }
+  html.theme--documenter-dark .content h5 {
+    font-size: 1.125em;
+    margin-bottom: 0.8888em; }
+  html.theme--documenter-dark .content h6 {
+    font-size: 1em;
+    margin-bottom: 1em; }
+  html.theme--documenter-dark .content blockquote {
+    background-color: #282f2f;
+    border-left: 5px solid #5e6d6f;
+    padding: 1.25em 1.5em; }
+  html.theme--documenter-dark .content ol {
+    list-style-position: outside;
+    margin-left: 2em;
+    margin-top: 1em; }
+    html.theme--documenter-dark .content ol:not([type]) {
+      list-style-type: decimal; }
+      html.theme--documenter-dark .content ol:not([type]).is-lower-alpha {
+        list-style-type: lower-alpha; }
+      html.theme--documenter-dark .content ol:not([type]).is-lower-roman {
+        list-style-type: lower-roman; }
+      html.theme--documenter-dark .content ol:not([type]).is-upper-alpha {
+        list-style-type: upper-alpha; }
+      html.theme--documenter-dark .content ol:not([type]).is-upper-roman {
+        list-style-type: upper-roman; }
+  html.theme--documenter-dark .content ul {
+    list-style: disc outside;
+    margin-left: 2em;
+    margin-top: 1em; }
+    html.theme--documenter-dark .content ul ul {
+      list-style-type: circle;
+      margin-top: 0.5em; }
+      html.theme--documenter-dark .content ul ul ul {
+        list-style-type: square; }
+  html.theme--documenter-dark .content dd {
+    margin-left: 2em; }
+  html.theme--documenter-dark .content figure {
+    margin-left: 2em;
+    margin-right: 2em;
+    text-align: center; }
+    html.theme--documenter-dark .content figure:not(:first-child) {
+      margin-top: 2em; }
+    html.theme--documenter-dark .content figure:not(:last-child) {
+      margin-bottom: 2em; }
+    html.theme--documenter-dark .content figure img {
+      display: inline-block; }
+    html.theme--documenter-dark .content figure figcaption {
+      font-style: italic; }
+  html.theme--documenter-dark .content pre {
+    -webkit-overflow-scrolling: touch;
+    overflow-x: auto;
+    padding: 0.7rem 0.5rem;
+    white-space: pre;
+    word-wrap: normal; }
+  html.theme--documenter-dark .content sup,
+  html.theme--documenter-dark .content sub {
+    font-size: 75%; }
+  html.theme--documenter-dark .content table {
+    width: 100%; }
+    html.theme--documenter-dark .content table td,
+    html.theme--documenter-dark .content table th {
+      border: 1px solid #5e6d6f;
+      border-width: 0 0 1px;
+      padding: 0.5em 0.75em;
+      vertical-align: top; }
+    html.theme--documenter-dark .content table th {
+      color: #f2f2f2; }
+      html.theme--documenter-dark .content table th:not([align]) {
+        text-align: left; }
+    html.theme--documenter-dark .content table thead td,
+    html.theme--documenter-dark .content table thead th {
+      border-width: 0 0 2px;
+      color: #f2f2f2; }
+    html.theme--documenter-dark .content table tfoot td,
+    html.theme--documenter-dark .content table tfoot th {
+      border-width: 2px 0 0;
+      color: #f2f2f2; }
+    html.theme--documenter-dark .content table tbody tr:last-child td,
+    html.theme--documenter-dark .content table tbody tr:last-child th {
+      border-bottom-width: 0; }
+  html.theme--documenter-dark .content .tabs li + li {
+    margin-top: 0; }
+  html.theme--documenter-dark .content.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.content {
+    font-size: 0.85em; }
+  html.theme--documenter-dark .content.is-medium {
+    font-size: 1.25rem; }
+  html.theme--documenter-dark .content.is-large {
+    font-size: 1.5rem; }
+  html.theme--documenter-dark .icon {
+    align-items: center;
+    display: inline-flex;
+    justify-content: center;
+    height: 1.5rem;
+    width: 1.5rem; }
+    html.theme--documenter-dark .icon.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.icon {
+      height: 1rem;
+      width: 1rem; }
+    html.theme--documenter-dark .icon.is-medium {
+      height: 2rem;
+      width: 2rem; }
+    html.theme--documenter-dark .icon.is-large {
+      height: 3rem;
+      width: 3rem; }
+  html.theme--documenter-dark .image, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img {
+    display: block;
+    position: relative; }
+    html.theme--documenter-dark .image img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img img {
+      display: block;
+      height: auto;
+      width: 100%; }
+      html.theme--documenter-dark .image img.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img img.is-rounded {
+        border-radius: 290486px; }
+    html.theme--documenter-dark .image.is-square img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square img,
+    html.theme--documenter-dark .image.is-square .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, html.theme--documenter-dark .image.is-1by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 img,
+    html.theme--documenter-dark .image.is-1by1 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, html.theme--documenter-dark .image.is-5by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 img,
+    html.theme--documenter-dark .image.is-5by4 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, html.theme--documenter-dark .image.is-4by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 img,
+    html.theme--documenter-dark .image.is-4by3 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, html.theme--documenter-dark .image.is-3by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 img,
+    html.theme--documenter-dark .image.is-3by2 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, html.theme--documenter-dark .image.is-5by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 img,
+    html.theme--documenter-dark .image.is-5by3 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, html.theme--documenter-dark .image.is-16by9 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 img,
+    html.theme--documenter-dark .image.is-16by9 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, html.theme--documenter-dark .image.is-2by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 img,
+    html.theme--documenter-dark .image.is-2by1 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, html.theme--documenter-dark .image.is-3by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 img,
+    html.theme--documenter-dark .image.is-3by1 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, html.theme--documenter-dark .image.is-4by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 img,
+    html.theme--documenter-dark .image.is-4by5 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, html.theme--documenter-dark .image.is-3by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 img,
+    html.theme--documenter-dark .image.is-3by4 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, html.theme--documenter-dark .image.is-2by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 img,
+    html.theme--documenter-dark .image.is-2by3 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, html.theme--documenter-dark .image.is-3by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 img,
+    html.theme--documenter-dark .image.is-3by5 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, html.theme--documenter-dark .image.is-9by16 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 img,
+    html.theme--documenter-dark .image.is-9by16 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, html.theme--documenter-dark .image.is-1by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 img,
+    html.theme--documenter-dark .image.is-1by2 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, html.theme--documenter-dark .image.is-1by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 img,
+    html.theme--documenter-dark .image.is-1by3 .has-ratio,
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio {
+      height: 100%;
+      width: 100%; }
+    html.theme--documenter-dark .image.is-square, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square, html.theme--documenter-dark .image.is-1by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 {
+      padding-top: 100%; }
+    html.theme--documenter-dark .image.is-5by4, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 {
+      padding-top: 80%; }
+    html.theme--documenter-dark .image.is-4by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 {
+      padding-top: 75%; }
+    html.theme--documenter-dark .image.is-3by2, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 {
+      padding-top: 66.6666%; }
+    html.theme--documenter-dark .image.is-5by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 {
+      padding-top: 60%; }
+    html.theme--documenter-dark .image.is-16by9, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 {
+      padding-top: 56.25%; }
+    html.theme--documenter-dark .image.is-2by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 {
+      padding-top: 50%; }
+    html.theme--documenter-dark .image.is-3by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 {
+      padding-top: 33.3333%; }
+    html.theme--documenter-dark .image.is-4by5, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 {
+      padding-top: 125%; }
+    html.theme--documenter-dark .image.is-3by4, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 {
+      padding-top: 133.3333%; }
+    html.theme--documenter-dark .image.is-2by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 {
+      padding-top: 150%; }
+    html.theme--documenter-dark .image.is-3by5, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 {
+      padding-top: 166.6666%; }
+    html.theme--documenter-dark .image.is-9by16, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 {
+      padding-top: 177.7777%; }
+    html.theme--documenter-dark .image.is-1by2, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 {
+      padding-top: 200%; }
+    html.theme--documenter-dark .image.is-1by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 {
+      padding-top: 300%; }
+    html.theme--documenter-dark .image.is-16x16, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16x16 {
+      height: 16px;
+      width: 16px; }
+    html.theme--documenter-dark .image.is-24x24, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-24x24 {
+      height: 24px;
+      width: 24px; }
+    html.theme--documenter-dark .image.is-32x32, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-32x32 {
+      height: 32px;
+      width: 32px; }
+    html.theme--documenter-dark .image.is-48x48, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-48x48 {
+      height: 48px;
+      width: 48px; }
+    html.theme--documenter-dark .image.is-64x64, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-64x64 {
+      height: 64px;
+      width: 64px; }
+    html.theme--documenter-dark .image.is-96x96, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-96x96 {
+      height: 96px;
+      width: 96px; }
+    html.theme--documenter-dark .image.is-128x128, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-128x128 {
+      height: 128px;
+      width: 128px; }
+  html.theme--documenter-dark .notification {
+    background-color: #282f2f;
+    border-radius: 0.4em;
+    padding: 1.25rem 2.5rem 1.25rem 1.5rem;
+    position: relative; }
+    html.theme--documenter-dark .notification a:not(.button):not(.dropdown-item) {
+      color: currentColor;
+      text-decoration: underline; }
+    html.theme--documenter-dark .notification strong {
+      color: currentColor; }
+    html.theme--documenter-dark .notification code,
+    html.theme--documenter-dark .notification pre {
+      background: white; }
+    html.theme--documenter-dark .notification pre code {
+      background: transparent; }
+    html.theme--documenter-dark .notification > .delete {
+      position: absolute;
+      right: 0.5rem;
+      top: 0.5rem; }
+    html.theme--documenter-dark .notification .title,
+    html.theme--documenter-dark .notification .subtitle,
+    html.theme--documenter-dark .notification .content {
+      color: currentColor; }
+    html.theme--documenter-dark .notification.is-white {
+      background-color: white;
+      color: #0a0a0a; }
+    html.theme--documenter-dark .notification.is-black {
+      background-color: #0a0a0a;
+      color: white; }
+    html.theme--documenter-dark .notification.is-light {
+      background-color: #ecf0f1;
+      color: #282f2f; }
+    html.theme--documenter-dark .notification.is-dark, html.theme--documenter-dark .content kbd.notification {
+      background-color: #282f2f;
+      color: #ecf0f1; }
+    html.theme--documenter-dark .notification.is-primary, html.theme--documenter-dark .docstring > section > a.notification.docs-sourcelink {
+      background-color: #375a7f;
+      color: #fff; }
+    html.theme--documenter-dark .notification.is-link {
+      background-color: #1abc9c;
+      color: #fff; }
+    html.theme--documenter-dark .notification.is-info {
+      background-color: #024c7d;
+      color: #fff; }
+    html.theme--documenter-dark .notification.is-success {
+      background-color: #008438;
+      color: #fff; }
+    html.theme--documenter-dark .notification.is-warning {
+      background-color: #ad8100;
+      color: #fff; }
+    html.theme--documenter-dark .notification.is-danger {
+      background-color: #9e1b0d;
+      color: #fff; }
+  html.theme--documenter-dark .progress {
+    -moz-appearance: none;
+    -webkit-appearance: none;
+    border: none;
+    border-radius: 290486px;
+    display: block;
+    height: 15px;
+    overflow: hidden;
+    padding: 0;
+    width: 100%; }
+    html.theme--documenter-dark .progress::-webkit-progress-bar {
+      background-color: #5e6d6f; }
+    html.theme--documenter-dark .progress::-webkit-progress-value {
+      background-color: #dbdee0; }
+    html.theme--documenter-dark .progress::-moz-progress-bar {
+      background-color: #dbdee0; }
+    html.theme--documenter-dark .progress::-ms-fill {
+      background-color: #dbdee0;
+      border: none; }
+    html.theme--documenter-dark .progress.is-white::-webkit-progress-value {
+      background-color: white; }
+    html.theme--documenter-dark .progress.is-white::-moz-progress-bar {
+      background-color: white; }
+    html.theme--documenter-dark .progress.is-white::-ms-fill {
+      background-color: white; }
+    html.theme--documenter-dark .progress.is-white:indeterminate {
+      background-image: linear-gradient(to right, white 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress.is-black::-webkit-progress-value {
+      background-color: #0a0a0a; }
+    html.theme--documenter-dark .progress.is-black::-moz-progress-bar {
+      background-color: #0a0a0a; }
+    html.theme--documenter-dark .progress.is-black::-ms-fill {
+      background-color: #0a0a0a; }
+    html.theme--documenter-dark .progress.is-black:indeterminate {
+      background-image: linear-gradient(to right, #0a0a0a 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress.is-light::-webkit-progress-value {
+      background-color: #ecf0f1; }
+    html.theme--documenter-dark .progress.is-light::-moz-progress-bar {
+      background-color: #ecf0f1; }
+    html.theme--documenter-dark .progress.is-light::-ms-fill {
+      background-color: #ecf0f1; }
+    html.theme--documenter-dark .progress.is-light:indeterminate {
+      background-image: linear-gradient(to right, #ecf0f1 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress.is-dark::-webkit-progress-value, html.theme--documenter-dark .content kbd.progress::-webkit-progress-value {
+      background-color: #282f2f; }
+    html.theme--documenter-dark .progress.is-dark::-moz-progress-bar, html.theme--documenter-dark .content kbd.progress::-moz-progress-bar {
+      background-color: #282f2f; }
+    html.theme--documenter-dark .progress.is-dark::-ms-fill, html.theme--documenter-dark .content kbd.progress::-ms-fill {
+      background-color: #282f2f; }
+    html.theme--documenter-dark .progress.is-dark:indeterminate, html.theme--documenter-dark .content kbd.progress:indeterminate {
+      background-image: linear-gradient(to right, #282f2f 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress.is-primary::-webkit-progress-value, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-webkit-progress-value {
+      background-color: #375a7f; }
+    html.theme--documenter-dark .progress.is-primary::-moz-progress-bar, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-moz-progress-bar {
+      background-color: #375a7f; }
+    html.theme--documenter-dark .progress.is-primary::-ms-fill, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-ms-fill {
+      background-color: #375a7f; }
+    html.theme--documenter-dark .progress.is-primary:indeterminate, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink:indeterminate {
+      background-image: linear-gradient(to right, #375a7f 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress.is-link::-webkit-progress-value {
+      background-color: #1abc9c; }
+    html.theme--documenter-dark .progress.is-link::-moz-progress-bar {
+      background-color: #1abc9c; }
+    html.theme--documenter-dark .progress.is-link::-ms-fill {
+      background-color: #1abc9c; }
+    html.theme--documenter-dark .progress.is-link:indeterminate {
+      background-image: linear-gradient(to right, #1abc9c 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress.is-info::-webkit-progress-value {
+      background-color: #024c7d; }
+    html.theme--documenter-dark .progress.is-info::-moz-progress-bar {
+      background-color: #024c7d; }
+    html.theme--documenter-dark .progress.is-info::-ms-fill {
+      background-color: #024c7d; }
+    html.theme--documenter-dark .progress.is-info:indeterminate {
+      background-image: linear-gradient(to right, #024c7d 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress.is-success::-webkit-progress-value {
+      background-color: #008438; }
+    html.theme--documenter-dark .progress.is-success::-moz-progress-bar {
+      background-color: #008438; }
+    html.theme--documenter-dark .progress.is-success::-ms-fill {
+      background-color: #008438; }
+    html.theme--documenter-dark .progress.is-success:indeterminate {
+      background-image: linear-gradient(to right, #008438 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress.is-warning::-webkit-progress-value {
+      background-color: #ad8100; }
+    html.theme--documenter-dark .progress.is-warning::-moz-progress-bar {
+      background-color: #ad8100; }
+    html.theme--documenter-dark .progress.is-warning::-ms-fill {
+      background-color: #ad8100; }
+    html.theme--documenter-dark .progress.is-warning:indeterminate {
+      background-image: linear-gradient(to right, #ad8100 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress.is-danger::-webkit-progress-value {
+      background-color: #9e1b0d; }
+    html.theme--documenter-dark .progress.is-danger::-moz-progress-bar {
+      background-color: #9e1b0d; }
+    html.theme--documenter-dark .progress.is-danger::-ms-fill {
+      background-color: #9e1b0d; }
+    html.theme--documenter-dark .progress.is-danger:indeterminate {
+      background-image: linear-gradient(to right, #9e1b0d 30%, #5e6d6f 30%); }
+    html.theme--documenter-dark .progress:indeterminate {
+      animation-duration: 1.5s;
+      animation-iteration-count: infinite;
+      animation-name: moveIndeterminate;
+      animation-timing-function: linear;
+      background-color: #5e6d6f;
+      background-image: linear-gradient(to right, #fff 30%, #5e6d6f 30%);
+      background-position: top left;
+      background-repeat: no-repeat;
+      background-size: 150% 150%; }
+      html.theme--documenter-dark .progress:indeterminate::-webkit-progress-bar {
+        background-color: transparent; }
+      html.theme--documenter-dark .progress:indeterminate::-moz-progress-bar {
+        background-color: transparent; }
+    html.theme--documenter-dark .progress.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.progress {
+      height: 0.85em; }
+    html.theme--documenter-dark .progress.is-medium {
+      height: 1.25rem; }
+    html.theme--documenter-dark .progress.is-large {
+      height: 1.5rem; }
+
+@keyframes moveIndeterminate {
+  from {
+    background-position: 200% 0; }
+  to {
+    background-position: -200% 0; } }
+  html.theme--documenter-dark .table {
+    background-color: #343c3d;
+    color: #fff; }
+    html.theme--documenter-dark .table td,
+    html.theme--documenter-dark .table th {
+      border: 1px solid #5e6d6f;
+      border-width: 0 0 1px;
+      padding: 0.5em 0.75em;
+      vertical-align: top; }
+      html.theme--documenter-dark .table td.is-white,
+      html.theme--documenter-dark .table th.is-white {
+        background-color: white;
+        border-color: white;
+        color: #0a0a0a; }
+      html.theme--documenter-dark .table td.is-black,
+      html.theme--documenter-dark .table th.is-black {
+        background-color: #0a0a0a;
+        border-color: #0a0a0a;
+        color: white; }
+      html.theme--documenter-dark .table td.is-light,
+      html.theme--documenter-dark .table th.is-light {
+        background-color: #ecf0f1;
+        border-color: #ecf0f1;
+        color: #282f2f; }
+      html.theme--documenter-dark .table td.is-dark,
+      html.theme--documenter-dark .table th.is-dark {
+        background-color: #282f2f;
+        border-color: #282f2f;
+        color: #ecf0f1; }
+      html.theme--documenter-dark .table td.is-primary,
+      html.theme--documenter-dark .table th.is-primary {
+        background-color: #375a7f;
+        border-color: #375a7f;
+        color: #fff; }
+      html.theme--documenter-dark .table td.is-link,
+      html.theme--documenter-dark .table th.is-link {
+        background-color: #1abc9c;
+        border-color: #1abc9c;
+        color: #fff; }
+      html.theme--documenter-dark .table td.is-info,
+      html.theme--documenter-dark .table th.is-info {
+        background-color: #024c7d;
+        border-color: #024c7d;
+        color: #fff; }
+      html.theme--documenter-dark .table td.is-success,
+      html.theme--documenter-dark .table th.is-success {
+        background-color: #008438;
+        border-color: #008438;
+        color: #fff; }
+      html.theme--documenter-dark .table td.is-warning,
+      html.theme--documenter-dark .table th.is-warning {
+        background-color: #ad8100;
+        border-color: #ad8100;
+        color: #fff; }
+      html.theme--documenter-dark .table td.is-danger,
+      html.theme--documenter-dark .table th.is-danger {
+        background-color: #9e1b0d;
+        border-color: #9e1b0d;
+        color: #fff; }
+      html.theme--documenter-dark .table td.is-narrow,
+      html.theme--documenter-dark .table th.is-narrow {
+        white-space: nowrap;
+        width: 1%; }
+      html.theme--documenter-dark .table td.is-selected,
+      html.theme--documenter-dark .table th.is-selected {
+        background-color: #375a7f;
+        color: #fff; }
+        html.theme--documenter-dark .table td.is-selected a,
+        html.theme--documenter-dark .table td.is-selected strong,
+        html.theme--documenter-dark .table th.is-selected a,
+        html.theme--documenter-dark .table th.is-selected strong {
+          color: currentColor; }
+    html.theme--documenter-dark .table th {
+      color: #f2f2f2; }
+      html.theme--documenter-dark .table th:not([align]) {
+        text-align: left; }
+    html.theme--documenter-dark .table tr.is-selected {
+      background-color: #375a7f;
+      color: #fff; }
+      html.theme--documenter-dark .table tr.is-selected a,
+      html.theme--documenter-dark .table tr.is-selected strong {
+        color: currentColor; }
+      html.theme--documenter-dark .table tr.is-selected td,
+      html.theme--documenter-dark .table tr.is-selected th {
+        border-color: #fff;
+        color: currentColor; }
+    html.theme--documenter-dark .table thead {
+      background-color: transparent; }
+      html.theme--documenter-dark .table thead td,
+      html.theme--documenter-dark .table thead th {
+        border-width: 0 0 2px;
+        color: #f2f2f2; }
+    html.theme--documenter-dark .table tfoot {
+      background-color: transparent; }
+      html.theme--documenter-dark .table tfoot td,
+      html.theme--documenter-dark .table tfoot th {
+        border-width: 2px 0 0;
+        color: #f2f2f2; }
+    html.theme--documenter-dark .table tbody {
+      background-color: transparent; }
+      html.theme--documenter-dark .table tbody tr:last-child td,
+      html.theme--documenter-dark .table tbody tr:last-child th {
+        border-bottom-width: 0; }
+    html.theme--documenter-dark .table.is-bordered td,
+    html.theme--documenter-dark .table.is-bordered th {
+      border-width: 1px; }
+    html.theme--documenter-dark .table.is-bordered tr:last-child td,
+    html.theme--documenter-dark .table.is-bordered tr:last-child th {
+      border-bottom-width: 1px; }
+    html.theme--documenter-dark .table.is-fullwidth {
+      width: 100%; }
+    html.theme--documenter-dark .table.is-hoverable tbody tr:not(.is-selected):hover {
+      background-color: #282f2f; }
+    html.theme--documenter-dark .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover {
+      background-color: #282f2f; }
+      html.theme--documenter-dark .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover:nth-child(even) {
+        background-color: #2d3435; }
+    html.theme--documenter-dark .table.is-narrow td,
+    html.theme--documenter-dark .table.is-narrow th {
+      padding: 0.25em 0.5em; }
+    html.theme--documenter-dark .table.is-striped tbody tr:not(.is-selected):nth-child(even) {
+      background-color: #282f2f; }
+  html.theme--documenter-dark .table-container {
+    -webkit-overflow-scrolling: touch;
+    overflow: auto;
+    overflow-y: hidden;
+    max-width: 100%; }
+  html.theme--documenter-dark .tags {
+    align-items: center;
+    display: flex;
+    flex-wrap: wrap;
+    justify-content: flex-start; }
+    html.theme--documenter-dark .tags .tag, html.theme--documenter-dark .tags .docstring > section > a.docs-sourcelink, html.theme--documenter-dark .tags .content kbd, html.theme--documenter-dark .content .tags kbd {
+      margin-bottom: 0.5rem; }
+      html.theme--documenter-dark .tags .tag:not(:last-child), html.theme--documenter-dark .tags .docstring > section > a.docs-sourcelink:not(:last-child), html.theme--documenter-dark .tags .content kbd:not(:last-child), html.theme--documenter-dark .content .tags kbd:not(:last-child) {
+        margin-right: 0.5rem; }
+    html.theme--documenter-dark .tags:last-child {
+      margin-bottom: -0.5rem; }
+    html.theme--documenter-dark .tags:not(:last-child) {
+      margin-bottom: 1rem; }
+    html.theme--documenter-dark .tags.are-medium .tag:not(.is-normal):not(.is-large), html.theme--documenter-dark .tags.are-medium .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-large), html.theme--documenter-dark .tags.are-medium .content kbd:not(.is-normal):not(.is-large), html.theme--documenter-dark .content .tags.are-medium kbd:not(.is-normal):not(.is-large) {
+      font-size: 15px; }
+    html.theme--documenter-dark .tags.are-large .tag:not(.is-normal):not(.is-medium), html.theme--documenter-dark .tags.are-large .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-medium), html.theme--documenter-dark .tags.are-large .content kbd:not(.is-normal):not(.is-medium), html.theme--documenter-dark .content .tags.are-large kbd:not(.is-normal):not(.is-medium) {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .tags.is-centered {
+      justify-content: center; }
+      html.theme--documenter-dark .tags.is-centered .tag, html.theme--documenter-dark .tags.is-centered .docstring > section > a.docs-sourcelink, html.theme--documenter-dark .tags.is-centered .content kbd, html.theme--documenter-dark .content .tags.is-centered kbd {
+        margin-right: 0.25rem;
+        margin-left: 0.25rem; }
+    html.theme--documenter-dark .tags.is-right {
+      justify-content: flex-end; }
+      html.theme--documenter-dark .tags.is-right .tag:not(:first-child), html.theme--documenter-dark .tags.is-right .docstring > section > a.docs-sourcelink:not(:first-child), html.theme--documenter-dark .tags.is-right .content kbd:not(:first-child), html.theme--documenter-dark .content .tags.is-right kbd:not(:first-child) {
+        margin-left: 0.5rem; }
+      html.theme--documenter-dark .tags.is-right .tag:not(:last-child), html.theme--documenter-dark .tags.is-right .docstring > section > a.docs-sourcelink:not(:last-child), html.theme--documenter-dark .tags.is-right .content kbd:not(:last-child), html.theme--documenter-dark .content .tags.is-right kbd:not(:last-child) {
+        margin-right: 0; }
+    html.theme--documenter-dark .tags.has-addons .tag, html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink, html.theme--documenter-dark .tags.has-addons .content kbd, html.theme--documenter-dark .content .tags.has-addons kbd {
+      margin-right: 0; }
+      html.theme--documenter-dark .tags.has-addons .tag:not(:first-child), html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink:not(:first-child), html.theme--documenter-dark .tags.has-addons .content kbd:not(:first-child), html.theme--documenter-dark .content .tags.has-addons kbd:not(:first-child) {
+        margin-left: 0;
+        border-bottom-left-radius: 0;
+        border-top-left-radius: 0; }
+      html.theme--documenter-dark .tags.has-addons .tag:not(:last-child), html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink:not(:last-child), html.theme--documenter-dark .tags.has-addons .content kbd:not(:last-child), html.theme--documenter-dark .content .tags.has-addons kbd:not(:last-child) {
+        border-bottom-right-radius: 0;
+        border-top-right-radius: 0; }
+  html.theme--documenter-dark .tag:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body), html.theme--documenter-dark .content kbd:not(body) {
+    align-items: center;
+    background-color: #282f2f;
+    border-radius: 0.4em;
+    color: #fff;
+    display: inline-flex;
+    font-size: 0.85em;
+    height: 2em;
+    justify-content: center;
+    line-height: 1.5;
+    padding-left: 0.75em;
+    padding-right: 0.75em;
+    white-space: nowrap; }
+    html.theme--documenter-dark .tag:not(body) .delete, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .delete, html.theme--documenter-dark .content kbd:not(body) .delete {
+      margin-left: 0.25rem;
+      margin-right: -0.375rem; }
+    html.theme--documenter-dark .tag:not(body).is-white, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-white, html.theme--documenter-dark .content kbd:not(body).is-white {
+      background-color: white;
+      color: #0a0a0a; }
+    html.theme--documenter-dark .tag:not(body).is-black, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-black, html.theme--documenter-dark .content kbd:not(body).is-black {
+      background-color: #0a0a0a;
+      color: white; }
+    html.theme--documenter-dark .tag:not(body).is-light, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-light, html.theme--documenter-dark .content kbd:not(body).is-light {
+      background-color: #ecf0f1;
+      color: #282f2f; }
+    html.theme--documenter-dark .tag:not(body).is-dark, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-dark, html.theme--documenter-dark .content kbd:not(body) {
+      background-color: #282f2f;
+      color: #ecf0f1; }
+    html.theme--documenter-dark .tag:not(body).is-primary, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body), html.theme--documenter-dark .content kbd:not(body).is-primary {
+      background-color: #375a7f;
+      color: #fff; }
+    html.theme--documenter-dark .tag:not(body).is-link, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-link, html.theme--documenter-dark .content kbd:not(body).is-link {
+      background-color: #1abc9c;
+      color: #fff; }
+    html.theme--documenter-dark .tag:not(body).is-info, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-info, html.theme--documenter-dark .content kbd:not(body).is-info {
+      background-color: #024c7d;
+      color: #fff; }
+    html.theme--documenter-dark .tag:not(body).is-success, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-success, html.theme--documenter-dark .content kbd:not(body).is-success {
+      background-color: #008438;
+      color: #fff; }
+    html.theme--documenter-dark .tag:not(body).is-warning, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-warning, html.theme--documenter-dark .content kbd:not(body).is-warning {
+      background-color: #ad8100;
+      color: #fff; }
+    html.theme--documenter-dark .tag:not(body).is-danger, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-danger, html.theme--documenter-dark .content kbd:not(body).is-danger {
+      background-color: #9e1b0d;
+      color: #fff; }
+    html.theme--documenter-dark .tag:not(body).is-normal, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-normal, html.theme--documenter-dark .content kbd:not(body).is-normal {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .tag:not(body).is-medium, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-medium, html.theme--documenter-dark .content kbd:not(body).is-medium {
+      font-size: 15px; }
+    html.theme--documenter-dark .tag:not(body).is-large, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-large, html.theme--documenter-dark .content kbd:not(body).is-large {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .tag:not(body) .icon:first-child:not(:last-child), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:not(:last-child), html.theme--documenter-dark .content kbd:not(body) .icon:first-child:not(:last-child) {
+      margin-left: -0.375em;
+      margin-right: 0.1875em; }
+    html.theme--documenter-dark .tag:not(body) .icon:last-child:not(:first-child), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:last-child:not(:first-child), html.theme--documenter-dark .content kbd:not(body) .icon:last-child:not(:first-child) {
+      margin-left: 0.1875em;
+      margin-right: -0.375em; }
+    html.theme--documenter-dark .tag:not(body) .icon:first-child:last-child, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:last-child, html.theme--documenter-dark .content kbd:not(body) .icon:first-child:last-child {
+      margin-left: -0.375em;
+      margin-right: -0.375em; }
+    html.theme--documenter-dark .tag:not(body).is-delete, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-delete, html.theme--documenter-dark .content kbd:not(body).is-delete {
+      margin-left: 1px;
+      padding: 0;
+      position: relative;
+      width: 2em; }
+      html.theme--documenter-dark .tag:not(body).is-delete::before, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-delete::before, html.theme--documenter-dark .content kbd:not(body).is-delete::before, html.theme--documenter-dark .tag:not(body).is-delete::after, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-delete::after, html.theme--documenter-dark .content kbd:not(body).is-delete::after {
+        background-color: currentColor;
+        content: "";
+        display: block;
+        left: 50%;
+        position: absolute;
+        top: 50%;
+        transform: translateX(-50%) translateY(-50%) rotate(45deg);
+        transform-origin: center center; }
+      html.theme--documenter-dark .tag:not(body).is-delete::before, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-delete::before, html.theme--documenter-dark .content kbd:not(body).is-delete::before {
+        height: 1px;
+        width: 50%; }
+      html.theme--documenter-dark .tag:not(body).is-delete::after, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-delete::after, html.theme--documenter-dark .content kbd:not(body).is-delete::after {
+        height: 50%;
+        width: 1px; }
+      html.theme--documenter-dark .tag:not(body).is-delete:hover, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-delete:hover, html.theme--documenter-dark .content kbd:not(body).is-delete:hover, html.theme--documenter-dark .tag:not(body).is-delete:focus, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-delete:focus, html.theme--documenter-dark .content kbd:not(body).is-delete:focus {
+        background-color: #1d2122; }
+      html.theme--documenter-dark .tag:not(body).is-delete:active, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-delete:active, html.theme--documenter-dark .content kbd:not(body).is-delete:active {
+        background-color: #111414; }
+    html.theme--documenter-dark .tag:not(body).is-rounded, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body).is-rounded, html.theme--documenter-dark .content kbd:not(body).is-rounded, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.tag:not(body) {
+      border-radius: 290486px; }
+  html.theme--documenter-dark a.tag:hover, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:hover {
+    text-decoration: underline; }
+  html.theme--documenter-dark .title,
+  html.theme--documenter-dark .subtitle {
+    word-break: break-word; }
+    html.theme--documenter-dark .title em,
+    html.theme--documenter-dark .title span,
+    html.theme--documenter-dark .subtitle em,
+    html.theme--documenter-dark .subtitle span {
+      font-weight: inherit; }
+    html.theme--documenter-dark .title sub,
+    html.theme--documenter-dark .subtitle sub {
+      font-size: 0.75em; }
+    html.theme--documenter-dark .title sup,
+    html.theme--documenter-dark .subtitle sup {
+      font-size: 0.75em; }
+    html.theme--documenter-dark .title .tag, html.theme--documenter-dark .title .docstring > section > a.docs-sourcelink, html.theme--documenter-dark .title .content kbd, html.theme--documenter-dark .content .title kbd,
+    html.theme--documenter-dark .subtitle .tag,
+    html.theme--documenter-dark .subtitle .docstring > section > a.docs-sourcelink,
+    html.theme--documenter-dark .subtitle .content kbd,
+    html.theme--documenter-dark .content .subtitle kbd {
+      vertical-align: middle; }
+  html.theme--documenter-dark .title {
+    color: #fff;
+    font-size: 2rem;
+    font-weight: 500;
+    line-height: 1.125; }
+    html.theme--documenter-dark .title strong {
+      color: inherit;
+      font-weight: inherit; }
+    html.theme--documenter-dark .title + .highlight {
+      margin-top: -0.75rem; }
+    html.theme--documenter-dark .title:not(.is-spaced) + .subtitle {
+      margin-top: -1.25rem; }
+    html.theme--documenter-dark .title.is-1 {
+      font-size: 3rem; }
+    html.theme--documenter-dark .title.is-2 {
+      font-size: 2.5rem; }
+    html.theme--documenter-dark .title.is-3 {
+      font-size: 2rem; }
+    html.theme--documenter-dark .title.is-4 {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .title.is-5 {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .title.is-6 {
+      font-size: 15px; }
+    html.theme--documenter-dark .title.is-7 {
+      font-size: 0.85em; }
+  html.theme--documenter-dark .subtitle {
+    color: #8c9b9d;
+    font-size: 1.25rem;
+    font-weight: 400;
+    line-height: 1.25; }
+    html.theme--documenter-dark .subtitle strong {
+      color: #8c9b9d;
+      font-weight: 600; }
+    html.theme--documenter-dark .subtitle:not(.is-spaced) + .title {
+      margin-top: -1.25rem; }
+    html.theme--documenter-dark .subtitle.is-1 {
+      font-size: 3rem; }
+    html.theme--documenter-dark .subtitle.is-2 {
+      font-size: 2.5rem; }
+    html.theme--documenter-dark .subtitle.is-3 {
+      font-size: 2rem; }
+    html.theme--documenter-dark .subtitle.is-4 {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .subtitle.is-5 {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .subtitle.is-6 {
+      font-size: 15px; }
+    html.theme--documenter-dark .subtitle.is-7 {
+      font-size: 0.85em; }
+  html.theme--documenter-dark .heading {
+    display: block;
+    font-size: 11px;
+    letter-spacing: 1px;
+    margin-bottom: 5px;
+    text-transform: uppercase; }
+  html.theme--documenter-dark .highlight {
+    font-weight: 400;
+    max-width: 100%;
+    overflow: hidden;
+    padding: 0; }
+    html.theme--documenter-dark .highlight pre {
+      overflow: auto;
+      max-width: 100%; }
+  html.theme--documenter-dark .number {
+    align-items: center;
+    background-color: #282f2f;
+    border-radius: 290486px;
+    display: inline-flex;
+    font-size: 1.25rem;
+    height: 2em;
+    justify-content: center;
+    margin-right: 1.5rem;
+    min-width: 2.5em;
+    padding: 0.25rem 0.5rem;
+    text-align: center;
+    vertical-align: top; }
+  html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark .textarea, html.theme--documenter-dark .select select {
+    background-color: #1f2424;
+    border-color: #5e6d6f;
+    border-radius: 0.4em;
+    color: #dbdee0; }
+    html.theme--documenter-dark .input::-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-moz-placeholder, html.theme--documenter-dark .textarea::-moz-placeholder, html.theme--documenter-dark .select select::-moz-placeholder {
+      color: rgba(219, 222, 224, 0.3); }
+    html.theme--documenter-dark .input::-webkit-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder, html.theme--documenter-dark .textarea::-webkit-input-placeholder, html.theme--documenter-dark .select select::-webkit-input-placeholder {
+      color: rgba(219, 222, 224, 0.3); }
+    html.theme--documenter-dark .input:-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-moz-placeholder, html.theme--documenter-dark .textarea:-moz-placeholder, html.theme--documenter-dark .select select:-moz-placeholder {
+      color: rgba(219, 222, 224, 0.3); }
+    html.theme--documenter-dark .input:-ms-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder, html.theme--documenter-dark .textarea:-ms-input-placeholder, html.theme--documenter-dark .select select:-ms-input-placeholder {
+      color: rgba(219, 222, 224, 0.3); }
+    html.theme--documenter-dark .input:hover, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:hover, html.theme--documenter-dark .textarea:hover, html.theme--documenter-dark .select select:hover, html.theme--documenter-dark .is-hovered.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-hovered, html.theme--documenter-dark .is-hovered.textarea, html.theme--documenter-dark .select select.is-hovered {
+      border-color: #8c9b9d; }
+    html.theme--documenter-dark .input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:focus, html.theme--documenter-dark .textarea:focus, html.theme--documenter-dark .select select:focus, html.theme--documenter-dark .is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-focused.textarea, html.theme--documenter-dark .select select.is-focused, html.theme--documenter-dark .input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:active, html.theme--documenter-dark .textarea:active, html.theme--documenter-dark .select select:active, html.theme--documenter-dark .is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active, html.theme--documenter-dark .is-active.textarea, html.theme--documenter-dark .select select.is-active {
+      border-color: #1abc9c;
+      box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); }
+    html.theme--documenter-dark .input[disabled], html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled], html.theme--documenter-dark .textarea[disabled], html.theme--documenter-dark .select select[disabled],
+    fieldset[disabled] html.theme--documenter-dark .input,
+    fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input,
+    fieldset[disabled] html.theme--documenter-dark .textarea,
+    fieldset[disabled] html.theme--documenter-dark .select select {
+      background-color: #8c9b9d;
+      border-color: #282f2f;
+      box-shadow: none;
+      color: white; }
+      html.theme--documenter-dark .input[disabled]::-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]::-moz-placeholder, html.theme--documenter-dark .textarea[disabled]::-moz-placeholder, html.theme--documenter-dark .select select[disabled]::-moz-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .input::-moz-placeholder,
+      fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-moz-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .textarea::-moz-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .select select::-moz-placeholder {
+        color: rgba(255, 255, 255, 0.3); }
+      html.theme--documenter-dark .input[disabled]::-webkit-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]::-webkit-input-placeholder, html.theme--documenter-dark .textarea[disabled]::-webkit-input-placeholder, html.theme--documenter-dark .select select[disabled]::-webkit-input-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .input::-webkit-input-placeholder,
+      fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .textarea::-webkit-input-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .select select::-webkit-input-placeholder {
+        color: rgba(255, 255, 255, 0.3); }
+      html.theme--documenter-dark .input[disabled]:-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]:-moz-placeholder, html.theme--documenter-dark .textarea[disabled]:-moz-placeholder, html.theme--documenter-dark .select select[disabled]:-moz-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .input:-moz-placeholder,
+      fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-moz-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .textarea:-moz-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .select select:-moz-placeholder {
+        color: rgba(255, 255, 255, 0.3); }
+      html.theme--documenter-dark .input[disabled]:-ms-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]:-ms-input-placeholder, html.theme--documenter-dark .textarea[disabled]:-ms-input-placeholder, html.theme--documenter-dark .select select[disabled]:-ms-input-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .input:-ms-input-placeholder,
+      fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .textarea:-ms-input-placeholder,
+      fieldset[disabled] html.theme--documenter-dark .select select:-ms-input-placeholder {
+        color: rgba(255, 255, 255, 0.3); }
+  html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark .textarea {
+    box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.1);
+    max-width: 100%;
+    width: 100%; }
+    html.theme--documenter-dark .input[readonly], html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[readonly], html.theme--documenter-dark .textarea[readonly] {
+      box-shadow: none; }
+    html.theme--documenter-dark .is-white.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white, html.theme--documenter-dark .is-white.textarea {
+      border-color: white; }
+      html.theme--documenter-dark .is-white.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white:focus, html.theme--documenter-dark .is-white.textarea:focus, html.theme--documenter-dark .is-white.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white.is-focused, html.theme--documenter-dark .is-white.is-focused.textarea, html.theme--documenter-dark .is-white.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white:active, html.theme--documenter-dark .is-white.textarea:active, html.theme--documenter-dark .is-white.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white.is-active, html.theme--documenter-dark .is-white.is-active.textarea {
+        box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); }
+    html.theme--documenter-dark .is-black.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black, html.theme--documenter-dark .is-black.textarea {
+      border-color: #0a0a0a; }
+      html.theme--documenter-dark .is-black.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black:focus, html.theme--documenter-dark .is-black.textarea:focus, html.theme--documenter-dark .is-black.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black.is-focused, html.theme--documenter-dark .is-black.is-focused.textarea, html.theme--documenter-dark .is-black.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black:active, html.theme--documenter-dark .is-black.textarea:active, html.theme--documenter-dark .is-black.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black.is-active, html.theme--documenter-dark .is-black.is-active.textarea {
+        box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); }
+    html.theme--documenter-dark .is-light.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light, html.theme--documenter-dark .is-light.textarea {
+      border-color: #ecf0f1; }
+      html.theme--documenter-dark .is-light.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light:focus, html.theme--documenter-dark .is-light.textarea:focus, html.theme--documenter-dark .is-light.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light.is-focused, html.theme--documenter-dark .is-light.is-focused.textarea, html.theme--documenter-dark .is-light.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light:active, html.theme--documenter-dark .is-light.textarea:active, html.theme--documenter-dark .is-light.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light.is-active, html.theme--documenter-dark .is-light.is-active.textarea {
+        box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); }
+    html.theme--documenter-dark .is-dark.input, html.theme--documenter-dark .content kbd.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark, html.theme--documenter-dark .is-dark.textarea, html.theme--documenter-dark .content kbd.textarea {
+      border-color: #282f2f; }
+      html.theme--documenter-dark .is-dark.input:focus, html.theme--documenter-dark .content kbd.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark:focus, html.theme--documenter-dark .is-dark.textarea:focus, html.theme--documenter-dark .content kbd.textarea:focus, html.theme--documenter-dark .is-dark.is-focused.input, html.theme--documenter-dark .content kbd.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark.is-focused, html.theme--documenter-dark .is-dark.is-focused.textarea, html.theme--documenter-dark .content kbd.is-focused.textarea, html.theme--documenter-dark .is-dark.input:active, html.theme--documenter-dark .content kbd.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark:active, html.theme--documenter-dark .is-dark.textarea:active, html.theme--documenter-dark .content kbd.textarea:active, html.theme--documenter-dark .is-dark.is-active.input, html.theme--documenter-dark .content kbd.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark.is-active, html.theme--documenter-dark .is-dark.is-active.textarea, html.theme--documenter-dark .content kbd.is-active.textarea {
+        box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); }
+    html.theme--documenter-dark .is-primary.input, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary, html.theme--documenter-dark .is-primary.textarea, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink {
+      border-color: #375a7f; }
+      html.theme--documenter-dark .is-primary.input:focus, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary:focus, html.theme--documenter-dark .is-primary.textarea:focus, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink:focus, html.theme--documenter-dark .is-primary.is-focused.input, html.theme--documenter-dark .docstring > section > a.is-focused.input.docs-sourcelink, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary.is-focused, html.theme--documenter-dark .is-primary.is-focused.textarea, html.theme--documenter-dark .docstring > section > a.is-focused.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.input:active, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary:active, html.theme--documenter-dark .is-primary.textarea:active, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink:active, html.theme--documenter-dark .is-primary.is-active.input, html.theme--documenter-dark .docstring > section > a.is-active.input.docs-sourcelink, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary.is-active, html.theme--documenter-dark .is-primary.is-active.textarea, html.theme--documenter-dark .docstring > section > a.is-active.textarea.docs-sourcelink {
+        box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); }
+    html.theme--documenter-dark .is-link.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link, html.theme--documenter-dark .is-link.textarea {
+      border-color: #1abc9c; }
+      html.theme--documenter-dark .is-link.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link:focus, html.theme--documenter-dark .is-link.textarea:focus, html.theme--documenter-dark .is-link.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link.is-focused, html.theme--documenter-dark .is-link.is-focused.textarea, html.theme--documenter-dark .is-link.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link:active, html.theme--documenter-dark .is-link.textarea:active, html.theme--documenter-dark .is-link.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link.is-active, html.theme--documenter-dark .is-link.is-active.textarea {
+        box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); }
+    html.theme--documenter-dark .is-info.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info, html.theme--documenter-dark .is-info.textarea {
+      border-color: #024c7d; }
+      html.theme--documenter-dark .is-info.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info:focus, html.theme--documenter-dark .is-info.textarea:focus, html.theme--documenter-dark .is-info.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info.is-focused, html.theme--documenter-dark .is-info.is-focused.textarea, html.theme--documenter-dark .is-info.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info:active, html.theme--documenter-dark .is-info.textarea:active, html.theme--documenter-dark .is-info.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info.is-active, html.theme--documenter-dark .is-info.is-active.textarea {
+        box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); }
+    html.theme--documenter-dark .is-success.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success, html.theme--documenter-dark .is-success.textarea {
+      border-color: #008438; }
+      html.theme--documenter-dark .is-success.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success:focus, html.theme--documenter-dark .is-success.textarea:focus, html.theme--documenter-dark .is-success.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success.is-focused, html.theme--documenter-dark .is-success.is-focused.textarea, html.theme--documenter-dark .is-success.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success:active, html.theme--documenter-dark .is-success.textarea:active, html.theme--documenter-dark .is-success.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success.is-active, html.theme--documenter-dark .is-success.is-active.textarea {
+        box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); }
+    html.theme--documenter-dark .is-warning.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning, html.theme--documenter-dark .is-warning.textarea {
+      border-color: #ad8100; }
+      html.theme--documenter-dark .is-warning.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning:focus, html.theme--documenter-dark .is-warning.textarea:focus, html.theme--documenter-dark .is-warning.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning.is-focused, html.theme--documenter-dark .is-warning.is-focused.textarea, html.theme--documenter-dark .is-warning.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning:active, html.theme--documenter-dark .is-warning.textarea:active, html.theme--documenter-dark .is-warning.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning.is-active, html.theme--documenter-dark .is-warning.is-active.textarea {
+        box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); }
+    html.theme--documenter-dark .is-danger.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger, html.theme--documenter-dark .is-danger.textarea {
+      border-color: #9e1b0d; }
+      html.theme--documenter-dark .is-danger.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger:focus, html.theme--documenter-dark .is-danger.textarea:focus, html.theme--documenter-dark .is-danger.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger.is-focused, html.theme--documenter-dark .is-danger.is-focused.textarea, html.theme--documenter-dark .is-danger.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger:active, html.theme--documenter-dark .is-danger.textarea:active, html.theme--documenter-dark .is-danger.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger.is-active, html.theme--documenter-dark .is-danger.is-active.textarea {
+        box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); }
+    html.theme--documenter-dark .is-small.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark .is-small.textarea {
+      border-radius: 3px;
+      font-size: 0.85em; }
+    html.theme--documenter-dark .is-medium.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-medium, html.theme--documenter-dark .is-medium.textarea {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .is-large.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-large, html.theme--documenter-dark .is-large.textarea {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .is-fullwidth.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-fullwidth, html.theme--documenter-dark .is-fullwidth.textarea {
+      display: block;
+      width: 100%; }
+    html.theme--documenter-dark .is-inline.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-inline, html.theme--documenter-dark .is-inline.textarea {
+      display: inline;
+      width: auto; }
+  html.theme--documenter-dark .input.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input {
+    border-radius: 290486px;
+    padding-left: 1em;
+    padding-right: 1em; }
+  html.theme--documenter-dark .input.is-static, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-static {
+    background-color: transparent;
+    border-color: transparent;
+    box-shadow: none;
+    padding-left: 0;
+    padding-right: 0; }
+  html.theme--documenter-dark .textarea {
+    display: block;
+    max-width: 100%;
+    min-width: 100%;
+    padding: 0.625em;
+    resize: vertical; }
+    html.theme--documenter-dark .textarea:not([rows]) {
+      max-height: 600px;
+      min-height: 120px; }
+    html.theme--documenter-dark .textarea[rows] {
+      height: initial; }
+    html.theme--documenter-dark .textarea.has-fixed-size {
+      resize: none; }
+  html.theme--documenter-dark .checkbox, html.theme--documenter-dark .radio {
+    cursor: pointer;
+    display: inline-block;
+    line-height: 1.25;
+    position: relative; }
+    html.theme--documenter-dark .checkbox input, html.theme--documenter-dark .radio input {
+      cursor: pointer; }
+    html.theme--documenter-dark .checkbox:hover, html.theme--documenter-dark .radio:hover {
+      color: #8c9b9d; }
+    html.theme--documenter-dark .checkbox[disabled], html.theme--documenter-dark .radio[disabled],
+    fieldset[disabled] html.theme--documenter-dark .checkbox,
+    fieldset[disabled] html.theme--documenter-dark .radio {
+      color: white;
+      cursor: not-allowed; }
+  html.theme--documenter-dark .radio + .radio {
+    margin-left: 0.5em; }
+  html.theme--documenter-dark .select {
+    display: inline-block;
+    max-width: 100%;
+    position: relative;
+    vertical-align: top; }
+    html.theme--documenter-dark .select:not(.is-multiple) {
+      height: 2.25em; }
+    html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after {
+      border-color: #1abc9c;
+      right: 1.125em;
+      z-index: 4; }
+    html.theme--documenter-dark .select.is-rounded select, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.select select {
+      border-radius: 290486px;
+      padding-left: 1em; }
+    html.theme--documenter-dark .select select {
+      cursor: pointer;
+      display: block;
+      font-size: 1em;
+      max-width: 100%;
+      outline: none; }
+      html.theme--documenter-dark .select select::-ms-expand {
+        display: none; }
+      html.theme--documenter-dark .select select[disabled]:hover,
+      fieldset[disabled] html.theme--documenter-dark .select select:hover {
+        border-color: #282f2f; }
+      html.theme--documenter-dark .select select:not([multiple]) {
+        padding-right: 2.5em; }
+      html.theme--documenter-dark .select select[multiple] {
+        height: auto;
+        padding: 0; }
+        html.theme--documenter-dark .select select[multiple] option {
+          padding: 0.5em 1em; }
+    html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading):hover::after {
+      border-color: #8c9b9d; }
+    html.theme--documenter-dark .select.is-white:not(:hover)::after {
+      border-color: white; }
+    html.theme--documenter-dark .select.is-white select {
+      border-color: white; }
+      html.theme--documenter-dark .select.is-white select:hover, html.theme--documenter-dark .select.is-white select.is-hovered {
+        border-color: #f2f2f2; }
+      html.theme--documenter-dark .select.is-white select:focus, html.theme--documenter-dark .select.is-white select.is-focused, html.theme--documenter-dark .select.is-white select:active, html.theme--documenter-dark .select.is-white select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); }
+    html.theme--documenter-dark .select.is-black:not(:hover)::after {
+      border-color: #0a0a0a; }
+    html.theme--documenter-dark .select.is-black select {
+      border-color: #0a0a0a; }
+      html.theme--documenter-dark .select.is-black select:hover, html.theme--documenter-dark .select.is-black select.is-hovered {
+        border-color: black; }
+      html.theme--documenter-dark .select.is-black select:focus, html.theme--documenter-dark .select.is-black select.is-focused, html.theme--documenter-dark .select.is-black select:active, html.theme--documenter-dark .select.is-black select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); }
+    html.theme--documenter-dark .select.is-light:not(:hover)::after {
+      border-color: #ecf0f1; }
+    html.theme--documenter-dark .select.is-light select {
+      border-color: #ecf0f1; }
+      html.theme--documenter-dark .select.is-light select:hover, html.theme--documenter-dark .select.is-light select.is-hovered {
+        border-color: #dde4e6; }
+      html.theme--documenter-dark .select.is-light select:focus, html.theme--documenter-dark .select.is-light select.is-focused, html.theme--documenter-dark .select.is-light select:active, html.theme--documenter-dark .select.is-light select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); }
+    html.theme--documenter-dark .select.is-dark:not(:hover)::after, html.theme--documenter-dark .content kbd.select:not(:hover)::after {
+      border-color: #282f2f; }
+    html.theme--documenter-dark .select.is-dark select, html.theme--documenter-dark .content kbd.select select {
+      border-color: #282f2f; }
+      html.theme--documenter-dark .select.is-dark select:hover, html.theme--documenter-dark .content kbd.select select:hover, html.theme--documenter-dark .select.is-dark select.is-hovered, html.theme--documenter-dark .content kbd.select select.is-hovered {
+        border-color: #1d2122; }
+      html.theme--documenter-dark .select.is-dark select:focus, html.theme--documenter-dark .content kbd.select select:focus, html.theme--documenter-dark .select.is-dark select.is-focused, html.theme--documenter-dark .content kbd.select select.is-focused, html.theme--documenter-dark .select.is-dark select:active, html.theme--documenter-dark .content kbd.select select:active, html.theme--documenter-dark .select.is-dark select.is-active, html.theme--documenter-dark .content kbd.select select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); }
+    html.theme--documenter-dark .select.is-primary:not(:hover)::after, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink:not(:hover)::after {
+      border-color: #375a7f; }
+    html.theme--documenter-dark .select.is-primary select, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select {
+      border-color: #375a7f; }
+      html.theme--documenter-dark .select.is-primary select:hover, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:hover, html.theme--documenter-dark .select.is-primary select.is-hovered, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-hovered {
+        border-color: #2f4d6d; }
+      html.theme--documenter-dark .select.is-primary select:focus, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:focus, html.theme--documenter-dark .select.is-primary select.is-focused, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-focused, html.theme--documenter-dark .select.is-primary select:active, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:active, html.theme--documenter-dark .select.is-primary select.is-active, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); }
+    html.theme--documenter-dark .select.is-link:not(:hover)::after {
+      border-color: #1abc9c; }
+    html.theme--documenter-dark .select.is-link select {
+      border-color: #1abc9c; }
+      html.theme--documenter-dark .select.is-link select:hover, html.theme--documenter-dark .select.is-link select.is-hovered {
+        border-color: #17a689; }
+      html.theme--documenter-dark .select.is-link select:focus, html.theme--documenter-dark .select.is-link select.is-focused, html.theme--documenter-dark .select.is-link select:active, html.theme--documenter-dark .select.is-link select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); }
+    html.theme--documenter-dark .select.is-info:not(:hover)::after {
+      border-color: #024c7d; }
+    html.theme--documenter-dark .select.is-info select {
+      border-color: #024c7d; }
+      html.theme--documenter-dark .select.is-info select:hover, html.theme--documenter-dark .select.is-info select.is-hovered {
+        border-color: #023d64; }
+      html.theme--documenter-dark .select.is-info select:focus, html.theme--documenter-dark .select.is-info select.is-focused, html.theme--documenter-dark .select.is-info select:active, html.theme--documenter-dark .select.is-info select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); }
+    html.theme--documenter-dark .select.is-success:not(:hover)::after {
+      border-color: #008438; }
+    html.theme--documenter-dark .select.is-success select {
+      border-color: #008438; }
+      html.theme--documenter-dark .select.is-success select:hover, html.theme--documenter-dark .select.is-success select.is-hovered {
+        border-color: #006b2d; }
+      html.theme--documenter-dark .select.is-success select:focus, html.theme--documenter-dark .select.is-success select.is-focused, html.theme--documenter-dark .select.is-success select:active, html.theme--documenter-dark .select.is-success select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); }
+    html.theme--documenter-dark .select.is-warning:not(:hover)::after {
+      border-color: #ad8100; }
+    html.theme--documenter-dark .select.is-warning select {
+      border-color: #ad8100; }
+      html.theme--documenter-dark .select.is-warning select:hover, html.theme--documenter-dark .select.is-warning select.is-hovered {
+        border-color: #946e00; }
+      html.theme--documenter-dark .select.is-warning select:focus, html.theme--documenter-dark .select.is-warning select.is-focused, html.theme--documenter-dark .select.is-warning select:active, html.theme--documenter-dark .select.is-warning select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); }
+    html.theme--documenter-dark .select.is-danger:not(:hover)::after {
+      border-color: #9e1b0d; }
+    html.theme--documenter-dark .select.is-danger select {
+      border-color: #9e1b0d; }
+      html.theme--documenter-dark .select.is-danger select:hover, html.theme--documenter-dark .select.is-danger select.is-hovered {
+        border-color: #86170b; }
+      html.theme--documenter-dark .select.is-danger select:focus, html.theme--documenter-dark .select.is-danger select.is-focused, html.theme--documenter-dark .select.is-danger select:active, html.theme--documenter-dark .select.is-danger select.is-active {
+        box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); }
+    html.theme--documenter-dark .select.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.select {
+      border-radius: 3px;
+      font-size: 0.85em; }
+    html.theme--documenter-dark .select.is-medium {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .select.is-large {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .select.is-disabled::after {
+      border-color: white; }
+    html.theme--documenter-dark .select.is-fullwidth {
+      width: 100%; }
+      html.theme--documenter-dark .select.is-fullwidth select {
+        width: 100%; }
+    html.theme--documenter-dark .select.is-loading::after {
+      margin-top: 0;
+      position: absolute;
+      right: 0.625em;
+      top: 0.625em;
+      transform: none; }
+    html.theme--documenter-dark .select.is-loading.is-small:after, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.select.is-loading:after {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .select.is-loading.is-medium:after {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .select.is-loading.is-large:after {
+      font-size: 1.5rem; }
+  html.theme--documenter-dark .file {
+    align-items: stretch;
+    display: flex;
+    justify-content: flex-start;
+    position: relative; }
+    html.theme--documenter-dark .file.is-white .file-cta {
+      background-color: white;
+      border-color: transparent;
+      color: #0a0a0a; }
+    html.theme--documenter-dark .file.is-white:hover .file-cta, html.theme--documenter-dark .file.is-white.is-hovered .file-cta {
+      background-color: #f9f9f9;
+      border-color: transparent;
+      color: #0a0a0a; }
+    html.theme--documenter-dark .file.is-white:focus .file-cta, html.theme--documenter-dark .file.is-white.is-focused .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(255, 255, 255, 0.25);
+      color: #0a0a0a; }
+    html.theme--documenter-dark .file.is-white:active .file-cta, html.theme--documenter-dark .file.is-white.is-active .file-cta {
+      background-color: #f2f2f2;
+      border-color: transparent;
+      color: #0a0a0a; }
+    html.theme--documenter-dark .file.is-black .file-cta {
+      background-color: #0a0a0a;
+      border-color: transparent;
+      color: white; }
+    html.theme--documenter-dark .file.is-black:hover .file-cta, html.theme--documenter-dark .file.is-black.is-hovered .file-cta {
+      background-color: #040404;
+      border-color: transparent;
+      color: white; }
+    html.theme--documenter-dark .file.is-black:focus .file-cta, html.theme--documenter-dark .file.is-black.is-focused .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(10, 10, 10, 0.25);
+      color: white; }
+    html.theme--documenter-dark .file.is-black:active .file-cta, html.theme--documenter-dark .file.is-black.is-active .file-cta {
+      background-color: black;
+      border-color: transparent;
+      color: white; }
+    html.theme--documenter-dark .file.is-light .file-cta {
+      background-color: #ecf0f1;
+      border-color: transparent;
+      color: #282f2f; }
+    html.theme--documenter-dark .file.is-light:hover .file-cta, html.theme--documenter-dark .file.is-light.is-hovered .file-cta {
+      background-color: #e5eaec;
+      border-color: transparent;
+      color: #282f2f; }
+    html.theme--documenter-dark .file.is-light:focus .file-cta, html.theme--documenter-dark .file.is-light.is-focused .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(236, 240, 241, 0.25);
+      color: #282f2f; }
+    html.theme--documenter-dark .file.is-light:active .file-cta, html.theme--documenter-dark .file.is-light.is-active .file-cta {
+      background-color: #dde4e6;
+      border-color: transparent;
+      color: #282f2f; }
+    html.theme--documenter-dark .file.is-dark .file-cta, html.theme--documenter-dark .content kbd.file .file-cta {
+      background-color: #282f2f;
+      border-color: transparent;
+      color: #ecf0f1; }
+    html.theme--documenter-dark .file.is-dark:hover .file-cta, html.theme--documenter-dark .content kbd.file:hover .file-cta, html.theme--documenter-dark .file.is-dark.is-hovered .file-cta, html.theme--documenter-dark .content kbd.file.is-hovered .file-cta {
+      background-color: #232829;
+      border-color: transparent;
+      color: #ecf0f1; }
+    html.theme--documenter-dark .file.is-dark:focus .file-cta, html.theme--documenter-dark .content kbd.file:focus .file-cta, html.theme--documenter-dark .file.is-dark.is-focused .file-cta, html.theme--documenter-dark .content kbd.file.is-focused .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(40, 47, 47, 0.25);
+      color: #ecf0f1; }
+    html.theme--documenter-dark .file.is-dark:active .file-cta, html.theme--documenter-dark .content kbd.file:active .file-cta, html.theme--documenter-dark .file.is-dark.is-active .file-cta, html.theme--documenter-dark .content kbd.file.is-active .file-cta {
+      background-color: #1d2122;
+      border-color: transparent;
+      color: #ecf0f1; }
+    html.theme--documenter-dark .file.is-primary .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink .file-cta {
+      background-color: #375a7f;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-primary:hover .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:hover .file-cta, html.theme--documenter-dark .file.is-primary.is-hovered .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-hovered.docs-sourcelink .file-cta {
+      background-color: #335476;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-primary:focus .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:focus .file-cta, html.theme--documenter-dark .file.is-primary.is-focused .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-focused.docs-sourcelink .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(55, 90, 127, 0.25);
+      color: #fff; }
+    html.theme--documenter-dark .file.is-primary:active .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:active .file-cta, html.theme--documenter-dark .file.is-primary.is-active .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-active.docs-sourcelink .file-cta {
+      background-color: #2f4d6d;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-link .file-cta {
+      background-color: #1abc9c;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-link:hover .file-cta, html.theme--documenter-dark .file.is-link.is-hovered .file-cta {
+      background-color: #18b193;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-link:focus .file-cta, html.theme--documenter-dark .file.is-link.is-focused .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(26, 188, 156, 0.25);
+      color: #fff; }
+    html.theme--documenter-dark .file.is-link:active .file-cta, html.theme--documenter-dark .file.is-link.is-active .file-cta {
+      background-color: #17a689;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-info .file-cta {
+      background-color: #024c7d;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-info:hover .file-cta, html.theme--documenter-dark .file.is-info.is-hovered .file-cta {
+      background-color: #024470;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-info:focus .file-cta, html.theme--documenter-dark .file.is-info.is-focused .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(2, 76, 125, 0.25);
+      color: #fff; }
+    html.theme--documenter-dark .file.is-info:active .file-cta, html.theme--documenter-dark .file.is-info.is-active .file-cta {
+      background-color: #023d64;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-success .file-cta {
+      background-color: #008438;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-success:hover .file-cta, html.theme--documenter-dark .file.is-success.is-hovered .file-cta {
+      background-color: #007733;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-success:focus .file-cta, html.theme--documenter-dark .file.is-success.is-focused .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(0, 132, 56, 0.25);
+      color: #fff; }
+    html.theme--documenter-dark .file.is-success:active .file-cta, html.theme--documenter-dark .file.is-success.is-active .file-cta {
+      background-color: #006b2d;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-warning .file-cta {
+      background-color: #ad8100;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-warning:hover .file-cta, html.theme--documenter-dark .file.is-warning.is-hovered .file-cta {
+      background-color: #a07700;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-warning:focus .file-cta, html.theme--documenter-dark .file.is-warning.is-focused .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(173, 129, 0, 0.25);
+      color: #fff; }
+    html.theme--documenter-dark .file.is-warning:active .file-cta, html.theme--documenter-dark .file.is-warning.is-active .file-cta {
+      background-color: #946e00;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-danger .file-cta {
+      background-color: #9e1b0d;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-danger:hover .file-cta, html.theme--documenter-dark .file.is-danger.is-hovered .file-cta {
+      background-color: #92190c;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-danger:focus .file-cta, html.theme--documenter-dark .file.is-danger.is-focused .file-cta {
+      border-color: transparent;
+      box-shadow: 0 0 0.5em rgba(158, 27, 13, 0.25);
+      color: #fff; }
+    html.theme--documenter-dark .file.is-danger:active .file-cta, html.theme--documenter-dark .file.is-danger.is-active .file-cta {
+      background-color: #86170b;
+      border-color: transparent;
+      color: #fff; }
+    html.theme--documenter-dark .file.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.file {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .file.is-medium {
+      font-size: 1.25rem; }
+      html.theme--documenter-dark .file.is-medium .file-icon .fa {
+        font-size: 21px; }
+    html.theme--documenter-dark .file.is-large {
+      font-size: 1.5rem; }
+      html.theme--documenter-dark .file.is-large .file-icon .fa {
+        font-size: 28px; }
+    html.theme--documenter-dark .file.has-name .file-cta {
+      border-bottom-right-radius: 0;
+      border-top-right-radius: 0; }
+    html.theme--documenter-dark .file.has-name .file-name {
+      border-bottom-left-radius: 0;
+      border-top-left-radius: 0; }
+    html.theme--documenter-dark .file.has-name.is-empty .file-cta {
+      border-radius: 0.4em; }
+    html.theme--documenter-dark .file.has-name.is-empty .file-name {
+      display: none; }
+    html.theme--documenter-dark .file.is-boxed .file-label {
+      flex-direction: column; }
+    html.theme--documenter-dark .file.is-boxed .file-cta {
+      flex-direction: column;
+      height: auto;
+      padding: 1em 3em; }
+    html.theme--documenter-dark .file.is-boxed .file-name {
+      border-width: 0 1px 1px; }
+    html.theme--documenter-dark .file.is-boxed .file-icon {
+      height: 1.5em;
+      width: 1.5em; }
+      html.theme--documenter-dark .file.is-boxed .file-icon .fa {
+        font-size: 21px; }
+    html.theme--documenter-dark .file.is-boxed.is-small .file-icon .fa, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.file.is-boxed .file-icon .fa {
+      font-size: 14px; }
+    html.theme--documenter-dark .file.is-boxed.is-medium .file-icon .fa {
+      font-size: 28px; }
+    html.theme--documenter-dark .file.is-boxed.is-large .file-icon .fa {
+      font-size: 35px; }
+    html.theme--documenter-dark .file.is-boxed.has-name .file-cta {
+      border-radius: 0.4em 0.4em 0 0; }
+    html.theme--documenter-dark .file.is-boxed.has-name .file-name {
+      border-radius: 0 0 0.4em 0.4em;
+      border-width: 0 1px 1px; }
+    html.theme--documenter-dark .file.is-centered {
+      justify-content: center; }
+    html.theme--documenter-dark .file.is-fullwidth .file-label {
+      width: 100%; }
+    html.theme--documenter-dark .file.is-fullwidth .file-name {
+      flex-grow: 1;
+      max-width: none; }
+    html.theme--documenter-dark .file.is-right {
+      justify-content: flex-end; }
+      html.theme--documenter-dark .file.is-right .file-cta {
+        border-radius: 0 0.4em 0.4em 0; }
+      html.theme--documenter-dark .file.is-right .file-name {
+        border-radius: 0.4em 0 0 0.4em;
+        border-width: 1px 0 1px 1px;
+        order: -1; }
+  html.theme--documenter-dark .file-label {
+    align-items: stretch;
+    display: flex;
+    cursor: pointer;
+    justify-content: flex-start;
+    overflow: hidden;
+    position: relative; }
+    html.theme--documenter-dark .file-label:hover .file-cta {
+      background-color: #e5eaec;
+      color: #282f2f; }
+    html.theme--documenter-dark .file-label:hover .file-name {
+      border-color: #596668; }
+    html.theme--documenter-dark .file-label:active .file-cta {
+      background-color: #dde4e6;
+      color: #282f2f; }
+    html.theme--documenter-dark .file-label:active .file-name {
+      border-color: #535f61; }
+  html.theme--documenter-dark .file-input {
+    height: 100%;
+    left: 0;
+    opacity: 0;
+    outline: none;
+    position: absolute;
+    top: 0;
+    width: 100%; }
+  html.theme--documenter-dark .file-cta,
+  html.theme--documenter-dark .file-name {
+    border-color: #5e6d6f;
+    border-radius: 0.4em;
+    font-size: 1em;
+    padding-left: 1em;
+    padding-right: 1em;
+    white-space: nowrap; }
+  html.theme--documenter-dark .file-cta {
+    background-color: #ecf0f1;
+    color: #343c3d; }
+  html.theme--documenter-dark .file-name {
+    border-color: #5e6d6f;
+    border-style: solid;
+    border-width: 1px 1px 1px 0;
+    display: block;
+    max-width: 16em;
+    overflow: hidden;
+    text-align: left;
+    text-overflow: ellipsis; }
+  html.theme--documenter-dark .file-icon {
+    align-items: center;
+    display: flex;
+    height: 1em;
+    justify-content: center;
+    margin-right: 0.5em;
+    width: 1em; }
+    html.theme--documenter-dark .file-icon .fa {
+      font-size: 14px; }
+  html.theme--documenter-dark .label {
+    color: #282f2f;
+    display: block;
+    font-size: 15px;
+    font-weight: 700; }
+    html.theme--documenter-dark .label:not(:last-child) {
+      margin-bottom: 0.5em; }
+    html.theme--documenter-dark .label.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.label {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .label.is-medium {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .label.is-large {
+      font-size: 1.5rem; }
+  html.theme--documenter-dark .help {
+    display: block;
+    font-size: 0.85em;
+    margin-top: 0.25rem; }
+    html.theme--documenter-dark .help.is-white {
+      color: white; }
+    html.theme--documenter-dark .help.is-black {
+      color: #0a0a0a; }
+    html.theme--documenter-dark .help.is-light {
+      color: #ecf0f1; }
+    html.theme--documenter-dark .help.is-dark, html.theme--documenter-dark .content kbd.help {
+      color: #282f2f; }
+    html.theme--documenter-dark .help.is-primary, html.theme--documenter-dark .docstring > section > a.help.docs-sourcelink {
+      color: #375a7f; }
+    html.theme--documenter-dark .help.is-link {
+      color: #1abc9c; }
+    html.theme--documenter-dark .help.is-info {
+      color: #024c7d; }
+    html.theme--documenter-dark .help.is-success {
+      color: #008438; }
+    html.theme--documenter-dark .help.is-warning {
+      color: #ad8100; }
+    html.theme--documenter-dark .help.is-danger {
+      color: #9e1b0d; }
+  html.theme--documenter-dark .field:not(:last-child) {
+    margin-bottom: 0.75rem; }
+  html.theme--documenter-dark .field.has-addons {
+    display: flex;
+    justify-content: flex-start; }
+    html.theme--documenter-dark .field.has-addons .control:not(:last-child) {
+      margin-right: -1px; }
+    html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .button,
+    html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .input,
+    html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search > input,
+    html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search > input,
+    html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .select select {
+      border-radius: 0; }
+    html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .button,
+    html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .input,
+    html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input,
+    html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search > input,
+    html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .select select {
+      border-bottom-right-radius: 0;
+      border-top-right-radius: 0; }
+    html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .button,
+    html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .input,
+    html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input,
+    html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search > input,
+    html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .select select {
+      border-bottom-left-radius: 0;
+      border-top-left-radius: 0; }
+    html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):hover, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]).is-hovered,
+    html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):hover,
+    html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):hover,
+    html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):hover,
+    html.theme--documenter-dark .field.has-addons .control .input:not([disabled]).is-hovered,
+    html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-hovered,
+    html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-hovered,
+    html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):hover,
+    html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]).is-hovered {
+      z-index: 2; }
+    html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):focus, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]).is-focused, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):active, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]).is-active,
+    html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):focus,
+    html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus,
+    html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus,
+    html.theme--documenter-dark .field.has-addons .control .input:not([disabled]).is-focused,
+    html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-focused,
+    html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-focused,
+    html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):active,
+    html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active,
+    html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active,
+    html.theme--documenter-dark .field.has-addons .control .input:not([disabled]).is-active,
+    html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-active,
+    html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-active,
+    html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):focus,
+    html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]).is-focused,
+    html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):active,
+    html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]).is-active {
+      z-index: 3; }
+      html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):focus:hover, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]).is-focused:hover, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):active:hover, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]).is-active:hover,
+      html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):focus:hover,
+      html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus:hover,
+      html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus:hover,
+      html.theme--documenter-dark .field.has-addons .control .input:not([disabled]).is-focused:hover,
+      html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-focused:hover,
+      html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-focused:hover,
+      html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):active:hover,
+      html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active:hover,
+      html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active:hover,
+      html.theme--documenter-dark .field.has-addons .control .input:not([disabled]).is-active:hover,
+      html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-active:hover,
+      html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-active:hover,
+      html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):focus:hover,
+      html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]).is-focused:hover,
+      html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):active:hover,
+      html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]).is-active:hover {
+        z-index: 4; }
+    html.theme--documenter-dark .field.has-addons .control.is-expanded {
+      flex-grow: 1;
+      flex-shrink: 1; }
+    html.theme--documenter-dark .field.has-addons.has-addons-centered {
+      justify-content: center; }
+    html.theme--documenter-dark .field.has-addons.has-addons-right {
+      justify-content: flex-end; }
+    html.theme--documenter-dark .field.has-addons.has-addons-fullwidth .control {
+      flex-grow: 1;
+      flex-shrink: 0; }
+  html.theme--documenter-dark .field.is-grouped {
+    display: flex;
+    justify-content: flex-start; }
+    html.theme--documenter-dark .field.is-grouped > .control {
+      flex-shrink: 0; }
+      html.theme--documenter-dark .field.is-grouped > .control:not(:last-child) {
+        margin-bottom: 0;
+        margin-right: 0.75rem; }
+      html.theme--documenter-dark .field.is-grouped > .control.is-expanded {
+        flex-grow: 1;
+        flex-shrink: 1; }
+    html.theme--documenter-dark .field.is-grouped.is-grouped-centered {
+      justify-content: center; }
+    html.theme--documenter-dark .field.is-grouped.is-grouped-right {
+      justify-content: flex-end; }
+    html.theme--documenter-dark .field.is-grouped.is-grouped-multiline {
+      flex-wrap: wrap; }
+      html.theme--documenter-dark .field.is-grouped.is-grouped-multiline > .control:last-child, html.theme--documenter-dark .field.is-grouped.is-grouped-multiline > .control:not(:last-child) {
+        margin-bottom: 0.75rem; }
+      html.theme--documenter-dark .field.is-grouped.is-grouped-multiline:last-child {
+        margin-bottom: -0.75rem; }
+      html.theme--documenter-dark .field.is-grouped.is-grouped-multiline:not(:last-child) {
+        margin-bottom: 0; }
+  @media screen and (min-width: 769px), print {
+    html.theme--documenter-dark .field.is-horizontal {
+      display: flex; } }
+  html.theme--documenter-dark .field-label .label {
+    font-size: inherit; }
+  @media screen and (max-width: 768px) {
+    html.theme--documenter-dark .field-label {
+      margin-bottom: 0.5rem; } }
+  @media screen and (min-width: 769px), print {
+    html.theme--documenter-dark .field-label {
+      flex-basis: 0;
+      flex-grow: 1;
+      flex-shrink: 0;
+      margin-right: 1.5rem;
+      text-align: right; }
+      html.theme--documenter-dark .field-label.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.field-label {
+        font-size: 0.85em;
+        padding-top: 0.375em; }
+      html.theme--documenter-dark .field-label.is-normal {
+        padding-top: 0.375em; }
+      html.theme--documenter-dark .field-label.is-medium {
+        font-size: 1.25rem;
+        padding-top: 0.375em; }
+      html.theme--documenter-dark .field-label.is-large {
+        font-size: 1.5rem;
+        padding-top: 0.375em; } }
+  html.theme--documenter-dark .field-body .field .field {
+    margin-bottom: 0; }
+  @media screen and (min-width: 769px), print {
+    html.theme--documenter-dark .field-body {
+      display: flex;
+      flex-basis: 0;
+      flex-grow: 5;
+      flex-shrink: 1; }
+      html.theme--documenter-dark .field-body .field {
+        margin-bottom: 0; }
+      html.theme--documenter-dark .field-body > .field {
+        flex-shrink: 1; }
+        html.theme--documenter-dark .field-body > .field:not(.is-narrow) {
+          flex-grow: 1; }
+        html.theme--documenter-dark .field-body > .field:not(:last-child) {
+          margin-right: 0.75rem; } }
+  html.theme--documenter-dark .control {
+    box-sizing: border-box;
+    clear: both;
+    font-size: 15px;
+    position: relative;
+    text-align: left; }
+    html.theme--documenter-dark .control.has-icons-left .input:focus ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input:focus ~ .icon,
+    html.theme--documenter-dark .control.has-icons-left .select:focus ~ .icon, html.theme--documenter-dark .control.has-icons-right .input:focus ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input:focus ~ .icon,
+    html.theme--documenter-dark .control.has-icons-right .select:focus ~ .icon {
+      color: #5e6d6f; }
+    html.theme--documenter-dark .control.has-icons-left .input.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input ~ .icon,
+    html.theme--documenter-dark .control.has-icons-left .select.is-small ~ .icon,
+    html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.select ~ .icon,
+    html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.select ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input ~ .icon,
+    html.theme--documenter-dark .control.has-icons-right .select.is-small ~ .icon,
+    html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.select ~ .icon,
+    html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.select ~ .icon {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .control.has-icons-left .input.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-medium ~ .icon,
+    html.theme--documenter-dark .control.has-icons-left .select.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-medium ~ .icon,
+    html.theme--documenter-dark .control.has-icons-right .select.is-medium ~ .icon {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .control.has-icons-left .input.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-large ~ .icon,
+    html.theme--documenter-dark .control.has-icons-left .select.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-large ~ .icon,
+    html.theme--documenter-dark .control.has-icons-right .select.is-large ~ .icon {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .control.has-icons-left .icon, html.theme--documenter-dark .control.has-icons-right .icon {
+      color: #dbdee0;
+      height: 2.25em;
+      pointer-events: none;
+      position: absolute;
+      top: 0;
+      width: 2.25em;
+      z-index: 4; }
+    html.theme--documenter-dark .control.has-icons-left .input, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input,
+    html.theme--documenter-dark .control.has-icons-left .select select {
+      padding-left: 2.25em; }
+    html.theme--documenter-dark .control.has-icons-left .icon.is-left {
+      left: 0; }
+    html.theme--documenter-dark .control.has-icons-right .input, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input,
+    html.theme--documenter-dark .control.has-icons-right .select select {
+      padding-right: 2.25em; }
+    html.theme--documenter-dark .control.has-icons-right .icon.is-right {
+      right: 0; }
+    html.theme--documenter-dark .control.is-loading::after {
+      position: absolute !important;
+      right: 0.625em;
+      top: 0.625em;
+      z-index: 4; }
+    html.theme--documenter-dark .control.is-loading.is-small:after, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.control.is-loading:after {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .control.is-loading.is-medium:after {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .control.is-loading.is-large:after {
+      font-size: 1.5rem; }
+  html.theme--documenter-dark .breadcrumb {
+    font-size: 15px;
+    white-space: nowrap; }
+    html.theme--documenter-dark .breadcrumb a {
+      align-items: center;
+      color: #1abc9c;
+      display: flex;
+      justify-content: center;
+      padding: 0 0.75em; }
+      html.theme--documenter-dark .breadcrumb a:hover {
+        color: #1dd2af; }
+    html.theme--documenter-dark .breadcrumb li {
+      align-items: center;
+      display: flex; }
+      html.theme--documenter-dark .breadcrumb li:first-child a {
+        padding-left: 0; }
+      html.theme--documenter-dark .breadcrumb li.is-active a {
+        color: #f2f2f2;
+        cursor: default;
+        pointer-events: none; }
+      html.theme--documenter-dark .breadcrumb li + li::before {
+        color: #8c9b9d;
+        content: "\0002f"; }
+    html.theme--documenter-dark .breadcrumb ul,
+    html.theme--documenter-dark .breadcrumb ol {
+      align-items: flex-start;
+      display: flex;
+      flex-wrap: wrap;
+      justify-content: flex-start; }
+    html.theme--documenter-dark .breadcrumb .icon:first-child {
+      margin-right: 0.5em; }
+    html.theme--documenter-dark .breadcrumb .icon:last-child {
+      margin-left: 0.5em; }
+    html.theme--documenter-dark .breadcrumb.is-centered ol,
+    html.theme--documenter-dark .breadcrumb.is-centered ul {
+      justify-content: center; }
+    html.theme--documenter-dark .breadcrumb.is-right ol,
+    html.theme--documenter-dark .breadcrumb.is-right ul {
+      justify-content: flex-end; }
+    html.theme--documenter-dark .breadcrumb.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.breadcrumb {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .breadcrumb.is-medium {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .breadcrumb.is-large {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .breadcrumb.has-arrow-separator li + li::before {
+      content: "\02192"; }
+    html.theme--documenter-dark .breadcrumb.has-bullet-separator li + li::before {
+      content: "\02022"; }
+    html.theme--documenter-dark .breadcrumb.has-dot-separator li + li::before {
+      content: "\000b7"; }
+    html.theme--documenter-dark .breadcrumb.has-succeeds-separator li + li::before {
+      content: "\0227B"; }
+  html.theme--documenter-dark .card {
+    background-color: white;
+    box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1);
+    color: #fff;
+    max-width: 100%;
+    position: relative; }
+  html.theme--documenter-dark .card-header {
+    background-color: transparent;
+    align-items: stretch;
+    box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1);
+    display: flex; }
+  html.theme--documenter-dark .card-header-title {
+    align-items: center;
+    color: #f2f2f2;
+    display: flex;
+    flex-grow: 1;
+    font-weight: 700;
+    padding: 0.75rem; }
+    html.theme--documenter-dark .card-header-title.is-centered {
+      justify-content: center; }
+  html.theme--documenter-dark .card-header-icon {
+    align-items: center;
+    cursor: pointer;
+    display: flex;
+    justify-content: center;
+    padding: 0.75rem; }
+  html.theme--documenter-dark .card-image {
+    display: block;
+    position: relative; }
+  html.theme--documenter-dark .card-content {
+    background-color: transparent;
+    padding: 1rem 1.25rem; }
+  html.theme--documenter-dark .card-footer {
+    background-color: transparent;
+    border-top: 1px solid #5e6d6f;
+    align-items: stretch;
+    display: flex; }
+  html.theme--documenter-dark .card-footer-item {
+    align-items: center;
+    display: flex;
+    flex-basis: 0;
+    flex-grow: 1;
+    flex-shrink: 0;
+    justify-content: center;
+    padding: 0.75rem; }
+    html.theme--documenter-dark .card-footer-item:not(:last-child) {
+      border-right: 1px solid #5e6d6f; }
+  html.theme--documenter-dark .card .media:not(:last-child) {
+    margin-bottom: 1.5rem; }
+  html.theme--documenter-dark .dropdown {
+    display: inline-flex;
+    position: relative;
+    vertical-align: top; }
+    html.theme--documenter-dark .dropdown.is-active .dropdown-menu, html.theme--documenter-dark .dropdown.is-hoverable:hover .dropdown-menu {
+      display: block; }
+    html.theme--documenter-dark .dropdown.is-right .dropdown-menu {
+      left: auto;
+      right: 0; }
+    html.theme--documenter-dark .dropdown.is-up .dropdown-menu {
+      bottom: 100%;
+      padding-bottom: 4px;
+      padding-top: initial;
+      top: auto; }
+  html.theme--documenter-dark .dropdown-menu {
+    display: none;
+    left: 0;
+    min-width: 12rem;
+    padding-top: 4px;
+    position: absolute;
+    top: 100%;
+    z-index: 20; }
+  html.theme--documenter-dark .dropdown-content {
+    background-color: #282f2f;
+    border-radius: 0.4em;
+    box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1);
+    padding-bottom: 0.5rem;
+    padding-top: 0.5rem; }
+  html.theme--documenter-dark .dropdown-item {
+    color: #fff;
+    display: block;
+    font-size: 0.875rem;
+    line-height: 1.5;
+    padding: 0.375rem 1rem;
+    position: relative; }
+  html.theme--documenter-dark a.dropdown-item,
+  html.theme--documenter-dark button.dropdown-item {
+    padding-right: 3rem;
+    text-align: left;
+    white-space: nowrap;
+    width: 100%; }
+    html.theme--documenter-dark a.dropdown-item:hover,
+    html.theme--documenter-dark button.dropdown-item:hover {
+      background-color: #282f2f;
+      color: #0a0a0a; }
+    html.theme--documenter-dark a.dropdown-item.is-active,
+    html.theme--documenter-dark button.dropdown-item.is-active {
+      background-color: #1abc9c;
+      color: #fff; }
+  html.theme--documenter-dark .dropdown-divider {
+    background-color: #5e6d6f;
+    border: none;
+    display: block;
+    height: 1px;
+    margin: 0.5rem 0; }
+  html.theme--documenter-dark .level {
+    align-items: center;
+    justify-content: space-between; }
+    html.theme--documenter-dark .level code {
+      border-radius: 0.4em; }
+    html.theme--documenter-dark .level img {
+      display: inline-block;
+      vertical-align: top; }
+    html.theme--documenter-dark .level.is-mobile {
+      display: flex; }
+      html.theme--documenter-dark .level.is-mobile .level-left,
+      html.theme--documenter-dark .level.is-mobile .level-right {
+        display: flex; }
+      html.theme--documenter-dark .level.is-mobile .level-left + .level-right {
+        margin-top: 0; }
+      html.theme--documenter-dark .level.is-mobile .level-item:not(:last-child) {
+        margin-bottom: 0;
+        margin-right: 0.75rem; }
+      html.theme--documenter-dark .level.is-mobile .level-item:not(.is-narrow) {
+        flex-grow: 1; }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .level {
+        display: flex; }
+        html.theme--documenter-dark .level > .level-item:not(.is-narrow) {
+          flex-grow: 1; } }
+  html.theme--documenter-dark .level-item {
+    align-items: center;
+    display: flex;
+    flex-basis: auto;
+    flex-grow: 0;
+    flex-shrink: 0;
+    justify-content: center; }
+    html.theme--documenter-dark .level-item .title,
+    html.theme--documenter-dark .level-item .subtitle {
+      margin-bottom: 0; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .level-item:not(:last-child) {
+        margin-bottom: 0.75rem; } }
+  html.theme--documenter-dark .level-left,
+  html.theme--documenter-dark .level-right {
+    flex-basis: auto;
+    flex-grow: 0;
+    flex-shrink: 0; }
+    html.theme--documenter-dark .level-left .level-item.is-flexible,
+    html.theme--documenter-dark .level-right .level-item.is-flexible {
+      flex-grow: 1; }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .level-left .level-item:not(:last-child),
+      html.theme--documenter-dark .level-right .level-item:not(:last-child) {
+        margin-right: 0.75rem; } }
+  html.theme--documenter-dark .level-left {
+    align-items: center;
+    justify-content: flex-start; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .level-left + .level-right {
+        margin-top: 1.5rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .level-left {
+        display: flex; } }
+  html.theme--documenter-dark .level-right {
+    align-items: center;
+    justify-content: flex-end; }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .level-right {
+        display: flex; } }
+  html.theme--documenter-dark .list {
+    background-color: white;
+    border-radius: 0.4em;
+    box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); }
+  html.theme--documenter-dark .list-item {
+    display: block;
+    padding: 0.5em 1em; }
+    html.theme--documenter-dark .list-item:not(a) {
+      color: #fff; }
+    html.theme--documenter-dark .list-item:first-child {
+      border-top-left-radius: 0.4em;
+      border-top-right-radius: 0.4em; }
+    html.theme--documenter-dark .list-item:last-child {
+      border-bottom-left-radius: 0.4em;
+      border-bottom-right-radius: 0.4em; }
+    html.theme--documenter-dark .list-item:not(:last-child) {
+      border-bottom: 1px solid #5e6d6f; }
+    html.theme--documenter-dark .list-item.is-active {
+      background-color: #1abc9c;
+      color: #fff; }
+  html.theme--documenter-dark a.list-item {
+    background-color: #282f2f;
+    cursor: pointer; }
+  html.theme--documenter-dark .media {
+    align-items: flex-start;
+    display: flex;
+    text-align: left; }
+    html.theme--documenter-dark .media .content:not(:last-child) {
+      margin-bottom: 0.75rem; }
+    html.theme--documenter-dark .media .media {
+      border-top: 1px solid rgba(94, 109, 111, 0.5);
+      display: flex;
+      padding-top: 0.75rem; }
+      html.theme--documenter-dark .media .media .content:not(:last-child),
+      html.theme--documenter-dark .media .media .control:not(:last-child) {
+        margin-bottom: 0.5rem; }
+      html.theme--documenter-dark .media .media .media {
+        padding-top: 0.5rem; }
+        html.theme--documenter-dark .media .media .media + .media {
+          margin-top: 0.5rem; }
+    html.theme--documenter-dark .media + .media {
+      border-top: 1px solid rgba(94, 109, 111, 0.5);
+      margin-top: 1rem;
+      padding-top: 1rem; }
+    html.theme--documenter-dark .media.is-large + .media {
+      margin-top: 1.5rem;
+      padding-top: 1.5rem; }
+  html.theme--documenter-dark .media-left,
+  html.theme--documenter-dark .media-right {
+    flex-basis: auto;
+    flex-grow: 0;
+    flex-shrink: 0; }
+  html.theme--documenter-dark .media-left {
+    margin-right: 1rem; }
+  html.theme--documenter-dark .media-right {
+    margin-left: 1rem; }
+  html.theme--documenter-dark .media-content {
+    flex-basis: auto;
+    flex-grow: 1;
+    flex-shrink: 1;
+    text-align: left; }
+  @media screen and (max-width: 768px) {
+    html.theme--documenter-dark .media-content {
+      overflow-x: auto; } }
+  html.theme--documenter-dark .menu {
+    font-size: 15px; }
+    html.theme--documenter-dark .menu.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.menu {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .menu.is-medium {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .menu.is-large {
+      font-size: 1.5rem; }
+  html.theme--documenter-dark .menu-list {
+    line-height: 1.25; }
+    html.theme--documenter-dark .menu-list a {
+      border-radius: 3px;
+      color: #fff;
+      display: block;
+      padding: 0.5em 0.75em; }
+      html.theme--documenter-dark .menu-list a:hover {
+        background-color: #282f2f;
+        color: #f2f2f2; }
+      html.theme--documenter-dark .menu-list a.is-active {
+        background-color: #1abc9c;
+        color: #fff; }
+    html.theme--documenter-dark .menu-list li ul {
+      border-left: 1px solid #5e6d6f;
+      margin: 0.75em;
+      padding-left: 0.75em; }
+  html.theme--documenter-dark .menu-label {
+    color: white;
+    font-size: 0.75em;
+    letter-spacing: 0.1em;
+    text-transform: uppercase; }
+    html.theme--documenter-dark .menu-label:not(:first-child) {
+      margin-top: 1em; }
+    html.theme--documenter-dark .menu-label:not(:last-child) {
+      margin-bottom: 1em; }
+  html.theme--documenter-dark .message {
+    background-color: #282f2f;
+    border-radius: 0.4em;
+    font-size: 15px; }
+    html.theme--documenter-dark .message strong {
+      color: currentColor; }
+    html.theme--documenter-dark .message a:not(.button):not(.tag):not(.dropdown-item) {
+      color: currentColor;
+      text-decoration: underline; }
+    html.theme--documenter-dark .message.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.message {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .message.is-medium {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .message.is-large {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .message.is-white {
+      background-color: white; }
+      html.theme--documenter-dark .message.is-white .message-header {
+        background-color: white;
+        color: #0a0a0a; }
+      html.theme--documenter-dark .message.is-white .message-body {
+        border-color: white;
+        color: #4d4d4d; }
+    html.theme--documenter-dark .message.is-black {
+      background-color: #fafafa; }
+      html.theme--documenter-dark .message.is-black .message-header {
+        background-color: #0a0a0a;
+        color: white; }
+      html.theme--documenter-dark .message.is-black .message-body {
+        border-color: #0a0a0a;
+        color: #090909; }
+    html.theme--documenter-dark .message.is-light {
+      background-color: #f9fafb; }
+      html.theme--documenter-dark .message.is-light .message-header {
+        background-color: #ecf0f1;
+        color: #282f2f; }
+      html.theme--documenter-dark .message.is-light .message-body {
+        border-color: #ecf0f1;
+        color: #505050; }
+    html.theme--documenter-dark .message.is-dark, html.theme--documenter-dark .content kbd.message {
+      background-color: #f9fafa; }
+      html.theme--documenter-dark .message.is-dark .message-header, html.theme--documenter-dark .content kbd.message .message-header {
+        background-color: #282f2f;
+        color: #ecf0f1; }
+      html.theme--documenter-dark .message.is-dark .message-body, html.theme--documenter-dark .content kbd.message .message-body {
+        border-color: #282f2f;
+        color: #212526; }
+    html.theme--documenter-dark .message.is-primary, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink {
+      background-color: #f8fafc; }
+      html.theme--documenter-dark .message.is-primary .message-header, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink .message-header {
+        background-color: #375a7f;
+        color: #fff; }
+      html.theme--documenter-dark .message.is-primary .message-body, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink .message-body {
+        border-color: #375a7f;
+        color: #2b4159; }
+    html.theme--documenter-dark .message.is-link {
+      background-color: #f6fefc; }
+      html.theme--documenter-dark .message.is-link .message-header {
+        background-color: #1abc9c;
+        color: #fff; }
+      html.theme--documenter-dark .message.is-link .message-body {
+        border-color: #1abc9c;
+        color: #0b2f28; }
+    html.theme--documenter-dark .message.is-info {
+      background-color: #f5fbff; }
+      html.theme--documenter-dark .message.is-info .message-header {
+        background-color: #024c7d;
+        color: #fff; }
+      html.theme--documenter-dark .message.is-info .message-body {
+        border-color: #024c7d;
+        color: #033659; }
+    html.theme--documenter-dark .message.is-success {
+      background-color: #f5fff9; }
+      html.theme--documenter-dark .message.is-success .message-header {
+        background-color: #008438;
+        color: #fff; }
+      html.theme--documenter-dark .message.is-success .message-body {
+        border-color: #008438;
+        color: #023518; }
+    html.theme--documenter-dark .message.is-warning {
+      background-color: #fffcf5; }
+      html.theme--documenter-dark .message.is-warning .message-header {
+        background-color: #ad8100;
+        color: #fff; }
+      html.theme--documenter-dark .message.is-warning .message-body {
+        border-color: #ad8100;
+        color: #3d2e03; }
+    html.theme--documenter-dark .message.is-danger {
+      background-color: #fef6f6; }
+      html.theme--documenter-dark .message.is-danger .message-header {
+        background-color: #9e1b0d;
+        color: #fff; }
+      html.theme--documenter-dark .message.is-danger .message-body {
+        border-color: #9e1b0d;
+        color: #7a170c; }
+  html.theme--documenter-dark .message-header {
+    align-items: center;
+    background-color: #fff;
+    border-radius: 0.4em 0.4em 0 0;
+    color: rgba(0, 0, 0, 0.7);
+    display: flex;
+    font-weight: 700;
+    justify-content: space-between;
+    line-height: 1.25;
+    padding: 0.75em;
+    position: relative; }
+    html.theme--documenter-dark .message-header .delete {
+      flex-grow: 0;
+      flex-shrink: 0;
+      margin-left: 0.75em; }
+    html.theme--documenter-dark .message-header + .message-body {
+      border-width: 0;
+      border-top-left-radius: 0;
+      border-top-right-radius: 0; }
+  html.theme--documenter-dark .message-body {
+    border-color: #5e6d6f;
+    border-radius: 0.4em;
+    border-style: solid;
+    border-width: 0 0 0 4px;
+    color: #fff;
+    padding: 1em 1.25em; }
+    html.theme--documenter-dark .message-body code,
+    html.theme--documenter-dark .message-body pre {
+      background-color: white; }
+    html.theme--documenter-dark .message-body pre code {
+      background-color: transparent; }
+  html.theme--documenter-dark .modal {
+    align-items: center;
+    display: none;
+    flex-direction: column;
+    justify-content: center;
+    overflow: hidden;
+    position: fixed;
+    z-index: 40; }
+    html.theme--documenter-dark .modal.is-active {
+      display: flex; }
+  html.theme--documenter-dark .modal-background {
+    background-color: rgba(10, 10, 10, 0.86); }
+  html.theme--documenter-dark .modal-content,
+  html.theme--documenter-dark .modal-card {
+    margin: 0 20px;
+    max-height: calc(100vh - 160px);
+    overflow: auto;
+    position: relative;
+    width: 100%; }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .modal-content,
+      html.theme--documenter-dark .modal-card {
+        margin: 0 auto;
+        max-height: calc(100vh - 40px);
+        width: 640px; } }
+  html.theme--documenter-dark .modal-close {
+    background: none;
+    height: 40px;
+    position: fixed;
+    right: 20px;
+    top: 20px;
+    width: 40px; }
+  html.theme--documenter-dark .modal-card {
+    display: flex;
+    flex-direction: column;
+    max-height: calc(100vh - 40px);
+    overflow: hidden;
+    -ms-overflow-y: visible; }
+  html.theme--documenter-dark .modal-card-head,
+  html.theme--documenter-dark .modal-card-foot {
+    align-items: center;
+    background-color: #282f2f;
+    display: flex;
+    flex-shrink: 0;
+    justify-content: flex-start;
+    padding: 20px;
+    position: relative; }
+  html.theme--documenter-dark .modal-card-head {
+    border-bottom: 1px solid #5e6d6f;
+    border-top-left-radius: 8px;
+    border-top-right-radius: 8px; }
+  html.theme--documenter-dark .modal-card-title {
+    color: #f2f2f2;
+    flex-grow: 1;
+    flex-shrink: 0;
+    font-size: 1.5rem;
+    line-height: 1; }
+  html.theme--documenter-dark .modal-card-foot {
+    border-bottom-left-radius: 8px;
+    border-bottom-right-radius: 8px;
+    border-top: 1px solid #5e6d6f; }
+    html.theme--documenter-dark .modal-card-foot .button:not(:last-child) {
+      margin-right: 0.5em; }
+  html.theme--documenter-dark .modal-card-body {
+    -webkit-overflow-scrolling: touch;
+    background-color: white;
+    flex-grow: 1;
+    flex-shrink: 1;
+    overflow: auto;
+    padding: 20px; }
+  html.theme--documenter-dark .navbar {
+    background-color: #375a7f;
+    min-height: 4rem;
+    position: relative;
+    z-index: 30; }
+    html.theme--documenter-dark .navbar.is-white {
+      background-color: white;
+      color: #0a0a0a; }
+      html.theme--documenter-dark .navbar.is-white .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link {
+        color: #0a0a0a; }
+      html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link.is-active {
+        background-color: #f2f2f2;
+        color: #0a0a0a; }
+      html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link::after {
+        border-color: #0a0a0a; }
+      html.theme--documenter-dark .navbar.is-white .navbar-burger {
+        color: #0a0a0a; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-white .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-white .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link {
+          color: #0a0a0a; }
+        html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link.is-active {
+          background-color: #f2f2f2;
+          color: #0a0a0a; }
+        html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link::after {
+          border-color: #0a0a0a; }
+        html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: #f2f2f2;
+          color: #0a0a0a; }
+        html.theme--documenter-dark .navbar.is-white .navbar-dropdown a.navbar-item.is-active {
+          background-color: white;
+          color: #0a0a0a; } }
+    html.theme--documenter-dark .navbar.is-black {
+      background-color: #0a0a0a;
+      color: white; }
+      html.theme--documenter-dark .navbar.is-black .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link {
+        color: white; }
+      html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link.is-active {
+        background-color: black;
+        color: white; }
+      html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link::after {
+        border-color: white; }
+      html.theme--documenter-dark .navbar.is-black .navbar-burger {
+        color: white; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-black .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-black .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link {
+          color: white; }
+        html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link.is-active {
+          background-color: black;
+          color: white; }
+        html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link::after {
+          border-color: white; }
+        html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: black;
+          color: white; }
+        html.theme--documenter-dark .navbar.is-black .navbar-dropdown a.navbar-item.is-active {
+          background-color: #0a0a0a;
+          color: white; } }
+    html.theme--documenter-dark .navbar.is-light {
+      background-color: #ecf0f1;
+      color: #282f2f; }
+      html.theme--documenter-dark .navbar.is-light .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link {
+        color: #282f2f; }
+      html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link.is-active {
+        background-color: #dde4e6;
+        color: #282f2f; }
+      html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link::after {
+        border-color: #282f2f; }
+      html.theme--documenter-dark .navbar.is-light .navbar-burger {
+        color: #282f2f; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-light .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-light .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link {
+          color: #282f2f; }
+        html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link.is-active {
+          background-color: #dde4e6;
+          color: #282f2f; }
+        html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link::after {
+          border-color: #282f2f; }
+        html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: #dde4e6;
+          color: #282f2f; }
+        html.theme--documenter-dark .navbar.is-light .navbar-dropdown a.navbar-item.is-active {
+          background-color: #ecf0f1;
+          color: #282f2f; } }
+    html.theme--documenter-dark .navbar.is-dark, html.theme--documenter-dark .content kbd.navbar {
+      background-color: #282f2f;
+      color: #ecf0f1; }
+      html.theme--documenter-dark .navbar.is-dark .navbar-brand > .navbar-item, html.theme--documenter-dark .content kbd.navbar .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link,
+      html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link {
+        color: #ecf0f1; }
+      html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link.is-active,
+      html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link.is-active {
+        background-color: #1d2122;
+        color: #ecf0f1; }
+      html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link::after, html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link::after {
+        border-color: #ecf0f1; }
+      html.theme--documenter-dark .navbar.is-dark .navbar-burger, html.theme--documenter-dark .content kbd.navbar .navbar-burger {
+        color: #ecf0f1; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-dark .navbar-start > .navbar-item, html.theme--documenter-dark .content kbd.navbar .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link,
+        html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-dark .navbar-end > .navbar-item,
+        html.theme--documenter-dark .content kbd.navbar .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link,
+        html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link {
+          color: #ecf0f1; }
+        html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link.is-active,
+        html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link.is-active {
+          background-color: #1d2122;
+          color: #ecf0f1; }
+        html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link::after, html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link::after,
+        html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link::after {
+          border-color: #ecf0f1; }
+        html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown.is-active .navbar-link,
+        html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: #1d2122;
+          color: #ecf0f1; }
+        html.theme--documenter-dark .navbar.is-dark .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-dropdown a.navbar-item.is-active {
+          background-color: #282f2f;
+          color: #ecf0f1; } }
+    html.theme--documenter-dark .navbar.is-primary, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink {
+      background-color: #375a7f;
+      color: #fff; }
+      html.theme--documenter-dark .navbar.is-primary .navbar-brand > .navbar-item, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link,
+      html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link {
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link.is-active,
+      html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link.is-active {
+        background-color: #2f4d6d;
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link::after, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link::after {
+        border-color: #fff; }
+      html.theme--documenter-dark .navbar.is-primary .navbar-burger, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-burger {
+        color: #fff; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-primary .navbar-start > .navbar-item, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-primary .navbar-end > .navbar-item,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link {
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link.is-active,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link.is-active {
+          background-color: #2f4d6d;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link::after, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link::after,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link::after {
+          border-color: #fff; }
+        html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown.is-active .navbar-link,
+        html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: #2f4d6d;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active {
+          background-color: #375a7f;
+          color: #fff; } }
+    html.theme--documenter-dark .navbar.is-link {
+      background-color: #1abc9c;
+      color: #fff; }
+      html.theme--documenter-dark .navbar.is-link .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link {
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link.is-active {
+        background-color: #17a689;
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link::after {
+        border-color: #fff; }
+      html.theme--documenter-dark .navbar.is-link .navbar-burger {
+        color: #fff; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-link .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-link .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link {
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link.is-active {
+          background-color: #17a689;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link::after {
+          border-color: #fff; }
+        html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: #17a689;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-link .navbar-dropdown a.navbar-item.is-active {
+          background-color: #1abc9c;
+          color: #fff; } }
+    html.theme--documenter-dark .navbar.is-info {
+      background-color: #024c7d;
+      color: #fff; }
+      html.theme--documenter-dark .navbar.is-info .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link {
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link.is-active {
+        background-color: #023d64;
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link::after {
+        border-color: #fff; }
+      html.theme--documenter-dark .navbar.is-info .navbar-burger {
+        color: #fff; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-info .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-info .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link {
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link.is-active {
+          background-color: #023d64;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link::after {
+          border-color: #fff; }
+        html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: #023d64;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-info .navbar-dropdown a.navbar-item.is-active {
+          background-color: #024c7d;
+          color: #fff; } }
+    html.theme--documenter-dark .navbar.is-success {
+      background-color: #008438;
+      color: #fff; }
+      html.theme--documenter-dark .navbar.is-success .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link {
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link.is-active {
+        background-color: #006b2d;
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link::after {
+        border-color: #fff; }
+      html.theme--documenter-dark .navbar.is-success .navbar-burger {
+        color: #fff; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-success .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-success .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link {
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link.is-active {
+          background-color: #006b2d;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link::after {
+          border-color: #fff; }
+        html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: #006b2d;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-success .navbar-dropdown a.navbar-item.is-active {
+          background-color: #008438;
+          color: #fff; } }
+    html.theme--documenter-dark .navbar.is-warning {
+      background-color: #ad8100;
+      color: #fff; }
+      html.theme--documenter-dark .navbar.is-warning .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link {
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link.is-active {
+        background-color: #946e00;
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link::after {
+        border-color: #fff; }
+      html.theme--documenter-dark .navbar.is-warning .navbar-burger {
+        color: #fff; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-warning .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-warning .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link {
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link.is-active {
+          background-color: #946e00;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link::after {
+          border-color: #fff; }
+        html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: #946e00;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-warning .navbar-dropdown a.navbar-item.is-active {
+          background-color: #ad8100;
+          color: #fff; } }
+    html.theme--documenter-dark .navbar.is-danger {
+      background-color: #9e1b0d;
+      color: #fff; }
+      html.theme--documenter-dark .navbar.is-danger .navbar-brand > .navbar-item,
+      html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link {
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link.is-active {
+        background-color: #86170b;
+        color: #fff; }
+      html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link::after {
+        border-color: #fff; }
+      html.theme--documenter-dark .navbar.is-danger .navbar-burger {
+        color: #fff; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark .navbar.is-danger .navbar-start > .navbar-item,
+        html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link,
+        html.theme--documenter-dark .navbar.is-danger .navbar-end > .navbar-item,
+        html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link {
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link.is-active,
+        html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item:focus,
+        html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item:hover,
+        html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item.is-active,
+        html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link:focus,
+        html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link:hover,
+        html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link.is-active {
+          background-color: #86170b;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link::after,
+        html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link::after {
+          border-color: #fff; }
+        html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown:focus .navbar-link,
+        html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown:hover .navbar-link,
+        html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown.is-active .navbar-link {
+          background-color: #86170b;
+          color: #fff; }
+        html.theme--documenter-dark .navbar.is-danger .navbar-dropdown a.navbar-item.is-active {
+          background-color: #9e1b0d;
+          color: #fff; } }
+    html.theme--documenter-dark .navbar > .container {
+      align-items: stretch;
+      display: flex;
+      min-height: 4rem;
+      width: 100%; }
+    html.theme--documenter-dark .navbar.has-shadow {
+      box-shadow: 0 2px 0 0 #282f2f; }
+    html.theme--documenter-dark .navbar.is-fixed-bottom, html.theme--documenter-dark .navbar.is-fixed-top {
+      left: 0;
+      position: fixed;
+      right: 0;
+      z-index: 30; }
+    html.theme--documenter-dark .navbar.is-fixed-bottom {
+      bottom: 0; }
+      html.theme--documenter-dark .navbar.is-fixed-bottom.has-shadow {
+        box-shadow: 0 -2px 0 0 #282f2f; }
+    html.theme--documenter-dark .navbar.is-fixed-top {
+      top: 0; }
+  html.theme--documenter-dark html.has-navbar-fixed-top,
+  html.theme--documenter-dark body.has-navbar-fixed-top {
+    padding-top: 4rem; }
+  html.theme--documenter-dark html.has-navbar-fixed-bottom,
+  html.theme--documenter-dark body.has-navbar-fixed-bottom {
+    padding-bottom: 4rem; }
+  html.theme--documenter-dark .navbar-brand,
+  html.theme--documenter-dark .navbar-tabs {
+    align-items: stretch;
+    display: flex;
+    flex-shrink: 0;
+    min-height: 4rem; }
+  html.theme--documenter-dark .navbar-brand a.navbar-item:focus, html.theme--documenter-dark .navbar-brand a.navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .navbar-tabs {
+    -webkit-overflow-scrolling: touch;
+    max-width: 100vw;
+    overflow-x: auto;
+    overflow-y: hidden; }
+  html.theme--documenter-dark .navbar-burger {
+    color: #fff;
+    cursor: pointer;
+    display: block;
+    height: 4rem;
+    position: relative;
+    width: 4rem;
+    margin-left: auto; }
+    html.theme--documenter-dark .navbar-burger span {
+      background-color: currentColor;
+      display: block;
+      height: 1px;
+      left: calc(50% - 8px);
+      position: absolute;
+      transform-origin: center;
+      transition-duration: 86ms;
+      transition-property: background-color, opacity, transform;
+      transition-timing-function: ease-out;
+      width: 16px; }
+      html.theme--documenter-dark .navbar-burger span:nth-child(1) {
+        top: calc(50% - 6px); }
+      html.theme--documenter-dark .navbar-burger span:nth-child(2) {
+        top: calc(50% - 1px); }
+      html.theme--documenter-dark .navbar-burger span:nth-child(3) {
+        top: calc(50% + 4px); }
+    html.theme--documenter-dark .navbar-burger:hover {
+      background-color: rgba(0, 0, 0, 0.05); }
+    html.theme--documenter-dark .navbar-burger.is-active span:nth-child(1) {
+      transform: translateY(5px) rotate(45deg); }
+    html.theme--documenter-dark .navbar-burger.is-active span:nth-child(2) {
+      opacity: 0; }
+    html.theme--documenter-dark .navbar-burger.is-active span:nth-child(3) {
+      transform: translateY(-5px) rotate(-45deg); }
+  html.theme--documenter-dark .navbar-menu {
+    display: none; }
+  html.theme--documenter-dark .navbar-item,
+  html.theme--documenter-dark .navbar-link {
+    color: #fff;
+    display: block;
+    line-height: 1.5;
+    padding: 0.5rem 0.75rem;
+    position: relative; }
+    html.theme--documenter-dark .navbar-item .icon:only-child,
+    html.theme--documenter-dark .navbar-link .icon:only-child {
+      margin-left: -0.25rem;
+      margin-right: -0.25rem; }
+  html.theme--documenter-dark a.navbar-item,
+  html.theme--documenter-dark .navbar-link {
+    cursor: pointer; }
+    html.theme--documenter-dark a.navbar-item:focus, html.theme--documenter-dark a.navbar-item:focus-within, html.theme--documenter-dark a.navbar-item:hover, html.theme--documenter-dark a.navbar-item.is-active,
+    html.theme--documenter-dark .navbar-link:focus,
+    html.theme--documenter-dark .navbar-link:focus-within,
+    html.theme--documenter-dark .navbar-link:hover,
+    html.theme--documenter-dark .navbar-link.is-active {
+      background-color: transparent;
+      color: #1abc9c; }
+  html.theme--documenter-dark .navbar-item {
+    display: block;
+    flex-grow: 0;
+    flex-shrink: 0; }
+    html.theme--documenter-dark .navbar-item img {
+      max-height: 1.75rem; }
+    html.theme--documenter-dark .navbar-item.has-dropdown {
+      padding: 0; }
+    html.theme--documenter-dark .navbar-item.is-expanded {
+      flex-grow: 1;
+      flex-shrink: 1; }
+    html.theme--documenter-dark .navbar-item.is-tab {
+      border-bottom: 1px solid transparent;
+      min-height: 4rem;
+      padding-bottom: calc(0.5rem - 1px); }
+      html.theme--documenter-dark .navbar-item.is-tab:focus, html.theme--documenter-dark .navbar-item.is-tab:hover {
+        background-color: transparent;
+        border-bottom-color: #1abc9c; }
+      html.theme--documenter-dark .navbar-item.is-tab.is-active {
+        background-color: transparent;
+        border-bottom-color: #1abc9c;
+        border-bottom-style: solid;
+        border-bottom-width: 3px;
+        color: #1abc9c;
+        padding-bottom: calc(0.5rem - 3px); }
+  html.theme--documenter-dark .navbar-content {
+    flex-grow: 1;
+    flex-shrink: 1; }
+  html.theme--documenter-dark .navbar-link:not(.is-arrowless) {
+    padding-right: 2.5em; }
+    html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after {
+      border-color: #fff;
+      margin-top: -0.375em;
+      right: 1.125em; }
+  html.theme--documenter-dark .navbar-dropdown {
+    font-size: 0.875rem;
+    padding-bottom: 0.5rem;
+    padding-top: 0.5rem; }
+    html.theme--documenter-dark .navbar-dropdown .navbar-item {
+      padding-left: 1.5rem;
+      padding-right: 1.5rem; }
+  html.theme--documenter-dark .navbar-divider {
+    background-color: rgba(0, 0, 0, 0.2);
+    border: none;
+    display: none;
+    height: 2px;
+    margin: 0.5rem 0; }
+  @media screen and (max-width: 1055px) {
+    html.theme--documenter-dark .navbar > .container {
+      display: block; }
+    html.theme--documenter-dark .navbar-brand .navbar-item,
+    html.theme--documenter-dark .navbar-tabs .navbar-item {
+      align-items: center;
+      display: flex; }
+    html.theme--documenter-dark .navbar-link::after {
+      display: none; }
+    html.theme--documenter-dark .navbar-menu {
+      background-color: #375a7f;
+      box-shadow: 0 8px 16px rgba(10, 10, 10, 0.1);
+      padding: 0.5rem 0; }
+      html.theme--documenter-dark .navbar-menu.is-active {
+        display: block; }
+    html.theme--documenter-dark .navbar.is-fixed-bottom-touch, html.theme--documenter-dark .navbar.is-fixed-top-touch {
+      left: 0;
+      position: fixed;
+      right: 0;
+      z-index: 30; }
+    html.theme--documenter-dark .navbar.is-fixed-bottom-touch {
+      bottom: 0; }
+      html.theme--documenter-dark .navbar.is-fixed-bottom-touch.has-shadow {
+        box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); }
+    html.theme--documenter-dark .navbar.is-fixed-top-touch {
+      top: 0; }
+    html.theme--documenter-dark .navbar.is-fixed-top .navbar-menu, html.theme--documenter-dark .navbar.is-fixed-top-touch .navbar-menu {
+      -webkit-overflow-scrolling: touch;
+      max-height: calc(100vh - 4rem);
+      overflow: auto; }
+    html.theme--documenter-dark html.has-navbar-fixed-top-touch,
+    html.theme--documenter-dark body.has-navbar-fixed-top-touch {
+      padding-top: 4rem; }
+    html.theme--documenter-dark html.has-navbar-fixed-bottom-touch,
+    html.theme--documenter-dark body.has-navbar-fixed-bottom-touch {
+      padding-bottom: 4rem; } }
+  @media screen and (min-width: 1056px) {
+    html.theme--documenter-dark .navbar,
+    html.theme--documenter-dark .navbar-menu,
+    html.theme--documenter-dark .navbar-start,
+    html.theme--documenter-dark .navbar-end {
+      align-items: stretch;
+      display: flex; }
+    html.theme--documenter-dark .navbar {
+      min-height: 4rem; }
+      html.theme--documenter-dark .navbar.is-spaced {
+        padding: 1rem 2rem; }
+        html.theme--documenter-dark .navbar.is-spaced .navbar-start,
+        html.theme--documenter-dark .navbar.is-spaced .navbar-end {
+          align-items: center; }
+        html.theme--documenter-dark .navbar.is-spaced a.navbar-item,
+        html.theme--documenter-dark .navbar.is-spaced .navbar-link {
+          border-radius: 0.4em; }
+      html.theme--documenter-dark .navbar.is-transparent a.navbar-item:focus, html.theme--documenter-dark .navbar.is-transparent a.navbar-item:hover, html.theme--documenter-dark .navbar.is-transparent a.navbar-item.is-active,
+      html.theme--documenter-dark .navbar.is-transparent .navbar-link:focus,
+      html.theme--documenter-dark .navbar.is-transparent .navbar-link:hover,
+      html.theme--documenter-dark .navbar.is-transparent .navbar-link.is-active {
+        background-color: transparent !important; }
+      html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:hover .navbar-link {
+        background-color: transparent !important; }
+      html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item:focus, html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item:hover {
+        background-color: transparent;
+        color: #dbdee0; }
+      html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item.is-active {
+        background-color: transparent;
+        color: #1abc9c; }
+    html.theme--documenter-dark .navbar-burger {
+      display: none; }
+    html.theme--documenter-dark .navbar-item,
+    html.theme--documenter-dark .navbar-link {
+      align-items: center;
+      display: flex; }
+    html.theme--documenter-dark .navbar-item {
+      display: flex; }
+      html.theme--documenter-dark .navbar-item.has-dropdown {
+        align-items: stretch; }
+      html.theme--documenter-dark .navbar-item.has-dropdown-up .navbar-link::after {
+        transform: rotate(135deg) translate(0.25em, -0.25em); }
+      html.theme--documenter-dark .navbar-item.has-dropdown-up .navbar-dropdown {
+        border-bottom: 1px solid rgba(0, 0, 0, 0.2);
+        border-radius: 8px 8px 0 0;
+        border-top: none;
+        bottom: 100%;
+        box-shadow: 0 -8px 8px rgba(10, 10, 10, 0.1);
+        top: auto; }
+      html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown {
+        display: block; }
+        .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed {
+          opacity: 1;
+          pointer-events: auto;
+          transform: translateY(0); }
+    html.theme--documenter-dark .navbar-menu {
+      flex-grow: 1;
+      flex-shrink: 0; }
+    html.theme--documenter-dark .navbar-start {
+      justify-content: flex-start;
+      margin-right: auto; }
+    html.theme--documenter-dark .navbar-end {
+      justify-content: flex-end;
+      margin-left: auto; }
+    html.theme--documenter-dark .navbar-dropdown {
+      background-color: #375a7f;
+      border-bottom-left-radius: 8px;
+      border-bottom-right-radius: 8px;
+      border-top: 1px solid rgba(0, 0, 0, 0.2);
+      box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1);
+      display: none;
+      font-size: 0.875rem;
+      left: 0;
+      min-width: 100%;
+      position: absolute;
+      top: 100%;
+      z-index: 20; }
+      html.theme--documenter-dark .navbar-dropdown .navbar-item {
+        padding: 0.375rem 1rem;
+        white-space: nowrap; }
+      html.theme--documenter-dark .navbar-dropdown a.navbar-item {
+        padding-right: 3rem; }
+        html.theme--documenter-dark .navbar-dropdown a.navbar-item:focus, html.theme--documenter-dark .navbar-dropdown a.navbar-item:hover {
+          background-color: transparent;
+          color: #dbdee0; }
+        html.theme--documenter-dark .navbar-dropdown a.navbar-item.is-active {
+          background-color: transparent;
+          color: #1abc9c; }
+      .navbar.is-spaced html.theme--documenter-dark .navbar-dropdown, html.theme--documenter-dark .navbar-dropdown.is-boxed {
+        border-radius: 8px;
+        border-top: none;
+        box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1);
+        display: block;
+        opacity: 0;
+        pointer-events: none;
+        top: calc(100% + (-4px));
+        transform: translateY(-5px);
+        transition-duration: 86ms;
+        transition-property: opacity, transform; }
+      html.theme--documenter-dark .navbar-dropdown.is-right {
+        left: auto;
+        right: 0; }
+    html.theme--documenter-dark .navbar-divider {
+      display: block; }
+    html.theme--documenter-dark .navbar > .container .navbar-brand,
+    html.theme--documenter-dark .container > .navbar .navbar-brand {
+      margin-left: -.75rem; }
+    html.theme--documenter-dark .navbar > .container .navbar-menu,
+    html.theme--documenter-dark .container > .navbar .navbar-menu {
+      margin-right: -.75rem; }
+    html.theme--documenter-dark .navbar.is-fixed-bottom-desktop, html.theme--documenter-dark .navbar.is-fixed-top-desktop {
+      left: 0;
+      position: fixed;
+      right: 0;
+      z-index: 30; }
+    html.theme--documenter-dark .navbar.is-fixed-bottom-desktop {
+      bottom: 0; }
+      html.theme--documenter-dark .navbar.is-fixed-bottom-desktop.has-shadow {
+        box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); }
+    html.theme--documenter-dark .navbar.is-fixed-top-desktop {
+      top: 0; }
+    html.theme--documenter-dark html.has-navbar-fixed-top-desktop,
+    html.theme--documenter-dark body.has-navbar-fixed-top-desktop {
+      padding-top: 4rem; }
+    html.theme--documenter-dark html.has-navbar-fixed-bottom-desktop,
+    html.theme--documenter-dark body.has-navbar-fixed-bottom-desktop {
+      padding-bottom: 4rem; }
+    html.theme--documenter-dark html.has-spaced-navbar-fixed-top,
+    html.theme--documenter-dark body.has-spaced-navbar-fixed-top {
+      padding-top: 6rem; }
+    html.theme--documenter-dark html.has-spaced-navbar-fixed-bottom,
+    html.theme--documenter-dark body.has-spaced-navbar-fixed-bottom {
+      padding-bottom: 6rem; }
+    html.theme--documenter-dark a.navbar-item.is-active,
+    html.theme--documenter-dark .navbar-link.is-active {
+      color: #1abc9c; }
+    html.theme--documenter-dark a.navbar-item.is-active:not(:focus):not(:hover),
+    html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover) {
+      background-color: transparent; }
+    html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link, html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link {
+      background-color: transparent; } }
+  html.theme--documenter-dark .hero.is-fullheight-with-navbar {
+    min-height: calc(100vh - 4rem); }
+  html.theme--documenter-dark .pagination {
+    font-size: 15px;
+    margin: -0.25rem; }
+    html.theme--documenter-dark .pagination.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .pagination.is-medium {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .pagination.is-large {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .pagination.is-rounded .pagination-previous, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-previous,
+    html.theme--documenter-dark .pagination.is-rounded .pagination-next,
+    html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-next {
+      padding-left: 1em;
+      padding-right: 1em;
+      border-radius: 290486px; }
+    html.theme--documenter-dark .pagination.is-rounded .pagination-link, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-link {
+      border-radius: 290486px; }
+  html.theme--documenter-dark .pagination,
+  html.theme--documenter-dark .pagination-list {
+    align-items: center;
+    display: flex;
+    justify-content: center;
+    text-align: center; }
+  html.theme--documenter-dark .pagination-previous,
+  html.theme--documenter-dark .pagination-next,
+  html.theme--documenter-dark .pagination-link,
+  html.theme--documenter-dark .pagination-ellipsis {
+    font-size: 1em;
+    justify-content: center;
+    margin: 0.25rem;
+    padding-left: 0.5em;
+    padding-right: 0.5em;
+    text-align: center; }
+  html.theme--documenter-dark .pagination-previous,
+  html.theme--documenter-dark .pagination-next,
+  html.theme--documenter-dark .pagination-link {
+    border-color: #5e6d6f;
+    color: #1abc9c;
+    min-width: 2.25em; }
+    html.theme--documenter-dark .pagination-previous:hover,
+    html.theme--documenter-dark .pagination-next:hover,
+    html.theme--documenter-dark .pagination-link:hover {
+      border-color: #8c9b9d;
+      color: #1dd2af; }
+    html.theme--documenter-dark .pagination-previous:focus,
+    html.theme--documenter-dark .pagination-next:focus,
+    html.theme--documenter-dark .pagination-link:focus {
+      border-color: #8c9b9d; }
+    html.theme--documenter-dark .pagination-previous:active,
+    html.theme--documenter-dark .pagination-next:active,
+    html.theme--documenter-dark .pagination-link:active {
+      box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2); }
+    html.theme--documenter-dark .pagination-previous[disabled],
+    html.theme--documenter-dark .pagination-next[disabled],
+    html.theme--documenter-dark .pagination-link[disabled] {
+      background-color: #dbdee0;
+      border-color: #dbdee0;
+      box-shadow: none;
+      color: #5e6d6f;
+      opacity: 0.5; }
+  html.theme--documenter-dark .pagination-previous,
+  html.theme--documenter-dark .pagination-next {
+    padding-left: 0.75em;
+    padding-right: 0.75em;
+    white-space: nowrap; }
+  html.theme--documenter-dark .pagination-link.is-current {
+    background-color: #1abc9c;
+    border-color: #1abc9c;
+    color: #fff; }
+  html.theme--documenter-dark .pagination-ellipsis {
+    color: #8c9b9d;
+    pointer-events: none; }
+  html.theme--documenter-dark .pagination-list {
+    flex-wrap: wrap; }
+  @media screen and (max-width: 768px) {
+    html.theme--documenter-dark .pagination {
+      flex-wrap: wrap; }
+    html.theme--documenter-dark .pagination-previous,
+    html.theme--documenter-dark .pagination-next {
+      flex-grow: 1;
+      flex-shrink: 1; }
+    html.theme--documenter-dark .pagination-list li {
+      flex-grow: 1;
+      flex-shrink: 1; } }
+  @media screen and (min-width: 769px), print {
+    html.theme--documenter-dark .pagination-list {
+      flex-grow: 1;
+      flex-shrink: 1;
+      justify-content: flex-start;
+      order: 1; }
+    html.theme--documenter-dark .pagination-previous {
+      order: 2; }
+    html.theme--documenter-dark .pagination-next {
+      order: 3; }
+    html.theme--documenter-dark .pagination {
+      justify-content: space-between; }
+      html.theme--documenter-dark .pagination.is-centered .pagination-previous {
+        order: 1; }
+      html.theme--documenter-dark .pagination.is-centered .pagination-list {
+        justify-content: center;
+        order: 2; }
+      html.theme--documenter-dark .pagination.is-centered .pagination-next {
+        order: 3; }
+      html.theme--documenter-dark .pagination.is-right .pagination-previous {
+        order: 1; }
+      html.theme--documenter-dark .pagination.is-right .pagination-next {
+        order: 2; }
+      html.theme--documenter-dark .pagination.is-right .pagination-list {
+        justify-content: flex-end;
+        order: 3; } }
+  html.theme--documenter-dark .panel {
+    font-size: 15px; }
+    html.theme--documenter-dark .panel:not(:last-child) {
+      margin-bottom: 1.5rem; }
+  html.theme--documenter-dark .panel-heading,
+  html.theme--documenter-dark .panel-tabs,
+  html.theme--documenter-dark .panel-block {
+    border-bottom: 1px solid #5e6d6f;
+    border-left: 1px solid #5e6d6f;
+    border-right: 1px solid #5e6d6f; }
+    html.theme--documenter-dark .panel-heading:first-child,
+    html.theme--documenter-dark .panel-tabs:first-child,
+    html.theme--documenter-dark .panel-block:first-child {
+      border-top: 1px solid #5e6d6f; }
+  html.theme--documenter-dark .panel-heading {
+    background-color: #282f2f;
+    border-radius: 0.4em 0.4em 0 0;
+    color: #f2f2f2;
+    font-size: 1.25em;
+    font-weight: 300;
+    line-height: 1.25;
+    padding: 0.5em 0.75em; }
+  html.theme--documenter-dark .panel-tabs {
+    align-items: flex-end;
+    display: flex;
+    font-size: 0.875em;
+    justify-content: center; }
+    html.theme--documenter-dark .panel-tabs a {
+      border-bottom: 1px solid #5e6d6f;
+      margin-bottom: -1px;
+      padding: 0.5em; }
+      html.theme--documenter-dark .panel-tabs a.is-active {
+        border-bottom-color: #343c3d;
+        color: #17a689; }
+  html.theme--documenter-dark .panel-list a {
+    color: #fff; }
+    html.theme--documenter-dark .panel-list a:hover {
+      color: #1abc9c; }
+  html.theme--documenter-dark .panel-block {
+    align-items: center;
+    color: #f2f2f2;
+    display: flex;
+    justify-content: flex-start;
+    padding: 0.5em 0.75em; }
+    html.theme--documenter-dark .panel-block input[type="checkbox"] {
+      margin-right: 0.75em; }
+    html.theme--documenter-dark .panel-block > .control {
+      flex-grow: 1;
+      flex-shrink: 1;
+      width: 100%; }
+    html.theme--documenter-dark .panel-block.is-wrapped {
+      flex-wrap: wrap; }
+    html.theme--documenter-dark .panel-block.is-active {
+      border-left-color: #1abc9c;
+      color: #17a689; }
+      html.theme--documenter-dark .panel-block.is-active .panel-icon {
+        color: #1abc9c; }
+  html.theme--documenter-dark a.panel-block,
+  html.theme--documenter-dark label.panel-block {
+    cursor: pointer; }
+    html.theme--documenter-dark a.panel-block:hover,
+    html.theme--documenter-dark label.panel-block:hover {
+      background-color: #282f2f; }
+  html.theme--documenter-dark .panel-icon {
+    display: inline-block;
+    font-size: 14px;
+    height: 1em;
+    line-height: 1em;
+    text-align: center;
+    vertical-align: top;
+    width: 1em;
+    color: white;
+    margin-right: 0.75em; }
+    html.theme--documenter-dark .panel-icon .fa {
+      font-size: inherit;
+      line-height: inherit; }
+  html.theme--documenter-dark .tabs {
+    -webkit-overflow-scrolling: touch;
+    align-items: stretch;
+    display: flex;
+    font-size: 15px;
+    justify-content: space-between;
+    overflow: hidden;
+    overflow-x: auto;
+    white-space: nowrap; }
+    html.theme--documenter-dark .tabs a {
+      align-items: center;
+      border-bottom-color: #5e6d6f;
+      border-bottom-style: solid;
+      border-bottom-width: 1px;
+      color: #fff;
+      display: flex;
+      justify-content: center;
+      margin-bottom: -1px;
+      padding: 0.5em 1em;
+      vertical-align: top; }
+      html.theme--documenter-dark .tabs a:hover {
+        border-bottom-color: #f2f2f2;
+        color: #f2f2f2; }
+    html.theme--documenter-dark .tabs li {
+      display: block; }
+      html.theme--documenter-dark .tabs li.is-active a {
+        border-bottom-color: #1abc9c;
+        color: #1abc9c; }
+    html.theme--documenter-dark .tabs ul {
+      align-items: center;
+      border-bottom-color: #5e6d6f;
+      border-bottom-style: solid;
+      border-bottom-width: 1px;
+      display: flex;
+      flex-grow: 1;
+      flex-shrink: 0;
+      justify-content: flex-start; }
+      html.theme--documenter-dark .tabs ul.is-left {
+        padding-right: 0.75em; }
+      html.theme--documenter-dark .tabs ul.is-center {
+        flex: none;
+        justify-content: center;
+        padding-left: 0.75em;
+        padding-right: 0.75em; }
+      html.theme--documenter-dark .tabs ul.is-right {
+        justify-content: flex-end;
+        padding-left: 0.75em; }
+    html.theme--documenter-dark .tabs .icon:first-child {
+      margin-right: 0.5em; }
+    html.theme--documenter-dark .tabs .icon:last-child {
+      margin-left: 0.5em; }
+    html.theme--documenter-dark .tabs.is-centered ul {
+      justify-content: center; }
+    html.theme--documenter-dark .tabs.is-right ul {
+      justify-content: flex-end; }
+    html.theme--documenter-dark .tabs.is-boxed a {
+      border: 1px solid transparent;
+      border-radius: 0.4em 0.4em 0 0; }
+      html.theme--documenter-dark .tabs.is-boxed a:hover {
+        background-color: #282f2f;
+        border-bottom-color: #5e6d6f; }
+    html.theme--documenter-dark .tabs.is-boxed li.is-active a {
+      background-color: white;
+      border-color: #5e6d6f;
+      border-bottom-color: transparent !important; }
+    html.theme--documenter-dark .tabs.is-fullwidth li {
+      flex-grow: 1;
+      flex-shrink: 0; }
+    html.theme--documenter-dark .tabs.is-toggle a {
+      border-color: #5e6d6f;
+      border-style: solid;
+      border-width: 1px;
+      margin-bottom: 0;
+      position: relative; }
+      html.theme--documenter-dark .tabs.is-toggle a:hover {
+        background-color: #282f2f;
+        border-color: #8c9b9d;
+        z-index: 2; }
+    html.theme--documenter-dark .tabs.is-toggle li + li {
+      margin-left: -1px; }
+    html.theme--documenter-dark .tabs.is-toggle li:first-child a {
+      border-radius: 0.4em 0 0 0.4em; }
+    html.theme--documenter-dark .tabs.is-toggle li:last-child a {
+      border-radius: 0 0.4em 0.4em 0; }
+    html.theme--documenter-dark .tabs.is-toggle li.is-active a {
+      background-color: #1abc9c;
+      border-color: #1abc9c;
+      color: #fff;
+      z-index: 1; }
+    html.theme--documenter-dark .tabs.is-toggle ul {
+      border-bottom: none; }
+    html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:first-child a {
+      border-bottom-left-radius: 290486px;
+      border-top-left-radius: 290486px;
+      padding-left: 1.25em; }
+    html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:last-child a {
+      border-bottom-right-radius: 290486px;
+      border-top-right-radius: 290486px;
+      padding-right: 1.25em; }
+    html.theme--documenter-dark .tabs.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.tabs {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .tabs.is-medium {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .tabs.is-large {
+      font-size: 1.5rem; }
+  html.theme--documenter-dark .column {
+    display: block;
+    flex-basis: 0;
+    flex-grow: 1;
+    flex-shrink: 1;
+    padding: 0.75rem; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-narrow {
+      flex: none; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-full {
+      flex: none;
+      width: 100%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-three-quarters {
+      flex: none;
+      width: 75%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-two-thirds {
+      flex: none;
+      width: 66.6666%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-half {
+      flex: none;
+      width: 50%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-one-third {
+      flex: none;
+      width: 33.3333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-one-quarter {
+      flex: none;
+      width: 25%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-one-fifth {
+      flex: none;
+      width: 20%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-two-fifths {
+      flex: none;
+      width: 40%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-three-fifths {
+      flex: none;
+      width: 60%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-four-fifths {
+      flex: none;
+      width: 80%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-three-quarters {
+      margin-left: 75%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-two-thirds {
+      margin-left: 66.6666%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-half {
+      margin-left: 50%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-third {
+      margin-left: 33.3333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-quarter {
+      margin-left: 25%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-fifth {
+      margin-left: 20%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-two-fifths {
+      margin-left: 40%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-three-fifths {
+      margin-left: 60%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-four-fifths {
+      margin-left: 80%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-0 {
+      flex: none;
+      width: 0%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-0 {
+      margin-left: 0%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-1 {
+      flex: none;
+      width: 8.33333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-1 {
+      margin-left: 8.33333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-2 {
+      flex: none;
+      width: 16.66667%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-2 {
+      margin-left: 16.66667%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-3 {
+      flex: none;
+      width: 25%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-3 {
+      margin-left: 25%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-4 {
+      flex: none;
+      width: 33.33333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-4 {
+      margin-left: 33.33333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-5 {
+      flex: none;
+      width: 41.66667%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-5 {
+      margin-left: 41.66667%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-6 {
+      flex: none;
+      width: 50%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-6 {
+      margin-left: 50%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-7 {
+      flex: none;
+      width: 58.33333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-7 {
+      margin-left: 58.33333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-8 {
+      flex: none;
+      width: 66.66667%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-8 {
+      margin-left: 66.66667%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-9 {
+      flex: none;
+      width: 75%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-9 {
+      margin-left: 75%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-10 {
+      flex: none;
+      width: 83.33333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-10 {
+      margin-left: 83.33333%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-11 {
+      flex: none;
+      width: 91.66667%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-11 {
+      margin-left: 91.66667%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-12 {
+      flex: none;
+      width: 100%; }
+    .columns.is-mobile > html.theme--documenter-dark .column.is-offset-12 {
+      margin-left: 100%; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .column.is-narrow-mobile {
+        flex: none; }
+      html.theme--documenter-dark .column.is-full-mobile {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-three-quarters-mobile {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-two-thirds-mobile {
+        flex: none;
+        width: 66.6666%; }
+      html.theme--documenter-dark .column.is-half-mobile {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-one-third-mobile {
+        flex: none;
+        width: 33.3333%; }
+      html.theme--documenter-dark .column.is-one-quarter-mobile {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-one-fifth-mobile {
+        flex: none;
+        width: 20%; }
+      html.theme--documenter-dark .column.is-two-fifths-mobile {
+        flex: none;
+        width: 40%; }
+      html.theme--documenter-dark .column.is-three-fifths-mobile {
+        flex: none;
+        width: 60%; }
+      html.theme--documenter-dark .column.is-four-fifths-mobile {
+        flex: none;
+        width: 80%; }
+      html.theme--documenter-dark .column.is-offset-three-quarters-mobile {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-offset-two-thirds-mobile {
+        margin-left: 66.6666%; }
+      html.theme--documenter-dark .column.is-offset-half-mobile {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-offset-one-third-mobile {
+        margin-left: 33.3333%; }
+      html.theme--documenter-dark .column.is-offset-one-quarter-mobile {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-offset-one-fifth-mobile {
+        margin-left: 20%; }
+      html.theme--documenter-dark .column.is-offset-two-fifths-mobile {
+        margin-left: 40%; }
+      html.theme--documenter-dark .column.is-offset-three-fifths-mobile {
+        margin-left: 60%; }
+      html.theme--documenter-dark .column.is-offset-four-fifths-mobile {
+        margin-left: 80%; }
+      html.theme--documenter-dark .column.is-0-mobile {
+        flex: none;
+        width: 0%; }
+      html.theme--documenter-dark .column.is-offset-0-mobile {
+        margin-left: 0%; }
+      html.theme--documenter-dark .column.is-1-mobile {
+        flex: none;
+        width: 8.33333%; }
+      html.theme--documenter-dark .column.is-offset-1-mobile {
+        margin-left: 8.33333%; }
+      html.theme--documenter-dark .column.is-2-mobile {
+        flex: none;
+        width: 16.66667%; }
+      html.theme--documenter-dark .column.is-offset-2-mobile {
+        margin-left: 16.66667%; }
+      html.theme--documenter-dark .column.is-3-mobile {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-offset-3-mobile {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-4-mobile {
+        flex: none;
+        width: 33.33333%; }
+      html.theme--documenter-dark .column.is-offset-4-mobile {
+        margin-left: 33.33333%; }
+      html.theme--documenter-dark .column.is-5-mobile {
+        flex: none;
+        width: 41.66667%; }
+      html.theme--documenter-dark .column.is-offset-5-mobile {
+        margin-left: 41.66667%; }
+      html.theme--documenter-dark .column.is-6-mobile {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-offset-6-mobile {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-7-mobile {
+        flex: none;
+        width: 58.33333%; }
+      html.theme--documenter-dark .column.is-offset-7-mobile {
+        margin-left: 58.33333%; }
+      html.theme--documenter-dark .column.is-8-mobile {
+        flex: none;
+        width: 66.66667%; }
+      html.theme--documenter-dark .column.is-offset-8-mobile {
+        margin-left: 66.66667%; }
+      html.theme--documenter-dark .column.is-9-mobile {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-offset-9-mobile {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-10-mobile {
+        flex: none;
+        width: 83.33333%; }
+      html.theme--documenter-dark .column.is-offset-10-mobile {
+        margin-left: 83.33333%; }
+      html.theme--documenter-dark .column.is-11-mobile {
+        flex: none;
+        width: 91.66667%; }
+      html.theme--documenter-dark .column.is-offset-11-mobile {
+        margin-left: 91.66667%; }
+      html.theme--documenter-dark .column.is-12-mobile {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-offset-12-mobile {
+        margin-left: 100%; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .column.is-narrow, html.theme--documenter-dark .column.is-narrow-tablet {
+        flex: none; }
+      html.theme--documenter-dark .column.is-full, html.theme--documenter-dark .column.is-full-tablet {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-three-quarters, html.theme--documenter-dark .column.is-three-quarters-tablet {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-two-thirds, html.theme--documenter-dark .column.is-two-thirds-tablet {
+        flex: none;
+        width: 66.6666%; }
+      html.theme--documenter-dark .column.is-half, html.theme--documenter-dark .column.is-half-tablet {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-one-third, html.theme--documenter-dark .column.is-one-third-tablet {
+        flex: none;
+        width: 33.3333%; }
+      html.theme--documenter-dark .column.is-one-quarter, html.theme--documenter-dark .column.is-one-quarter-tablet {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-one-fifth, html.theme--documenter-dark .column.is-one-fifth-tablet {
+        flex: none;
+        width: 20%; }
+      html.theme--documenter-dark .column.is-two-fifths, html.theme--documenter-dark .column.is-two-fifths-tablet {
+        flex: none;
+        width: 40%; }
+      html.theme--documenter-dark .column.is-three-fifths, html.theme--documenter-dark .column.is-three-fifths-tablet {
+        flex: none;
+        width: 60%; }
+      html.theme--documenter-dark .column.is-four-fifths, html.theme--documenter-dark .column.is-four-fifths-tablet {
+        flex: none;
+        width: 80%; }
+      html.theme--documenter-dark .column.is-offset-three-quarters, html.theme--documenter-dark .column.is-offset-three-quarters-tablet {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-offset-two-thirds, html.theme--documenter-dark .column.is-offset-two-thirds-tablet {
+        margin-left: 66.6666%; }
+      html.theme--documenter-dark .column.is-offset-half, html.theme--documenter-dark .column.is-offset-half-tablet {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-offset-one-third, html.theme--documenter-dark .column.is-offset-one-third-tablet {
+        margin-left: 33.3333%; }
+      html.theme--documenter-dark .column.is-offset-one-quarter, html.theme--documenter-dark .column.is-offset-one-quarter-tablet {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-offset-one-fifth, html.theme--documenter-dark .column.is-offset-one-fifth-tablet {
+        margin-left: 20%; }
+      html.theme--documenter-dark .column.is-offset-two-fifths, html.theme--documenter-dark .column.is-offset-two-fifths-tablet {
+        margin-left: 40%; }
+      html.theme--documenter-dark .column.is-offset-three-fifths, html.theme--documenter-dark .column.is-offset-three-fifths-tablet {
+        margin-left: 60%; }
+      html.theme--documenter-dark .column.is-offset-four-fifths, html.theme--documenter-dark .column.is-offset-four-fifths-tablet {
+        margin-left: 80%; }
+      html.theme--documenter-dark .column.is-0, html.theme--documenter-dark .column.is-0-tablet {
+        flex: none;
+        width: 0%; }
+      html.theme--documenter-dark .column.is-offset-0, html.theme--documenter-dark .column.is-offset-0-tablet {
+        margin-left: 0%; }
+      html.theme--documenter-dark .column.is-1, html.theme--documenter-dark .column.is-1-tablet {
+        flex: none;
+        width: 8.33333%; }
+      html.theme--documenter-dark .column.is-offset-1, html.theme--documenter-dark .column.is-offset-1-tablet {
+        margin-left: 8.33333%; }
+      html.theme--documenter-dark .column.is-2, html.theme--documenter-dark .column.is-2-tablet {
+        flex: none;
+        width: 16.66667%; }
+      html.theme--documenter-dark .column.is-offset-2, html.theme--documenter-dark .column.is-offset-2-tablet {
+        margin-left: 16.66667%; }
+      html.theme--documenter-dark .column.is-3, html.theme--documenter-dark .column.is-3-tablet {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-offset-3, html.theme--documenter-dark .column.is-offset-3-tablet {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-4, html.theme--documenter-dark .column.is-4-tablet {
+        flex: none;
+        width: 33.33333%; }
+      html.theme--documenter-dark .column.is-offset-4, html.theme--documenter-dark .column.is-offset-4-tablet {
+        margin-left: 33.33333%; }
+      html.theme--documenter-dark .column.is-5, html.theme--documenter-dark .column.is-5-tablet {
+        flex: none;
+        width: 41.66667%; }
+      html.theme--documenter-dark .column.is-offset-5, html.theme--documenter-dark .column.is-offset-5-tablet {
+        margin-left: 41.66667%; }
+      html.theme--documenter-dark .column.is-6, html.theme--documenter-dark .column.is-6-tablet {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-offset-6, html.theme--documenter-dark .column.is-offset-6-tablet {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-7, html.theme--documenter-dark .column.is-7-tablet {
+        flex: none;
+        width: 58.33333%; }
+      html.theme--documenter-dark .column.is-offset-7, html.theme--documenter-dark .column.is-offset-7-tablet {
+        margin-left: 58.33333%; }
+      html.theme--documenter-dark .column.is-8, html.theme--documenter-dark .column.is-8-tablet {
+        flex: none;
+        width: 66.66667%; }
+      html.theme--documenter-dark .column.is-offset-8, html.theme--documenter-dark .column.is-offset-8-tablet {
+        margin-left: 66.66667%; }
+      html.theme--documenter-dark .column.is-9, html.theme--documenter-dark .column.is-9-tablet {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-offset-9, html.theme--documenter-dark .column.is-offset-9-tablet {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-10, html.theme--documenter-dark .column.is-10-tablet {
+        flex: none;
+        width: 83.33333%; }
+      html.theme--documenter-dark .column.is-offset-10, html.theme--documenter-dark .column.is-offset-10-tablet {
+        margin-left: 83.33333%; }
+      html.theme--documenter-dark .column.is-11, html.theme--documenter-dark .column.is-11-tablet {
+        flex: none;
+        width: 91.66667%; }
+      html.theme--documenter-dark .column.is-offset-11, html.theme--documenter-dark .column.is-offset-11-tablet {
+        margin-left: 91.66667%; }
+      html.theme--documenter-dark .column.is-12, html.theme--documenter-dark .column.is-12-tablet {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-offset-12, html.theme--documenter-dark .column.is-offset-12-tablet {
+        margin-left: 100%; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .column.is-narrow-touch {
+        flex: none; }
+      html.theme--documenter-dark .column.is-full-touch {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-three-quarters-touch {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-two-thirds-touch {
+        flex: none;
+        width: 66.6666%; }
+      html.theme--documenter-dark .column.is-half-touch {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-one-third-touch {
+        flex: none;
+        width: 33.3333%; }
+      html.theme--documenter-dark .column.is-one-quarter-touch {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-one-fifth-touch {
+        flex: none;
+        width: 20%; }
+      html.theme--documenter-dark .column.is-two-fifths-touch {
+        flex: none;
+        width: 40%; }
+      html.theme--documenter-dark .column.is-three-fifths-touch {
+        flex: none;
+        width: 60%; }
+      html.theme--documenter-dark .column.is-four-fifths-touch {
+        flex: none;
+        width: 80%; }
+      html.theme--documenter-dark .column.is-offset-three-quarters-touch {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-offset-two-thirds-touch {
+        margin-left: 66.6666%; }
+      html.theme--documenter-dark .column.is-offset-half-touch {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-offset-one-third-touch {
+        margin-left: 33.3333%; }
+      html.theme--documenter-dark .column.is-offset-one-quarter-touch {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-offset-one-fifth-touch {
+        margin-left: 20%; }
+      html.theme--documenter-dark .column.is-offset-two-fifths-touch {
+        margin-left: 40%; }
+      html.theme--documenter-dark .column.is-offset-three-fifths-touch {
+        margin-left: 60%; }
+      html.theme--documenter-dark .column.is-offset-four-fifths-touch {
+        margin-left: 80%; }
+      html.theme--documenter-dark .column.is-0-touch {
+        flex: none;
+        width: 0%; }
+      html.theme--documenter-dark .column.is-offset-0-touch {
+        margin-left: 0%; }
+      html.theme--documenter-dark .column.is-1-touch {
+        flex: none;
+        width: 8.33333%; }
+      html.theme--documenter-dark .column.is-offset-1-touch {
+        margin-left: 8.33333%; }
+      html.theme--documenter-dark .column.is-2-touch {
+        flex: none;
+        width: 16.66667%; }
+      html.theme--documenter-dark .column.is-offset-2-touch {
+        margin-left: 16.66667%; }
+      html.theme--documenter-dark .column.is-3-touch {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-offset-3-touch {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-4-touch {
+        flex: none;
+        width: 33.33333%; }
+      html.theme--documenter-dark .column.is-offset-4-touch {
+        margin-left: 33.33333%; }
+      html.theme--documenter-dark .column.is-5-touch {
+        flex: none;
+        width: 41.66667%; }
+      html.theme--documenter-dark .column.is-offset-5-touch {
+        margin-left: 41.66667%; }
+      html.theme--documenter-dark .column.is-6-touch {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-offset-6-touch {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-7-touch {
+        flex: none;
+        width: 58.33333%; }
+      html.theme--documenter-dark .column.is-offset-7-touch {
+        margin-left: 58.33333%; }
+      html.theme--documenter-dark .column.is-8-touch {
+        flex: none;
+        width: 66.66667%; }
+      html.theme--documenter-dark .column.is-offset-8-touch {
+        margin-left: 66.66667%; }
+      html.theme--documenter-dark .column.is-9-touch {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-offset-9-touch {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-10-touch {
+        flex: none;
+        width: 83.33333%; }
+      html.theme--documenter-dark .column.is-offset-10-touch {
+        margin-left: 83.33333%; }
+      html.theme--documenter-dark .column.is-11-touch {
+        flex: none;
+        width: 91.66667%; }
+      html.theme--documenter-dark .column.is-offset-11-touch {
+        margin-left: 91.66667%; }
+      html.theme--documenter-dark .column.is-12-touch {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-offset-12-touch {
+        margin-left: 100%; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .column.is-narrow-desktop {
+        flex: none; }
+      html.theme--documenter-dark .column.is-full-desktop {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-three-quarters-desktop {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-two-thirds-desktop {
+        flex: none;
+        width: 66.6666%; }
+      html.theme--documenter-dark .column.is-half-desktop {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-one-third-desktop {
+        flex: none;
+        width: 33.3333%; }
+      html.theme--documenter-dark .column.is-one-quarter-desktop {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-one-fifth-desktop {
+        flex: none;
+        width: 20%; }
+      html.theme--documenter-dark .column.is-two-fifths-desktop {
+        flex: none;
+        width: 40%; }
+      html.theme--documenter-dark .column.is-three-fifths-desktop {
+        flex: none;
+        width: 60%; }
+      html.theme--documenter-dark .column.is-four-fifths-desktop {
+        flex: none;
+        width: 80%; }
+      html.theme--documenter-dark .column.is-offset-three-quarters-desktop {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-offset-two-thirds-desktop {
+        margin-left: 66.6666%; }
+      html.theme--documenter-dark .column.is-offset-half-desktop {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-offset-one-third-desktop {
+        margin-left: 33.3333%; }
+      html.theme--documenter-dark .column.is-offset-one-quarter-desktop {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-offset-one-fifth-desktop {
+        margin-left: 20%; }
+      html.theme--documenter-dark .column.is-offset-two-fifths-desktop {
+        margin-left: 40%; }
+      html.theme--documenter-dark .column.is-offset-three-fifths-desktop {
+        margin-left: 60%; }
+      html.theme--documenter-dark .column.is-offset-four-fifths-desktop {
+        margin-left: 80%; }
+      html.theme--documenter-dark .column.is-0-desktop {
+        flex: none;
+        width: 0%; }
+      html.theme--documenter-dark .column.is-offset-0-desktop {
+        margin-left: 0%; }
+      html.theme--documenter-dark .column.is-1-desktop {
+        flex: none;
+        width: 8.33333%; }
+      html.theme--documenter-dark .column.is-offset-1-desktop {
+        margin-left: 8.33333%; }
+      html.theme--documenter-dark .column.is-2-desktop {
+        flex: none;
+        width: 16.66667%; }
+      html.theme--documenter-dark .column.is-offset-2-desktop {
+        margin-left: 16.66667%; }
+      html.theme--documenter-dark .column.is-3-desktop {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-offset-3-desktop {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-4-desktop {
+        flex: none;
+        width: 33.33333%; }
+      html.theme--documenter-dark .column.is-offset-4-desktop {
+        margin-left: 33.33333%; }
+      html.theme--documenter-dark .column.is-5-desktop {
+        flex: none;
+        width: 41.66667%; }
+      html.theme--documenter-dark .column.is-offset-5-desktop {
+        margin-left: 41.66667%; }
+      html.theme--documenter-dark .column.is-6-desktop {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-offset-6-desktop {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-7-desktop {
+        flex: none;
+        width: 58.33333%; }
+      html.theme--documenter-dark .column.is-offset-7-desktop {
+        margin-left: 58.33333%; }
+      html.theme--documenter-dark .column.is-8-desktop {
+        flex: none;
+        width: 66.66667%; }
+      html.theme--documenter-dark .column.is-offset-8-desktop {
+        margin-left: 66.66667%; }
+      html.theme--documenter-dark .column.is-9-desktop {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-offset-9-desktop {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-10-desktop {
+        flex: none;
+        width: 83.33333%; }
+      html.theme--documenter-dark .column.is-offset-10-desktop {
+        margin-left: 83.33333%; }
+      html.theme--documenter-dark .column.is-11-desktop {
+        flex: none;
+        width: 91.66667%; }
+      html.theme--documenter-dark .column.is-offset-11-desktop {
+        margin-left: 91.66667%; }
+      html.theme--documenter-dark .column.is-12-desktop {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-offset-12-desktop {
+        margin-left: 100%; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .column.is-narrow-widescreen {
+        flex: none; }
+      html.theme--documenter-dark .column.is-full-widescreen {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-three-quarters-widescreen {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-two-thirds-widescreen {
+        flex: none;
+        width: 66.6666%; }
+      html.theme--documenter-dark .column.is-half-widescreen {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-one-third-widescreen {
+        flex: none;
+        width: 33.3333%; }
+      html.theme--documenter-dark .column.is-one-quarter-widescreen {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-one-fifth-widescreen {
+        flex: none;
+        width: 20%; }
+      html.theme--documenter-dark .column.is-two-fifths-widescreen {
+        flex: none;
+        width: 40%; }
+      html.theme--documenter-dark .column.is-three-fifths-widescreen {
+        flex: none;
+        width: 60%; }
+      html.theme--documenter-dark .column.is-four-fifths-widescreen {
+        flex: none;
+        width: 80%; }
+      html.theme--documenter-dark .column.is-offset-three-quarters-widescreen {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-offset-two-thirds-widescreen {
+        margin-left: 66.6666%; }
+      html.theme--documenter-dark .column.is-offset-half-widescreen {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-offset-one-third-widescreen {
+        margin-left: 33.3333%; }
+      html.theme--documenter-dark .column.is-offset-one-quarter-widescreen {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-offset-one-fifth-widescreen {
+        margin-left: 20%; }
+      html.theme--documenter-dark .column.is-offset-two-fifths-widescreen {
+        margin-left: 40%; }
+      html.theme--documenter-dark .column.is-offset-three-fifths-widescreen {
+        margin-left: 60%; }
+      html.theme--documenter-dark .column.is-offset-four-fifths-widescreen {
+        margin-left: 80%; }
+      html.theme--documenter-dark .column.is-0-widescreen {
+        flex: none;
+        width: 0%; }
+      html.theme--documenter-dark .column.is-offset-0-widescreen {
+        margin-left: 0%; }
+      html.theme--documenter-dark .column.is-1-widescreen {
+        flex: none;
+        width: 8.33333%; }
+      html.theme--documenter-dark .column.is-offset-1-widescreen {
+        margin-left: 8.33333%; }
+      html.theme--documenter-dark .column.is-2-widescreen {
+        flex: none;
+        width: 16.66667%; }
+      html.theme--documenter-dark .column.is-offset-2-widescreen {
+        margin-left: 16.66667%; }
+      html.theme--documenter-dark .column.is-3-widescreen {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-offset-3-widescreen {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-4-widescreen {
+        flex: none;
+        width: 33.33333%; }
+      html.theme--documenter-dark .column.is-offset-4-widescreen {
+        margin-left: 33.33333%; }
+      html.theme--documenter-dark .column.is-5-widescreen {
+        flex: none;
+        width: 41.66667%; }
+      html.theme--documenter-dark .column.is-offset-5-widescreen {
+        margin-left: 41.66667%; }
+      html.theme--documenter-dark .column.is-6-widescreen {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-offset-6-widescreen {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-7-widescreen {
+        flex: none;
+        width: 58.33333%; }
+      html.theme--documenter-dark .column.is-offset-7-widescreen {
+        margin-left: 58.33333%; }
+      html.theme--documenter-dark .column.is-8-widescreen {
+        flex: none;
+        width: 66.66667%; }
+      html.theme--documenter-dark .column.is-offset-8-widescreen {
+        margin-left: 66.66667%; }
+      html.theme--documenter-dark .column.is-9-widescreen {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-offset-9-widescreen {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-10-widescreen {
+        flex: none;
+        width: 83.33333%; }
+      html.theme--documenter-dark .column.is-offset-10-widescreen {
+        margin-left: 83.33333%; }
+      html.theme--documenter-dark .column.is-11-widescreen {
+        flex: none;
+        width: 91.66667%; }
+      html.theme--documenter-dark .column.is-offset-11-widescreen {
+        margin-left: 91.66667%; }
+      html.theme--documenter-dark .column.is-12-widescreen {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-offset-12-widescreen {
+        margin-left: 100%; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .column.is-narrow-fullhd {
+        flex: none; }
+      html.theme--documenter-dark .column.is-full-fullhd {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-three-quarters-fullhd {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-two-thirds-fullhd {
+        flex: none;
+        width: 66.6666%; }
+      html.theme--documenter-dark .column.is-half-fullhd {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-one-third-fullhd {
+        flex: none;
+        width: 33.3333%; }
+      html.theme--documenter-dark .column.is-one-quarter-fullhd {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-one-fifth-fullhd {
+        flex: none;
+        width: 20%; }
+      html.theme--documenter-dark .column.is-two-fifths-fullhd {
+        flex: none;
+        width: 40%; }
+      html.theme--documenter-dark .column.is-three-fifths-fullhd {
+        flex: none;
+        width: 60%; }
+      html.theme--documenter-dark .column.is-four-fifths-fullhd {
+        flex: none;
+        width: 80%; }
+      html.theme--documenter-dark .column.is-offset-three-quarters-fullhd {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-offset-two-thirds-fullhd {
+        margin-left: 66.6666%; }
+      html.theme--documenter-dark .column.is-offset-half-fullhd {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-offset-one-third-fullhd {
+        margin-left: 33.3333%; }
+      html.theme--documenter-dark .column.is-offset-one-quarter-fullhd {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-offset-one-fifth-fullhd {
+        margin-left: 20%; }
+      html.theme--documenter-dark .column.is-offset-two-fifths-fullhd {
+        margin-left: 40%; }
+      html.theme--documenter-dark .column.is-offset-three-fifths-fullhd {
+        margin-left: 60%; }
+      html.theme--documenter-dark .column.is-offset-four-fifths-fullhd {
+        margin-left: 80%; }
+      html.theme--documenter-dark .column.is-0-fullhd {
+        flex: none;
+        width: 0%; }
+      html.theme--documenter-dark .column.is-offset-0-fullhd {
+        margin-left: 0%; }
+      html.theme--documenter-dark .column.is-1-fullhd {
+        flex: none;
+        width: 8.33333%; }
+      html.theme--documenter-dark .column.is-offset-1-fullhd {
+        margin-left: 8.33333%; }
+      html.theme--documenter-dark .column.is-2-fullhd {
+        flex: none;
+        width: 16.66667%; }
+      html.theme--documenter-dark .column.is-offset-2-fullhd {
+        margin-left: 16.66667%; }
+      html.theme--documenter-dark .column.is-3-fullhd {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .column.is-offset-3-fullhd {
+        margin-left: 25%; }
+      html.theme--documenter-dark .column.is-4-fullhd {
+        flex: none;
+        width: 33.33333%; }
+      html.theme--documenter-dark .column.is-offset-4-fullhd {
+        margin-left: 33.33333%; }
+      html.theme--documenter-dark .column.is-5-fullhd {
+        flex: none;
+        width: 41.66667%; }
+      html.theme--documenter-dark .column.is-offset-5-fullhd {
+        margin-left: 41.66667%; }
+      html.theme--documenter-dark .column.is-6-fullhd {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .column.is-offset-6-fullhd {
+        margin-left: 50%; }
+      html.theme--documenter-dark .column.is-7-fullhd {
+        flex: none;
+        width: 58.33333%; }
+      html.theme--documenter-dark .column.is-offset-7-fullhd {
+        margin-left: 58.33333%; }
+      html.theme--documenter-dark .column.is-8-fullhd {
+        flex: none;
+        width: 66.66667%; }
+      html.theme--documenter-dark .column.is-offset-8-fullhd {
+        margin-left: 66.66667%; }
+      html.theme--documenter-dark .column.is-9-fullhd {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .column.is-offset-9-fullhd {
+        margin-left: 75%; }
+      html.theme--documenter-dark .column.is-10-fullhd {
+        flex: none;
+        width: 83.33333%; }
+      html.theme--documenter-dark .column.is-offset-10-fullhd {
+        margin-left: 83.33333%; }
+      html.theme--documenter-dark .column.is-11-fullhd {
+        flex: none;
+        width: 91.66667%; }
+      html.theme--documenter-dark .column.is-offset-11-fullhd {
+        margin-left: 91.66667%; }
+      html.theme--documenter-dark .column.is-12-fullhd {
+        flex: none;
+        width: 100%; }
+      html.theme--documenter-dark .column.is-offset-12-fullhd {
+        margin-left: 100%; } }
+  html.theme--documenter-dark .columns {
+    margin-left: -0.75rem;
+    margin-right: -0.75rem;
+    margin-top: -0.75rem; }
+    html.theme--documenter-dark .columns:last-child {
+      margin-bottom: -0.75rem; }
+    html.theme--documenter-dark .columns:not(:last-child) {
+      margin-bottom: calc(1.5rem - 0.75rem); }
+    html.theme--documenter-dark .columns.is-centered {
+      justify-content: center; }
+    html.theme--documenter-dark .columns.is-gapless {
+      margin-left: 0;
+      margin-right: 0;
+      margin-top: 0; }
+      html.theme--documenter-dark .columns.is-gapless > .column {
+        margin: 0;
+        padding: 0 !important; }
+      html.theme--documenter-dark .columns.is-gapless:not(:last-child) {
+        margin-bottom: 1.5rem; }
+      html.theme--documenter-dark .columns.is-gapless:last-child {
+        margin-bottom: 0; }
+    html.theme--documenter-dark .columns.is-mobile {
+      display: flex; }
+    html.theme--documenter-dark .columns.is-multiline {
+      flex-wrap: wrap; }
+    html.theme--documenter-dark .columns.is-vcentered {
+      align-items: center; }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns:not(.is-desktop) {
+        display: flex; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-desktop {
+        display: flex; } }
+  html.theme--documenter-dark .columns.is-variable {
+    --columnGap: 0.75rem;
+    margin-left: calc(-1 * var(--columnGap));
+    margin-right: calc(-1 * var(--columnGap)); }
+    html.theme--documenter-dark .columns.is-variable .column {
+      padding-left: var(--columnGap);
+      padding-right: var(--columnGap); }
+    html.theme--documenter-dark .columns.is-variable.is-0 {
+      --columnGap: 0rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .columns.is-variable.is-0-mobile {
+        --columnGap: 0rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns.is-variable.is-0-tablet {
+        --columnGap: 0rem; } }
+    @media screen and (min-width: 769px) and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-0-tablet-only {
+        --columnGap: 0rem; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-0-touch {
+        --columnGap: 0rem; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-variable.is-0-desktop {
+        --columnGap: 0rem; } }
+    @media screen and (min-width: 1056px) and (max-width: 1215px) {
+      html.theme--documenter-dark .columns.is-variable.is-0-desktop-only {
+        --columnGap: 0rem; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .columns.is-variable.is-0-widescreen {
+        --columnGap: 0rem; } }
+    @media screen and (min-width: 1216px) and (max-width: 1407px) {
+      html.theme--documenter-dark .columns.is-variable.is-0-widescreen-only {
+        --columnGap: 0rem; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .columns.is-variable.is-0-fullhd {
+        --columnGap: 0rem; } }
+    html.theme--documenter-dark .columns.is-variable.is-1 {
+      --columnGap: 0.25rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .columns.is-variable.is-1-mobile {
+        --columnGap: 0.25rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns.is-variable.is-1-tablet {
+        --columnGap: 0.25rem; } }
+    @media screen and (min-width: 769px) and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-1-tablet-only {
+        --columnGap: 0.25rem; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-1-touch {
+        --columnGap: 0.25rem; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-variable.is-1-desktop {
+        --columnGap: 0.25rem; } }
+    @media screen and (min-width: 1056px) and (max-width: 1215px) {
+      html.theme--documenter-dark .columns.is-variable.is-1-desktop-only {
+        --columnGap: 0.25rem; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .columns.is-variable.is-1-widescreen {
+        --columnGap: 0.25rem; } }
+    @media screen and (min-width: 1216px) and (max-width: 1407px) {
+      html.theme--documenter-dark .columns.is-variable.is-1-widescreen-only {
+        --columnGap: 0.25rem; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .columns.is-variable.is-1-fullhd {
+        --columnGap: 0.25rem; } }
+    html.theme--documenter-dark .columns.is-variable.is-2 {
+      --columnGap: 0.5rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .columns.is-variable.is-2-mobile {
+        --columnGap: 0.5rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns.is-variable.is-2-tablet {
+        --columnGap: 0.5rem; } }
+    @media screen and (min-width: 769px) and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-2-tablet-only {
+        --columnGap: 0.5rem; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-2-touch {
+        --columnGap: 0.5rem; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-variable.is-2-desktop {
+        --columnGap: 0.5rem; } }
+    @media screen and (min-width: 1056px) and (max-width: 1215px) {
+      html.theme--documenter-dark .columns.is-variable.is-2-desktop-only {
+        --columnGap: 0.5rem; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .columns.is-variable.is-2-widescreen {
+        --columnGap: 0.5rem; } }
+    @media screen and (min-width: 1216px) and (max-width: 1407px) {
+      html.theme--documenter-dark .columns.is-variable.is-2-widescreen-only {
+        --columnGap: 0.5rem; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .columns.is-variable.is-2-fullhd {
+        --columnGap: 0.5rem; } }
+    html.theme--documenter-dark .columns.is-variable.is-3 {
+      --columnGap: 0.75rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .columns.is-variable.is-3-mobile {
+        --columnGap: 0.75rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns.is-variable.is-3-tablet {
+        --columnGap: 0.75rem; } }
+    @media screen and (min-width: 769px) and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-3-tablet-only {
+        --columnGap: 0.75rem; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-3-touch {
+        --columnGap: 0.75rem; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-variable.is-3-desktop {
+        --columnGap: 0.75rem; } }
+    @media screen and (min-width: 1056px) and (max-width: 1215px) {
+      html.theme--documenter-dark .columns.is-variable.is-3-desktop-only {
+        --columnGap: 0.75rem; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .columns.is-variable.is-3-widescreen {
+        --columnGap: 0.75rem; } }
+    @media screen and (min-width: 1216px) and (max-width: 1407px) {
+      html.theme--documenter-dark .columns.is-variable.is-3-widescreen-only {
+        --columnGap: 0.75rem; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .columns.is-variable.is-3-fullhd {
+        --columnGap: 0.75rem; } }
+    html.theme--documenter-dark .columns.is-variable.is-4 {
+      --columnGap: 1rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .columns.is-variable.is-4-mobile {
+        --columnGap: 1rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns.is-variable.is-4-tablet {
+        --columnGap: 1rem; } }
+    @media screen and (min-width: 769px) and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-4-tablet-only {
+        --columnGap: 1rem; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-4-touch {
+        --columnGap: 1rem; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-variable.is-4-desktop {
+        --columnGap: 1rem; } }
+    @media screen and (min-width: 1056px) and (max-width: 1215px) {
+      html.theme--documenter-dark .columns.is-variable.is-4-desktop-only {
+        --columnGap: 1rem; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .columns.is-variable.is-4-widescreen {
+        --columnGap: 1rem; } }
+    @media screen and (min-width: 1216px) and (max-width: 1407px) {
+      html.theme--documenter-dark .columns.is-variable.is-4-widescreen-only {
+        --columnGap: 1rem; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .columns.is-variable.is-4-fullhd {
+        --columnGap: 1rem; } }
+    html.theme--documenter-dark .columns.is-variable.is-5 {
+      --columnGap: 1.25rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .columns.is-variable.is-5-mobile {
+        --columnGap: 1.25rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns.is-variable.is-5-tablet {
+        --columnGap: 1.25rem; } }
+    @media screen and (min-width: 769px) and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-5-tablet-only {
+        --columnGap: 1.25rem; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-5-touch {
+        --columnGap: 1.25rem; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-variable.is-5-desktop {
+        --columnGap: 1.25rem; } }
+    @media screen and (min-width: 1056px) and (max-width: 1215px) {
+      html.theme--documenter-dark .columns.is-variable.is-5-desktop-only {
+        --columnGap: 1.25rem; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .columns.is-variable.is-5-widescreen {
+        --columnGap: 1.25rem; } }
+    @media screen and (min-width: 1216px) and (max-width: 1407px) {
+      html.theme--documenter-dark .columns.is-variable.is-5-widescreen-only {
+        --columnGap: 1.25rem; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .columns.is-variable.is-5-fullhd {
+        --columnGap: 1.25rem; } }
+    html.theme--documenter-dark .columns.is-variable.is-6 {
+      --columnGap: 1.5rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .columns.is-variable.is-6-mobile {
+        --columnGap: 1.5rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns.is-variable.is-6-tablet {
+        --columnGap: 1.5rem; } }
+    @media screen and (min-width: 769px) and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-6-tablet-only {
+        --columnGap: 1.5rem; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-6-touch {
+        --columnGap: 1.5rem; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-variable.is-6-desktop {
+        --columnGap: 1.5rem; } }
+    @media screen and (min-width: 1056px) and (max-width: 1215px) {
+      html.theme--documenter-dark .columns.is-variable.is-6-desktop-only {
+        --columnGap: 1.5rem; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .columns.is-variable.is-6-widescreen {
+        --columnGap: 1.5rem; } }
+    @media screen and (min-width: 1216px) and (max-width: 1407px) {
+      html.theme--documenter-dark .columns.is-variable.is-6-widescreen-only {
+        --columnGap: 1.5rem; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .columns.is-variable.is-6-fullhd {
+        --columnGap: 1.5rem; } }
+    html.theme--documenter-dark .columns.is-variable.is-7 {
+      --columnGap: 1.75rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .columns.is-variable.is-7-mobile {
+        --columnGap: 1.75rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns.is-variable.is-7-tablet {
+        --columnGap: 1.75rem; } }
+    @media screen and (min-width: 769px) and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-7-tablet-only {
+        --columnGap: 1.75rem; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-7-touch {
+        --columnGap: 1.75rem; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-variable.is-7-desktop {
+        --columnGap: 1.75rem; } }
+    @media screen and (min-width: 1056px) and (max-width: 1215px) {
+      html.theme--documenter-dark .columns.is-variable.is-7-desktop-only {
+        --columnGap: 1.75rem; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .columns.is-variable.is-7-widescreen {
+        --columnGap: 1.75rem; } }
+    @media screen and (min-width: 1216px) and (max-width: 1407px) {
+      html.theme--documenter-dark .columns.is-variable.is-7-widescreen-only {
+        --columnGap: 1.75rem; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .columns.is-variable.is-7-fullhd {
+        --columnGap: 1.75rem; } }
+    html.theme--documenter-dark .columns.is-variable.is-8 {
+      --columnGap: 2rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .columns.is-variable.is-8-mobile {
+        --columnGap: 2rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .columns.is-variable.is-8-tablet {
+        --columnGap: 2rem; } }
+    @media screen and (min-width: 769px) and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-8-tablet-only {
+        --columnGap: 2rem; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .columns.is-variable.is-8-touch {
+        --columnGap: 2rem; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .columns.is-variable.is-8-desktop {
+        --columnGap: 2rem; } }
+    @media screen and (min-width: 1056px) and (max-width: 1215px) {
+      html.theme--documenter-dark .columns.is-variable.is-8-desktop-only {
+        --columnGap: 2rem; } }
+    @media screen and (min-width: 1216px) {
+      html.theme--documenter-dark .columns.is-variable.is-8-widescreen {
+        --columnGap: 2rem; } }
+    @media screen and (min-width: 1216px) and (max-width: 1407px) {
+      html.theme--documenter-dark .columns.is-variable.is-8-widescreen-only {
+        --columnGap: 2rem; } }
+    @media screen and (min-width: 1408px) {
+      html.theme--documenter-dark .columns.is-variable.is-8-fullhd {
+        --columnGap: 2rem; } }
+  html.theme--documenter-dark .tile {
+    align-items: stretch;
+    display: block;
+    flex-basis: 0;
+    flex-grow: 1;
+    flex-shrink: 1;
+    min-height: min-content; }
+    html.theme--documenter-dark .tile.is-ancestor {
+      margin-left: -0.75rem;
+      margin-right: -0.75rem;
+      margin-top: -0.75rem; }
+      html.theme--documenter-dark .tile.is-ancestor:last-child {
+        margin-bottom: -0.75rem; }
+      html.theme--documenter-dark .tile.is-ancestor:not(:last-child) {
+        margin-bottom: 0.75rem; }
+    html.theme--documenter-dark .tile.is-child {
+      margin: 0 !important; }
+    html.theme--documenter-dark .tile.is-parent {
+      padding: 0.75rem; }
+    html.theme--documenter-dark .tile.is-vertical {
+      flex-direction: column; }
+      html.theme--documenter-dark .tile.is-vertical > .tile.is-child:not(:last-child) {
+        margin-bottom: 1.5rem !important; }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .tile:not(.is-child) {
+        display: flex; }
+      html.theme--documenter-dark .tile.is-1 {
+        flex: none;
+        width: 8.33333%; }
+      html.theme--documenter-dark .tile.is-2 {
+        flex: none;
+        width: 16.66667%; }
+      html.theme--documenter-dark .tile.is-3 {
+        flex: none;
+        width: 25%; }
+      html.theme--documenter-dark .tile.is-4 {
+        flex: none;
+        width: 33.33333%; }
+      html.theme--documenter-dark .tile.is-5 {
+        flex: none;
+        width: 41.66667%; }
+      html.theme--documenter-dark .tile.is-6 {
+        flex: none;
+        width: 50%; }
+      html.theme--documenter-dark .tile.is-7 {
+        flex: none;
+        width: 58.33333%; }
+      html.theme--documenter-dark .tile.is-8 {
+        flex: none;
+        width: 66.66667%; }
+      html.theme--documenter-dark .tile.is-9 {
+        flex: none;
+        width: 75%; }
+      html.theme--documenter-dark .tile.is-10 {
+        flex: none;
+        width: 83.33333%; }
+      html.theme--documenter-dark .tile.is-11 {
+        flex: none;
+        width: 91.66667%; }
+      html.theme--documenter-dark .tile.is-12 {
+        flex: none;
+        width: 100%; } }
+  html.theme--documenter-dark .hero {
+    align-items: stretch;
+    display: flex;
+    flex-direction: column;
+    justify-content: space-between; }
+    html.theme--documenter-dark .hero .navbar {
+      background: none; }
+    html.theme--documenter-dark .hero .tabs ul {
+      border-bottom: none; }
+    html.theme--documenter-dark .hero.is-white {
+      background-color: white;
+      color: #0a0a0a; }
+      html.theme--documenter-dark .hero.is-white a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-white strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-white .title {
+        color: #0a0a0a; }
+      html.theme--documenter-dark .hero.is-white .subtitle {
+        color: rgba(10, 10, 10, 0.9); }
+        html.theme--documenter-dark .hero.is-white .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-white .subtitle strong {
+          color: #0a0a0a; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-white .navbar-menu {
+          background-color: white; } }
+      html.theme--documenter-dark .hero.is-white .navbar-item,
+      html.theme--documenter-dark .hero.is-white .navbar-link {
+        color: rgba(10, 10, 10, 0.7); }
+      html.theme--documenter-dark .hero.is-white a.navbar-item:hover, html.theme--documenter-dark .hero.is-white a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-white .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-white .navbar-link.is-active {
+        background-color: #f2f2f2;
+        color: #0a0a0a; }
+      html.theme--documenter-dark .hero.is-white .tabs a {
+        color: #0a0a0a;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-white .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-white .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-white .tabs.is-boxed a, html.theme--documenter-dark .hero.is-white .tabs.is-toggle a {
+        color: #0a0a0a; }
+        html.theme--documenter-dark .hero.is-white .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-white .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-white .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-white .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-white .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-white .tabs.is-toggle li.is-active a:hover {
+        background-color: #0a0a0a;
+        border-color: #0a0a0a;
+        color: white; }
+      html.theme--documenter-dark .hero.is-white.is-bold {
+        background-image: linear-gradient(141deg, #e6e6e6 0%, white 71%, white 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-white.is-bold .navbar-menu {
+            background-image: linear-gradient(141deg, #e6e6e6 0%, white 71%, white 100%); } }
+    html.theme--documenter-dark .hero.is-black {
+      background-color: #0a0a0a;
+      color: white; }
+      html.theme--documenter-dark .hero.is-black a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-black strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-black .title {
+        color: white; }
+      html.theme--documenter-dark .hero.is-black .subtitle {
+        color: rgba(255, 255, 255, 0.9); }
+        html.theme--documenter-dark .hero.is-black .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-black .subtitle strong {
+          color: white; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-black .navbar-menu {
+          background-color: #0a0a0a; } }
+      html.theme--documenter-dark .hero.is-black .navbar-item,
+      html.theme--documenter-dark .hero.is-black .navbar-link {
+        color: rgba(255, 255, 255, 0.7); }
+      html.theme--documenter-dark .hero.is-black a.navbar-item:hover, html.theme--documenter-dark .hero.is-black a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-black .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-black .navbar-link.is-active {
+        background-color: black;
+        color: white; }
+      html.theme--documenter-dark .hero.is-black .tabs a {
+        color: white;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-black .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-black .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-black .tabs.is-boxed a, html.theme--documenter-dark .hero.is-black .tabs.is-toggle a {
+        color: white; }
+        html.theme--documenter-dark .hero.is-black .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-black .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-black .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-black .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-black .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-black .tabs.is-toggle li.is-active a:hover {
+        background-color: white;
+        border-color: white;
+        color: #0a0a0a; }
+      html.theme--documenter-dark .hero.is-black.is-bold {
+        background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-black.is-bold .navbar-menu {
+            background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } }
+    html.theme--documenter-dark .hero.is-light {
+      background-color: #ecf0f1;
+      color: #282f2f; }
+      html.theme--documenter-dark .hero.is-light a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-light strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-light .title {
+        color: #282f2f; }
+      html.theme--documenter-dark .hero.is-light .subtitle {
+        color: rgba(40, 47, 47, 0.9); }
+        html.theme--documenter-dark .hero.is-light .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-light .subtitle strong {
+          color: #282f2f; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-light .navbar-menu {
+          background-color: #ecf0f1; } }
+      html.theme--documenter-dark .hero.is-light .navbar-item,
+      html.theme--documenter-dark .hero.is-light .navbar-link {
+        color: rgba(40, 47, 47, 0.7); }
+      html.theme--documenter-dark .hero.is-light a.navbar-item:hover, html.theme--documenter-dark .hero.is-light a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-light .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-light .navbar-link.is-active {
+        background-color: #dde4e6;
+        color: #282f2f; }
+      html.theme--documenter-dark .hero.is-light .tabs a {
+        color: #282f2f;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-light .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-light .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-light .tabs.is-boxed a, html.theme--documenter-dark .hero.is-light .tabs.is-toggle a {
+        color: #282f2f; }
+        html.theme--documenter-dark .hero.is-light .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-light .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-light .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-light .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-light .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-light .tabs.is-toggle li.is-active a:hover {
+        background-color: #282f2f;
+        border-color: #282f2f;
+        color: #ecf0f1; }
+      html.theme--documenter-dark .hero.is-light.is-bold {
+        background-image: linear-gradient(141deg, #cadfe0 0%, #ecf0f1 71%, #fafbfc 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-light.is-bold .navbar-menu {
+            background-image: linear-gradient(141deg, #cadfe0 0%, #ecf0f1 71%, #fafbfc 100%); } }
+    html.theme--documenter-dark .hero.is-dark, html.theme--documenter-dark .content kbd.hero {
+      background-color: #282f2f;
+      color: #ecf0f1; }
+      html.theme--documenter-dark .hero.is-dark a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), html.theme--documenter-dark .content kbd.hero a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-dark strong,
+      html.theme--documenter-dark .content kbd.hero strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-dark .title, html.theme--documenter-dark .content kbd.hero .title {
+        color: #ecf0f1; }
+      html.theme--documenter-dark .hero.is-dark .subtitle, html.theme--documenter-dark .content kbd.hero .subtitle {
+        color: rgba(236, 240, 241, 0.9); }
+        html.theme--documenter-dark .hero.is-dark .subtitle a:not(.button), html.theme--documenter-dark .content kbd.hero .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-dark .subtitle strong,
+        html.theme--documenter-dark .content kbd.hero .subtitle strong {
+          color: #ecf0f1; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-dark .navbar-menu, html.theme--documenter-dark .content kbd.hero .navbar-menu {
+          background-color: #282f2f; } }
+      html.theme--documenter-dark .hero.is-dark .navbar-item, html.theme--documenter-dark .content kbd.hero .navbar-item,
+      html.theme--documenter-dark .hero.is-dark .navbar-link,
+      html.theme--documenter-dark .content kbd.hero .navbar-link {
+        color: rgba(236, 240, 241, 0.7); }
+      html.theme--documenter-dark .hero.is-dark a.navbar-item:hover, html.theme--documenter-dark .content kbd.hero a.navbar-item:hover, html.theme--documenter-dark .hero.is-dark a.navbar-item.is-active, html.theme--documenter-dark .content kbd.hero a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-dark .navbar-link:hover,
+      html.theme--documenter-dark .content kbd.hero .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-dark .navbar-link.is-active,
+      html.theme--documenter-dark .content kbd.hero .navbar-link.is-active {
+        background-color: #1d2122;
+        color: #ecf0f1; }
+      html.theme--documenter-dark .hero.is-dark .tabs a, html.theme--documenter-dark .content kbd.hero .tabs a {
+        color: #ecf0f1;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-dark .tabs a:hover, html.theme--documenter-dark .content kbd.hero .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-dark .tabs li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-dark .tabs.is-boxed a, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed a, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle a, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle a {
+        color: #ecf0f1; }
+        html.theme--documenter-dark .hero.is-dark .tabs.is-boxed a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-dark .tabs.is-boxed li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-dark .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle li.is-active a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle li.is-active a:hover {
+        background-color: #ecf0f1;
+        border-color: #ecf0f1;
+        color: #282f2f; }
+      html.theme--documenter-dark .hero.is-dark.is-bold, html.theme--documenter-dark .content kbd.hero.is-bold {
+        background-image: linear-gradient(141deg, #0f1615 0%, #282f2f 71%, #313c40 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-dark.is-bold .navbar-menu, html.theme--documenter-dark .content kbd.hero.is-bold .navbar-menu {
+            background-image: linear-gradient(141deg, #0f1615 0%, #282f2f 71%, #313c40 100%); } }
+    html.theme--documenter-dark .hero.is-primary, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink {
+      background-color: #375a7f;
+      color: #fff; }
+      html.theme--documenter-dark .hero.is-primary a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-primary strong,
+      html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-primary .title, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .title {
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-primary .subtitle, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle {
+        color: rgba(255, 255, 255, 0.9); }
+        html.theme--documenter-dark .hero.is-primary .subtitle a:not(.button), html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-primary .subtitle strong,
+        html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle strong {
+          color: #fff; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-primary .navbar-menu, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-menu {
+          background-color: #375a7f; } }
+      html.theme--documenter-dark .hero.is-primary .navbar-item, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-item,
+      html.theme--documenter-dark .hero.is-primary .navbar-link,
+      html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link {
+        color: rgba(255, 255, 255, 0.7); }
+      html.theme--documenter-dark .hero.is-primary a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a.navbar-item:hover, html.theme--documenter-dark .hero.is-primary a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-primary .navbar-link:hover,
+      html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-primary .navbar-link.is-active,
+      html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link.is-active {
+        background-color: #2f4d6d;
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-primary .tabs a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs a {
+        color: #fff;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-primary .tabs a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-primary .tabs li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-primary .tabs.is-boxed a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a {
+        color: #fff; }
+        html.theme--documenter-dark .hero.is-primary .tabs.is-boxed a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-primary .tabs.is-boxed li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-primary .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle li.is-active a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a:hover {
+        background-color: #fff;
+        border-color: #fff;
+        color: #375a7f; }
+      html.theme--documenter-dark .hero.is-primary.is-bold, html.theme--documenter-dark .docstring > section > a.hero.is-bold.docs-sourcelink {
+        background-image: linear-gradient(141deg, #214b62 0%, #375a7f 71%, #3a5796 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-primary.is-bold .navbar-menu, html.theme--documenter-dark .docstring > section > a.hero.is-bold.docs-sourcelink .navbar-menu {
+            background-image: linear-gradient(141deg, #214b62 0%, #375a7f 71%, #3a5796 100%); } }
+    html.theme--documenter-dark .hero.is-link {
+      background-color: #1abc9c;
+      color: #fff; }
+      html.theme--documenter-dark .hero.is-link a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-link strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-link .title {
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-link .subtitle {
+        color: rgba(255, 255, 255, 0.9); }
+        html.theme--documenter-dark .hero.is-link .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-link .subtitle strong {
+          color: #fff; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-link .navbar-menu {
+          background-color: #1abc9c; } }
+      html.theme--documenter-dark .hero.is-link .navbar-item,
+      html.theme--documenter-dark .hero.is-link .navbar-link {
+        color: rgba(255, 255, 255, 0.7); }
+      html.theme--documenter-dark .hero.is-link a.navbar-item:hover, html.theme--documenter-dark .hero.is-link a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-link .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-link .navbar-link.is-active {
+        background-color: #17a689;
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-link .tabs a {
+        color: #fff;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-link .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-link .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-link .tabs.is-boxed a, html.theme--documenter-dark .hero.is-link .tabs.is-toggle a {
+        color: #fff; }
+        html.theme--documenter-dark .hero.is-link .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-link .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-link .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-link .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-link .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-link .tabs.is-toggle li.is-active a:hover {
+        background-color: #fff;
+        border-color: #fff;
+        color: #1abc9c; }
+      html.theme--documenter-dark .hero.is-link.is-bold {
+        background-image: linear-gradient(141deg, #0c9764 0%, #1abc9c 71%, #17d8d2 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-link.is-bold .navbar-menu {
+            background-image: linear-gradient(141deg, #0c9764 0%, #1abc9c 71%, #17d8d2 100%); } }
+    html.theme--documenter-dark .hero.is-info {
+      background-color: #024c7d;
+      color: #fff; }
+      html.theme--documenter-dark .hero.is-info a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-info strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-info .title {
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-info .subtitle {
+        color: rgba(255, 255, 255, 0.9); }
+        html.theme--documenter-dark .hero.is-info .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-info .subtitle strong {
+          color: #fff; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-info .navbar-menu {
+          background-color: #024c7d; } }
+      html.theme--documenter-dark .hero.is-info .navbar-item,
+      html.theme--documenter-dark .hero.is-info .navbar-link {
+        color: rgba(255, 255, 255, 0.7); }
+      html.theme--documenter-dark .hero.is-info a.navbar-item:hover, html.theme--documenter-dark .hero.is-info a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-info .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-info .navbar-link.is-active {
+        background-color: #023d64;
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-info .tabs a {
+        color: #fff;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-info .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-info .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-info .tabs.is-boxed a, html.theme--documenter-dark .hero.is-info .tabs.is-toggle a {
+        color: #fff; }
+        html.theme--documenter-dark .hero.is-info .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-info .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-info .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-info .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-info .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-info .tabs.is-toggle li.is-active a:hover {
+        background-color: #fff;
+        border-color: #fff;
+        color: #024c7d; }
+      html.theme--documenter-dark .hero.is-info.is-bold {
+        background-image: linear-gradient(141deg, #003a4c 0%, #024c7d 71%, #004299 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-info.is-bold .navbar-menu {
+            background-image: linear-gradient(141deg, #003a4c 0%, #024c7d 71%, #004299 100%); } }
+    html.theme--documenter-dark .hero.is-success {
+      background-color: #008438;
+      color: #fff; }
+      html.theme--documenter-dark .hero.is-success a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-success strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-success .title {
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-success .subtitle {
+        color: rgba(255, 255, 255, 0.9); }
+        html.theme--documenter-dark .hero.is-success .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-success .subtitle strong {
+          color: #fff; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-success .navbar-menu {
+          background-color: #008438; } }
+      html.theme--documenter-dark .hero.is-success .navbar-item,
+      html.theme--documenter-dark .hero.is-success .navbar-link {
+        color: rgba(255, 255, 255, 0.7); }
+      html.theme--documenter-dark .hero.is-success a.navbar-item:hover, html.theme--documenter-dark .hero.is-success a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-success .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-success .navbar-link.is-active {
+        background-color: #006b2d;
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-success .tabs a {
+        color: #fff;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-success .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-success .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-success .tabs.is-boxed a, html.theme--documenter-dark .hero.is-success .tabs.is-toggle a {
+        color: #fff; }
+        html.theme--documenter-dark .hero.is-success .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-success .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-success .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-success .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-success .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-success .tabs.is-toggle li.is-active a:hover {
+        background-color: #fff;
+        border-color: #fff;
+        color: #008438; }
+      html.theme--documenter-dark .hero.is-success.is-bold {
+        background-image: linear-gradient(141deg, #005115 0%, #008438 71%, #009e5d 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-success.is-bold .navbar-menu {
+            background-image: linear-gradient(141deg, #005115 0%, #008438 71%, #009e5d 100%); } }
+    html.theme--documenter-dark .hero.is-warning {
+      background-color: #ad8100;
+      color: #fff; }
+      html.theme--documenter-dark .hero.is-warning a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-warning strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-warning .title {
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-warning .subtitle {
+        color: rgba(255, 255, 255, 0.9); }
+        html.theme--documenter-dark .hero.is-warning .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-warning .subtitle strong {
+          color: #fff; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-warning .navbar-menu {
+          background-color: #ad8100; } }
+      html.theme--documenter-dark .hero.is-warning .navbar-item,
+      html.theme--documenter-dark .hero.is-warning .navbar-link {
+        color: rgba(255, 255, 255, 0.7); }
+      html.theme--documenter-dark .hero.is-warning a.navbar-item:hover, html.theme--documenter-dark .hero.is-warning a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-warning .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-warning .navbar-link.is-active {
+        background-color: #946e00;
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-warning .tabs a {
+        color: #fff;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-warning .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-warning .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-warning .tabs.is-boxed a, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle a {
+        color: #fff; }
+        html.theme--documenter-dark .hero.is-warning .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-warning .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-warning .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle li.is-active a:hover {
+        background-color: #fff;
+        border-color: #fff;
+        color: #ad8100; }
+      html.theme--documenter-dark .hero.is-warning.is-bold {
+        background-image: linear-gradient(141deg, #7a4700 0%, #ad8100 71%, #c7b500 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-warning.is-bold .navbar-menu {
+            background-image: linear-gradient(141deg, #7a4700 0%, #ad8100 71%, #c7b500 100%); } }
+    html.theme--documenter-dark .hero.is-danger {
+      background-color: #9e1b0d;
+      color: #fff; }
+      html.theme--documenter-dark .hero.is-danger a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+      html.theme--documenter-dark .hero.is-danger strong {
+        color: inherit; }
+      html.theme--documenter-dark .hero.is-danger .title {
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-danger .subtitle {
+        color: rgba(255, 255, 255, 0.9); }
+        html.theme--documenter-dark .hero.is-danger .subtitle a:not(.button),
+        html.theme--documenter-dark .hero.is-danger .subtitle strong {
+          color: #fff; }
+      @media screen and (max-width: 1055px) {
+        html.theme--documenter-dark .hero.is-danger .navbar-menu {
+          background-color: #9e1b0d; } }
+      html.theme--documenter-dark .hero.is-danger .navbar-item,
+      html.theme--documenter-dark .hero.is-danger .navbar-link {
+        color: rgba(255, 255, 255, 0.7); }
+      html.theme--documenter-dark .hero.is-danger a.navbar-item:hover, html.theme--documenter-dark .hero.is-danger a.navbar-item.is-active,
+      html.theme--documenter-dark .hero.is-danger .navbar-link:hover,
+      html.theme--documenter-dark .hero.is-danger .navbar-link.is-active {
+        background-color: #86170b;
+        color: #fff; }
+      html.theme--documenter-dark .hero.is-danger .tabs a {
+        color: #fff;
+        opacity: 0.9; }
+        html.theme--documenter-dark .hero.is-danger .tabs a:hover {
+          opacity: 1; }
+      html.theme--documenter-dark .hero.is-danger .tabs li.is-active a {
+        opacity: 1; }
+      html.theme--documenter-dark .hero.is-danger .tabs.is-boxed a, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle a {
+        color: #fff; }
+        html.theme--documenter-dark .hero.is-danger .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle a:hover {
+          background-color: rgba(10, 10, 10, 0.1); }
+      html.theme--documenter-dark .hero.is-danger .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-danger .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle li.is-active a:hover {
+        background-color: #fff;
+        border-color: #fff;
+        color: #9e1b0d; }
+      html.theme--documenter-dark .hero.is-danger.is-bold {
+        background-image: linear-gradient(141deg, #75030b 0%, #9e1b0d 71%, #ba380a 100%); }
+        @media screen and (max-width: 768px) {
+          html.theme--documenter-dark .hero.is-danger.is-bold .navbar-menu {
+            background-image: linear-gradient(141deg, #75030b 0%, #9e1b0d 71%, #ba380a 100%); } }
+    html.theme--documenter-dark .hero.is-small .hero-body, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.hero .hero-body {
+      padding-bottom: 1.5rem;
+      padding-top: 1.5rem; }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .hero.is-medium .hero-body {
+        padding-bottom: 9rem;
+        padding-top: 9rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .hero.is-large .hero-body {
+        padding-bottom: 18rem;
+        padding-top: 18rem; } }
+    html.theme--documenter-dark .hero.is-halfheight .hero-body, html.theme--documenter-dark .hero.is-fullheight .hero-body, html.theme--documenter-dark .hero.is-fullheight-with-navbar .hero-body {
+      align-items: center;
+      display: flex; }
+      html.theme--documenter-dark .hero.is-halfheight .hero-body > .container, html.theme--documenter-dark .hero.is-fullheight .hero-body > .container, html.theme--documenter-dark .hero.is-fullheight-with-navbar .hero-body > .container {
+        flex-grow: 1;
+        flex-shrink: 1; }
+    html.theme--documenter-dark .hero.is-halfheight {
+      min-height: 50vh; }
+    html.theme--documenter-dark .hero.is-fullheight {
+      min-height: 100vh; }
+  html.theme--documenter-dark .hero-video {
+    overflow: hidden; }
+    html.theme--documenter-dark .hero-video video {
+      left: 50%;
+      min-height: 100%;
+      min-width: 100%;
+      position: absolute;
+      top: 50%;
+      transform: translate3d(-50%, -50%, 0); }
+    html.theme--documenter-dark .hero-video.is-transparent {
+      opacity: 0.3; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .hero-video {
+        display: none; } }
+  html.theme--documenter-dark .hero-buttons {
+    margin-top: 1.5rem; }
+    @media screen and (max-width: 768px) {
+      html.theme--documenter-dark .hero-buttons .button {
+        display: flex; }
+        html.theme--documenter-dark .hero-buttons .button:not(:last-child) {
+          margin-bottom: 0.75rem; } }
+    @media screen and (min-width: 769px), print {
+      html.theme--documenter-dark .hero-buttons {
+        display: flex;
+        justify-content: center; }
+        html.theme--documenter-dark .hero-buttons .button:not(:last-child) {
+          margin-right: 1.5rem; } }
+  html.theme--documenter-dark .hero-head,
+  html.theme--documenter-dark .hero-foot {
+    flex-grow: 0;
+    flex-shrink: 0; }
+  html.theme--documenter-dark .hero-body {
+    flex-grow: 1;
+    flex-shrink: 0;
+    padding: 3rem 1.5rem; }
+  html.theme--documenter-dark .section {
+    padding: 3rem 1.5rem; }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark .section.is-medium {
+        padding: 9rem 1.5rem; }
+      html.theme--documenter-dark .section.is-large {
+        padding: 18rem 1.5rem; } }
+  html.theme--documenter-dark .footer {
+    background-color: #282f2f;
+    padding: 3rem 1.5rem 6rem; }
+  html.theme--documenter-dark hr {
+    height: 1px; }
+  html.theme--documenter-dark h6 {
+    text-transform: uppercase;
+    letter-spacing: 0.5px; }
+  html.theme--documenter-dark .hero {
+    background-color: #343c3d; }
+  html.theme--documenter-dark a {
+    transition: all 200ms ease; }
+  html.theme--documenter-dark .button {
+    transition: all 200ms ease;
+    border-width: 1px;
+    color: white; }
+    html.theme--documenter-dark .button.is-active, html.theme--documenter-dark .button.is-focused, html.theme--documenter-dark .button:active, html.theme--documenter-dark .button:focus {
+      box-shadow: 0 0 0 2px rgba(140, 155, 157, 0.5); }
+    html.theme--documenter-dark .button.is-white.is-hovered, html.theme--documenter-dark .button.is-white:hover {
+      background-color: white; }
+    html.theme--documenter-dark .button.is-white.is-active, html.theme--documenter-dark .button.is-white.is-focused, html.theme--documenter-dark .button.is-white:active, html.theme--documenter-dark .button.is-white:focus {
+      border-color: white;
+      box-shadow: 0 0 0 2px rgba(255, 255, 255, 0.5); }
+    html.theme--documenter-dark .button.is-black.is-hovered, html.theme--documenter-dark .button.is-black:hover {
+      background-color: #1d1d1d; }
+    html.theme--documenter-dark .button.is-black.is-active, html.theme--documenter-dark .button.is-black.is-focused, html.theme--documenter-dark .button.is-black:active, html.theme--documenter-dark .button.is-black:focus {
+      border-color: #0a0a0a;
+      box-shadow: 0 0 0 2px rgba(10, 10, 10, 0.5); }
+    html.theme--documenter-dark .button.is-light.is-hovered, html.theme--documenter-dark .button.is-light:hover {
+      background-color: white; }
+    html.theme--documenter-dark .button.is-light.is-active, html.theme--documenter-dark .button.is-light.is-focused, html.theme--documenter-dark .button.is-light:active, html.theme--documenter-dark .button.is-light:focus {
+      border-color: #ecf0f1;
+      box-shadow: 0 0 0 2px rgba(236, 240, 241, 0.5); }
+    html.theme--documenter-dark .button.is-dark.is-hovered, html.theme--documenter-dark .content kbd.button.is-hovered, html.theme--documenter-dark .button.is-dark:hover, html.theme--documenter-dark .content kbd.button:hover {
+      background-color: #3a4344; }
+    html.theme--documenter-dark .button.is-dark.is-active, html.theme--documenter-dark .content kbd.button.is-active, html.theme--documenter-dark .button.is-dark.is-focused, html.theme--documenter-dark .content kbd.button.is-focused, html.theme--documenter-dark .button.is-dark:active, html.theme--documenter-dark .content kbd.button:active, html.theme--documenter-dark .button.is-dark:focus, html.theme--documenter-dark .content kbd.button:focus {
+      border-color: #282f2f;
+      box-shadow: 0 0 0 2px rgba(40, 47, 47, 0.5); }
+    html.theme--documenter-dark .button.is-primary.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary:hover, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:hover {
+      background-color: #436d9a; }
+    html.theme--documenter-dark .button.is-primary.is-active, html.theme--documenter-dark .docstring > section > a.button.is-active.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink, html.theme--documenter-dark .button.is-primary:active, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:active, html.theme--documenter-dark .button.is-primary:focus, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus {
+      border-color: #375a7f;
+      box-shadow: 0 0 0 2px rgba(55, 90, 127, 0.5); }
+    html.theme--documenter-dark .button.is-link.is-hovered, html.theme--documenter-dark .button.is-link:hover {
+      background-color: #1fdeb8; }
+    html.theme--documenter-dark .button.is-link.is-active, html.theme--documenter-dark .button.is-link.is-focused, html.theme--documenter-dark .button.is-link:active, html.theme--documenter-dark .button.is-link:focus {
+      border-color: #1abc9c;
+      box-shadow: 0 0 0 2px rgba(26, 188, 156, 0.5); }
+    html.theme--documenter-dark .button.is-info.is-hovered, html.theme--documenter-dark .button.is-info:hover {
+      background-color: #0363a3; }
+    html.theme--documenter-dark .button.is-info.is-active, html.theme--documenter-dark .button.is-info.is-focused, html.theme--documenter-dark .button.is-info:active, html.theme--documenter-dark .button.is-info:focus {
+      border-color: #024c7d;
+      box-shadow: 0 0 0 2px rgba(2, 76, 125, 0.5); }
+    html.theme--documenter-dark .button.is-success.is-hovered, html.theme--documenter-dark .button.is-success:hover {
+      background-color: #00aa48; }
+    html.theme--documenter-dark .button.is-success.is-active, html.theme--documenter-dark .button.is-success.is-focused, html.theme--documenter-dark .button.is-success:active, html.theme--documenter-dark .button.is-success:focus {
+      border-color: #008438;
+      box-shadow: 0 0 0 2px rgba(0, 132, 56, 0.5); }
+    html.theme--documenter-dark .button.is-warning.is-hovered, html.theme--documenter-dark .button.is-warning:hover {
+      background-color: #d39e00; }
+    html.theme--documenter-dark .button.is-warning.is-active, html.theme--documenter-dark .button.is-warning.is-focused, html.theme--documenter-dark .button.is-warning:active, html.theme--documenter-dark .button.is-warning:focus {
+      border-color: #ad8100;
+      box-shadow: 0 0 0 2px rgba(173, 129, 0, 0.5); }
+    html.theme--documenter-dark .button.is-danger.is-hovered, html.theme--documenter-dark .button.is-danger:hover {
+      background-color: #c12110; }
+    html.theme--documenter-dark .button.is-danger.is-active, html.theme--documenter-dark .button.is-danger.is-focused, html.theme--documenter-dark .button.is-danger:active, html.theme--documenter-dark .button.is-danger:focus {
+      border-color: #9e1b0d;
+      box-shadow: 0 0 0 2px rgba(158, 27, 13, 0.5); }
+  html.theme--documenter-dark .label {
+    color: #dbdee0; }
+  html.theme--documenter-dark .button,
+  html.theme--documenter-dark .control.has-icons-left .icon,
+  html.theme--documenter-dark .control.has-icons-right .icon,
+  html.theme--documenter-dark .input,
+  html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input,
+  html.theme--documenter-dark .pagination-ellipsis,
+  html.theme--documenter-dark .pagination-link,
+  html.theme--documenter-dark .pagination-next,
+  html.theme--documenter-dark .pagination-previous,
+  html.theme--documenter-dark .select,
+  html.theme--documenter-dark .select select,
+  html.theme--documenter-dark .textarea {
+    height: 2.5em; }
+  
+  html.theme--documenter-dark .input,
+  html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input,
+  html.theme--documenter-dark .textarea {
+    transition: all 200ms ease;
+    box-shadow: none;
+    border-width: 1px;
+    padding-left: 1em;
+    padding-right: 1em; }
+  html.theme--documenter-dark .select:after,
+  html.theme--documenter-dark .select select {
+    border-width: 1px; }
+  html.theme--documenter-dark .control.has-addons .button,
+  html.theme--documenter-dark .control.has-addons .input,
+  html.theme--documenter-dark .control.has-addons #documenter .docs-sidebar form.docs-search > input,
+  html.theme--documenter-dark #documenter .docs-sidebar .control.has-addons form.docs-search > input,
+  html.theme--documenter-dark .control.has-addons .select {
+    margin-right: -1px; }
+  html.theme--documenter-dark .notification {
+    background-color: #343c3d; }
+  html.theme--documenter-dark .card {
+    box-shadow: none;
+    border: 1px solid #343c3d;
+    background-color: #282f2f;
+    border-radius: 0.4em; }
+    html.theme--documenter-dark .card .card-image img {
+      border-radius: 0.4em 0.4em 0 0; }
+    html.theme--documenter-dark .card .card-header {
+      box-shadow: none;
+      background-color: rgba(18, 18, 18, 0.2);
+      border-radius: 0.4em 0.4em 0 0; }
+    html.theme--documenter-dark .card .card-footer {
+      background-color: rgba(18, 18, 18, 0.2); }
+    html.theme--documenter-dark .card .card-footer,
+    html.theme--documenter-dark .card .card-footer-item {
+      border-width: 1px;
+      border-color: #343c3d; }
+  html.theme--documenter-dark .notification.is-white a:not(.button) {
+    color: #0a0a0a;
+    text-decoration: underline; }
+  html.theme--documenter-dark .notification.is-black a:not(.button) {
+    color: white;
+    text-decoration: underline; }
+  html.theme--documenter-dark .notification.is-light a:not(.button) {
+    color: #282f2f;
+    text-decoration: underline; }
+  html.theme--documenter-dark .notification.is-dark a:not(.button), html.theme--documenter-dark .content kbd.notification a:not(.button) {
+    color: #ecf0f1;
+    text-decoration: underline; }
+  html.theme--documenter-dark .notification.is-primary a:not(.button), html.theme--documenter-dark .docstring > section > a.notification.docs-sourcelink a:not(.button) {
+    color: #fff;
+    text-decoration: underline; }
+  html.theme--documenter-dark .notification.is-link a:not(.button) {
+    color: #fff;
+    text-decoration: underline; }
+  html.theme--documenter-dark .notification.is-info a:not(.button) {
+    color: #fff;
+    text-decoration: underline; }
+  html.theme--documenter-dark .notification.is-success a:not(.button) {
+    color: #fff;
+    text-decoration: underline; }
+  html.theme--documenter-dark .notification.is-warning a:not(.button) {
+    color: #fff;
+    text-decoration: underline; }
+  html.theme--documenter-dark .notification.is-danger a:not(.button) {
+    color: #fff;
+    text-decoration: underline; }
+  html.theme--documenter-dark .tag, html.theme--documenter-dark .docstring > section > a.docs-sourcelink, html.theme--documenter-dark .content kbd {
+    border-radius: 0.4em; }
+  html.theme--documenter-dark .menu-list a {
+    transition: all 300ms ease; }
+  html.theme--documenter-dark .modal-card-body {
+    background-color: #282f2f; }
+  html.theme--documenter-dark .modal-card-foot,
+  html.theme--documenter-dark .modal-card-head {
+    border-color: #343c3d; }
+  html.theme--documenter-dark .message-header {
+    font-weight: 700;
+    background-color: #343c3d;
+    color: white; }
+  html.theme--documenter-dark .message-body {
+    border-width: 1px;
+    border-color: #343c3d; }
+  html.theme--documenter-dark .navbar {
+    border-radius: 0.4em; }
+    html.theme--documenter-dark .navbar.is-transparent {
+      background: none; }
+    html.theme--documenter-dark .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active {
+      background-color: #1abc9c; }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark .navbar .navbar-menu {
+        background-color: #375a7f;
+        border-radius: 0 0 0.4em 0.4em; } }
+  html.theme--documenter-dark .hero .navbar,
+  html.theme--documenter-dark body > .navbar {
+    border-radius: 0; }
+  html.theme--documenter-dark .pagination-link,
+  html.theme--documenter-dark .pagination-next,
+  html.theme--documenter-dark .pagination-previous {
+    border-width: 1px; }
+  html.theme--documenter-dark .panel-block,
+  html.theme--documenter-dark .panel-heading,
+  html.theme--documenter-dark .panel-tabs {
+    border-width: 1px; }
+    html.theme--documenter-dark .panel-block:first-child,
+    html.theme--documenter-dark .panel-heading:first-child,
+    html.theme--documenter-dark .panel-tabs:first-child {
+      border-top-width: 1px; }
+  html.theme--documenter-dark .panel-heading {
+    font-weight: 700; }
+  html.theme--documenter-dark .panel-tabs a {
+    border-width: 1px;
+    margin-bottom: -1px; }
+    html.theme--documenter-dark .panel-tabs a.is-active {
+      border-bottom-color: #17a689; }
+  html.theme--documenter-dark .panel-block:hover {
+    color: #1dd2af; }
+    html.theme--documenter-dark .panel-block:hover .panel-icon {
+      color: #1dd2af; }
+  html.theme--documenter-dark .panel-block.is-active .panel-icon {
+    color: #17a689; }
+  html.theme--documenter-dark .tabs a {
+    border-bottom-width: 1px;
+    margin-bottom: -1px; }
+  html.theme--documenter-dark .tabs ul {
+    border-bottom-width: 1px; }
+  html.theme--documenter-dark .tabs.is-boxed a {
+    border-width: 1px; }
+  html.theme--documenter-dark .tabs.is-boxed li.is-active a {
+    background-color: #1f2424; }
+  html.theme--documenter-dark .tabs.is-toggle li a {
+    border-width: 1px;
+    margin-bottom: 0; }
+  html.theme--documenter-dark .tabs.is-toggle li + li {
+    margin-left: -1px; }
+  html.theme--documenter-dark .hero.is-white .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .hero.is-black .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .hero.is-light .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .hero.is-dark .navbar .navbar-dropdown .navbar-item:hover, html.theme--documenter-dark .content kbd.hero .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .hero.is-primary .navbar .navbar-dropdown .navbar-item:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .hero.is-link .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .hero.is-info .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .hero.is-success .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .hero.is-warning .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark .hero.is-danger .navbar .navbar-dropdown .navbar-item:hover {
+    background-color: transparent; }
+  html.theme--documenter-dark h1 .docs-heading-anchor, html.theme--documenter-dark h1 .docs-heading-anchor:hover, html.theme--documenter-dark h1 .docs-heading-anchor:visited, html.theme--documenter-dark h2 .docs-heading-anchor, html.theme--documenter-dark h2 .docs-heading-anchor:hover, html.theme--documenter-dark h2 .docs-heading-anchor:visited, html.theme--documenter-dark h3 .docs-heading-anchor, html.theme--documenter-dark h3 .docs-heading-anchor:hover, html.theme--documenter-dark h3 .docs-heading-anchor:visited, html.theme--documenter-dark h4 .docs-heading-anchor, html.theme--documenter-dark h4 .docs-heading-anchor:hover, html.theme--documenter-dark h4 .docs-heading-anchor:visited, html.theme--documenter-dark h5 .docs-heading-anchor, html.theme--documenter-dark h5 .docs-heading-anchor:hover, html.theme--documenter-dark h5 .docs-heading-anchor:visited, html.theme--documenter-dark h6 .docs-heading-anchor, html.theme--documenter-dark h6 .docs-heading-anchor:hover, html.theme--documenter-dark h6 .docs-heading-anchor:visited {
+    color: #f2f2f2; }
+  html.theme--documenter-dark h1 .docs-heading-anchor-permalink, html.theme--documenter-dark h2 .docs-heading-anchor-permalink, html.theme--documenter-dark h3 .docs-heading-anchor-permalink, html.theme--documenter-dark h4 .docs-heading-anchor-permalink, html.theme--documenter-dark h5 .docs-heading-anchor-permalink, html.theme--documenter-dark h6 .docs-heading-anchor-permalink {
+    visibility: hidden;
+    vertical-align: middle;
+    margin-left: 0.5em;
+    font-size: 0.7rem; }
+    html.theme--documenter-dark h1 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h2 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h3 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h4 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h5 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h6 .docs-heading-anchor-permalink::before {
+      font-family: "Font Awesome 5 Free";
+      font-weight: 900;
+      content: "\f0c1"; }
+  html.theme--documenter-dark h1:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h2:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h3:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h4:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h5:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h6:hover .docs-heading-anchor-permalink {
+    visibility: visible; }
+  html.theme--documenter-dark .docs-light-only {
+    display: none !important; }
+  html.theme--documenter-dark .admonition {
+    background-color: #282f2f;
+    border-style: solid;
+    border-width: 1px;
+    border-color: #5e6d6f;
+    border-radius: 0.4em;
+    font-size: 15px; }
+    html.theme--documenter-dark .admonition strong {
+      color: currentColor; }
+    html.theme--documenter-dark .admonition.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.admonition {
+      font-size: 0.85em; }
+    html.theme--documenter-dark .admonition.is-medium {
+      font-size: 1.25rem; }
+    html.theme--documenter-dark .admonition.is-large {
+      font-size: 1.5rem; }
+    html.theme--documenter-dark .admonition.is-default {
+      background-color: #282f2f;
+      border-color: #5e6d6f; }
+      html.theme--documenter-dark .admonition.is-default > .admonition-header {
+        background-color: #5e6d6f; }
+    html.theme--documenter-dark .admonition.is-info {
+      background-color: #282f2f;
+      border-color: #024c7d; }
+      html.theme--documenter-dark .admonition.is-info > .admonition-header {
+        background-color: #024c7d; }
+    html.theme--documenter-dark .admonition.is-success {
+      background-color: #282f2f;
+      border-color: #008438; }
+      html.theme--documenter-dark .admonition.is-success > .admonition-header {
+        background-color: #008438; }
+    html.theme--documenter-dark .admonition.is-warning {
+      background-color: #282f2f;
+      border-color: #ad8100; }
+      html.theme--documenter-dark .admonition.is-warning > .admonition-header {
+        background-color: #ad8100; }
+    html.theme--documenter-dark .admonition.is-danger {
+      background-color: #282f2f;
+      border-color: #9e1b0d; }
+      html.theme--documenter-dark .admonition.is-danger > .admonition-header {
+        background-color: #9e1b0d; }
+    html.theme--documenter-dark .admonition.is-compat {
+      background-color: #282f2f;
+      border-color: #137886; }
+      html.theme--documenter-dark .admonition.is-compat > .admonition-header {
+        background-color: #137886; }
+  html.theme--documenter-dark .admonition-header {
+    background-color: #5e6d6f;
+    align-items: center;
+    font-weight: 700;
+    justify-content: space-between;
+    line-height: 1.25;
+    padding: 0.75em;
+    position: relative; }
+    html.theme--documenter-dark .admonition-header:before {
+      font-family: "Font Awesome 5 Free";
+      font-weight: 900;
+      margin-right: 0.75em;
+      content: "\f06a"; }
+  html.theme--documenter-dark .admonition-body {
+    color: #fff;
+    padding: 1em 1.25em; }
+    html.theme--documenter-dark .admonition-body pre {
+      background-color: #282f2f; }
+    html.theme--documenter-dark .admonition-body code {
+      background-color: rgba(255, 255, 255, 0.05); }
+  html.theme--documenter-dark .docstring {
+    margin-bottom: 1em;
+    background-color: transparent;
+    border: 1px solid #5e6d6f;
+    box-shadow: none;
+    max-width: 100%; }
+    html.theme--documenter-dark .docstring > header {
+      display: flex;
+      flex-grow: 1;
+      align-items: stretch;
+      padding: 0.75rem;
+      background-color: #282f2f;
+      box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1);
+      box-shadow: none;
+      border-bottom: 1px solid #5e6d6f; }
+      html.theme--documenter-dark .docstring > header code {
+        background-color: transparent; }
+      html.theme--documenter-dark .docstring > header .docstring-binding {
+        margin-right: 0.3em; }
+      html.theme--documenter-dark .docstring > header .docstring-category {
+        margin-left: 0.3em; }
+    html.theme--documenter-dark .docstring > section {
+      position: relative;
+      padding: 1rem 1.25rem;
+      border-bottom: 1px solid #5e6d6f; }
+      html.theme--documenter-dark .docstring > section:last-child {
+        border-bottom: none; }
+      html.theme--documenter-dark .docstring > section > a.docs-sourcelink {
+        transition: opacity 0.3s;
+        opacity: 0;
+        position: absolute;
+        right: 0.625rem;
+        bottom: 0.5rem; }
+    html.theme--documenter-dark .docstring:hover > section > a.docs-sourcelink {
+      opacity: 0.2; }
+    html.theme--documenter-dark .docstring > section:hover a.docs-sourcelink {
+      opacity: 1; }
+  html.theme--documenter-dark .documenter-example-output {
+    background-color: #1f2424; }
+  html.theme--documenter-dark .content pre {
+    border: 1px solid #5e6d6f; }
+  html.theme--documenter-dark .content code {
+    font-weight: inherit; }
+  html.theme--documenter-dark .content a code {
+    color: #1abc9c; }
+  html.theme--documenter-dark .content h1 code, html.theme--documenter-dark .content h2 code, html.theme--documenter-dark .content h3 code, html.theme--documenter-dark .content h4 code, html.theme--documenter-dark .content h5 code, html.theme--documenter-dark .content h6 code {
+    color: #f2f2f2; }
+  html.theme--documenter-dark .content table {
+    display: block;
+    width: initial;
+    max-width: 100%;
+    overflow-x: auto; }
+  html.theme--documenter-dark .content blockquote > ul:first-child, html.theme--documenter-dark .content blockquote > ol:first-child, html.theme--documenter-dark .content .admonition-body > ul:first-child, html.theme--documenter-dark .content .admonition-body > ol:first-child {
+    margin-top: 0; }
+  html.theme--documenter-dark .breadcrumb a.is-disabled {
+    cursor: default;
+    pointer-events: none; }
+    html.theme--documenter-dark .breadcrumb a.is-disabled, html.theme--documenter-dark .breadcrumb a.is-disabled:hover {
+      color: #f2f2f2; }
+  html.theme--documenter-dark .hljs {
+    background: initial !important;
+    padding: initial !important; }
+  html.theme--documenter-dark .katex .katex-mathml {
+    top: 0;
+    right: 0; }
+  html.theme--documenter-dark .katex-display, html.theme--documenter-dark mjx-container, html.theme--documenter-dark .MathJax_Display {
+    margin: 0.5em 0 !important; }
+  html.theme--documenter-dark html {
+    -moz-osx-font-smoothing: auto;
+    -webkit-font-smoothing: auto; }
+  html.theme--documenter-dark #documenter .docs-main > article {
+    overflow-wrap: break-word; }
+    html.theme--documenter-dark #documenter .docs-main > article .math-container {
+      overflow-x: auto; }
+  @media screen and (min-width: 1056px) {
+    html.theme--documenter-dark #documenter .docs-main {
+      max-width: 52rem;
+      margin-left: 20rem;
+      padding-right: 1rem; } }
+  @media screen and (max-width: 1055px) {
+    html.theme--documenter-dark #documenter .docs-main {
+      width: 100%; }
+      html.theme--documenter-dark #documenter .docs-main > article {
+        max-width: 52rem;
+        margin-left: auto;
+        margin-right: auto;
+        margin-bottom: 1rem;
+        padding: 0 1rem; }
+      html.theme--documenter-dark #documenter .docs-main > header, html.theme--documenter-dark #documenter .docs-main > nav {
+        max-width: 100%;
+        width: 100%;
+        margin: 0; } }
+  html.theme--documenter-dark #documenter .docs-main header.docs-navbar {
+    background-color: #1f2424;
+    border-bottom: 1px solid #5e6d6f;
+    z-index: 2;
+    min-height: 4rem;
+    margin-bottom: 1rem;
+    display: flex; }
+    html.theme--documenter-dark #documenter .docs-main header.docs-navbar .breadcrumb {
+      flex-grow: 1; }
+    html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right {
+      display: flex;
+      white-space: nowrap; }
+      html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-icon, html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-label, html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button {
+        display: inline-block; }
+      html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-label {
+        padding: 0;
+        margin-left: 0.3em; }
+      html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-settings-button {
+        margin: auto 0 auto 1rem; }
+      html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button {
+        font-size: 1.5rem;
+        margin: auto 0 auto 1rem; }
+    html.theme--documenter-dark #documenter .docs-main header.docs-navbar > * {
+      margin: auto 0; }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark #documenter .docs-main header.docs-navbar {
+        position: sticky;
+        top: 0;
+        padding: 0 1rem;
+        /* For Headroom.js */
+        transition-property: top, box-shadow;
+        -webkit-transition-property: top, box-shadow;
+        /* Safari */
+        transition-duration: 0.3s;
+        -webkit-transition-duration: 0.3s;
+        /* Safari */ }
+        html.theme--documenter-dark #documenter .docs-main header.docs-navbar.headroom--not-top {
+          box-shadow: 0.2rem 0rem 0.4rem #171717;
+          transition-duration: 0.7s;
+          -webkit-transition-duration: 0.7s;
+          /* Safari */ }
+        html.theme--documenter-dark #documenter .docs-main header.docs-navbar.headroom--unpinned.headroom--not-top.headroom--not-bottom {
+          top: -4.5rem;
+          transition-duration: 0.7s;
+          -webkit-transition-duration: 0.7s;
+          /* Safari */ } }
+  html.theme--documenter-dark #documenter .docs-main section.footnotes {
+    border-top: 1px solid #5e6d6f; }
+    html.theme--documenter-dark #documenter .docs-main section.footnotes li .tag:first-child, html.theme--documenter-dark #documenter .docs-main section.footnotes li .docstring > section > a.docs-sourcelink:first-child, html.theme--documenter-dark #documenter .docs-main section.footnotes li .content kbd:first-child, html.theme--documenter-dark .content #documenter .docs-main section.footnotes li kbd:first-child {
+      margin-right: 1em;
+      margin-bottom: 0.4em; }
+  html.theme--documenter-dark #documenter .docs-main .docs-footer {
+    display: flex;
+    flex-wrap: wrap;
+    margin-left: 0;
+    margin-right: 0;
+    border-top: 1px solid #5e6d6f;
+    padding-top: 1rem;
+    padding-bottom: 1rem; }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark #documenter .docs-main .docs-footer {
+        padding-left: 1rem;
+        padding-right: 1rem; } }
+    html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-nextpage, html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-prevpage {
+      flex-grow: 1; }
+    html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-nextpage {
+      text-align: right; }
+    html.theme--documenter-dark #documenter .docs-main .docs-footer .flexbox-break {
+      flex-basis: 100%;
+      height: 0; }
+    html.theme--documenter-dark #documenter .docs-main .docs-footer .footer-message {
+      font-size: 0.8em;
+      margin: 0.5em auto 0 auto;
+      text-align: center; }
+  html.theme--documenter-dark #documenter .docs-sidebar {
+    display: flex;
+    flex-direction: column;
+    color: #fff;
+    background-color: #282f2f;
+    border-right: 1px solid #5e6d6f;
+    padding: 0;
+    flex: 0 0 18rem;
+    z-index: 5;
+    font-size: 15px;
+    position: fixed;
+    left: -18rem;
+    width: 18rem;
+    height: 100%;
+    transition: left 0.3s;
+    /* Setting up a nicer theme style for the scrollbar */ }
+    html.theme--documenter-dark #documenter .docs-sidebar.visible {
+      left: 0;
+      box-shadow: 0.4rem 0rem 0.8rem #171717; }
+      @media screen and (min-width: 1056px) {
+        html.theme--documenter-dark #documenter .docs-sidebar.visible {
+          box-shadow: none; } }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark #documenter .docs-sidebar {
+        left: 0;
+        top: 0; } }
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-logo {
+      margin-top: 1rem;
+      padding: 0 1rem; }
+      html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img {
+        max-height: 6rem;
+        margin: auto; }
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name {
+      flex-shrink: 0;
+      font-size: 1.5rem;
+      font-weight: 700;
+      text-align: center;
+      white-space: nowrap;
+      overflow: hidden;
+      padding: 0.5rem 0; }
+      html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name .docs-autofit {
+        max-width: 16.2rem; }
+    html.theme--documenter-dark #documenter .docs-sidebar .docs-version-selector {
+      border-top: 1px solid #5e6d6f;
+      display: none;
+      padding: 0.5rem; }
+      html.theme--documenter-dark #documenter .docs-sidebar .docs-version-selector.visible {
+        display: flex; }
+    html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu {
+      flex-grow: 1;
+      user-select: none;
+      border-top: 1px solid #5e6d6f;
+      padding-bottom: 1.5rem;
+      /* Managing collapsible submenus */ }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li > .tocitem {
+        font-weight: bold; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li li {
+        font-size: 14.25px;
+        margin-left: 1em;
+        border-left: 1px solid #5e6d6f; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input.collapse-toggle {
+        display: none; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.collapsed {
+        display: none; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input:checked ~ ul.collapsed {
+        display: block; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem {
+        display: flex; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-label {
+          flex-grow: 2; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron {
+          display: inline-block;
+          font-style: normal;
+          font-variant: normal;
+          text-rendering: auto;
+          line-height: 1;
+          font-size: 11.25px;
+          margin-left: 1rem;
+          margin-top: auto;
+          margin-bottom: auto; }
+          html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron::before {
+            font-family: "Font Awesome 5 Free";
+            font-weight: 900;
+            content: "\f054"; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input:checked ~ label.tocitem .docs-chevron::before {
+        content: "\f078"; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem {
+        display: block;
+        padding: 0.5rem 0.5rem; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem:hover {
+          color: #fff;
+          background: #282f2f; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu a.tocitem:hover, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem:hover {
+        color: #fff;
+        background-color: #32393a; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active {
+        border-top: 1px solid #5e6d6f;
+        border-bottom: 1px solid #5e6d6f;
+        background-color: #1f2424; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem:hover {
+          background-color: #1f2424;
+          color: #fff; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active ul.internal .tocitem:hover {
+          background-color: #32393a;
+          color: #fff; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li.is-active:first-child {
+        border-top: none; }
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal {
+        margin: 0 0.5rem 0.5rem;
+        border-top: 1px solid #5e6d6f; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal li {
+          font-size: 12.75px;
+          border-left: none;
+          margin-left: 0;
+          margin-top: 0.5rem; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem {
+          width: 100%;
+          padding: 0; }
+          html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem::before {
+            content: "⚬";
+            margin-right: 0.4em; }
+    html.theme--documenter-dark #documenter .docs-sidebar form.docs-search {
+      margin: auto;
+      margin-top: 0.5rem;
+      margin-bottom: 0.5rem; }
+      html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input {
+        width: 14.4rem; }
+    @media screen and (min-width: 1056px) {
+      html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu {
+        overflow-y: auto;
+        -webkit-overflow-scroll: touch; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar {
+          width: .3rem;
+          background: none; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb {
+          border-radius: 5px 0px 0px 5px;
+          background: #3b4445; }
+        html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb:hover {
+          background: #4e5a5c; } }
+    @media screen and (max-width: 1055px) {
+      html.theme--documenter-dark #documenter .docs-sidebar {
+        overflow-y: auto;
+        -webkit-overflow-scroll: touch; }
+        html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar {
+          width: .3rem;
+          background: none; }
+        html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar-thumb {
+          border-radius: 5px 0px 0px 5px;
+          background: #3b4445; }
+        html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar-thumb:hover {
+          background: #4e5a5c; } }
+  html.theme--documenter-dark #documenter .docs-main #documenter-search-info {
+    margin-bottom: 1rem; }
+  html.theme--documenter-dark #documenter .docs-main #documenter-search-results {
+    list-style-type: circle;
+    list-style-position: outside; }
+    html.theme--documenter-dark #documenter .docs-main #documenter-search-results li {
+      margin-left: 2rem; }
+    html.theme--documenter-dark #documenter .docs-main #documenter-search-results .docs-highlight {
+      background-color: yellow; }
+  html.theme--documenter-dark {
+    background-color: #1f2424;
+    font-size: 16px;
+    min-width: 300px;
+    overflow-x: auto;
+    overflow-y: scroll;
+    text-rendering: optimizeLegibility;
+    text-size-adjust: 100%; }
+  html.theme--documenter-dark .hljs-comment,
+  html.theme--documenter-dark .hljs-quote {
+    color: #d4d0ab; }
+  html.theme--documenter-dark .hljs-variable,
+  html.theme--documenter-dark .hljs-template-variable,
+  html.theme--documenter-dark .hljs-tag,
+  html.theme--documenter-dark .hljs-name,
+  html.theme--documenter-dark .hljs-selector-id,
+  html.theme--documenter-dark .hljs-selector-class,
+  html.theme--documenter-dark .hljs-regexp,
+  html.theme--documenter-dark .hljs-deletion {
+    color: #ffa07a; }
+  html.theme--documenter-dark .hljs-number,
+  html.theme--documenter-dark .hljs-built_in,
+  html.theme--documenter-dark .hljs-builtin-name,
+  html.theme--documenter-dark .hljs-literal,
+  html.theme--documenter-dark .hljs-type,
+  html.theme--documenter-dark .hljs-params,
+  html.theme--documenter-dark .hljs-meta,
+  html.theme--documenter-dark .hljs-link {
+    color: #f5ab35; }
+  html.theme--documenter-dark .hljs-attribute {
+    color: #ffd700; }
+  html.theme--documenter-dark .hljs-string,
+  html.theme--documenter-dark .hljs-symbol,
+  html.theme--documenter-dark .hljs-bullet,
+  html.theme--documenter-dark .hljs-addition {
+    color: #abe338; }
+  html.theme--documenter-dark .hljs-title,
+  html.theme--documenter-dark .hljs-section {
+    color: #00e0e0; }
+  html.theme--documenter-dark .hljs-keyword,
+  html.theme--documenter-dark .hljs-selector-tag {
+    color: #dcc6e0; }
+  html.theme--documenter-dark .hljs {
+    display: block;
+    overflow-x: auto;
+    background: #2b2b2b;
+    color: #f8f8f2;
+    padding: 0.5em; }
+  html.theme--documenter-dark .hljs-emphasis {
+    font-style: italic; }
+  html.theme--documenter-dark .hljs-strong {
+    font-weight: bold; }
+  @media screen and (-ms-high-contrast: active) {
+    html.theme--documenter-dark .hljs-addition,
+    html.theme--documenter-dark .hljs-attribute,
+    html.theme--documenter-dark .hljs-built_in,
+    html.theme--documenter-dark .hljs-builtin-name,
+    html.theme--documenter-dark .hljs-bullet,
+    html.theme--documenter-dark .hljs-comment,
+    html.theme--documenter-dark .hljs-link,
+    html.theme--documenter-dark .hljs-literal,
+    html.theme--documenter-dark .hljs-meta,
+    html.theme--documenter-dark .hljs-number,
+    html.theme--documenter-dark .hljs-params,
+    html.theme--documenter-dark .hljs-string,
+    html.theme--documenter-dark .hljs-symbol,
+    html.theme--documenter-dark .hljs-type,
+    html.theme--documenter-dark .hljs-quote {
+      color: highlight; }
+    html.theme--documenter-dark .hljs-keyword,
+    html.theme--documenter-dark .hljs-selector-tag {
+      font-weight: bold; } }
+  html.theme--documenter-dark .hljs-subst {
+    color: #f8f8f2; }
diff --git a/docs/assets/themes/documenter-light.css b/docs/assets/themes/documenter-light.css
index e000447..57fdf94 100644
--- a/docs/assets/themes/documenter-light.css
+++ b/docs/assets/themes/documenter-light.css
@@ -1,9 +1,7630 @@
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.filter-tabs::-webkit-scrollbar-thumb{background-color:gray;border-radius:1rem}#search-modal .modal-card-body::-webkit-scrollbar-track,#search-modal .filter-tabs::-webkit-scrollbar-track{-webkit-box-shadow:inset 0 0 6px rgba(0,0,0,0.6);background-color:transparent}.w-100{width:100%}.gap-2{gap:0.5rem}.gap-4{gap:1rem}.gap-8{gap:2rem}.ansi span.sgr1{font-weight:bolder}.ansi span.sgr2{font-weight:lighter}.ansi span.sgr3{font-style:italic}.ansi span.sgr4{text-decoration:underline}.ansi span.sgr7{color:#fff;background-color:#222}.ansi span.sgr8{color:transparent}.ansi span.sgr8 span{color:transparent}.ansi span.sgr9{text-decoration:line-through}.ansi span.sgr30{color:#242424}.ansi span.sgr31{color:#a7201f}.ansi span.sgr32{color:#066f00}.ansi span.sgr33{color:#856b00}.ansi span.sgr34{color:#2149b0}.ansi span.sgr35{color:#7d4498}.ansi span.sgr36{color:#007989}.ansi span.sgr37{color:gray}.ansi span.sgr40{background-color:#242424}.ansi span.sgr41{background-color:#a7201f}.ansi span.sgr42{background-color:#066f00}.ansi span.sgr43{background-color:#856b00}.ansi span.sgr44{background-color:#2149b0}.ansi span.sgr45{background-color:#7d4498}.ansi span.sgr46{background-color:#007989}.ansi span.sgr47{background-color:gray}.ansi span.sgr90{color:#616161}.ansi span.sgr91{color:#cb3c33}.ansi span.sgr92{color:#0e8300}.ansi span.sgr93{color:#a98800}.ansi span.sgr94{color:#3c5dcd}.ansi span.sgr95{color:#9256af}.ansi span.sgr96{color:#008fa3}.ansi span.sgr97{color:#f5f5f5}.ansi span.sgr100{background-color:#616161}.ansi span.sgr101{background-color:#cb3c33}.ansi span.sgr102{background-color:#0e8300}.ansi span.sgr103{background-color:#a98800}.ansi span.sgr104{background-color:#3c5dcd}.ansi span.sgr105{background-color:#9256af}.ansi span.sgr106{background-color:#008fa3}.ansi span.sgr107{background-color:#f5f5f5}code.language-julia-repl>span.hljs-meta{color:#066f00;font-weight:bolder}/*!
-  Theme: Default
-  Description: Original highlight.js style
-  Author: (c) Ivan Sagalaev <maniac@softwaremaniacs.org>
-  Maintainer: @highlightjs/core-team
-  Website: https://highlightjs.org/
-  License: see project LICENSE
-  Touched: 2021
-*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem}
+@charset "UTF-8";
+/* Font Awesome 5 mixin. Can be included in any rule that should render Font Awesome icons. */
+@keyframes spinAround {
+  from {
+    transform: rotate(0deg); }
+  to {
+    transform: rotate(359deg); } }
+
+.delete, .modal-close, .is-unselectable, .button, .file, .breadcrumb, .pagination-previous,
+.pagination-next,
+.pagination-link,
+.pagination-ellipsis, .tabs {
+  -webkit-touch-callout: none;
+  -webkit-user-select: none;
+  -moz-user-select: none;
+  -ms-user-select: none;
+  user-select: none; }
+
+.select:not(.is-multiple):not(.is-loading)::after, .navbar-link:not(.is-arrowless)::after {
+  border: 3px solid transparent;
+  border-radius: 2px;
+  border-right: 0;
+  border-top: 0;
+  content: " ";
+  display: block;
+  height: 0.625em;
+  margin-top: -0.4375em;
+  pointer-events: none;
+  position: absolute;
+  top: 50%;
+  transform: rotate(-45deg);
+  transform-origin: center;
+  width: 0.625em; }
+
+.box:not(:last-child), .content:not(:last-child), .notification:not(:last-child), .progress:not(:last-child), .table:not(:last-child), .table-container:not(:last-child), .title:not(:last-child),
+.subtitle:not(:last-child), .block:not(:last-child), .highlight:not(:last-child), .breadcrumb:not(:last-child), .level:not(:last-child), .list:not(:last-child), .message:not(:last-child), .tabs:not(:last-child), .admonition:not(:last-child) {
+  margin-bottom: 1.5rem; }
+
+.delete, .modal-close {
+  -moz-appearance: none;
+  -webkit-appearance: none;
+  background-color: rgba(10, 10, 10, 0.2);
+  border: none;
+  border-radius: 290486px;
+  cursor: pointer;
+  pointer-events: auto;
+  display: inline-block;
+  flex-grow: 0;
+  flex-shrink: 0;
+  font-size: 0;
+  height: 20px;
+  max-height: 20px;
+  max-width: 20px;
+  min-height: 20px;
+  min-width: 20px;
+  outline: none;
+  position: relative;
+  vertical-align: top;
+  width: 20px; }
+  .delete::before, .modal-close::before, .delete::after, .modal-close::after {
+    background-color: white;
+    content: "";
+    display: block;
+    left: 50%;
+    position: absolute;
+    top: 50%;
+    transform: translateX(-50%) translateY(-50%) rotate(45deg);
+    transform-origin: center center; }
+  .delete::before, .modal-close::before {
+    height: 2px;
+    width: 50%; }
+  .delete::after, .modal-close::after {
+    height: 50%;
+    width: 2px; }
+  .delete:hover, .modal-close:hover, .delete:focus, .modal-close:focus {
+    background-color: rgba(10, 10, 10, 0.3); }
+  .delete:active, .modal-close:active {
+    background-color: rgba(10, 10, 10, 0.4); }
+  .is-small.delete, #documenter .docs-sidebar form.docs-search > input.delete, .is-small.modal-close, #documenter .docs-sidebar form.docs-search > input.modal-close {
+    height: 16px;
+    max-height: 16px;
+    max-width: 16px;
+    min-height: 16px;
+    min-width: 16px;
+    width: 16px; }
+  .is-medium.delete, .is-medium.modal-close {
+    height: 24px;
+    max-height: 24px;
+    max-width: 24px;
+    min-height: 24px;
+    min-width: 24px;
+    width: 24px; }
+  .is-large.delete, .is-large.modal-close {
+    height: 32px;
+    max-height: 32px;
+    max-width: 32px;
+    min-height: 32px;
+    min-width: 32px;
+    width: 32px; }
+
+.button.is-loading::after, .loader, .select.is-loading::after, .control.is-loading::after {
+  animation: spinAround 500ms infinite linear;
+  border: 2px solid #dbdbdb;
+  border-radius: 290486px;
+  border-right-color: transparent;
+  border-top-color: transparent;
+  content: "";
+  display: block;
+  height: 1em;
+  position: relative;
+  width: 1em; }
+
+.is-overlay, .image.is-square img, #documenter .docs-sidebar .docs-logo > img.is-square img,
+.image.is-square .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, .image.is-1by1 img, #documenter .docs-sidebar .docs-logo > img.is-1by1 img,
+.image.is-1by1 .has-ratio,
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+.image.is-5by4 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, .image.is-4by3 img, #documenter .docs-sidebar .docs-logo > img.is-4by3 img,
+.image.is-4by3 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, .image.is-3by2 img, #documenter .docs-sidebar .docs-logo > img.is-3by2 img,
+.image.is-3by2 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, .image.is-5by3 img, #documenter .docs-sidebar .docs-logo > img.is-5by3 img,
+.image.is-5by3 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, .image.is-16by9 img, #documenter .docs-sidebar .docs-logo > img.is-16by9 img,
+.image.is-16by9 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, .image.is-2by1 img, #documenter .docs-sidebar .docs-logo > img.is-2by1 img,
+.image.is-2by1 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, .image.is-3by1 img, #documenter .docs-sidebar .docs-logo > img.is-3by1 img,
+.image.is-3by1 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, .image.is-4by5 img, #documenter .docs-sidebar .docs-logo > img.is-4by5 img,
+.image.is-4by5 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, .image.is-3by4 img, #documenter .docs-sidebar .docs-logo > img.is-3by4 img,
+.image.is-3by4 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, .image.is-2by3 img, #documenter .docs-sidebar .docs-logo > img.is-2by3 img,
+.image.is-2by3 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, .image.is-3by5 img, #documenter .docs-sidebar .docs-logo > img.is-3by5 img,
+.image.is-3by5 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, .image.is-9by16 img, #documenter .docs-sidebar .docs-logo > img.is-9by16 img,
+.image.is-9by16 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, .image.is-1by2 img, #documenter .docs-sidebar .docs-logo > img.is-1by2 img,
+.image.is-1by2 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, .image.is-1by3 img, #documenter .docs-sidebar .docs-logo > img.is-1by3 img,
+.image.is-1by3 .has-ratio,
+#documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio, .modal, .modal-background, .hero-video {
+  bottom: 0;
+  left: 0;
+  position: absolute;
+  right: 0;
+  top: 0; }
+
+.button, .input, #documenter .docs-sidebar form.docs-search > input, .textarea, .select select, .file-cta,
+.file-name, .pagination-previous,
+.pagination-next,
+.pagination-link,
+.pagination-ellipsis {
+  -moz-appearance: none;
+  -webkit-appearance: none;
+  align-items: center;
+  border: 1px solid transparent;
+  border-radius: 4px;
+  box-shadow: none;
+  display: inline-flex;
+  font-size: 1rem;
+  height: 2.25em;
+  justify-content: flex-start;
+  line-height: 1.5;
+  padding-bottom: calc(0.375em - 1px);
+  padding-left: calc(0.625em - 1px);
+  padding-right: calc(0.625em - 1px);
+  padding-top: calc(0.375em - 1px);
+  position: relative;
+  vertical-align: top; }
+  .button:focus, .input:focus, #documenter .docs-sidebar form.docs-search > input:focus, .textarea:focus, .select select:focus, .file-cta:focus,
+  .file-name:focus, .pagination-previous:focus,
+  .pagination-next:focus,
+  .pagination-link:focus,
+  .pagination-ellipsis:focus, .is-focused.button, .is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-focused.textarea, .select select.is-focused, .is-focused.file-cta,
+  .is-focused.file-name, .is-focused.pagination-previous,
+  .is-focused.pagination-next,
+  .is-focused.pagination-link,
+  .is-focused.pagination-ellipsis, .button:active, .input:active, #documenter .docs-sidebar form.docs-search > input:active, .textarea:active, .select select:active, .file-cta:active,
+  .file-name:active, .pagination-previous:active,
+  .pagination-next:active,
+  .pagination-link:active,
+  .pagination-ellipsis:active, .is-active.button, .is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .is-active.textarea, .select select.is-active, .is-active.file-cta,
+  .is-active.file-name, .is-active.pagination-previous,
+  .is-active.pagination-next,
+  .is-active.pagination-link,
+  .is-active.pagination-ellipsis {
+    outline: none; }
+  .button[disabled], .input[disabled], #documenter .docs-sidebar form.docs-search > input[disabled], .textarea[disabled], .select select[disabled], .file-cta[disabled],
+  .file-name[disabled], .pagination-previous[disabled],
+  .pagination-next[disabled],
+  .pagination-link[disabled],
+  .pagination-ellipsis[disabled],
+  fieldset[disabled] .button,
+  fieldset[disabled] .input,
+  fieldset[disabled] #documenter .docs-sidebar form.docs-search > input,
+  #documenter .docs-sidebar fieldset[disabled] form.docs-search > input,
+  fieldset[disabled] .textarea,
+  fieldset[disabled] .select select,
+  .select fieldset[disabled] select,
+  fieldset[disabled] .file-cta,
+  fieldset[disabled] .file-name,
+  fieldset[disabled] .pagination-previous,
+  fieldset[disabled] .pagination-next,
+  fieldset[disabled] .pagination-link,
+  fieldset[disabled] .pagination-ellipsis {
+    cursor: not-allowed; }
+
+/*! minireset.css v0.0.4 | MIT License | github.com/jgthms/minireset.css */
+html,
+body,
+p,
+ol,
+ul,
+li,
+dl,
+dt,
+dd,
+blockquote,
+figure,
+fieldset,
+legend,
+textarea,
+pre,
+iframe,
+hr,
+h1,
+h2,
+h3,
+h4,
+h5,
+h6 {
+  margin: 0;
+  padding: 0; }
+
+h1,
+h2,
+h3,
+h4,
+h5,
+h6 {
+  font-size: 100%;
+  font-weight: normal; }
+
+ul {
+  list-style: none; }
+
+button,
+input,
+select,
+textarea {
+  margin: 0; }
+
+html {
+  box-sizing: border-box; }
+
+*, *::before, *::after {
+  box-sizing: inherit; }
+
+img,
+embed,
+iframe,
+object,
+video {
+  height: auto;
+  max-width: 100%; }
+
+audio {
+  max-width: 100%; }
+
+iframe {
+  border: 0; }
+
+table {
+  border-collapse: collapse;
+  border-spacing: 0; }
+
+td,
+th {
+  padding: 0; }
+  td:not([align]),
+  th:not([align]) {
+    text-align: left; }
+
+html {
+  background-color: white;
+  font-size: 16px;
+  -moz-osx-font-smoothing: grayscale;
+  -webkit-font-smoothing: antialiased;
+  min-width: 300px;
+  overflow-x: auto;
+  overflow-y: scroll;
+  text-rendering: optimizeLegibility;
+  text-size-adjust: 100%; }
+
+article,
+aside,
+figure,
+footer,
+header,
+hgroup,
+section {
+  display: block; }
+
+body,
+button,
+input,
+select,
+textarea {
+  font-family: "Lato", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif; }
+
+code,
+pre {
+  -moz-osx-font-smoothing: auto;
+  -webkit-font-smoothing: auto;
+  font-family: "Roboto Mono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace; }
+
+body {
+  color: #222222;
+  font-size: 1em;
+  font-weight: 400;
+  line-height: 1.5; }
+
+a {
+  color: #2e63b8;
+  cursor: pointer;
+  text-decoration: none; }
+  a strong {
+    color: currentColor; }
+  a:hover {
+    color: #363636; }
+
+code {
+  background-color: rgba(0, 0, 0, 0.05);
+  color: #000000;
+  font-size: 0.875em;
+  font-weight: normal;
+  padding: 0.1em; }
+
+hr {
+  background-color: whitesmoke;
+  border: none;
+  display: block;
+  height: 2px;
+  margin: 1.5rem 0; }
+
+img {
+  height: auto;
+  max-width: 100%; }
+
+input[type="checkbox"],
+input[type="radio"] {
+  vertical-align: baseline; }
+
+small {
+  font-size: 0.875em; }
+
+span {
+  font-style: inherit;
+  font-weight: inherit; }
+
+strong {
+  color: #222222;
+  font-weight: 700; }
+
+fieldset {
+  border: none; }
+
+pre {
+  -webkit-overflow-scrolling: touch;
+  background-color: whitesmoke;
+  color: #222222;
+  font-size: 0.875em;
+  overflow-x: auto;
+  padding: 1.25rem 1.5rem;
+  white-space: pre;
+  word-wrap: normal; }
+  pre code {
+    background-color: transparent;
+    color: currentColor;
+    font-size: 1em;
+    padding: 0; }
+
+table td,
+table th {
+  vertical-align: top; }
+  table td:not([align]),
+  table th:not([align]) {
+    text-align: left; }
+
+table th {
+  color: #222222; }
+
+.is-clearfix::after {
+  clear: both;
+  content: " ";
+  display: table; }
+
+.is-pulled-left {
+  float: left !important; }
+
+.is-pulled-right {
+  float: right !important; }
+
+.is-clipped {
+  overflow: hidden !important; }
+
+.is-size-1 {
+  font-size: 3rem !important; }
+
+.is-size-2 {
+  font-size: 2.5rem !important; }
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+.is-size-5 {
+  font-size: 1.25rem !important; }
+
+.is-size-6 {
+  font-size: 1rem !important; }
+
+.is-size-7, .docstring > section > a.docs-sourcelink {
+  font-size: 0.75rem !important; }
+
+@media screen and (max-width: 768px) {
+  .is-size-1-mobile {
+    font-size: 3rem !important; }
+  .is-size-2-mobile {
+    font-size: 2.5rem !important; }
+  .is-size-3-mobile {
+    font-size: 2rem !important; }
+  .is-size-4-mobile {
+    font-size: 1.5rem !important; }
+  .is-size-5-mobile {
+    font-size: 1.25rem !important; }
+  .is-size-6-mobile {
+    font-size: 1rem !important; }
+  .is-size-7-mobile {
+    font-size: 0.75rem !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-size-1-tablet {
+    font-size: 3rem !important; }
+  .is-size-2-tablet {
+    font-size: 2.5rem !important; }
+  .is-size-3-tablet {
+    font-size: 2rem !important; }
+  .is-size-4-tablet {
+    font-size: 1.5rem !important; }
+  .is-size-5-tablet {
+    font-size: 1.25rem !important; }
+  .is-size-6-tablet {
+    font-size: 1rem !important; }
+  .is-size-7-tablet {
+    font-size: 0.75rem !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-size-1-touch {
+    font-size: 3rem !important; }
+  .is-size-2-touch {
+    font-size: 2.5rem !important; }
+  .is-size-3-touch {
+    font-size: 2rem !important; }
+  .is-size-4-touch {
+    font-size: 1.5rem !important; }
+  .is-size-5-touch {
+    font-size: 1.25rem !important; }
+  .is-size-6-touch {
+    font-size: 1rem !important; }
+  .is-size-7-touch {
+    font-size: 0.75rem !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-size-1-desktop {
+    font-size: 3rem !important; }
+  .is-size-2-desktop {
+    font-size: 2.5rem !important; }
+  .is-size-3-desktop {
+    font-size: 2rem !important; }
+  .is-size-4-desktop {
+    font-size: 1.5rem !important; }
+  .is-size-5-desktop {
+    font-size: 1.25rem !important; }
+  .is-size-6-desktop {
+    font-size: 1rem !important; }
+  .is-size-7-desktop {
+    font-size: 0.75rem !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-size-1-widescreen {
+    font-size: 3rem !important; }
+  .is-size-2-widescreen {
+    font-size: 2.5rem !important; }
+  .is-size-3-widescreen {
+    font-size: 2rem !important; }
+  .is-size-4-widescreen {
+    font-size: 1.5rem !important; }
+  .is-size-5-widescreen {
+    font-size: 1.25rem !important; }
+  .is-size-6-widescreen {
+    font-size: 1rem !important; }
+  .is-size-7-widescreen {
+    font-size: 0.75rem !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-size-1-fullhd {
+    font-size: 3rem !important; }
+  .is-size-2-fullhd {
+    font-size: 2.5rem !important; }
+  .is-size-3-fullhd {
+    font-size: 2rem !important; }
+  .is-size-4-fullhd {
+    font-size: 1.5rem !important; }
+  .is-size-5-fullhd {
+    font-size: 1.25rem !important; }
+  .is-size-6-fullhd {
+    font-size: 1rem !important; }
+  .is-size-7-fullhd {
+    font-size: 0.75rem !important; } }
+
+.has-text-centered {
+  text-align: center !important; }
+
+.has-text-justified {
+  text-align: justify !important; }
+
+.has-text-left {
+  text-align: left !important; }
+
+.has-text-right {
+  text-align: right !important; }
+
+@media screen and (max-width: 768px) {
+  .has-text-centered-mobile {
+    text-align: center !important; } }
+
+@media screen and (min-width: 769px), print {
+  .has-text-centered-tablet {
+    text-align: center !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .has-text-centered-tablet-only {
+    text-align: center !important; } }
+
+@media screen and (max-width: 1055px) {
+  .has-text-centered-touch {
+    text-align: center !important; } }
+
+@media screen and (min-width: 1056px) {
+  .has-text-centered-desktop {
+    text-align: center !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .has-text-centered-desktop-only {
+    text-align: center !important; } }
+
+@media screen and (min-width: 1216px) {
+  .has-text-centered-widescreen {
+    text-align: center !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .has-text-centered-widescreen-only {
+    text-align: center !important; } }
+
+@media screen and (min-width: 1408px) {
+  .has-text-centered-fullhd {
+    text-align: center !important; } }
+
+@media screen and (max-width: 768px) {
+  .has-text-justified-mobile {
+    text-align: justify !important; } }
+
+@media screen and (min-width: 769px), print {
+  .has-text-justified-tablet {
+    text-align: justify !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .has-text-justified-tablet-only {
+    text-align: justify !important; } }
+
+@media screen and (max-width: 1055px) {
+  .has-text-justified-touch {
+    text-align: justify !important; } }
+
+@media screen and (min-width: 1056px) {
+  .has-text-justified-desktop {
+    text-align: justify !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .has-text-justified-desktop-only {
+    text-align: justify !important; } }
+
+@media screen and (min-width: 1216px) {
+  .has-text-justified-widescreen {
+    text-align: justify !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .has-text-justified-widescreen-only {
+    text-align: justify !important; } }
+
+@media screen and (min-width: 1408px) {
+  .has-text-justified-fullhd {
+    text-align: justify !important; } }
+
+@media screen and (max-width: 768px) {
+  .has-text-left-mobile {
+    text-align: left !important; } }
+
+@media screen and (min-width: 769px), print {
+  .has-text-left-tablet {
+    text-align: left !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .has-text-left-tablet-only {
+    text-align: left !important; } }
+
+@media screen and (max-width: 1055px) {
+  .has-text-left-touch {
+    text-align: left !important; } }
+
+@media screen and (min-width: 1056px) {
+  .has-text-left-desktop {
+    text-align: left !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .has-text-left-desktop-only {
+    text-align: left !important; } }
+
+@media screen and (min-width: 1216px) {
+  .has-text-left-widescreen {
+    text-align: left !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .has-text-left-widescreen-only {
+    text-align: left !important; } }
+
+@media screen and (min-width: 1408px) {
+  .has-text-left-fullhd {
+    text-align: left !important; } }
+
+@media screen and (max-width: 768px) {
+  .has-text-right-mobile {
+    text-align: right !important; } }
+
+@media screen and (min-width: 769px), print {
+  .has-text-right-tablet {
+    text-align: right !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .has-text-right-tablet-only {
+    text-align: right !important; } }
+
+@media screen and (max-width: 1055px) {
+  .has-text-right-touch {
+    text-align: right !important; } }
+
+@media screen and (min-width: 1056px) {
+  .has-text-right-desktop {
+    text-align: right !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .has-text-right-desktop-only {
+    text-align: right !important; } }
+
+@media screen and (min-width: 1216px) {
+  .has-text-right-widescreen {
+    text-align: right !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .has-text-right-widescreen-only {
+    text-align: right !important; } }
+
+@media screen and (min-width: 1408px) {
+  .has-text-right-fullhd {
+    text-align: right !important; } }
+
+.is-capitalized {
+  text-transform: capitalize !important; }
+
+.is-lowercase {
+  text-transform: lowercase !important; }
+
+.is-uppercase {
+  text-transform: uppercase !important; }
+
+.is-italic {
+  font-style: italic !important; }
+
+.has-text-white {
+  color: white !important; }
+
+a.has-text-white:hover, a.has-text-white:focus {
+  color: #e6e6e6 !important; }
+
+.has-background-white {
+  background-color: white !important; }
+
+.has-text-black {
+  color: #0a0a0a !important; }
+
+a.has-text-black:hover, a.has-text-black:focus {
+  color: black !important; }
+
+.has-background-black {
+  background-color: #0a0a0a !important; }
+
+.has-text-light {
+  color: whitesmoke !important; }
+
+a.has-text-light:hover, a.has-text-light:focus {
+  color: #dbdbdb !important; }
+
+.has-background-light {
+  background-color: whitesmoke !important; }
+
+.has-text-dark {
+  color: #363636 !important; }
+
+a.has-text-dark:hover, a.has-text-dark:focus {
+  color: #1c1c1c !important; }
+
+.has-background-dark {
+  background-color: #363636 !important; }
+
+.has-text-primary {
+  color: #4eb5de !important; }
+
+a.has-text-primary:hover, a.has-text-primary:focus {
+  color: #27a1d2 !important; }
+
+.has-background-primary {
+  background-color: #4eb5de !important; }
+
+.has-text-link {
+  color: #2e63b8 !important; }
+
+a.has-text-link:hover, a.has-text-link:focus {
+  color: #244d8f !important; }
+
+.has-background-link {
+  background-color: #2e63b8 !important; }
+
+.has-text-info {
+  color: #209cee !important; }
+
+a.has-text-info:hover, a.has-text-info:focus {
+  color: #0f81cc !important; }
+
+.has-background-info {
+  background-color: #209cee !important; }
+
+.has-text-success {
+  color: #22c35b !important; }
+
+a.has-text-success:hover, a.has-text-success:focus {
+  color: #1a9847 !important; }
+
+.has-background-success {
+  background-color: #22c35b !important; }
+
+.has-text-warning {
+  color: #ffdd57 !important; }
+
+a.has-text-warning:hover, a.has-text-warning:focus {
+  color: #ffd324 !important; }
+
+.has-background-warning {
+  background-color: #ffdd57 !important; }
+
+.has-text-danger {
+  color: #da0b00 !important; }
+
+a.has-text-danger:hover, a.has-text-danger:focus {
+  color: #a70800 !important; }
+
+.has-background-danger {
+  background-color: #da0b00 !important; }
+
+.has-text-black-bis {
+  color: #121212 !important; }
+
+.has-background-black-bis {
+  background-color: #121212 !important; }
+
+.has-text-black-ter {
+  color: #242424 !important; }
+
+.has-background-black-ter {
+  background-color: #242424 !important; }
+
+.has-text-grey-darker {
+  color: #363636 !important; }
+
+.has-background-grey-darker {
+  background-color: #363636 !important; }
+
+.has-text-grey-dark {
+  color: #4a4a4a !important; }
+
+.has-background-grey-dark {
+  background-color: #4a4a4a !important; }
+
+.has-text-grey {
+  color: #7a7a7a !important; }
+
+.has-background-grey {
+  background-color: #7a7a7a !important; }
+
+.has-text-grey-light {
+  color: #b5b5b5 !important; }
+
+.has-background-grey-light {
+  background-color: #b5b5b5 !important; }
+
+.has-text-grey-lighter {
+  color: #dbdbdb !important; }
+
+.has-background-grey-lighter {
+  background-color: #dbdbdb !important; }
+
+.has-text-white-ter {
+  color: whitesmoke !important; }
+
+.has-background-white-ter {
+  background-color: whitesmoke !important; }
+
+.has-text-white-bis {
+  color: #fafafa !important; }
+
+.has-background-white-bis {
+  background-color: #fafafa !important; }
+
+.has-text-weight-light {
+  font-weight: 300 !important; }
+
+.has-text-weight-normal {
+  font-weight: 400 !important; }
+
+.has-text-weight-medium {
+  font-weight: 500 !important; }
+
+.has-text-weight-semibold {
+  font-weight: 600 !important; }
+
+.has-text-weight-bold {
+  font-weight: 700 !important; }
+
+.is-family-primary {
+  font-family: "Lato", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif !important; }
+
+.is-family-secondary {
+  font-family: "Lato", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif !important; }
+
+.is-family-sans-serif {
+  font-family: "Lato", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif !important; }
+
+.is-family-monospace {
+  font-family: "Roboto Mono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace !important; }
+
+.is-family-code {
+  font-family: "Roboto Mono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace !important; }
+
+.is-block {
+  display: block !important; }
+
+@media screen and (max-width: 768px) {
+  .is-block-mobile {
+    display: block !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-block-tablet {
+    display: block !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-block-tablet-only {
+    display: block !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-block-touch {
+    display: block !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-block-desktop {
+    display: block !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-block-desktop-only {
+    display: block !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-block-widescreen {
+    display: block !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-block-widescreen-only {
+    display: block !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-block-fullhd {
+    display: block !important; } }
+
+.is-flex {
+  display: flex !important; }
+
+@media screen and (max-width: 768px) {
+  .is-flex-mobile {
+    display: flex !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-flex-tablet {
+    display: flex !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-flex-tablet-only {
+    display: flex !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-flex-touch {
+    display: flex !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-flex-desktop {
+    display: flex !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-flex-desktop-only {
+    display: flex !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-flex-widescreen {
+    display: flex !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-flex-widescreen-only {
+    display: flex !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-flex-fullhd {
+    display: flex !important; } }
+
+.is-inline {
+  display: inline !important; }
+
+@media screen and (max-width: 768px) {
+  .is-inline-mobile {
+    display: inline !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-inline-tablet {
+    display: inline !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-inline-tablet-only {
+    display: inline !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-inline-touch {
+    display: inline !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-inline-desktop {
+    display: inline !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-inline-desktop-only {
+    display: inline !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-inline-widescreen {
+    display: inline !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-inline-widescreen-only {
+    display: inline !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-inline-fullhd {
+    display: inline !important; } }
+
+.is-inline-block {
+  display: inline-block !important; }
+
+@media screen and (max-width: 768px) {
+  .is-inline-block-mobile {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-inline-block-tablet {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-inline-block-tablet-only {
+    display: inline-block !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-inline-block-touch {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-inline-block-desktop {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-inline-block-desktop-only {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-inline-block-widescreen {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-inline-block-widescreen-only {
+    display: inline-block !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-inline-block-fullhd {
+    display: inline-block !important; } }
+
+.is-inline-flex {
+  display: inline-flex !important; }
+
+@media screen and (max-width: 768px) {
+  .is-inline-flex-mobile {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-inline-flex-tablet {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-inline-flex-tablet-only {
+    display: inline-flex !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-inline-flex-touch {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-inline-flex-desktop {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-inline-flex-desktop-only {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-inline-flex-widescreen {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-inline-flex-widescreen-only {
+    display: inline-flex !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-inline-flex-fullhd {
+    display: inline-flex !important; } }
+
+.is-hidden {
+  display: none !important; }
+
+.is-sr-only {
+  border: none !important;
+  clip: rect(0, 0, 0, 0) !important;
+  height: 0.01em !important;
+  overflow: hidden !important;
+  padding: 0 !important;
+  position: absolute !important;
+  white-space: nowrap !important;
+  width: 0.01em !important; }
+
+@media screen and (max-width: 768px) {
+  .is-hidden-mobile {
+    display: none !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-hidden-tablet {
+    display: none !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-hidden-tablet-only {
+    display: none !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-hidden-touch {
+    display: none !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-hidden-desktop {
+    display: none !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-hidden-desktop-only {
+    display: none !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-hidden-widescreen {
+    display: none !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-hidden-widescreen-only {
+    display: none !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-hidden-fullhd {
+    display: none !important; } }
+
+.is-invisible {
+  visibility: hidden !important; }
+
+@media screen and (max-width: 768px) {
+  .is-invisible-mobile {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 769px), print {
+  .is-invisible-tablet {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 769px) and (max-width: 1055px) {
+  .is-invisible-tablet-only {
+    visibility: hidden !important; } }
+
+@media screen and (max-width: 1055px) {
+  .is-invisible-touch {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1056px) {
+  .is-invisible-desktop {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1056px) and (max-width: 1215px) {
+  .is-invisible-desktop-only {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1216px) {
+  .is-invisible-widescreen {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1216px) and (max-width: 1407px) {
+  .is-invisible-widescreen-only {
+    visibility: hidden !important; } }
+
+@media screen and (min-width: 1408px) {
+  .is-invisible-fullhd {
+    visibility: hidden !important; } }
+
+.is-marginless {
+  margin: 0 !important; }
+
+.is-paddingless {
+  padding: 0 !important; }
+
+.is-radiusless {
+  border-radius: 0 !important; }
+
+.is-shadowless {
+  box-shadow: none !important; }
+
+.is-relative {
+  position: relative !important; }
+
+.box {
+  background-color: white;
+  border-radius: 6px;
+  box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1);
+  color: #222222;
+  display: block;
+  padding: 1.25rem; }
+
+a.box:hover, a.box:focus {
+  box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px #2e63b8; }
+
+a.box:active {
+  box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2), 0 0 0 1px #2e63b8; }
+
+.button {
+  background-color: white;
+  border-color: #dbdbdb;
+  border-width: 1px;
+  color: #363636;
+  cursor: pointer;
+  justify-content: center;
+  padding-bottom: calc(0.375em - 1px);
+  padding-left: 0.75em;
+  padding-right: 0.75em;
+  padding-top: calc(0.375em - 1px);
+  text-align: center;
+  white-space: nowrap; }
+  .button strong {
+    color: inherit; }
+  .button .icon, .button .icon.is-small, .button #documenter .docs-sidebar form.docs-search > input.icon, #documenter .docs-sidebar .button form.docs-search > input.icon, .button .icon.is-medium, .button .icon.is-large {
+    height: 1.5em;
+    width: 1.5em; }
+  .button .icon:first-child:not(:last-child) {
+    margin-left: calc(-0.375em - 1px);
+    margin-right: 0.1875em; }
+  .button .icon:last-child:not(:first-child) {
+    margin-left: 0.1875em;
+    margin-right: calc(-0.375em - 1px); }
+  .button .icon:first-child:last-child {
+    margin-left: calc(-0.375em - 1px);
+    margin-right: calc(-0.375em - 1px); }
+  .button:hover, .button.is-hovered {
+    border-color: #b5b5b5;
+    color: #363636; }
+  .button:focus, .button.is-focused {
+    border-color: #2e63b8;
+    color: #363636; }
+    .button:focus:not(:active), .button.is-focused:not(:active) {
+      box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); }
+  .button:active, .button.is-active {
+    border-color: #4a4a4a;
+    color: #363636; }
+  .button.is-text {
+    background-color: transparent;
+    border-color: transparent;
+    color: #222222;
+    text-decoration: underline; }
+    .button.is-text:hover, .button.is-text.is-hovered, .button.is-text:focus, .button.is-text.is-focused {
+      background-color: whitesmoke;
+      color: #222222; }
+    .button.is-text:active, .button.is-text.is-active {
+      background-color: #e8e8e8;
+      color: #222222; }
+    .button.is-text[disabled],
+    fieldset[disabled] .button.is-text {
+      background-color: transparent;
+      border-color: transparent;
+      box-shadow: none; }
+  .button.is-white {
+    background-color: white;
+    border-color: transparent;
+    color: #0a0a0a; }
+    .button.is-white:hover, .button.is-white.is-hovered {
+      background-color: #f9f9f9;
+      border-color: transparent;
+      color: #0a0a0a; }
+    .button.is-white:focus, .button.is-white.is-focused {
+      border-color: transparent;
+      color: #0a0a0a; }
+      .button.is-white:focus:not(:active), .button.is-white.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); }
+    .button.is-white:active, .button.is-white.is-active {
+      background-color: #f2f2f2;
+      border-color: transparent;
+      color: #0a0a0a; }
+    .button.is-white[disabled],
+    fieldset[disabled] .button.is-white {
+      background-color: white;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-white.is-inverted {
+      background-color: #0a0a0a;
+      color: white; }
+      .button.is-white.is-inverted:hover, .button.is-white.is-inverted.is-hovered {
+        background-color: black; }
+      .button.is-white.is-inverted[disabled],
+      fieldset[disabled] .button.is-white.is-inverted {
+        background-color: #0a0a0a;
+        border-color: transparent;
+        box-shadow: none;
+        color: white; }
+    .button.is-white.is-loading::after {
+      border-color: transparent transparent #0a0a0a #0a0a0a !important; }
+    .button.is-white.is-outlined {
+      background-color: transparent;
+      border-color: white;
+      color: white; }
+      .button.is-white.is-outlined:hover, .button.is-white.is-outlined.is-hovered, .button.is-white.is-outlined:focus, .button.is-white.is-outlined.is-focused {
+        background-color: white;
+        border-color: white;
+        color: #0a0a0a; }
+      .button.is-white.is-outlined.is-loading::after {
+        border-color: transparent transparent white white !important; }
+      .button.is-white.is-outlined.is-loading:hover::after, .button.is-white.is-outlined.is-loading.is-hovered::after, .button.is-white.is-outlined.is-loading:focus::after, .button.is-white.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #0a0a0a #0a0a0a !important; }
+      .button.is-white.is-outlined[disabled],
+      fieldset[disabled] .button.is-white.is-outlined {
+        background-color: transparent;
+        border-color: white;
+        box-shadow: none;
+        color: white; }
+    .button.is-white.is-inverted.is-outlined {
+      background-color: transparent;
+      border-color: #0a0a0a;
+      color: #0a0a0a; }
+      .button.is-white.is-inverted.is-outlined:hover, .button.is-white.is-inverted.is-outlined.is-hovered, .button.is-white.is-inverted.is-outlined:focus, .button.is-white.is-inverted.is-outlined.is-focused {
+        background-color: #0a0a0a;
+        color: white; }
+      .button.is-white.is-inverted.is-outlined.is-loading:hover::after, .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-white.is-inverted.is-outlined.is-loading:focus::after, .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent white white !important; }
+      .button.is-white.is-inverted.is-outlined[disabled],
+      fieldset[disabled] .button.is-white.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #0a0a0a;
+        box-shadow: none;
+        color: #0a0a0a; }
+  .button.is-black {
+    background-color: #0a0a0a;
+    border-color: transparent;
+    color: white; }
+    .button.is-black:hover, .button.is-black.is-hovered {
+      background-color: #040404;
+      border-color: transparent;
+      color: white; }
+    .button.is-black:focus, .button.is-black.is-focused {
+      border-color: transparent;
+      color: white; }
+      .button.is-black:focus:not(:active), .button.is-black.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); }
+    .button.is-black:active, .button.is-black.is-active {
+      background-color: black;
+      border-color: transparent;
+      color: white; }
+    .button.is-black[disabled],
+    fieldset[disabled] .button.is-black {
+      background-color: #0a0a0a;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-black.is-inverted {
+      background-color: white;
+      color: #0a0a0a; }
+      .button.is-black.is-inverted:hover, .button.is-black.is-inverted.is-hovered {
+        background-color: #f2f2f2; }
+      .button.is-black.is-inverted[disabled],
+      fieldset[disabled] .button.is-black.is-inverted {
+        background-color: white;
+        border-color: transparent;
+        box-shadow: none;
+        color: #0a0a0a; }
+    .button.is-black.is-loading::after {
+      border-color: transparent transparent white white !important; }
+    .button.is-black.is-outlined {
+      background-color: transparent;
+      border-color: #0a0a0a;
+      color: #0a0a0a; }
+      .button.is-black.is-outlined:hover, .button.is-black.is-outlined.is-hovered, .button.is-black.is-outlined:focus, .button.is-black.is-outlined.is-focused {
+        background-color: #0a0a0a;
+        border-color: #0a0a0a;
+        color: white; }
+      .button.is-black.is-outlined.is-loading::after {
+        border-color: transparent transparent #0a0a0a #0a0a0a !important; }
+      .button.is-black.is-outlined.is-loading:hover::after, .button.is-black.is-outlined.is-loading.is-hovered::after, .button.is-black.is-outlined.is-loading:focus::after, .button.is-black.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent white white !important; }
+      .button.is-black.is-outlined[disabled],
+      fieldset[disabled] .button.is-black.is-outlined {
+        background-color: transparent;
+        border-color: #0a0a0a;
+        box-shadow: none;
+        color: #0a0a0a; }
+    .button.is-black.is-inverted.is-outlined {
+      background-color: transparent;
+      border-color: white;
+      color: white; }
+      .button.is-black.is-inverted.is-outlined:hover, .button.is-black.is-inverted.is-outlined.is-hovered, .button.is-black.is-inverted.is-outlined:focus, .button.is-black.is-inverted.is-outlined.is-focused {
+        background-color: white;
+        color: #0a0a0a; }
+      .button.is-black.is-inverted.is-outlined.is-loading:hover::after, .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-black.is-inverted.is-outlined.is-loading:focus::after, .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #0a0a0a #0a0a0a !important; }
+      .button.is-black.is-inverted.is-outlined[disabled],
+      fieldset[disabled] .button.is-black.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: white;
+        box-shadow: none;
+        color: white; }
+  .button.is-light {
+    background-color: whitesmoke;
+    border-color: transparent;
+    color: #363636; }
+    .button.is-light:hover, .button.is-light.is-hovered {
+      background-color: #eeeeee;
+      border-color: transparent;
+      color: #363636; }
+    .button.is-light:focus, .button.is-light.is-focused {
+      border-color: transparent;
+      color: #363636; }
+      .button.is-light:focus:not(:active), .button.is-light.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); }
+    .button.is-light:active, .button.is-light.is-active {
+      background-color: #e8e8e8;
+      border-color: transparent;
+      color: #363636; }
+    .button.is-light[disabled],
+    fieldset[disabled] .button.is-light {
+      background-color: whitesmoke;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-light.is-inverted {
+      background-color: #363636;
+      color: whitesmoke; }
+      .button.is-light.is-inverted:hover, .button.is-light.is-inverted.is-hovered {
+        background-color: #292929; }
+      .button.is-light.is-inverted[disabled],
+      fieldset[disabled] .button.is-light.is-inverted {
+        background-color: #363636;
+        border-color: transparent;
+        box-shadow: none;
+        color: whitesmoke; }
+    .button.is-light.is-loading::after {
+      border-color: transparent transparent #363636 #363636 !important; }
+    .button.is-light.is-outlined {
+      background-color: transparent;
+      border-color: whitesmoke;
+      color: whitesmoke; }
+      .button.is-light.is-outlined:hover, .button.is-light.is-outlined.is-hovered, .button.is-light.is-outlined:focus, .button.is-light.is-outlined.is-focused {
+        background-color: whitesmoke;
+        border-color: whitesmoke;
+        color: #363636; }
+      .button.is-light.is-outlined.is-loading::after {
+        border-color: transparent transparent whitesmoke whitesmoke !important; }
+      .button.is-light.is-outlined.is-loading:hover::after, .button.is-light.is-outlined.is-loading.is-hovered::after, .button.is-light.is-outlined.is-loading:focus::after, .button.is-light.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #363636 #363636 !important; }
+      .button.is-light.is-outlined[disabled],
+      fieldset[disabled] .button.is-light.is-outlined {
+        background-color: transparent;
+        border-color: whitesmoke;
+        box-shadow: none;
+        color: whitesmoke; }
+    .button.is-light.is-inverted.is-outlined {
+      background-color: transparent;
+      border-color: #363636;
+      color: #363636; }
+      .button.is-light.is-inverted.is-outlined:hover, .button.is-light.is-inverted.is-outlined.is-hovered, .button.is-light.is-inverted.is-outlined:focus, .button.is-light.is-inverted.is-outlined.is-focused {
+        background-color: #363636;
+        color: whitesmoke; }
+      .button.is-light.is-inverted.is-outlined.is-loading:hover::after, .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-light.is-inverted.is-outlined.is-loading:focus::after, .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent whitesmoke whitesmoke !important; }
+      .button.is-light.is-inverted.is-outlined[disabled],
+      fieldset[disabled] .button.is-light.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #363636;
+        box-shadow: none;
+        color: #363636; }
+  .button.is-dark, .content kbd.button {
+    background-color: #363636;
+    border-color: transparent;
+    color: whitesmoke; }
+    .button.is-dark:hover, .content kbd.button:hover, .button.is-dark.is-hovered, .content kbd.button.is-hovered {
+      background-color: #2f2f2f;
+      border-color: transparent;
+      color: whitesmoke; }
+    .button.is-dark:focus, .content kbd.button:focus, .button.is-dark.is-focused, .content kbd.button.is-focused {
+      border-color: transparent;
+      color: whitesmoke; }
+      .button.is-dark:focus:not(:active), .content kbd.button:focus:not(:active), .button.is-dark.is-focused:not(:active), .content kbd.button.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); }
+    .button.is-dark:active, .content kbd.button:active, .button.is-dark.is-active, .content kbd.button.is-active {
+      background-color: #292929;
+      border-color: transparent;
+      color: whitesmoke; }
+    .button.is-dark[disabled], .content kbd.button[disabled],
+    fieldset[disabled] .button.is-dark,
+    fieldset[disabled] .content kbd.button,
+    .content fieldset[disabled] kbd.button {
+      background-color: #363636;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-dark.is-inverted, .content kbd.button.is-inverted {
+      background-color: whitesmoke;
+      color: #363636; }
+      .button.is-dark.is-inverted:hover, .content kbd.button.is-inverted:hover, .button.is-dark.is-inverted.is-hovered, .content kbd.button.is-inverted.is-hovered {
+        background-color: #e8e8e8; }
+      .button.is-dark.is-inverted[disabled], .content kbd.button.is-inverted[disabled],
+      fieldset[disabled] .button.is-dark.is-inverted,
+      fieldset[disabled] .content kbd.button.is-inverted,
+      .content fieldset[disabled] kbd.button.is-inverted {
+        background-color: whitesmoke;
+        border-color: transparent;
+        box-shadow: none;
+        color: #363636; }
+    .button.is-dark.is-loading::after, .content kbd.button.is-loading::after {
+      border-color: transparent transparent whitesmoke whitesmoke !important; }
+    .button.is-dark.is-outlined, .content kbd.button.is-outlined {
+      background-color: transparent;
+      border-color: #363636;
+      color: #363636; }
+      .button.is-dark.is-outlined:hover, .content kbd.button.is-outlined:hover, .button.is-dark.is-outlined.is-hovered, .content kbd.button.is-outlined.is-hovered, .button.is-dark.is-outlined:focus, .content kbd.button.is-outlined:focus, .button.is-dark.is-outlined.is-focused, .content kbd.button.is-outlined.is-focused {
+        background-color: #363636;
+        border-color: #363636;
+        color: whitesmoke; }
+      .button.is-dark.is-outlined.is-loading::after, .content kbd.button.is-outlined.is-loading::after {
+        border-color: transparent transparent #363636 #363636 !important; }
+      .button.is-dark.is-outlined.is-loading:hover::after, .content kbd.button.is-outlined.is-loading:hover::after, .button.is-dark.is-outlined.is-loading.is-hovered::after, .content kbd.button.is-outlined.is-loading.is-hovered::after, .button.is-dark.is-outlined.is-loading:focus::after, .content kbd.button.is-outlined.is-loading:focus::after, .button.is-dark.is-outlined.is-loading.is-focused::after, .content kbd.button.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent whitesmoke whitesmoke !important; }
+      .button.is-dark.is-outlined[disabled], .content kbd.button.is-outlined[disabled],
+      fieldset[disabled] .button.is-dark.is-outlined,
+      fieldset[disabled] .content kbd.button.is-outlined,
+      .content fieldset[disabled] kbd.button.is-outlined {
+        background-color: transparent;
+        border-color: #363636;
+        box-shadow: none;
+        color: #363636; }
+    .button.is-dark.is-inverted.is-outlined, .content kbd.button.is-inverted.is-outlined {
+      background-color: transparent;
+      border-color: whitesmoke;
+      color: whitesmoke; }
+      .button.is-dark.is-inverted.is-outlined:hover, .content kbd.button.is-inverted.is-outlined:hover, .button.is-dark.is-inverted.is-outlined.is-hovered, .content kbd.button.is-inverted.is-outlined.is-hovered, .button.is-dark.is-inverted.is-outlined:focus, .content kbd.button.is-inverted.is-outlined:focus, .button.is-dark.is-inverted.is-outlined.is-focused, .content kbd.button.is-inverted.is-outlined.is-focused {
+        background-color: whitesmoke;
+        color: #363636; }
+      .button.is-dark.is-inverted.is-outlined.is-loading:hover::after, .content kbd.button.is-inverted.is-outlined.is-loading:hover::after, .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after, .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-dark.is-inverted.is-outlined.is-loading:focus::after, .content kbd.button.is-inverted.is-outlined.is-loading:focus::after, .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after, .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #363636 #363636 !important; }
+      .button.is-dark.is-inverted.is-outlined[disabled], .content kbd.button.is-inverted.is-outlined[disabled],
+      fieldset[disabled] .button.is-dark.is-inverted.is-outlined,
+      fieldset[disabled] .content kbd.button.is-inverted.is-outlined,
+      .content fieldset[disabled] kbd.button.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: whitesmoke;
+        box-shadow: none;
+        color: whitesmoke; }
+  .button.is-primary, .docstring > section > a.button.docs-sourcelink {
+    background-color: #4eb5de;
+    border-color: transparent;
+    color: #fff; }
+    .button.is-primary:hover, .docstring > section > a.button.docs-sourcelink:hover, .button.is-primary.is-hovered, .docstring > section > a.button.is-hovered.docs-sourcelink {
+      background-color: #43b1dc;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-primary:focus, .docstring > section > a.button.docs-sourcelink:focus, .button.is-primary.is-focused, .docstring > section > a.button.is-focused.docs-sourcelink {
+      border-color: transparent;
+      color: #fff; }
+      .button.is-primary:focus:not(:active), .docstring > section > a.button.docs-sourcelink:focus:not(:active), .button.is-primary.is-focused:not(:active), .docstring > section > a.button.is-focused.docs-sourcelink:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); }
+    .button.is-primary:active, .docstring > section > a.button.docs-sourcelink:active, .button.is-primary.is-active, .docstring > section > a.button.is-active.docs-sourcelink {
+      background-color: #39acda;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-primary[disabled], .docstring > section > a.button.docs-sourcelink[disabled],
+    fieldset[disabled] .button.is-primary,
+    fieldset[disabled] .docstring > section > a.button.docs-sourcelink {
+      background-color: #4eb5de;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-primary.is-inverted, .docstring > section > a.button.is-inverted.docs-sourcelink {
+      background-color: #fff;
+      color: #4eb5de; }
+      .button.is-primary.is-inverted:hover, .docstring > section > a.button.is-inverted.docs-sourcelink:hover, .button.is-primary.is-inverted.is-hovered, .docstring > section > a.button.is-inverted.is-hovered.docs-sourcelink {
+        background-color: #f2f2f2; }
+      .button.is-primary.is-inverted[disabled], .docstring > section > a.button.is-inverted.docs-sourcelink[disabled],
+      fieldset[disabled] .button.is-primary.is-inverted,
+      fieldset[disabled] .docstring > section > a.button.is-inverted.docs-sourcelink {
+        background-color: #fff;
+        border-color: transparent;
+        box-shadow: none;
+        color: #4eb5de; }
+    .button.is-primary.is-loading::after, .docstring > section > a.button.is-loading.docs-sourcelink::after {
+      border-color: transparent transparent #fff #fff !important; }
+    .button.is-primary.is-outlined, .docstring > section > a.button.is-outlined.docs-sourcelink {
+      background-color: transparent;
+      border-color: #4eb5de;
+      color: #4eb5de; }
+      .button.is-primary.is-outlined:hover, .docstring > section > a.button.is-outlined.docs-sourcelink:hover, .button.is-primary.is-outlined.is-hovered, .docstring > section > a.button.is-outlined.is-hovered.docs-sourcelink, .button.is-primary.is-outlined:focus, .docstring > section > a.button.is-outlined.docs-sourcelink:focus, .button.is-primary.is-outlined.is-focused, .docstring > section > a.button.is-outlined.is-focused.docs-sourcelink {
+        background-color: #4eb5de;
+        border-color: #4eb5de;
+        color: #fff; }
+      .button.is-primary.is-outlined.is-loading::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink::after {
+        border-color: transparent transparent #4eb5de #4eb5de !important; }
+      .button.is-primary.is-outlined.is-loading:hover::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:hover::after, .button.is-primary.is-outlined.is-loading.is-hovered::after, .docstring > section > a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after, .button.is-primary.is-outlined.is-loading:focus::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:focus::after, .button.is-primary.is-outlined.is-loading.is-focused::after, .docstring > section > a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after {
+        border-color: transparent transparent #fff #fff !important; }
+      .button.is-primary.is-outlined[disabled], .docstring > section > a.button.is-outlined.docs-sourcelink[disabled],
+      fieldset[disabled] .button.is-primary.is-outlined,
+      fieldset[disabled] .docstring > section > a.button.is-outlined.docs-sourcelink {
+        background-color: transparent;
+        border-color: #4eb5de;
+        box-shadow: none;
+        color: #4eb5de; }
+    .button.is-primary.is-inverted.is-outlined, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink {
+      background-color: transparent;
+      border-color: #fff;
+      color: #fff; }
+      .button.is-primary.is-inverted.is-outlined:hover, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:hover, .button.is-primary.is-inverted.is-outlined.is-hovered, .docstring > section > a.button.is-inverted.is-outlined.is-hovered.docs-sourcelink, .button.is-primary.is-inverted.is-outlined:focus, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:focus, .button.is-primary.is-inverted.is-outlined.is-focused, .docstring > section > a.button.is-inverted.is-outlined.is-focused.docs-sourcelink {
+        background-color: #fff;
+        color: #4eb5de; }
+      .button.is-primary.is-inverted.is-outlined.is-loading:hover::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after, .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after, .button.is-primary.is-inverted.is-outlined.is-loading:focus::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after, .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after {
+        border-color: transparent transparent #4eb5de #4eb5de !important; }
+      .button.is-primary.is-inverted.is-outlined[disabled], .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink[disabled],
+      fieldset[disabled] .button.is-primary.is-inverted.is-outlined,
+      fieldset[disabled] .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink {
+        background-color: transparent;
+        border-color: #fff;
+        box-shadow: none;
+        color: #fff; }
+  .button.is-link {
+    background-color: #2e63b8;
+    border-color: transparent;
+    color: #fff; }
+    .button.is-link:hover, .button.is-link.is-hovered {
+      background-color: #2b5eae;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-link:focus, .button.is-link.is-focused {
+      border-color: transparent;
+      color: #fff; }
+      .button.is-link:focus:not(:active), .button.is-link.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); }
+    .button.is-link:active, .button.is-link.is-active {
+      background-color: #2958a4;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-link[disabled],
+    fieldset[disabled] .button.is-link {
+      background-color: #2e63b8;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-link.is-inverted {
+      background-color: #fff;
+      color: #2e63b8; }
+      .button.is-link.is-inverted:hover, .button.is-link.is-inverted.is-hovered {
+        background-color: #f2f2f2; }
+      .button.is-link.is-inverted[disabled],
+      fieldset[disabled] .button.is-link.is-inverted {
+        background-color: #fff;
+        border-color: transparent;
+        box-shadow: none;
+        color: #2e63b8; }
+    .button.is-link.is-loading::after {
+      border-color: transparent transparent #fff #fff !important; }
+    .button.is-link.is-outlined {
+      background-color: transparent;
+      border-color: #2e63b8;
+      color: #2e63b8; }
+      .button.is-link.is-outlined:hover, .button.is-link.is-outlined.is-hovered, .button.is-link.is-outlined:focus, .button.is-link.is-outlined.is-focused {
+        background-color: #2e63b8;
+        border-color: #2e63b8;
+        color: #fff; }
+      .button.is-link.is-outlined.is-loading::after {
+        border-color: transparent transparent #2e63b8 #2e63b8 !important; }
+      .button.is-link.is-outlined.is-loading:hover::after, .button.is-link.is-outlined.is-loading.is-hovered::after, .button.is-link.is-outlined.is-loading:focus::after, .button.is-link.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #fff #fff !important; }
+      .button.is-link.is-outlined[disabled],
+      fieldset[disabled] .button.is-link.is-outlined {
+        background-color: transparent;
+        border-color: #2e63b8;
+        box-shadow: none;
+        color: #2e63b8; }
+    .button.is-link.is-inverted.is-outlined {
+      background-color: transparent;
+      border-color: #fff;
+      color: #fff; }
+      .button.is-link.is-inverted.is-outlined:hover, .button.is-link.is-inverted.is-outlined.is-hovered, .button.is-link.is-inverted.is-outlined:focus, .button.is-link.is-inverted.is-outlined.is-focused {
+        background-color: #fff;
+        color: #2e63b8; }
+      .button.is-link.is-inverted.is-outlined.is-loading:hover::after, .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-link.is-inverted.is-outlined.is-loading:focus::after, .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #2e63b8 #2e63b8 !important; }
+      .button.is-link.is-inverted.is-outlined[disabled],
+      fieldset[disabled] .button.is-link.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #fff;
+        box-shadow: none;
+        color: #fff; }
+  .button.is-info {
+    background-color: #209cee;
+    border-color: transparent;
+    color: #fff; }
+    .button.is-info:hover, .button.is-info.is-hovered {
+      background-color: #1496ed;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-info:focus, .button.is-info.is-focused {
+      border-color: transparent;
+      color: #fff; }
+      .button.is-info:focus:not(:active), .button.is-info.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); }
+    .button.is-info:active, .button.is-info.is-active {
+      background-color: #118fe4;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-info[disabled],
+    fieldset[disabled] .button.is-info {
+      background-color: #209cee;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-info.is-inverted {
+      background-color: #fff;
+      color: #209cee; }
+      .button.is-info.is-inverted:hover, .button.is-info.is-inverted.is-hovered {
+        background-color: #f2f2f2; }
+      .button.is-info.is-inverted[disabled],
+      fieldset[disabled] .button.is-info.is-inverted {
+        background-color: #fff;
+        border-color: transparent;
+        box-shadow: none;
+        color: #209cee; }
+    .button.is-info.is-loading::after {
+      border-color: transparent transparent #fff #fff !important; }
+    .button.is-info.is-outlined {
+      background-color: transparent;
+      border-color: #209cee;
+      color: #209cee; }
+      .button.is-info.is-outlined:hover, .button.is-info.is-outlined.is-hovered, .button.is-info.is-outlined:focus, .button.is-info.is-outlined.is-focused {
+        background-color: #209cee;
+        border-color: #209cee;
+        color: #fff; }
+      .button.is-info.is-outlined.is-loading::after {
+        border-color: transparent transparent #209cee #209cee !important; }
+      .button.is-info.is-outlined.is-loading:hover::after, .button.is-info.is-outlined.is-loading.is-hovered::after, .button.is-info.is-outlined.is-loading:focus::after, .button.is-info.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #fff #fff !important; }
+      .button.is-info.is-outlined[disabled],
+      fieldset[disabled] .button.is-info.is-outlined {
+        background-color: transparent;
+        border-color: #209cee;
+        box-shadow: none;
+        color: #209cee; }
+    .button.is-info.is-inverted.is-outlined {
+      background-color: transparent;
+      border-color: #fff;
+      color: #fff; }
+      .button.is-info.is-inverted.is-outlined:hover, .button.is-info.is-inverted.is-outlined.is-hovered, .button.is-info.is-inverted.is-outlined:focus, .button.is-info.is-inverted.is-outlined.is-focused {
+        background-color: #fff;
+        color: #209cee; }
+      .button.is-info.is-inverted.is-outlined.is-loading:hover::after, .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-info.is-inverted.is-outlined.is-loading:focus::after, .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #209cee #209cee !important; }
+      .button.is-info.is-inverted.is-outlined[disabled],
+      fieldset[disabled] .button.is-info.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #fff;
+        box-shadow: none;
+        color: #fff; }
+  .button.is-success {
+    background-color: #22c35b;
+    border-color: transparent;
+    color: #fff; }
+    .button.is-success:hover, .button.is-success.is-hovered {
+      background-color: #20b856;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-success:focus, .button.is-success.is-focused {
+      border-color: transparent;
+      color: #fff; }
+      .button.is-success:focus:not(:active), .button.is-success.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); }
+    .button.is-success:active, .button.is-success.is-active {
+      background-color: #1ead51;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-success[disabled],
+    fieldset[disabled] .button.is-success {
+      background-color: #22c35b;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-success.is-inverted {
+      background-color: #fff;
+      color: #22c35b; }
+      .button.is-success.is-inverted:hover, .button.is-success.is-inverted.is-hovered {
+        background-color: #f2f2f2; }
+      .button.is-success.is-inverted[disabled],
+      fieldset[disabled] .button.is-success.is-inverted {
+        background-color: #fff;
+        border-color: transparent;
+        box-shadow: none;
+        color: #22c35b; }
+    .button.is-success.is-loading::after {
+      border-color: transparent transparent #fff #fff !important; }
+    .button.is-success.is-outlined {
+      background-color: transparent;
+      border-color: #22c35b;
+      color: #22c35b; }
+      .button.is-success.is-outlined:hover, .button.is-success.is-outlined.is-hovered, .button.is-success.is-outlined:focus, .button.is-success.is-outlined.is-focused {
+        background-color: #22c35b;
+        border-color: #22c35b;
+        color: #fff; }
+      .button.is-success.is-outlined.is-loading::after {
+        border-color: transparent transparent #22c35b #22c35b !important; }
+      .button.is-success.is-outlined.is-loading:hover::after, .button.is-success.is-outlined.is-loading.is-hovered::after, .button.is-success.is-outlined.is-loading:focus::after, .button.is-success.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #fff #fff !important; }
+      .button.is-success.is-outlined[disabled],
+      fieldset[disabled] .button.is-success.is-outlined {
+        background-color: transparent;
+        border-color: #22c35b;
+        box-shadow: none;
+        color: #22c35b; }
+    .button.is-success.is-inverted.is-outlined {
+      background-color: transparent;
+      border-color: #fff;
+      color: #fff; }
+      .button.is-success.is-inverted.is-outlined:hover, .button.is-success.is-inverted.is-outlined.is-hovered, .button.is-success.is-inverted.is-outlined:focus, .button.is-success.is-inverted.is-outlined.is-focused {
+        background-color: #fff;
+        color: #22c35b; }
+      .button.is-success.is-inverted.is-outlined.is-loading:hover::after, .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-success.is-inverted.is-outlined.is-loading:focus::after, .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #22c35b #22c35b !important; }
+      .button.is-success.is-inverted.is-outlined[disabled],
+      fieldset[disabled] .button.is-success.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #fff;
+        box-shadow: none;
+        color: #fff; }
+  .button.is-warning {
+    background-color: #ffdd57;
+    border-color: transparent;
+    color: rgba(0, 0, 0, 0.7); }
+    .button.is-warning:hover, .button.is-warning.is-hovered {
+      background-color: #ffdb4a;
+      border-color: transparent;
+      color: rgba(0, 0, 0, 0.7); }
+    .button.is-warning:focus, .button.is-warning.is-focused {
+      border-color: transparent;
+      color: rgba(0, 0, 0, 0.7); }
+      .button.is-warning:focus:not(:active), .button.is-warning.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); }
+    .button.is-warning:active, .button.is-warning.is-active {
+      background-color: #ffd83d;
+      border-color: transparent;
+      color: rgba(0, 0, 0, 0.7); }
+    .button.is-warning[disabled],
+    fieldset[disabled] .button.is-warning {
+      background-color: #ffdd57;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-warning.is-inverted {
+      background-color: rgba(0, 0, 0, 0.7);
+      color: #ffdd57; }
+      .button.is-warning.is-inverted:hover, .button.is-warning.is-inverted.is-hovered {
+        background-color: rgba(0, 0, 0, 0.7); }
+      .button.is-warning.is-inverted[disabled],
+      fieldset[disabled] .button.is-warning.is-inverted {
+        background-color: rgba(0, 0, 0, 0.7);
+        border-color: transparent;
+        box-shadow: none;
+        color: #ffdd57; }
+    .button.is-warning.is-loading::after {
+      border-color: transparent transparent rgba(0, 0, 0, 0.7) rgba(0, 0, 0, 0.7) !important; }
+    .button.is-warning.is-outlined {
+      background-color: transparent;
+      border-color: #ffdd57;
+      color: #ffdd57; }
+      .button.is-warning.is-outlined:hover, .button.is-warning.is-outlined.is-hovered, .button.is-warning.is-outlined:focus, .button.is-warning.is-outlined.is-focused {
+        background-color: #ffdd57;
+        border-color: #ffdd57;
+        color: rgba(0, 0, 0, 0.7); }
+      .button.is-warning.is-outlined.is-loading::after {
+        border-color: transparent transparent #ffdd57 #ffdd57 !important; }
+      .button.is-warning.is-outlined.is-loading:hover::after, .button.is-warning.is-outlined.is-loading.is-hovered::after, .button.is-warning.is-outlined.is-loading:focus::after, .button.is-warning.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent rgba(0, 0, 0, 0.7) rgba(0, 0, 0, 0.7) !important; }
+      .button.is-warning.is-outlined[disabled],
+      fieldset[disabled] .button.is-warning.is-outlined {
+        background-color: transparent;
+        border-color: #ffdd57;
+        box-shadow: none;
+        color: #ffdd57; }
+    .button.is-warning.is-inverted.is-outlined {
+      background-color: transparent;
+      border-color: rgba(0, 0, 0, 0.7);
+      color: rgba(0, 0, 0, 0.7); }
+      .button.is-warning.is-inverted.is-outlined:hover, .button.is-warning.is-inverted.is-outlined.is-hovered, .button.is-warning.is-inverted.is-outlined:focus, .button.is-warning.is-inverted.is-outlined.is-focused {
+        background-color: rgba(0, 0, 0, 0.7);
+        color: #ffdd57; }
+      .button.is-warning.is-inverted.is-outlined.is-loading:hover::after, .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-warning.is-inverted.is-outlined.is-loading:focus::after, .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #ffdd57 #ffdd57 !important; }
+      .button.is-warning.is-inverted.is-outlined[disabled],
+      fieldset[disabled] .button.is-warning.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: rgba(0, 0, 0, 0.7);
+        box-shadow: none;
+        color: rgba(0, 0, 0, 0.7); }
+  .button.is-danger {
+    background-color: #da0b00;
+    border-color: transparent;
+    color: #fff; }
+    .button.is-danger:hover, .button.is-danger.is-hovered {
+      background-color: #cd0a00;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-danger:focus, .button.is-danger.is-focused {
+      border-color: transparent;
+      color: #fff; }
+      .button.is-danger:focus:not(:active), .button.is-danger.is-focused:not(:active) {
+        box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); }
+    .button.is-danger:active, .button.is-danger.is-active {
+      background-color: #c10a00;
+      border-color: transparent;
+      color: #fff; }
+    .button.is-danger[disabled],
+    fieldset[disabled] .button.is-danger {
+      background-color: #da0b00;
+      border-color: transparent;
+      box-shadow: none; }
+    .button.is-danger.is-inverted {
+      background-color: #fff;
+      color: #da0b00; }
+      .button.is-danger.is-inverted:hover, .button.is-danger.is-inverted.is-hovered {
+        background-color: #f2f2f2; }
+      .button.is-danger.is-inverted[disabled],
+      fieldset[disabled] .button.is-danger.is-inverted {
+        background-color: #fff;
+        border-color: transparent;
+        box-shadow: none;
+        color: #da0b00; }
+    .button.is-danger.is-loading::after {
+      border-color: transparent transparent #fff #fff !important; }
+    .button.is-danger.is-outlined {
+      background-color: transparent;
+      border-color: #da0b00;
+      color: #da0b00; }
+      .button.is-danger.is-outlined:hover, .button.is-danger.is-outlined.is-hovered, .button.is-danger.is-outlined:focus, .button.is-danger.is-outlined.is-focused {
+        background-color: #da0b00;
+        border-color: #da0b00;
+        color: #fff; }
+      .button.is-danger.is-outlined.is-loading::after {
+        border-color: transparent transparent #da0b00 #da0b00 !important; }
+      .button.is-danger.is-outlined.is-loading:hover::after, .button.is-danger.is-outlined.is-loading.is-hovered::after, .button.is-danger.is-outlined.is-loading:focus::after, .button.is-danger.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #fff #fff !important; }
+      .button.is-danger.is-outlined[disabled],
+      fieldset[disabled] .button.is-danger.is-outlined {
+        background-color: transparent;
+        border-color: #da0b00;
+        box-shadow: none;
+        color: #da0b00; }
+    .button.is-danger.is-inverted.is-outlined {
+      background-color: transparent;
+      border-color: #fff;
+      color: #fff; }
+      .button.is-danger.is-inverted.is-outlined:hover, .button.is-danger.is-inverted.is-outlined.is-hovered, .button.is-danger.is-inverted.is-outlined:focus, .button.is-danger.is-inverted.is-outlined.is-focused {
+        background-color: #fff;
+        color: #da0b00; }
+      .button.is-danger.is-inverted.is-outlined.is-loading:hover::after, .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-danger.is-inverted.is-outlined.is-loading:focus::after, .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after {
+        border-color: transparent transparent #da0b00 #da0b00 !important; }
+      .button.is-danger.is-inverted.is-outlined[disabled],
+      fieldset[disabled] .button.is-danger.is-inverted.is-outlined {
+        background-color: transparent;
+        border-color: #fff;
+        box-shadow: none;
+        color: #fff; }
+  .button.is-small, #documenter .docs-sidebar form.docs-search > input.button {
+    border-radius: 2px;
+    font-size: 0.75rem; }
+  .button.is-normal {
+    font-size: 1rem; }
+  .button.is-medium {
+    font-size: 1.25rem; }
+  .button.is-large {
+    font-size: 1.5rem; }
+  .button[disabled],
+  fieldset[disabled] .button {
+    background-color: white;
+    border-color: #dbdbdb;
+    box-shadow: none;
+    opacity: 0.5; }
+  .button.is-fullwidth {
+    display: flex;
+    width: 100%; }
+  .button.is-loading {
+    color: transparent !important;
+    pointer-events: none; }
+    .button.is-loading::after {
+      position: absolute;
+      left: calc(50% - (1em / 2));
+      top: calc(50% - (1em / 2));
+      position: absolute !important; }
+  .button.is-static {
+    background-color: whitesmoke;
+    border-color: #dbdbdb;
+    color: #7a7a7a;
+    box-shadow: none;
+    pointer-events: none; }
+  .button.is-rounded, #documenter .docs-sidebar form.docs-search > input.button {
+    border-radius: 290486px;
+    padding-left: 1em;
+    padding-right: 1em; }
+
+.buttons {
+  align-items: center;
+  display: flex;
+  flex-wrap: wrap;
+  justify-content: flex-start; }
+  .buttons .button {
+    margin-bottom: 0.5rem; }
+    .buttons .button:not(:last-child):not(.is-fullwidth) {
+      margin-right: 0.5rem; }
+  .buttons:last-child {
+    margin-bottom: -0.5rem; }
+  .buttons:not(:last-child) {
+    margin-bottom: 1rem; }
+  .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large) {
+    border-radius: 2px;
+    font-size: 0.75rem; }
+  .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large) {
+    font-size: 1.25rem; }
+  .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium) {
+    font-size: 1.5rem; }
+  .buttons.has-addons .button:not(:first-child) {
+    border-bottom-left-radius: 0;
+    border-top-left-radius: 0; }
+  .buttons.has-addons .button:not(:last-child) {
+    border-bottom-right-radius: 0;
+    border-top-right-radius: 0;
+    margin-right: -1px; }
+  .buttons.has-addons .button:last-child {
+    margin-right: 0; }
+  .buttons.has-addons .button:hover, .buttons.has-addons .button.is-hovered {
+    z-index: 2; }
+  .buttons.has-addons .button:focus, .buttons.has-addons .button.is-focused, .buttons.has-addons .button:active, .buttons.has-addons .button.is-active, .buttons.has-addons .button.is-selected {
+    z-index: 3; }
+    .buttons.has-addons .button:focus:hover, .buttons.has-addons .button.is-focused:hover, .buttons.has-addons .button:active:hover, .buttons.has-addons .button.is-active:hover, .buttons.has-addons .button.is-selected:hover {
+      z-index: 4; }
+  .buttons.has-addons .button.is-expanded {
+    flex-grow: 1;
+    flex-shrink: 1; }
+  .buttons.is-centered {
+    justify-content: center; }
+    .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth) {
+      margin-left: 0.25rem;
+      margin-right: 0.25rem; }
+  .buttons.is-right {
+    justify-content: flex-end; }
+    .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth) {
+      margin-left: 0.25rem;
+      margin-right: 0.25rem; }
+
+.container {
+  flex-grow: 1;
+  margin: 0 auto;
+  position: relative;
+  width: auto; }
+  @media screen and (min-width: 1056px) {
+    .container {
+      max-width: 992px; }
+      .container.is-fluid {
+        margin-left: 32px;
+        margin-right: 32px;
+        max-width: none; } }
+  @media screen and (max-width: 1215px) {
+    .container.is-widescreen {
+      max-width: 1152px; } }
+  @media screen and (max-width: 1407px) {
+    .container.is-fullhd {
+      max-width: 1344px; } }
+  @media screen and (min-width: 1216px) {
+    .container {
+      max-width: 1152px; } }
+  @media screen and (min-width: 1408px) {
+    .container {
+      max-width: 1344px; } }
+
+.content li + li {
+  margin-top: 0.25em; }
+
+.content p:not(:last-child),
+.content dl:not(:last-child),
+.content ol:not(:last-child),
+.content ul:not(:last-child),
+.content blockquote:not(:last-child),
+.content pre:not(:last-child),
+.content table:not(:last-child) {
+  margin-bottom: 1em; }
+
+.content h1,
+.content h2,
+.content h3,
+.content h4,
+.content h5,
+.content h6 {
+  color: #222222;
+  font-weight: 600;
+  line-height: 1.125; }
+
+.content h1 {
+  font-size: 2em;
+  margin-bottom: 0.5em; }
+  .content h1:not(:first-child) {
+    margin-top: 1em; }
+
+.content h2 {
+  font-size: 1.75em;
+  margin-bottom: 0.5714em; }
+  .content h2:not(:first-child) {
+    margin-top: 1.1428em; }
+
+.content h3 {
+  font-size: 1.5em;
+  margin-bottom: 0.6666em; }
+  .content h3:not(:first-child) {
+    margin-top: 1.3333em; }
+
+.content h4 {
+  font-size: 1.25em;
+  margin-bottom: 0.8em; }
+
+.content h5 {
+  font-size: 1.125em;
+  margin-bottom: 0.8888em; }
+
+.content h6 {
+  font-size: 1em;
+  margin-bottom: 1em; }
+
+.content blockquote {
+  background-color: whitesmoke;
+  border-left: 5px solid #dbdbdb;
+  padding: 1.25em 1.5em; }
+
+.content ol {
+  list-style-position: outside;
+  margin-left: 2em;
+  margin-top: 1em; }
+  .content ol:not([type]) {
+    list-style-type: decimal; }
+    .content ol:not([type]).is-lower-alpha {
+      list-style-type: lower-alpha; }
+    .content ol:not([type]).is-lower-roman {
+      list-style-type: lower-roman; }
+    .content ol:not([type]).is-upper-alpha {
+      list-style-type: upper-alpha; }
+    .content ol:not([type]).is-upper-roman {
+      list-style-type: upper-roman; }
+
+.content ul {
+  list-style: disc outside;
+  margin-left: 2em;
+  margin-top: 1em; }
+  .content ul ul {
+    list-style-type: circle;
+    margin-top: 0.5em; }
+    .content ul ul ul {
+      list-style-type: square; }
+
+.content dd {
+  margin-left: 2em; }
+
+.content figure {
+  margin-left: 2em;
+  margin-right: 2em;
+  text-align: center; }
+  .content figure:not(:first-child) {
+    margin-top: 2em; }
+  .content figure:not(:last-child) {
+    margin-bottom: 2em; }
+  .content figure img {
+    display: inline-block; }
+  .content figure figcaption {
+    font-style: italic; }
+
+.content pre {
+  -webkit-overflow-scrolling: touch;
+  overflow-x: auto;
+  padding: 0.7rem 0.5rem;
+  white-space: pre;
+  word-wrap: normal; }
+
+.content sup,
+.content sub {
+  font-size: 75%; }
+
+.content table {
+  width: 100%; }
+  .content table td,
+  .content table th {
+    border: 1px solid #dbdbdb;
+    border-width: 0 0 1px;
+    padding: 0.5em 0.75em;
+    vertical-align: top; }
+  .content table th {
+    color: #222222; }
+    .content table th:not([align]) {
+      text-align: left; }
+  .content table thead td,
+  .content table thead th {
+    border-width: 0 0 2px;
+    color: #222222; }
+  .content table tfoot td,
+  .content table tfoot th {
+    border-width: 2px 0 0;
+    color: #222222; }
+  .content table tbody tr:last-child td,
+  .content table tbody tr:last-child th {
+    border-bottom-width: 0; }
+
+.content .tabs li + li {
+  margin-top: 0; }
+
+.content.is-small, #documenter .docs-sidebar form.docs-search > input.content {
+  font-size: 0.75rem; }
+
+.content.is-medium {
+  font-size: 1.25rem; }
+
+.content.is-large {
+  font-size: 1.5rem; }
+
+.icon {
+  align-items: center;
+  display: inline-flex;
+  justify-content: center;
+  height: 1.5rem;
+  width: 1.5rem; }
+  .icon.is-small, #documenter .docs-sidebar form.docs-search > input.icon {
+    height: 1rem;
+    width: 1rem; }
+  .icon.is-medium {
+    height: 2rem;
+    width: 2rem; }
+  .icon.is-large {
+    height: 3rem;
+    width: 3rem; }
+
+.image, #documenter .docs-sidebar .docs-logo > img {
+  display: block;
+  position: relative; }
+  .image img, #documenter .docs-sidebar .docs-logo > img img {
+    display: block;
+    height: auto;
+    width: 100%; }
+    .image img.is-rounded, #documenter .docs-sidebar .docs-logo > img img.is-rounded {
+      border-radius: 290486px; }
+  .image.is-square img, #documenter .docs-sidebar .docs-logo > img.is-square img,
+  .image.is-square .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, .image.is-1by1 img, #documenter .docs-sidebar .docs-logo > img.is-1by1 img,
+  .image.is-1by1 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, .image.is-5by4 img, #documenter .docs-sidebar .docs-logo > img.is-5by4 img,
+  .image.is-5by4 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, .image.is-4by3 img, #documenter .docs-sidebar .docs-logo > img.is-4by3 img,
+  .image.is-4by3 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, .image.is-3by2 img, #documenter .docs-sidebar .docs-logo > img.is-3by2 img,
+  .image.is-3by2 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, .image.is-5by3 img, #documenter .docs-sidebar .docs-logo > img.is-5by3 img,
+  .image.is-5by3 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, .image.is-16by9 img, #documenter .docs-sidebar .docs-logo > img.is-16by9 img,
+  .image.is-16by9 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, .image.is-2by1 img, #documenter .docs-sidebar .docs-logo > img.is-2by1 img,
+  .image.is-2by1 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, .image.is-3by1 img, #documenter .docs-sidebar .docs-logo > img.is-3by1 img,
+  .image.is-3by1 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, .image.is-4by5 img, #documenter .docs-sidebar .docs-logo > img.is-4by5 img,
+  .image.is-4by5 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, .image.is-3by4 img, #documenter .docs-sidebar .docs-logo > img.is-3by4 img,
+  .image.is-3by4 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, .image.is-2by3 img, #documenter .docs-sidebar .docs-logo > img.is-2by3 img,
+  .image.is-2by3 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, .image.is-3by5 img, #documenter .docs-sidebar .docs-logo > img.is-3by5 img,
+  .image.is-3by5 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, .image.is-9by16 img, #documenter .docs-sidebar .docs-logo > img.is-9by16 img,
+  .image.is-9by16 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, .image.is-1by2 img, #documenter .docs-sidebar .docs-logo > img.is-1by2 img,
+  .image.is-1by2 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, .image.is-1by3 img, #documenter .docs-sidebar .docs-logo > img.is-1by3 img,
+  .image.is-1by3 .has-ratio,
+  #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio {
+    height: 100%;
+    width: 100%; }
+  .image.is-square, #documenter .docs-sidebar .docs-logo > img.is-square, .image.is-1by1, #documenter .docs-sidebar .docs-logo > img.is-1by1 {
+    padding-top: 100%; }
+  .image.is-5by4, #documenter .docs-sidebar .docs-logo > img.is-5by4 {
+    padding-top: 80%; }
+  .image.is-4by3, #documenter .docs-sidebar .docs-logo > img.is-4by3 {
+    padding-top: 75%; }
+  .image.is-3by2, #documenter .docs-sidebar .docs-logo > img.is-3by2 {
+    padding-top: 66.6666%; }
+  .image.is-5by3, #documenter .docs-sidebar .docs-logo > img.is-5by3 {
+    padding-top: 60%; }
+  .image.is-16by9, #documenter .docs-sidebar .docs-logo > img.is-16by9 {
+    padding-top: 56.25%; }
+  .image.is-2by1, #documenter .docs-sidebar .docs-logo > img.is-2by1 {
+    padding-top: 50%; }
+  .image.is-3by1, #documenter .docs-sidebar .docs-logo > img.is-3by1 {
+    padding-top: 33.3333%; }
+  .image.is-4by5, #documenter .docs-sidebar .docs-logo > img.is-4by5 {
+    padding-top: 125%; }
+  .image.is-3by4, #documenter .docs-sidebar .docs-logo > img.is-3by4 {
+    padding-top: 133.3333%; }
+  .image.is-2by3, #documenter .docs-sidebar .docs-logo > img.is-2by3 {
+    padding-top: 150%; }
+  .image.is-3by5, #documenter .docs-sidebar .docs-logo > img.is-3by5 {
+    padding-top: 166.6666%; }
+  .image.is-9by16, #documenter .docs-sidebar .docs-logo > img.is-9by16 {
+    padding-top: 177.7777%; }
+  .image.is-1by2, #documenter .docs-sidebar .docs-logo > img.is-1by2 {
+    padding-top: 200%; }
+  .image.is-1by3, #documenter .docs-sidebar .docs-logo > img.is-1by3 {
+    padding-top: 300%; }
+  .image.is-16x16, #documenter .docs-sidebar .docs-logo > img.is-16x16 {
+    height: 16px;
+    width: 16px; }
+  .image.is-24x24, #documenter .docs-sidebar .docs-logo > img.is-24x24 {
+    height: 24px;
+    width: 24px; }
+  .image.is-32x32, #documenter .docs-sidebar .docs-logo > img.is-32x32 {
+    height: 32px;
+    width: 32px; }
+  .image.is-48x48, #documenter .docs-sidebar .docs-logo > img.is-48x48 {
+    height: 48px;
+    width: 48px; }
+  .image.is-64x64, #documenter .docs-sidebar .docs-logo > img.is-64x64 {
+    height: 64px;
+    width: 64px; }
+  .image.is-96x96, #documenter .docs-sidebar .docs-logo > img.is-96x96 {
+    height: 96px;
+    width: 96px; }
+  .image.is-128x128, #documenter .docs-sidebar .docs-logo > img.is-128x128 {
+    height: 128px;
+    width: 128px; }
+
+.notification {
+  background-color: whitesmoke;
+  border-radius: 4px;
+  padding: 1.25rem 2.5rem 1.25rem 1.5rem;
+  position: relative; }
+  .notification a:not(.button):not(.dropdown-item) {
+    color: currentColor;
+    text-decoration: underline; }
+  .notification strong {
+    color: currentColor; }
+  .notification code,
+  .notification pre {
+    background: white; }
+  .notification pre code {
+    background: transparent; }
+  .notification > .delete {
+    position: absolute;
+    right: 0.5rem;
+    top: 0.5rem; }
+  .notification .title,
+  .notification .subtitle,
+  .notification .content {
+    color: currentColor; }
+  .notification.is-white {
+    background-color: white;
+    color: #0a0a0a; }
+  .notification.is-black {
+    background-color: #0a0a0a;
+    color: white; }
+  .notification.is-light {
+    background-color: whitesmoke;
+    color: #363636; }
+  .notification.is-dark, .content kbd.notification {
+    background-color: #363636;
+    color: whitesmoke; }
+  .notification.is-primary, .docstring > section > a.notification.docs-sourcelink {
+    background-color: #4eb5de;
+    color: #fff; }
+  .notification.is-link {
+    background-color: #2e63b8;
+    color: #fff; }
+  .notification.is-info {
+    background-color: #209cee;
+    color: #fff; }
+  .notification.is-success {
+    background-color: #22c35b;
+    color: #fff; }
+  .notification.is-warning {
+    background-color: #ffdd57;
+    color: rgba(0, 0, 0, 0.7); }
+  .notification.is-danger {
+    background-color: #da0b00;
+    color: #fff; }
+
+.progress {
+  -moz-appearance: none;
+  -webkit-appearance: none;
+  border: none;
+  border-radius: 290486px;
+  display: block;
+  height: 1rem;
+  overflow: hidden;
+  padding: 0;
+  width: 100%; }
+  .progress::-webkit-progress-bar {
+    background-color: #dbdbdb; }
+  .progress::-webkit-progress-value {
+    background-color: #222222; }
+  .progress::-moz-progress-bar {
+    background-color: #222222; }
+  .progress::-ms-fill {
+    background-color: #222222;
+    border: none; }
+  .progress.is-white::-webkit-progress-value {
+    background-color: white; }
+  .progress.is-white::-moz-progress-bar {
+    background-color: white; }
+  .progress.is-white::-ms-fill {
+    background-color: white; }
+  .progress.is-white:indeterminate {
+    background-image: linear-gradient(to right, white 30%, #dbdbdb 30%); }
+  .progress.is-black::-webkit-progress-value {
+    background-color: #0a0a0a; }
+  .progress.is-black::-moz-progress-bar {
+    background-color: #0a0a0a; }
+  .progress.is-black::-ms-fill {
+    background-color: #0a0a0a; }
+  .progress.is-black:indeterminate {
+    background-image: linear-gradient(to right, #0a0a0a 30%, #dbdbdb 30%); }
+  .progress.is-light::-webkit-progress-value {
+    background-color: whitesmoke; }
+  .progress.is-light::-moz-progress-bar {
+    background-color: whitesmoke; }
+  .progress.is-light::-ms-fill {
+    background-color: whitesmoke; }
+  .progress.is-light:indeterminate {
+    background-image: linear-gradient(to right, whitesmoke 30%, #dbdbdb 30%); }
+  .progress.is-dark::-webkit-progress-value, .content kbd.progress::-webkit-progress-value {
+    background-color: #363636; }
+  .progress.is-dark::-moz-progress-bar, .content kbd.progress::-moz-progress-bar {
+    background-color: #363636; }
+  .progress.is-dark::-ms-fill, .content kbd.progress::-ms-fill {
+    background-color: #363636; }
+  .progress.is-dark:indeterminate, .content kbd.progress:indeterminate {
+    background-image: linear-gradient(to right, #363636 30%, #dbdbdb 30%); }
+  .progress.is-primary::-webkit-progress-value, .docstring > section > a.progress.docs-sourcelink::-webkit-progress-value {
+    background-color: #4eb5de; }
+  .progress.is-primary::-moz-progress-bar, .docstring > section > a.progress.docs-sourcelink::-moz-progress-bar {
+    background-color: #4eb5de; }
+  .progress.is-primary::-ms-fill, .docstring > section > a.progress.docs-sourcelink::-ms-fill {
+    background-color: #4eb5de; }
+  .progress.is-primary:indeterminate, .docstring > section > a.progress.docs-sourcelink:indeterminate {
+    background-image: linear-gradient(to right, #4eb5de 30%, #dbdbdb 30%); }
+  .progress.is-link::-webkit-progress-value {
+    background-color: #2e63b8; }
+  .progress.is-link::-moz-progress-bar {
+    background-color: #2e63b8; }
+  .progress.is-link::-ms-fill {
+    background-color: #2e63b8; }
+  .progress.is-link:indeterminate {
+    background-image: linear-gradient(to right, #2e63b8 30%, #dbdbdb 30%); }
+  .progress.is-info::-webkit-progress-value {
+    background-color: #209cee; }
+  .progress.is-info::-moz-progress-bar {
+    background-color: #209cee; }
+  .progress.is-info::-ms-fill {
+    background-color: #209cee; }
+  .progress.is-info:indeterminate {
+    background-image: linear-gradient(to right, #209cee 30%, #dbdbdb 30%); }
+  .progress.is-success::-webkit-progress-value {
+    background-color: #22c35b; }
+  .progress.is-success::-moz-progress-bar {
+    background-color: #22c35b; }
+  .progress.is-success::-ms-fill {
+    background-color: #22c35b; }
+  .progress.is-success:indeterminate {
+    background-image: linear-gradient(to right, #22c35b 30%, #dbdbdb 30%); }
+  .progress.is-warning::-webkit-progress-value {
+    background-color: #ffdd57; }
+  .progress.is-warning::-moz-progress-bar {
+    background-color: #ffdd57; }
+  .progress.is-warning::-ms-fill {
+    background-color: #ffdd57; }
+  .progress.is-warning:indeterminate {
+    background-image: linear-gradient(to right, #ffdd57 30%, #dbdbdb 30%); }
+  .progress.is-danger::-webkit-progress-value {
+    background-color: #da0b00; }
+  .progress.is-danger::-moz-progress-bar {
+    background-color: #da0b00; }
+  .progress.is-danger::-ms-fill {
+    background-color: #da0b00; }
+  .progress.is-danger:indeterminate {
+    background-image: linear-gradient(to right, #da0b00 30%, #dbdbdb 30%); }
+  .progress:indeterminate {
+    animation-duration: 1.5s;
+    animation-iteration-count: infinite;
+    animation-name: moveIndeterminate;
+    animation-timing-function: linear;
+    background-color: #dbdbdb;
+    background-image: linear-gradient(to right, #222222 30%, #dbdbdb 30%);
+    background-position: top left;
+    background-repeat: no-repeat;
+    background-size: 150% 150%; }
+    .progress:indeterminate::-webkit-progress-bar {
+      background-color: transparent; }
+    .progress:indeterminate::-moz-progress-bar {
+      background-color: transparent; }
+  .progress.is-small, #documenter .docs-sidebar form.docs-search > input.progress {
+    height: 0.75rem; }
+  .progress.is-medium {
+    height: 1.25rem; }
+  .progress.is-large {
+    height: 1.5rem; }
+
+@keyframes moveIndeterminate {
+  from {
+    background-position: 200% 0; }
+  to {
+    background-position: -200% 0; } }
+
+.table {
+  background-color: white;
+  color: #363636; }
+  .table td,
+  .table th {
+    border: 1px solid #dbdbdb;
+    border-width: 0 0 1px;
+    padding: 0.5em 0.75em;
+    vertical-align: top; }
+    .table td.is-white,
+    .table th.is-white {
+      background-color: white;
+      border-color: white;
+      color: #0a0a0a; }
+    .table td.is-black,
+    .table th.is-black {
+      background-color: #0a0a0a;
+      border-color: #0a0a0a;
+      color: white; }
+    .table td.is-light,
+    .table th.is-light {
+      background-color: whitesmoke;
+      border-color: whitesmoke;
+      color: #363636; }
+    .table td.is-dark,
+    .table th.is-dark {
+      background-color: #363636;
+      border-color: #363636;
+      color: whitesmoke; }
+    .table td.is-primary,
+    .table th.is-primary {
+      background-color: #4eb5de;
+      border-color: #4eb5de;
+      color: #fff; }
+    .table td.is-link,
+    .table th.is-link {
+      background-color: #2e63b8;
+      border-color: #2e63b8;
+      color: #fff; }
+    .table td.is-info,
+    .table th.is-info {
+      background-color: #209cee;
+      border-color: #209cee;
+      color: #fff; }
+    .table td.is-success,
+    .table th.is-success {
+      background-color: #22c35b;
+      border-color: #22c35b;
+      color: #fff; }
+    .table td.is-warning,
+    .table th.is-warning {
+      background-color: #ffdd57;
+      border-color: #ffdd57;
+      color: rgba(0, 0, 0, 0.7); }
+    .table td.is-danger,
+    .table th.is-danger {
+      background-color: #da0b00;
+      border-color: #da0b00;
+      color: #fff; }
+    .table td.is-narrow,
+    .table th.is-narrow {
+      white-space: nowrap;
+      width: 1%; }
+    .table td.is-selected,
+    .table th.is-selected {
+      background-color: #4eb5de;
+      color: #fff; }
+      .table td.is-selected a,
+      .table td.is-selected strong,
+      .table th.is-selected a,
+      .table th.is-selected strong {
+        color: currentColor; }
+  .table th {
+    color: #222222; }
+    .table th:not([align]) {
+      text-align: left; }
+  .table tr.is-selected {
+    background-color: #4eb5de;
+    color: #fff; }
+    .table tr.is-selected a,
+    .table tr.is-selected strong {
+      color: currentColor; }
+    .table tr.is-selected td,
+    .table tr.is-selected th {
+      border-color: #fff;
+      color: currentColor; }
+  .table thead {
+    background-color: transparent; }
+    .table thead td,
+    .table thead th {
+      border-width: 0 0 2px;
+      color: #222222; }
+  .table tfoot {
+    background-color: transparent; }
+    .table tfoot td,
+    .table tfoot th {
+      border-width: 2px 0 0;
+      color: #222222; }
+  .table tbody {
+    background-color: transparent; }
+    .table tbody tr:last-child td,
+    .table tbody tr:last-child th {
+      border-bottom-width: 0; }
+  .table.is-bordered td,
+  .table.is-bordered th {
+    border-width: 1px; }
+  .table.is-bordered tr:last-child td,
+  .table.is-bordered tr:last-child th {
+    border-bottom-width: 1px; }
+  .table.is-fullwidth {
+    width: 100%; }
+  .table.is-hoverable tbody tr:not(.is-selected):hover {
+    background-color: #fafafa; }
+  .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover {
+    background-color: #fafafa; }
+    .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover:nth-child(even) {
+      background-color: whitesmoke; }
+  .table.is-narrow td,
+  .table.is-narrow th {
+    padding: 0.25em 0.5em; }
+  .table.is-striped tbody tr:not(.is-selected):nth-child(even) {
+    background-color: #fafafa; }
+
+.table-container {
+  -webkit-overflow-scrolling: touch;
+  overflow: auto;
+  overflow-y: hidden;
+  max-width: 100%; }
+
+.tags {
+  align-items: center;
+  display: flex;
+  flex-wrap: wrap;
+  justify-content: flex-start; }
+  .tags .tag, .tags .docstring > section > a.docs-sourcelink, .tags .content kbd, .content .tags kbd {
+    margin-bottom: 0.5rem; }
+    .tags .tag:not(:last-child), .tags .docstring > section > a.docs-sourcelink:not(:last-child), .tags .content kbd:not(:last-child), .content .tags kbd:not(:last-child) {
+      margin-right: 0.5rem; }
+  .tags:last-child {
+    margin-bottom: -0.5rem; }
+  .tags:not(:last-child) {
+    margin-bottom: 1rem; }
+  .tags.are-medium .tag:not(.is-normal):not(.is-large), .tags.are-medium .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-large), .tags.are-medium .content kbd:not(.is-normal):not(.is-large), .content .tags.are-medium kbd:not(.is-normal):not(.is-large) {
+    font-size: 1rem; }
+  .tags.are-large .tag:not(.is-normal):not(.is-medium), .tags.are-large .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-medium), .tags.are-large .content kbd:not(.is-normal):not(.is-medium), .content .tags.are-large kbd:not(.is-normal):not(.is-medium) {
+    font-size: 1.25rem; }
+  .tags.is-centered {
+    justify-content: center; }
+    .tags.is-centered .tag, .tags.is-centered .docstring > section > a.docs-sourcelink, .tags.is-centered .content kbd, .content .tags.is-centered kbd {
+      margin-right: 0.25rem;
+      margin-left: 0.25rem; }
+  .tags.is-right {
+    justify-content: flex-end; }
+    .tags.is-right .tag:not(:first-child), .tags.is-right .docstring > section > a.docs-sourcelink:not(:first-child), .tags.is-right .content kbd:not(:first-child), .content .tags.is-right kbd:not(:first-child) {
+      margin-left: 0.5rem; }
+    .tags.is-right .tag:not(:last-child), .tags.is-right .docstring > section > a.docs-sourcelink:not(:last-child), .tags.is-right .content kbd:not(:last-child), .content .tags.is-right kbd:not(:last-child) {
+      margin-right: 0; }
+  .tags.has-addons .tag, .tags.has-addons .docstring > section > a.docs-sourcelink, .tags.has-addons .content kbd, .content .tags.has-addons kbd {
+    margin-right: 0; }
+    .tags.has-addons .tag:not(:first-child), .tags.has-addons .docstring > section > a.docs-sourcelink:not(:first-child), .tags.has-addons .content kbd:not(:first-child), .content .tags.has-addons kbd:not(:first-child) {
+      margin-left: 0;
+      border-bottom-left-radius: 0;
+      border-top-left-radius: 0; }
+    .tags.has-addons .tag:not(:last-child), .tags.has-addons .docstring > section > a.docs-sourcelink:not(:last-child), .tags.has-addons .content kbd:not(:last-child), .content .tags.has-addons kbd:not(:last-child) {
+      border-bottom-right-radius: 0;
+      border-top-right-radius: 0; }
+
+.tag:not(body), .docstring > section > a.docs-sourcelink:not(body), .content kbd:not(body) {
+  align-items: center;
+  background-color: whitesmoke;
+  border-radius: 4px;
+  color: #222222;
+  display: inline-flex;
+  font-size: 0.75rem;
+  height: 2em;
+  justify-content: center;
+  line-height: 1.5;
+  padding-left: 0.75em;
+  padding-right: 0.75em;
+  white-space: nowrap; }
+  .tag:not(body) .delete, .docstring > section > a.docs-sourcelink:not(body) .delete, .content kbd:not(body) .delete {
+    margin-left: 0.25rem;
+    margin-right: -0.375rem; }
+  .tag:not(body).is-white, .docstring > section > a.docs-sourcelink:not(body).is-white, .content kbd:not(body).is-white {
+    background-color: white;
+    color: #0a0a0a; }
+  .tag:not(body).is-black, .docstring > section > a.docs-sourcelink:not(body).is-black, .content kbd:not(body).is-black {
+    background-color: #0a0a0a;
+    color: white; }
+  .tag:not(body).is-light, .docstring > section > a.docs-sourcelink:not(body).is-light, .content kbd:not(body).is-light {
+    background-color: whitesmoke;
+    color: #363636; }
+  .tag:not(body).is-dark, .docstring > section > a.docs-sourcelink:not(body).is-dark, .content kbd:not(body) {
+    background-color: #363636;
+    color: whitesmoke; }
+  .tag:not(body).is-primary, .docstring > section > a.docs-sourcelink:not(body), .content kbd:not(body).is-primary {
+    background-color: #4eb5de;
+    color: #fff; }
+  .tag:not(body).is-link, .docstring > section > a.docs-sourcelink:not(body).is-link, .content kbd:not(body).is-link {
+    background-color: #2e63b8;
+    color: #fff; }
+  .tag:not(body).is-info, .docstring > section > a.docs-sourcelink:not(body).is-info, .content kbd:not(body).is-info {
+    background-color: #209cee;
+    color: #fff; }
+  .tag:not(body).is-success, .docstring > section > a.docs-sourcelink:not(body).is-success, .content kbd:not(body).is-success {
+    background-color: #22c35b;
+    color: #fff; }
+  .tag:not(body).is-warning, .docstring > section > a.docs-sourcelink:not(body).is-warning, .content kbd:not(body).is-warning {
+    background-color: #ffdd57;
+    color: rgba(0, 0, 0, 0.7); }
+  .tag:not(body).is-danger, .docstring > section > a.docs-sourcelink:not(body).is-danger, .content kbd:not(body).is-danger {
+    background-color: #da0b00;
+    color: #fff; }
+  .tag:not(body).is-normal, .docstring > section > a.docs-sourcelink:not(body).is-normal, .content kbd:not(body).is-normal {
+    font-size: 0.75rem; }
+  .tag:not(body).is-medium, .docstring > section > a.docs-sourcelink:not(body).is-medium, .content kbd:not(body).is-medium {
+    font-size: 1rem; }
+  .tag:not(body).is-large, .docstring > section > a.docs-sourcelink:not(body).is-large, .content kbd:not(body).is-large {
+    font-size: 1.25rem; }
+  .tag:not(body) .icon:first-child:not(:last-child), .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:not(:last-child), .content kbd:not(body) .icon:first-child:not(:last-child) {
+    margin-left: -0.375em;
+    margin-right: 0.1875em; }
+  .tag:not(body) .icon:last-child:not(:first-child), .docstring > section > a.docs-sourcelink:not(body) .icon:last-child:not(:first-child), .content kbd:not(body) .icon:last-child:not(:first-child) {
+    margin-left: 0.1875em;
+    margin-right: -0.375em; }
+  .tag:not(body) .icon:first-child:last-child, .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:last-child, .content kbd:not(body) .icon:first-child:last-child {
+    margin-left: -0.375em;
+    margin-right: -0.375em; }
+  .tag:not(body).is-delete, .docstring > section > a.docs-sourcelink:not(body).is-delete, .content kbd:not(body).is-delete {
+    margin-left: 1px;
+    padding: 0;
+    position: relative;
+    width: 2em; }
+    .tag:not(body).is-delete::before, .docstring > section > a.docs-sourcelink:not(body).is-delete::before, .content kbd:not(body).is-delete::before, .tag:not(body).is-delete::after, .docstring > section > a.docs-sourcelink:not(body).is-delete::after, .content kbd:not(body).is-delete::after {
+      background-color: currentColor;
+      content: "";
+      display: block;
+      left: 50%;
+      position: absolute;
+      top: 50%;
+      transform: translateX(-50%) translateY(-50%) rotate(45deg);
+      transform-origin: center center; }
+    .tag:not(body).is-delete::before, .docstring > section > a.docs-sourcelink:not(body).is-delete::before, .content kbd:not(body).is-delete::before {
+      height: 1px;
+      width: 50%; }
+    .tag:not(body).is-delete::after, .docstring > section > a.docs-sourcelink:not(body).is-delete::after, .content kbd:not(body).is-delete::after {
+      height: 50%;
+      width: 1px; }
+    .tag:not(body).is-delete:hover, .docstring > section > a.docs-sourcelink:not(body).is-delete:hover, .content kbd:not(body).is-delete:hover, .tag:not(body).is-delete:focus, .docstring > section > a.docs-sourcelink:not(body).is-delete:focus, .content kbd:not(body).is-delete:focus {
+      background-color: #e8e8e8; }
+    .tag:not(body).is-delete:active, .docstring > section > a.docs-sourcelink:not(body).is-delete:active, .content kbd:not(body).is-delete:active {
+      background-color: #dbdbdb; }
+  .tag:not(body).is-rounded, .docstring > section > a.docs-sourcelink:not(body).is-rounded, .content kbd:not(body).is-rounded, #documenter .docs-sidebar form.docs-search > input.tag:not(body) {
+    border-radius: 290486px; }
+
+a.tag:hover, .docstring > section > a.docs-sourcelink:hover {
+  text-decoration: underline; }
+
+.title,
+.subtitle {
+  word-break: break-word; }
+  .title em,
+  .title span,
+  .subtitle em,
+  .subtitle span {
+    font-weight: inherit; }
+  .title sub,
+  .subtitle sub {
+    font-size: 0.75em; }
+  .title sup,
+  .subtitle sup {
+    font-size: 0.75em; }
+  .title .tag, .title .docstring > section > a.docs-sourcelink, .title .content kbd, .content .title kbd,
+  .subtitle .tag,
+  .subtitle .docstring > section > a.docs-sourcelink,
+  .subtitle .content kbd,
+  .content .subtitle kbd {
+    vertical-align: middle; }
+
+.title {
+  color: #363636;
+  font-size: 2rem;
+  font-weight: 600;
+  line-height: 1.125; }
+  .title strong {
+    color: inherit;
+    font-weight: inherit; }
+  .title + .highlight {
+    margin-top: -0.75rem; }
+  .title:not(.is-spaced) + .subtitle {
+    margin-top: -1.25rem; }
+  .title.is-1 {
+    font-size: 3rem; }
+  .title.is-2 {
+    font-size: 2.5rem; }
+  .title.is-3 {
+    font-size: 2rem; }
+  .title.is-4 {
+    font-size: 1.5rem; }
+  .title.is-5 {
+    font-size: 1.25rem; }
+  .title.is-6 {
+    font-size: 1rem; }
+  .title.is-7 {
+    font-size: 0.75rem; }
+
+.subtitle {
+  color: #4a4a4a;
+  font-size: 1.25rem;
+  font-weight: 400;
+  line-height: 1.25; }
+  .subtitle strong {
+    color: #363636;
+    font-weight: 600; }
+  .subtitle:not(.is-spaced) + .title {
+    margin-top: -1.25rem; }
+  .subtitle.is-1 {
+    font-size: 3rem; }
+  .subtitle.is-2 {
+    font-size: 2.5rem; }
+  .subtitle.is-3 {
+    font-size: 2rem; }
+  .subtitle.is-4 {
+    font-size: 1.5rem; }
+  .subtitle.is-5 {
+    font-size: 1.25rem; }
+  .subtitle.is-6 {
+    font-size: 1rem; }
+  .subtitle.is-7 {
+    font-size: 0.75rem; }
+
+.heading {
+  display: block;
+  font-size: 11px;
+  letter-spacing: 1px;
+  margin-bottom: 5px;
+  text-transform: uppercase; }
+
+.highlight {
+  font-weight: 400;
+  max-width: 100%;
+  overflow: hidden;
+  padding: 0; }
+  .highlight pre {
+    overflow: auto;
+    max-width: 100%; }
+
+.number {
+  align-items: center;
+  background-color: whitesmoke;
+  border-radius: 290486px;
+  display: inline-flex;
+  font-size: 1.25rem;
+  height: 2em;
+  justify-content: center;
+  margin-right: 1.5rem;
+  min-width: 2.5em;
+  padding: 0.25rem 0.5rem;
+  text-align: center;
+  vertical-align: top; }
+
+.input, #documenter .docs-sidebar form.docs-search > input, .textarea, .select select {
+  background-color: white;
+  border-color: #dbdbdb;
+  border-radius: 4px;
+  color: #363636; }
+  .input::-moz-placeholder, #documenter .docs-sidebar form.docs-search > input::-moz-placeholder, .textarea::-moz-placeholder, .select select::-moz-placeholder {
+    color: rgba(54, 54, 54, 0.3); }
+  .input::-webkit-input-placeholder, #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder, .textarea::-webkit-input-placeholder, .select select::-webkit-input-placeholder {
+    color: rgba(54, 54, 54, 0.3); }
+  .input:-moz-placeholder, #documenter .docs-sidebar form.docs-search > input:-moz-placeholder, .textarea:-moz-placeholder, .select select:-moz-placeholder {
+    color: rgba(54, 54, 54, 0.3); }
+  .input:-ms-input-placeholder, #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder, .textarea:-ms-input-placeholder, .select select:-ms-input-placeholder {
+    color: rgba(54, 54, 54, 0.3); }
+  .input:hover, #documenter .docs-sidebar form.docs-search > input:hover, .textarea:hover, .select select:hover, .is-hovered.input, #documenter .docs-sidebar form.docs-search > input.is-hovered, .is-hovered.textarea, .select select.is-hovered {
+    border-color: #b5b5b5; }
+  .input:focus, #documenter .docs-sidebar form.docs-search > input:focus, .textarea:focus, .select select:focus, .is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-focused.textarea, .select select.is-focused, .input:active, #documenter .docs-sidebar form.docs-search > input:active, .textarea:active, .select select:active, .is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .is-active.textarea, .select select.is-active {
+    border-color: #2e63b8;
+    box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); }
+  .input[disabled], #documenter .docs-sidebar form.docs-search > input[disabled], .textarea[disabled], .select select[disabled],
+  fieldset[disabled] .input,
+  fieldset[disabled] #documenter .docs-sidebar form.docs-search > input,
+  #documenter .docs-sidebar fieldset[disabled] form.docs-search > input,
+  fieldset[disabled] .textarea,
+  fieldset[disabled] .select select,
+  .select fieldset[disabled] select {
+    background-color: whitesmoke;
+    border-color: whitesmoke;
+    box-shadow: none;
+    color: #7a7a7a; }
+    .input[disabled]::-moz-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]::-moz-placeholder, .textarea[disabled]::-moz-placeholder, .select select[disabled]::-moz-placeholder,
+    fieldset[disabled] .input::-moz-placeholder,
+    fieldset[disabled] #documenter .docs-sidebar form.docs-search > input::-moz-placeholder,
+    #documenter .docs-sidebar fieldset[disabled] form.docs-search > input::-moz-placeholder,
+    fieldset[disabled] .textarea::-moz-placeholder,
+    fieldset[disabled] .select select::-moz-placeholder,
+    .select fieldset[disabled] select::-moz-placeholder {
+      color: rgba(122, 122, 122, 0.3); }
+    .input[disabled]::-webkit-input-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]::-webkit-input-placeholder, .textarea[disabled]::-webkit-input-placeholder, .select select[disabled]::-webkit-input-placeholder,
+    fieldset[disabled] .input::-webkit-input-placeholder,
+    fieldset[disabled] #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder,
+    #documenter .docs-sidebar fieldset[disabled] form.docs-search > input::-webkit-input-placeholder,
+    fieldset[disabled] .textarea::-webkit-input-placeholder,
+    fieldset[disabled] .select select::-webkit-input-placeholder,
+    .select fieldset[disabled] select::-webkit-input-placeholder {
+      color: rgba(122, 122, 122, 0.3); }
+    .input[disabled]:-moz-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]:-moz-placeholder, .textarea[disabled]:-moz-placeholder, .select select[disabled]:-moz-placeholder,
+    fieldset[disabled] .input:-moz-placeholder,
+    fieldset[disabled] #documenter .docs-sidebar form.docs-search > input:-moz-placeholder,
+    #documenter .docs-sidebar fieldset[disabled] form.docs-search > input:-moz-placeholder,
+    fieldset[disabled] .textarea:-moz-placeholder,
+    fieldset[disabled] .select select:-moz-placeholder,
+    .select fieldset[disabled] select:-moz-placeholder {
+      color: rgba(122, 122, 122, 0.3); }
+    .input[disabled]:-ms-input-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]:-ms-input-placeholder, .textarea[disabled]:-ms-input-placeholder, .select select[disabled]:-ms-input-placeholder,
+    fieldset[disabled] .input:-ms-input-placeholder,
+    fieldset[disabled] #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder,
+    #documenter .docs-sidebar fieldset[disabled] form.docs-search > input:-ms-input-placeholder,
+    fieldset[disabled] .textarea:-ms-input-placeholder,
+    fieldset[disabled] .select select:-ms-input-placeholder,
+    .select fieldset[disabled] select:-ms-input-placeholder {
+      color: rgba(122, 122, 122, 0.3); }
+
+.input, #documenter .docs-sidebar form.docs-search > input, .textarea {
+  box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.1);
+  max-width: 100%;
+  width: 100%; }
+  .input[readonly], #documenter .docs-sidebar form.docs-search > input[readonly], .textarea[readonly] {
+    box-shadow: none; }
+  .is-white.input, #documenter .docs-sidebar form.docs-search > input.is-white, .is-white.textarea {
+    border-color: white; }
+    .is-white.input:focus, #documenter .docs-sidebar form.docs-search > input.is-white:focus, .is-white.textarea:focus, .is-white.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-white.is-focused, .is-white.is-focused.textarea, .is-white.input:active, #documenter .docs-sidebar form.docs-search > input.is-white:active, .is-white.textarea:active, .is-white.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-white.is-active, .is-white.is-active.textarea {
+      box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); }
+  .is-black.input, #documenter .docs-sidebar form.docs-search > input.is-black, .is-black.textarea {
+    border-color: #0a0a0a; }
+    .is-black.input:focus, #documenter .docs-sidebar form.docs-search > input.is-black:focus, .is-black.textarea:focus, .is-black.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-black.is-focused, .is-black.is-focused.textarea, .is-black.input:active, #documenter .docs-sidebar form.docs-search > input.is-black:active, .is-black.textarea:active, .is-black.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-black.is-active, .is-black.is-active.textarea {
+      box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); }
+  .is-light.input, #documenter .docs-sidebar form.docs-search > input.is-light, .is-light.textarea {
+    border-color: whitesmoke; }
+    .is-light.input:focus, #documenter .docs-sidebar form.docs-search > input.is-light:focus, .is-light.textarea:focus, .is-light.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-light.is-focused, .is-light.is-focused.textarea, .is-light.input:active, #documenter .docs-sidebar form.docs-search > input.is-light:active, .is-light.textarea:active, .is-light.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-light.is-active, .is-light.is-active.textarea {
+      box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); }
+  .is-dark.input, .content kbd.input, #documenter .docs-sidebar form.docs-search > input.is-dark, .is-dark.textarea, .content kbd.textarea {
+    border-color: #363636; }
+    .is-dark.input:focus, .content kbd.input:focus, #documenter .docs-sidebar form.docs-search > input.is-dark:focus, .is-dark.textarea:focus, .content kbd.textarea:focus, .is-dark.is-focused.input, .content kbd.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-dark.is-focused, .is-dark.is-focused.textarea, .content kbd.is-focused.textarea, .is-dark.input:active, .content kbd.input:active, #documenter .docs-sidebar form.docs-search > input.is-dark:active, .is-dark.textarea:active, .content kbd.textarea:active, .is-dark.is-active.input, .content kbd.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-dark.is-active, .is-dark.is-active.textarea, .content kbd.is-active.textarea {
+      box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); }
+  .is-primary.input, .docstring > section > a.input.docs-sourcelink, #documenter .docs-sidebar form.docs-search > input.is-primary, .is-primary.textarea, .docstring > section > a.textarea.docs-sourcelink {
+    border-color: #4eb5de; }
+    .is-primary.input:focus, .docstring > section > a.input.docs-sourcelink:focus, #documenter .docs-sidebar form.docs-search > input.is-primary:focus, .is-primary.textarea:focus, .docstring > section > a.textarea.docs-sourcelink:focus, .is-primary.is-focused.input, .docstring > section > a.is-focused.input.docs-sourcelink, #documenter .docs-sidebar form.docs-search > input.is-primary.is-focused, .is-primary.is-focused.textarea, .docstring > section > a.is-focused.textarea.docs-sourcelink, .is-primary.input:active, .docstring > section > a.input.docs-sourcelink:active, #documenter .docs-sidebar form.docs-search > input.is-primary:active, .is-primary.textarea:active, .docstring > section > a.textarea.docs-sourcelink:active, .is-primary.is-active.input, .docstring > section > a.is-active.input.docs-sourcelink, #documenter .docs-sidebar form.docs-search > input.is-primary.is-active, .is-primary.is-active.textarea, .docstring > section > a.is-active.textarea.docs-sourcelink {
+      box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); }
+  .is-link.input, #documenter .docs-sidebar form.docs-search > input.is-link, .is-link.textarea {
+    border-color: #2e63b8; }
+    .is-link.input:focus, #documenter .docs-sidebar form.docs-search > input.is-link:focus, .is-link.textarea:focus, .is-link.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-link.is-focused, .is-link.is-focused.textarea, .is-link.input:active, #documenter .docs-sidebar form.docs-search > input.is-link:active, .is-link.textarea:active, .is-link.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-link.is-active, .is-link.is-active.textarea {
+      box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); }
+  .is-info.input, #documenter .docs-sidebar form.docs-search > input.is-info, .is-info.textarea {
+    border-color: #209cee; }
+    .is-info.input:focus, #documenter .docs-sidebar form.docs-search > input.is-info:focus, .is-info.textarea:focus, .is-info.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-info.is-focused, .is-info.is-focused.textarea, .is-info.input:active, #documenter .docs-sidebar form.docs-search > input.is-info:active, .is-info.textarea:active, .is-info.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-info.is-active, .is-info.is-active.textarea {
+      box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); }
+  .is-success.input, #documenter .docs-sidebar form.docs-search > input.is-success, .is-success.textarea {
+    border-color: #22c35b; }
+    .is-success.input:focus, #documenter .docs-sidebar form.docs-search > input.is-success:focus, .is-success.textarea:focus, .is-success.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-success.is-focused, .is-success.is-focused.textarea, .is-success.input:active, #documenter .docs-sidebar form.docs-search > input.is-success:active, .is-success.textarea:active, .is-success.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-success.is-active, .is-success.is-active.textarea {
+      box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); }
+  .is-warning.input, #documenter .docs-sidebar form.docs-search > input.is-warning, .is-warning.textarea {
+    border-color: #ffdd57; }
+    .is-warning.input:focus, #documenter .docs-sidebar form.docs-search > input.is-warning:focus, .is-warning.textarea:focus, .is-warning.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-warning.is-focused, .is-warning.is-focused.textarea, .is-warning.input:active, #documenter .docs-sidebar form.docs-search > input.is-warning:active, .is-warning.textarea:active, .is-warning.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-warning.is-active, .is-warning.is-active.textarea {
+      box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); }
+  .is-danger.input, #documenter .docs-sidebar form.docs-search > input.is-danger, .is-danger.textarea {
+    border-color: #da0b00; }
+    .is-danger.input:focus, #documenter .docs-sidebar form.docs-search > input.is-danger:focus, .is-danger.textarea:focus, .is-danger.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-danger.is-focused, .is-danger.is-focused.textarea, .is-danger.input:active, #documenter .docs-sidebar form.docs-search > input.is-danger:active, .is-danger.textarea:active, .is-danger.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-danger.is-active, .is-danger.is-active.textarea {
+      box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); }
+  .is-small.input, #documenter .docs-sidebar form.docs-search > input, .is-small.textarea {
+    border-radius: 2px;
+    font-size: 0.75rem; }
+  .is-medium.input, #documenter .docs-sidebar form.docs-search > input.is-medium, .is-medium.textarea {
+    font-size: 1.25rem; }
+  .is-large.input, #documenter .docs-sidebar form.docs-search > input.is-large, .is-large.textarea {
+    font-size: 1.5rem; }
+  .is-fullwidth.input, #documenter .docs-sidebar form.docs-search > input.is-fullwidth, .is-fullwidth.textarea {
+    display: block;
+    width: 100%; }
+  .is-inline.input, #documenter .docs-sidebar form.docs-search > input.is-inline, .is-inline.textarea {
+    display: inline;
+    width: auto; }
+
+.input.is-rounded, #documenter .docs-sidebar form.docs-search > input {
+  border-radius: 290486px;
+  padding-left: 1em;
+  padding-right: 1em; }
+
+.input.is-static, #documenter .docs-sidebar form.docs-search > input.is-static {
+  background-color: transparent;
+  border-color: transparent;
+  box-shadow: none;
+  padding-left: 0;
+  padding-right: 0; }
+
+.textarea {
+  display: block;
+  max-width: 100%;
+  min-width: 100%;
+  padding: 0.625em;
+  resize: vertical; }
+  .textarea:not([rows]) {
+    max-height: 600px;
+    min-height: 120px; }
+  .textarea[rows] {
+    height: initial; }
+  .textarea.has-fixed-size {
+    resize: none; }
+
+.checkbox, .radio {
+  cursor: pointer;
+  display: inline-block;
+  line-height: 1.25;
+  position: relative; }
+  .checkbox input, .radio input {
+    cursor: pointer; }
+  .checkbox:hover, .radio:hover {
+    color: #363636; }
+  .checkbox[disabled], .radio[disabled],
+  fieldset[disabled] .checkbox,
+  fieldset[disabled] .radio {
+    color: #7a7a7a;
+    cursor: not-allowed; }
+
+.radio + .radio {
+  margin-left: 0.5em; }
+
+.select {
+  display: inline-block;
+  max-width: 100%;
+  position: relative;
+  vertical-align: top; }
+  .select:not(.is-multiple) {
+    height: 2.25em; }
+  .select:not(.is-multiple):not(.is-loading)::after {
+    border-color: #2e63b8;
+    right: 1.125em;
+    z-index: 4; }
+  .select.is-rounded select, #documenter .docs-sidebar form.docs-search > input.select select {
+    border-radius: 290486px;
+    padding-left: 1em; }
+  .select select {
+    cursor: pointer;
+    display: block;
+    font-size: 1em;
+    max-width: 100%;
+    outline: none; }
+    .select select::-ms-expand {
+      display: none; }
+    .select select[disabled]:hover,
+    fieldset[disabled] .select select:hover {
+      border-color: whitesmoke; }
+    .select select:not([multiple]) {
+      padding-right: 2.5em; }
+    .select select[multiple] {
+      height: auto;
+      padding: 0; }
+      .select select[multiple] option {
+        padding: 0.5em 1em; }
+  .select:not(.is-multiple):not(.is-loading):hover::after {
+    border-color: #363636; }
+  .select.is-white:not(:hover)::after {
+    border-color: white; }
+  .select.is-white select {
+    border-color: white; }
+    .select.is-white select:hover, .select.is-white select.is-hovered {
+      border-color: #f2f2f2; }
+    .select.is-white select:focus, .select.is-white select.is-focused, .select.is-white select:active, .select.is-white select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); }
+  .select.is-black:not(:hover)::after {
+    border-color: #0a0a0a; }
+  .select.is-black select {
+    border-color: #0a0a0a; }
+    .select.is-black select:hover, .select.is-black select.is-hovered {
+      border-color: black; }
+    .select.is-black select:focus, .select.is-black select.is-focused, .select.is-black select:active, .select.is-black select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); }
+  .select.is-light:not(:hover)::after {
+    border-color: whitesmoke; }
+  .select.is-light select {
+    border-color: whitesmoke; }
+    .select.is-light select:hover, .select.is-light select.is-hovered {
+      border-color: #e8e8e8; }
+    .select.is-light select:focus, .select.is-light select.is-focused, .select.is-light select:active, .select.is-light select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); }
+  .select.is-dark:not(:hover)::after, .content kbd.select:not(:hover)::after {
+    border-color: #363636; }
+  .select.is-dark select, .content kbd.select select {
+    border-color: #363636; }
+    .select.is-dark select:hover, .content kbd.select select:hover, .select.is-dark select.is-hovered, .content kbd.select select.is-hovered {
+      border-color: #292929; }
+    .select.is-dark select:focus, .content kbd.select select:focus, .select.is-dark select.is-focused, .content kbd.select select.is-focused, .select.is-dark select:active, .content kbd.select select:active, .select.is-dark select.is-active, .content kbd.select select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); }
+  .select.is-primary:not(:hover)::after, .docstring > section > a.select.docs-sourcelink:not(:hover)::after {
+    border-color: #4eb5de; }
+  .select.is-primary select, .docstring > section > a.select.docs-sourcelink select {
+    border-color: #4eb5de; }
+    .select.is-primary select:hover, .docstring > section > a.select.docs-sourcelink select:hover, .select.is-primary select.is-hovered, .docstring > section > a.select.docs-sourcelink select.is-hovered {
+      border-color: #39acda; }
+    .select.is-primary select:focus, .docstring > section > a.select.docs-sourcelink select:focus, .select.is-primary select.is-focused, .docstring > section > a.select.docs-sourcelink select.is-focused, .select.is-primary select:active, .docstring > section > a.select.docs-sourcelink select:active, .select.is-primary select.is-active, .docstring > section > a.select.docs-sourcelink select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); }
+  .select.is-link:not(:hover)::after {
+    border-color: #2e63b8; }
+  .select.is-link select {
+    border-color: #2e63b8; }
+    .select.is-link select:hover, .select.is-link select.is-hovered {
+      border-color: #2958a4; }
+    .select.is-link select:focus, .select.is-link select.is-focused, .select.is-link select:active, .select.is-link select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); }
+  .select.is-info:not(:hover)::after {
+    border-color: #209cee; }
+  .select.is-info select {
+    border-color: #209cee; }
+    .select.is-info select:hover, .select.is-info select.is-hovered {
+      border-color: #118fe4; }
+    .select.is-info select:focus, .select.is-info select.is-focused, .select.is-info select:active, .select.is-info select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); }
+  .select.is-success:not(:hover)::after {
+    border-color: #22c35b; }
+  .select.is-success select {
+    border-color: #22c35b; }
+    .select.is-success select:hover, .select.is-success select.is-hovered {
+      border-color: #1ead51; }
+    .select.is-success select:focus, .select.is-success select.is-focused, .select.is-success select:active, .select.is-success select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); }
+  .select.is-warning:not(:hover)::after {
+    border-color: #ffdd57; }
+  .select.is-warning select {
+    border-color: #ffdd57; }
+    .select.is-warning select:hover, .select.is-warning select.is-hovered {
+      border-color: #ffd83d; }
+    .select.is-warning select:focus, .select.is-warning select.is-focused, .select.is-warning select:active, .select.is-warning select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); }
+  .select.is-danger:not(:hover)::after {
+    border-color: #da0b00; }
+  .select.is-danger select {
+    border-color: #da0b00; }
+    .select.is-danger select:hover, .select.is-danger select.is-hovered {
+      border-color: #c10a00; }
+    .select.is-danger select:focus, .select.is-danger select.is-focused, .select.is-danger select:active, .select.is-danger select.is-active {
+      box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); }
+  .select.is-small, #documenter .docs-sidebar form.docs-search > input.select {
+    border-radius: 2px;
+    font-size: 0.75rem; }
+  .select.is-medium {
+    font-size: 1.25rem; }
+  .select.is-large {
+    font-size: 1.5rem; }
+  .select.is-disabled::after {
+    border-color: #7a7a7a; }
+  .select.is-fullwidth {
+    width: 100%; }
+    .select.is-fullwidth select {
+      width: 100%; }
+  .select.is-loading::after {
+    margin-top: 0;
+    position: absolute;
+    right: 0.625em;
+    top: 0.625em;
+    transform: none; }
+  .select.is-loading.is-small:after, #documenter .docs-sidebar form.docs-search > input.select.is-loading:after {
+    font-size: 0.75rem; }
+  .select.is-loading.is-medium:after {
+    font-size: 1.25rem; }
+  .select.is-loading.is-large:after {
+    font-size: 1.5rem; }
+
+.file {
+  align-items: stretch;
+  display: flex;
+  justify-content: flex-start;
+  position: relative; }
+  .file.is-white .file-cta {
+    background-color: white;
+    border-color: transparent;
+    color: #0a0a0a; }
+  .file.is-white:hover .file-cta, .file.is-white.is-hovered .file-cta {
+    background-color: #f9f9f9;
+    border-color: transparent;
+    color: #0a0a0a; }
+  .file.is-white:focus .file-cta, .file.is-white.is-focused .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(255, 255, 255, 0.25);
+    color: #0a0a0a; }
+  .file.is-white:active .file-cta, .file.is-white.is-active .file-cta {
+    background-color: #f2f2f2;
+    border-color: transparent;
+    color: #0a0a0a; }
+  .file.is-black .file-cta {
+    background-color: #0a0a0a;
+    border-color: transparent;
+    color: white; }
+  .file.is-black:hover .file-cta, .file.is-black.is-hovered .file-cta {
+    background-color: #040404;
+    border-color: transparent;
+    color: white; }
+  .file.is-black:focus .file-cta, .file.is-black.is-focused .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(10, 10, 10, 0.25);
+    color: white; }
+  .file.is-black:active .file-cta, .file.is-black.is-active .file-cta {
+    background-color: black;
+    border-color: transparent;
+    color: white; }
+  .file.is-light .file-cta {
+    background-color: whitesmoke;
+    border-color: transparent;
+    color: #363636; }
+  .file.is-light:hover .file-cta, .file.is-light.is-hovered .file-cta {
+    background-color: #eeeeee;
+    border-color: transparent;
+    color: #363636; }
+  .file.is-light:focus .file-cta, .file.is-light.is-focused .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(245, 245, 245, 0.25);
+    color: #363636; }
+  .file.is-light:active .file-cta, .file.is-light.is-active .file-cta {
+    background-color: #e8e8e8;
+    border-color: transparent;
+    color: #363636; }
+  .file.is-dark .file-cta, .content kbd.file .file-cta {
+    background-color: #363636;
+    border-color: transparent;
+    color: whitesmoke; }
+  .file.is-dark:hover .file-cta, .content kbd.file:hover .file-cta, .file.is-dark.is-hovered .file-cta, .content kbd.file.is-hovered .file-cta {
+    background-color: #2f2f2f;
+    border-color: transparent;
+    color: whitesmoke; }
+  .file.is-dark:focus .file-cta, .content kbd.file:focus .file-cta, .file.is-dark.is-focused .file-cta, .content kbd.file.is-focused .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(54, 54, 54, 0.25);
+    color: whitesmoke; }
+  .file.is-dark:active .file-cta, .content kbd.file:active .file-cta, .file.is-dark.is-active .file-cta, .content kbd.file.is-active .file-cta {
+    background-color: #292929;
+    border-color: transparent;
+    color: whitesmoke; }
+  .file.is-primary .file-cta, .docstring > section > a.file.docs-sourcelink .file-cta {
+    background-color: #4eb5de;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-primary:hover .file-cta, .docstring > section > a.file.docs-sourcelink:hover .file-cta, .file.is-primary.is-hovered .file-cta, .docstring > section > a.file.is-hovered.docs-sourcelink .file-cta {
+    background-color: #43b1dc;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-primary:focus .file-cta, .docstring > section > a.file.docs-sourcelink:focus .file-cta, .file.is-primary.is-focused .file-cta, .docstring > section > a.file.is-focused.docs-sourcelink .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(78, 181, 222, 0.25);
+    color: #fff; }
+  .file.is-primary:active .file-cta, .docstring > section > a.file.docs-sourcelink:active .file-cta, .file.is-primary.is-active .file-cta, .docstring > section > a.file.is-active.docs-sourcelink .file-cta {
+    background-color: #39acda;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-link .file-cta {
+    background-color: #2e63b8;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-link:hover .file-cta, .file.is-link.is-hovered .file-cta {
+    background-color: #2b5eae;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-link:focus .file-cta, .file.is-link.is-focused .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(46, 99, 184, 0.25);
+    color: #fff; }
+  .file.is-link:active .file-cta, .file.is-link.is-active .file-cta {
+    background-color: #2958a4;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-info .file-cta {
+    background-color: #209cee;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-info:hover .file-cta, .file.is-info.is-hovered .file-cta {
+    background-color: #1496ed;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-info:focus .file-cta, .file.is-info.is-focused .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(32, 156, 238, 0.25);
+    color: #fff; }
+  .file.is-info:active .file-cta, .file.is-info.is-active .file-cta {
+    background-color: #118fe4;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-success .file-cta {
+    background-color: #22c35b;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-success:hover .file-cta, .file.is-success.is-hovered .file-cta {
+    background-color: #20b856;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-success:focus .file-cta, .file.is-success.is-focused .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(34, 195, 91, 0.25);
+    color: #fff; }
+  .file.is-success:active .file-cta, .file.is-success.is-active .file-cta {
+    background-color: #1ead51;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-warning .file-cta {
+    background-color: #ffdd57;
+    border-color: transparent;
+    color: rgba(0, 0, 0, 0.7); }
+  .file.is-warning:hover .file-cta, .file.is-warning.is-hovered .file-cta {
+    background-color: #ffdb4a;
+    border-color: transparent;
+    color: rgba(0, 0, 0, 0.7); }
+  .file.is-warning:focus .file-cta, .file.is-warning.is-focused .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(255, 221, 87, 0.25);
+    color: rgba(0, 0, 0, 0.7); }
+  .file.is-warning:active .file-cta, .file.is-warning.is-active .file-cta {
+    background-color: #ffd83d;
+    border-color: transparent;
+    color: rgba(0, 0, 0, 0.7); }
+  .file.is-danger .file-cta {
+    background-color: #da0b00;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-danger:hover .file-cta, .file.is-danger.is-hovered .file-cta {
+    background-color: #cd0a00;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-danger:focus .file-cta, .file.is-danger.is-focused .file-cta {
+    border-color: transparent;
+    box-shadow: 0 0 0.5em rgba(218, 11, 0, 0.25);
+    color: #fff; }
+  .file.is-danger:active .file-cta, .file.is-danger.is-active .file-cta {
+    background-color: #c10a00;
+    border-color: transparent;
+    color: #fff; }
+  .file.is-small, #documenter .docs-sidebar form.docs-search > input.file {
+    font-size: 0.75rem; }
+  .file.is-medium {
+    font-size: 1.25rem; }
+    .file.is-medium .file-icon .fa {
+      font-size: 21px; }
+  .file.is-large {
+    font-size: 1.5rem; }
+    .file.is-large .file-icon .fa {
+      font-size: 28px; }
+  .file.has-name .file-cta {
+    border-bottom-right-radius: 0;
+    border-top-right-radius: 0; }
+  .file.has-name .file-name {
+    border-bottom-left-radius: 0;
+    border-top-left-radius: 0; }
+  .file.has-name.is-empty .file-cta {
+    border-radius: 4px; }
+  .file.has-name.is-empty .file-name {
+    display: none; }
+  .file.is-boxed .file-label {
+    flex-direction: column; }
+  .file.is-boxed .file-cta {
+    flex-direction: column;
+    height: auto;
+    padding: 1em 3em; }
+  .file.is-boxed .file-name {
+    border-width: 0 1px 1px; }
+  .file.is-boxed .file-icon {
+    height: 1.5em;
+    width: 1.5em; }
+    .file.is-boxed .file-icon .fa {
+      font-size: 21px; }
+  .file.is-boxed.is-small .file-icon .fa, #documenter .docs-sidebar form.docs-search > input.file.is-boxed .file-icon .fa {
+    font-size: 14px; }
+  .file.is-boxed.is-medium .file-icon .fa {
+    font-size: 28px; }
+  .file.is-boxed.is-large .file-icon .fa {
+    font-size: 35px; }
+  .file.is-boxed.has-name .file-cta {
+    border-radius: 4px 4px 0 0; }
+  .file.is-boxed.has-name .file-name {
+    border-radius: 0 0 4px 4px;
+    border-width: 0 1px 1px; }
+  .file.is-centered {
+    justify-content: center; }
+  .file.is-fullwidth .file-label {
+    width: 100%; }
+  .file.is-fullwidth .file-name {
+    flex-grow: 1;
+    max-width: none; }
+  .file.is-right {
+    justify-content: flex-end; }
+    .file.is-right .file-cta {
+      border-radius: 0 4px 4px 0; }
+    .file.is-right .file-name {
+      border-radius: 4px 0 0 4px;
+      border-width: 1px 0 1px 1px;
+      order: -1; }
+
+.file-label {
+  align-items: stretch;
+  display: flex;
+  cursor: pointer;
+  justify-content: flex-start;
+  overflow: hidden;
+  position: relative; }
+  .file-label:hover .file-cta {
+    background-color: #eeeeee;
+    color: #363636; }
+  .file-label:hover .file-name {
+    border-color: #d5d5d5; }
+  .file-label:active .file-cta {
+    background-color: #e8e8e8;
+    color: #363636; }
+  .file-label:active .file-name {
+    border-color: #cfcfcf; }
+
+.file-input {
+  height: 100%;
+  left: 0;
+  opacity: 0;
+  outline: none;
+  position: absolute;
+  top: 0;
+  width: 100%; }
+
+.file-cta,
+.file-name {
+  border-color: #dbdbdb;
+  border-radius: 4px;
+  font-size: 1em;
+  padding-left: 1em;
+  padding-right: 1em;
+  white-space: nowrap; }
+
+.file-cta {
+  background-color: whitesmoke;
+  color: #4a4a4a; }
+
+.file-name {
+  border-color: #dbdbdb;
+  border-style: solid;
+  border-width: 1px 1px 1px 0;
+  display: block;
+  max-width: 16em;
+  overflow: hidden;
+  text-align: left;
+  text-overflow: ellipsis; }
+
+.file-icon {
+  align-items: center;
+  display: flex;
+  height: 1em;
+  justify-content: center;
+  margin-right: 0.5em;
+  width: 1em; }
+  .file-icon .fa {
+    font-size: 14px; }
+
+.label {
+  color: #363636;
+  display: block;
+  font-size: 1rem;
+  font-weight: 700; }
+  .label:not(:last-child) {
+    margin-bottom: 0.5em; }
+  .label.is-small, #documenter .docs-sidebar form.docs-search > input.label {
+    font-size: 0.75rem; }
+  .label.is-medium {
+    font-size: 1.25rem; }
+  .label.is-large {
+    font-size: 1.5rem; }
+
+.help {
+  display: block;
+  font-size: 0.75rem;
+  margin-top: 0.25rem; }
+  .help.is-white {
+    color: white; }
+  .help.is-black {
+    color: #0a0a0a; }
+  .help.is-light {
+    color: whitesmoke; }
+  .help.is-dark, .content kbd.help {
+    color: #363636; }
+  .help.is-primary, .docstring > section > a.help.docs-sourcelink {
+    color: #4eb5de; }
+  .help.is-link {
+    color: #2e63b8; }
+  .help.is-info {
+    color: #209cee; }
+  .help.is-success {
+    color: #22c35b; }
+  .help.is-warning {
+    color: #ffdd57; }
+  .help.is-danger {
+    color: #da0b00; }
+
+.field:not(:last-child) {
+  margin-bottom: 0.75rem; }
+
+.field.has-addons {
+  display: flex;
+  justify-content: flex-start; }
+  .field.has-addons .control:not(:last-child) {
+    margin-right: -1px; }
+  .field.has-addons .control:not(:first-child):not(:last-child) .button,
+  .field.has-addons .control:not(:first-child):not(:last-child) .input,
+  .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search > input,
+  #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search > input,
+  .field.has-addons .control:not(:first-child):not(:last-child) .select select {
+    border-radius: 0; }
+  .field.has-addons .control:first-child:not(:only-child) .button,
+  .field.has-addons .control:first-child:not(:only-child) .input,
+  .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input,
+  #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search > input,
+  .field.has-addons .control:first-child:not(:only-child) .select select {
+    border-bottom-right-radius: 0;
+    border-top-right-radius: 0; }
+  .field.has-addons .control:last-child:not(:only-child) .button,
+  .field.has-addons .control:last-child:not(:only-child) .input,
+  .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input,
+  #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search > input,
+  .field.has-addons .control:last-child:not(:only-child) .select select {
+    border-bottom-left-radius: 0;
+    border-top-left-radius: 0; }
+  .field.has-addons .control .button:not([disabled]):hover, .field.has-addons .control .button:not([disabled]).is-hovered,
+  .field.has-addons .control .input:not([disabled]):hover,
+  .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):hover,
+  #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):hover,
+  .field.has-addons .control .input:not([disabled]).is-hovered,
+  .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-hovered,
+  #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-hovered,
+  .field.has-addons .control .select select:not([disabled]):hover,
+  .field.has-addons .control .select select:not([disabled]).is-hovered {
+    z-index: 2; }
+  .field.has-addons .control .button:not([disabled]):focus, .field.has-addons .control .button:not([disabled]).is-focused, .field.has-addons .control .button:not([disabled]):active, .field.has-addons .control .button:not([disabled]).is-active,
+  .field.has-addons .control .input:not([disabled]):focus,
+  .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus,
+  #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus,
+  .field.has-addons .control .input:not([disabled]).is-focused,
+  .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-focused,
+  #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-focused,
+  .field.has-addons .control .input:not([disabled]):active,
+  .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active,
+  #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active,
+  .field.has-addons .control .input:not([disabled]).is-active,
+  .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-active,
+  #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-active,
+  .field.has-addons .control .select select:not([disabled]):focus,
+  .field.has-addons .control .select select:not([disabled]).is-focused,
+  .field.has-addons .control .select select:not([disabled]):active,
+  .field.has-addons .control .select select:not([disabled]).is-active {
+    z-index: 3; }
+    .field.has-addons .control .button:not([disabled]):focus:hover, .field.has-addons .control .button:not([disabled]).is-focused:hover, .field.has-addons .control .button:not([disabled]):active:hover, .field.has-addons .control .button:not([disabled]).is-active:hover,
+    .field.has-addons .control .input:not([disabled]):focus:hover,
+    .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus:hover,
+    #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus:hover,
+    .field.has-addons .control .input:not([disabled]).is-focused:hover,
+    .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-focused:hover,
+    #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-focused:hover,
+    .field.has-addons .control .input:not([disabled]):active:hover,
+    .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active:hover,
+    #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active:hover,
+    .field.has-addons .control .input:not([disabled]).is-active:hover,
+    .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]).is-active:hover,
+    #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]).is-active:hover,
+    .field.has-addons .control .select select:not([disabled]):focus:hover,
+    .field.has-addons .control .select select:not([disabled]).is-focused:hover,
+    .field.has-addons .control .select select:not([disabled]):active:hover,
+    .field.has-addons .control .select select:not([disabled]).is-active:hover {
+      z-index: 4; }
+  .field.has-addons .control.is-expanded {
+    flex-grow: 1;
+    flex-shrink: 1; }
+  .field.has-addons.has-addons-centered {
+    justify-content: center; }
+  .field.has-addons.has-addons-right {
+    justify-content: flex-end; }
+  .field.has-addons.has-addons-fullwidth .control {
+    flex-grow: 1;
+    flex-shrink: 0; }
+
+.field.is-grouped {
+  display: flex;
+  justify-content: flex-start; }
+  .field.is-grouped > .control {
+    flex-shrink: 0; }
+    .field.is-grouped > .control:not(:last-child) {
+      margin-bottom: 0;
+      margin-right: 0.75rem; }
+    .field.is-grouped > .control.is-expanded {
+      flex-grow: 1;
+      flex-shrink: 1; }
+  .field.is-grouped.is-grouped-centered {
+    justify-content: center; }
+  .field.is-grouped.is-grouped-right {
+    justify-content: flex-end; }
+  .field.is-grouped.is-grouped-multiline {
+    flex-wrap: wrap; }
+    .field.is-grouped.is-grouped-multiline > .control:last-child, .field.is-grouped.is-grouped-multiline > .control:not(:last-child) {
+      margin-bottom: 0.75rem; }
+    .field.is-grouped.is-grouped-multiline:last-child {
+      margin-bottom: -0.75rem; }
+    .field.is-grouped.is-grouped-multiline:not(:last-child) {
+      margin-bottom: 0; }
+
+@media screen and (min-width: 769px), print {
+  .field.is-horizontal {
+    display: flex; } }
+
+.field-label .label {
+  font-size: inherit; }
+
+@media screen and (max-width: 768px) {
+  .field-label {
+    margin-bottom: 0.5rem; } }
+
+@media screen and (min-width: 769px), print {
+  .field-label {
+    flex-basis: 0;
+    flex-grow: 1;
+    flex-shrink: 0;
+    margin-right: 1.5rem;
+    text-align: right; }
+    .field-label.is-small, #documenter .docs-sidebar form.docs-search > input.field-label {
+      font-size: 0.75rem;
+      padding-top: 0.375em; }
+    .field-label.is-normal {
+      padding-top: 0.375em; }
+    .field-label.is-medium {
+      font-size: 1.25rem;
+      padding-top: 0.375em; }
+    .field-label.is-large {
+      font-size: 1.5rem;
+      padding-top: 0.375em; } }
+
+.field-body .field .field {
+  margin-bottom: 0; }
+
+@media screen and (min-width: 769px), print {
+  .field-body {
+    display: flex;
+    flex-basis: 0;
+    flex-grow: 5;
+    flex-shrink: 1; }
+    .field-body .field {
+      margin-bottom: 0; }
+    .field-body > .field {
+      flex-shrink: 1; }
+      .field-body > .field:not(.is-narrow) {
+        flex-grow: 1; }
+      .field-body > .field:not(:last-child) {
+        margin-right: 0.75rem; } }
+
+.control {
+  box-sizing: border-box;
+  clear: both;
+  font-size: 1rem;
+  position: relative;
+  text-align: left; }
+  .control.has-icons-left .input:focus ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input:focus ~ .icon,
+  .control.has-icons-left .select:focus ~ .icon, .control.has-icons-right .input:focus ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input:focus ~ .icon,
+  .control.has-icons-right .select:focus ~ .icon {
+    color: #7a7a7a; }
+  .control.has-icons-left .input.is-small ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input ~ .icon,
+  .control.has-icons-left .select.is-small ~ .icon,
+  .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.select ~ .icon,
+  #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.select ~ .icon, .control.has-icons-right .input.is-small ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input ~ .icon,
+  .control.has-icons-right .select.is-small ~ .icon,
+  .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.select ~ .icon,
+  #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.select ~ .icon {
+    font-size: 0.75rem; }
+  .control.has-icons-left .input.is-medium ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-medium ~ .icon,
+  .control.has-icons-left .select.is-medium ~ .icon, .control.has-icons-right .input.is-medium ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-medium ~ .icon,
+  .control.has-icons-right .select.is-medium ~ .icon {
+    font-size: 1.25rem; }
+  .control.has-icons-left .input.is-large ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-large ~ .icon,
+  .control.has-icons-left .select.is-large ~ .icon, .control.has-icons-right .input.is-large ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-large ~ .icon,
+  .control.has-icons-right .select.is-large ~ .icon {
+    font-size: 1.5rem; }
+  .control.has-icons-left .icon, .control.has-icons-right .icon {
+    color: #dbdbdb;
+    height: 2.25em;
+    pointer-events: none;
+    position: absolute;
+    top: 0;
+    width: 2.25em;
+    z-index: 4; }
+  .control.has-icons-left .input, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input,
+  .control.has-icons-left .select select {
+    padding-left: 2.25em; }
+  .control.has-icons-left .icon.is-left {
+    left: 0; }
+  .control.has-icons-right .input, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input,
+  .control.has-icons-right .select select {
+    padding-right: 2.25em; }
+  .control.has-icons-right .icon.is-right {
+    right: 0; }
+  .control.is-loading::after {
+    position: absolute !important;
+    right: 0.625em;
+    top: 0.625em;
+    z-index: 4; }
+  .control.is-loading.is-small:after, #documenter .docs-sidebar form.docs-search > input.control.is-loading:after {
+    font-size: 0.75rem; }
+  .control.is-loading.is-medium:after {
+    font-size: 1.25rem; }
+  .control.is-loading.is-large:after {
+    font-size: 1.5rem; }
+
+.breadcrumb {
+  font-size: 1rem;
+  white-space: nowrap; }
+  .breadcrumb a {
+    align-items: center;
+    color: #2e63b8;
+    display: flex;
+    justify-content: center;
+    padding: 0 0.75em; }
+    .breadcrumb a:hover {
+      color: #363636; }
+  .breadcrumb li {
+    align-items: center;
+    display: flex; }
+    .breadcrumb li:first-child a {
+      padding-left: 0; }
+    .breadcrumb li.is-active a {
+      color: #222222;
+      cursor: default;
+      pointer-events: none; }
+    .breadcrumb li + li::before {
+      color: #b5b5b5;
+      content: "\0002f"; }
+  .breadcrumb ul,
+  .breadcrumb ol {
+    align-items: flex-start;
+    display: flex;
+    flex-wrap: wrap;
+    justify-content: flex-start; }
+  .breadcrumb .icon:first-child {
+    margin-right: 0.5em; }
+  .breadcrumb .icon:last-child {
+    margin-left: 0.5em; }
+  .breadcrumb.is-centered ol,
+  .breadcrumb.is-centered ul {
+    justify-content: center; }
+  .breadcrumb.is-right ol,
+  .breadcrumb.is-right ul {
+    justify-content: flex-end; }
+  .breadcrumb.is-small, #documenter .docs-sidebar form.docs-search > input.breadcrumb {
+    font-size: 0.75rem; }
+  .breadcrumb.is-medium {
+    font-size: 1.25rem; }
+  .breadcrumb.is-large {
+    font-size: 1.5rem; }
+  .breadcrumb.has-arrow-separator li + li::before {
+    content: "\02192"; }
+  .breadcrumb.has-bullet-separator li + li::before {
+    content: "\02022"; }
+  .breadcrumb.has-dot-separator li + li::before {
+    content: "\000b7"; }
+  .breadcrumb.has-succeeds-separator li + li::before {
+    content: "\0227B"; }
+
+.card {
+  background-color: white;
+  box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1);
+  color: #222222;
+  max-width: 100%;
+  position: relative; }
+
+.card-header {
+  background-color: transparent;
+  align-items: stretch;
+  box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1);
+  display: flex; }
+
+.card-header-title {
+  align-items: center;
+  color: #222222;
+  display: flex;
+  flex-grow: 1;
+  font-weight: 700;
+  padding: 0.75rem; }
+  .card-header-title.is-centered {
+    justify-content: center; }
+
+.card-header-icon {
+  align-items: center;
+  cursor: pointer;
+  display: flex;
+  justify-content: center;
+  padding: 0.75rem; }
+
+.card-image {
+  display: block;
+  position: relative; }
+
+.card-content {
+  background-color: transparent;
+  padding: 1rem 1.25rem; }
+
+.card-footer {
+  background-color: transparent;
+  border-top: 1px solid #dbdbdb;
+  align-items: stretch;
+  display: flex; }
+
+.card-footer-item {
+  align-items: center;
+  display: flex;
+  flex-basis: 0;
+  flex-grow: 1;
+  flex-shrink: 0;
+  justify-content: center;
+  padding: 0.75rem; }
+  .card-footer-item:not(:last-child) {
+    border-right: 1px solid #dbdbdb; }
+
+.card .media:not(:last-child) {
+  margin-bottom: 1.5rem; }
+
+.dropdown {
+  display: inline-flex;
+  position: relative;
+  vertical-align: top; }
+  .dropdown.is-active .dropdown-menu, .dropdown.is-hoverable:hover .dropdown-menu {
+    display: block; }
+  .dropdown.is-right .dropdown-menu {
+    left: auto;
+    right: 0; }
+  .dropdown.is-up .dropdown-menu {
+    bottom: 100%;
+    padding-bottom: 4px;
+    padding-top: initial;
+    top: auto; }
+
+.dropdown-menu {
+  display: none;
+  left: 0;
+  min-width: 12rem;
+  padding-top: 4px;
+  position: absolute;
+  top: 100%;
+  z-index: 20; }
+
+.dropdown-content {
+  background-color: white;
+  border-radius: 4px;
+  box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1);
+  padding-bottom: 0.5rem;
+  padding-top: 0.5rem; }
+
+.dropdown-item {
+  color: #4a4a4a;
+  display: block;
+  font-size: 0.875rem;
+  line-height: 1.5;
+  padding: 0.375rem 1rem;
+  position: relative; }
+
+a.dropdown-item,
+button.dropdown-item {
+  padding-right: 3rem;
+  text-align: left;
+  white-space: nowrap;
+  width: 100%; }
+  a.dropdown-item:hover,
+  button.dropdown-item:hover {
+    background-color: whitesmoke;
+    color: #0a0a0a; }
+  a.dropdown-item.is-active,
+  button.dropdown-item.is-active {
+    background-color: #2e63b8;
+    color: #fff; }
+
+.dropdown-divider {
+  background-color: #dbdbdb;
+  border: none;
+  display: block;
+  height: 1px;
+  margin: 0.5rem 0; }
+
+.level {
+  align-items: center;
+  justify-content: space-between; }
+  .level code {
+    border-radius: 4px; }
+  .level img {
+    display: inline-block;
+    vertical-align: top; }
+  .level.is-mobile {
+    display: flex; }
+    .level.is-mobile .level-left,
+    .level.is-mobile .level-right {
+      display: flex; }
+    .level.is-mobile .level-left + .level-right {
+      margin-top: 0; }
+    .level.is-mobile .level-item:not(:last-child) {
+      margin-bottom: 0;
+      margin-right: 0.75rem; }
+    .level.is-mobile .level-item:not(.is-narrow) {
+      flex-grow: 1; }
+  @media screen and (min-width: 769px), print {
+    .level {
+      display: flex; }
+      .level > .level-item:not(.is-narrow) {
+        flex-grow: 1; } }
+
+.level-item {
+  align-items: center;
+  display: flex;
+  flex-basis: auto;
+  flex-grow: 0;
+  flex-shrink: 0;
+  justify-content: center; }
+  .level-item .title,
+  .level-item .subtitle {
+    margin-bottom: 0; }
+  @media screen and (max-width: 768px) {
+    .level-item:not(:last-child) {
+      margin-bottom: 0.75rem; } }
+
+.level-left,
+.level-right {
+  flex-basis: auto;
+  flex-grow: 0;
+  flex-shrink: 0; }
+  .level-left .level-item.is-flexible,
+  .level-right .level-item.is-flexible {
+    flex-grow: 1; }
+  @media screen and (min-width: 769px), print {
+    .level-left .level-item:not(:last-child),
+    .level-right .level-item:not(:last-child) {
+      margin-right: 0.75rem; } }
+
+.level-left {
+  align-items: center;
+  justify-content: flex-start; }
+  @media screen and (max-width: 768px) {
+    .level-left + .level-right {
+      margin-top: 1.5rem; } }
+  @media screen and (min-width: 769px), print {
+    .level-left {
+      display: flex; } }
+
+.level-right {
+  align-items: center;
+  justify-content: flex-end; }
+  @media screen and (min-width: 769px), print {
+    .level-right {
+      display: flex; } }
+
+.list {
+  background-color: white;
+  border-radius: 4px;
+  box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); }
+
+.list-item {
+  display: block;
+  padding: 0.5em 1em; }
+  .list-item:not(a) {
+    color: #222222; }
+  .list-item:first-child {
+    border-top-left-radius: 4px;
+    border-top-right-radius: 4px; }
+  .list-item:last-child {
+    border-bottom-left-radius: 4px;
+    border-bottom-right-radius: 4px; }
+  .list-item:not(:last-child) {
+    border-bottom: 1px solid #dbdbdb; }
+  .list-item.is-active {
+    background-color: #2e63b8;
+    color: #fff; }
+
+a.list-item {
+  background-color: whitesmoke;
+  cursor: pointer; }
+
+.media {
+  align-items: flex-start;
+  display: flex;
+  text-align: left; }
+  .media .content:not(:last-child) {
+    margin-bottom: 0.75rem; }
+  .media .media {
+    border-top: 1px solid rgba(219, 219, 219, 0.5);
+    display: flex;
+    padding-top: 0.75rem; }
+    .media .media .content:not(:last-child),
+    .media .media .control:not(:last-child) {
+      margin-bottom: 0.5rem; }
+    .media .media .media {
+      padding-top: 0.5rem; }
+      .media .media .media + .media {
+        margin-top: 0.5rem; }
+  .media + .media {
+    border-top: 1px solid rgba(219, 219, 219, 0.5);
+    margin-top: 1rem;
+    padding-top: 1rem; }
+  .media.is-large + .media {
+    margin-top: 1.5rem;
+    padding-top: 1.5rem; }
+
+.media-left,
+.media-right {
+  flex-basis: auto;
+  flex-grow: 0;
+  flex-shrink: 0; }
+
+.media-left {
+  margin-right: 1rem; }
+
+.media-right {
+  margin-left: 1rem; }
+
+.media-content {
+  flex-basis: auto;
+  flex-grow: 1;
+  flex-shrink: 1;
+  text-align: left; }
+
+@media screen and (max-width: 768px) {
+  .media-content {
+    overflow-x: auto; } }
+
+.menu {
+  font-size: 1rem; }
+  .menu.is-small, #documenter .docs-sidebar form.docs-search > input.menu {
+    font-size: 0.75rem; }
+  .menu.is-medium {
+    font-size: 1.25rem; }
+  .menu.is-large {
+    font-size: 1.5rem; }
+
+.menu-list {
+  line-height: 1.25; }
+  .menu-list a {
+    border-radius: 2px;
+    color: #222222;
+    display: block;
+    padding: 0.5em 0.75em; }
+    .menu-list a:hover {
+      background-color: whitesmoke;
+      color: #222222; }
+    .menu-list a.is-active {
+      background-color: #2e63b8;
+      color: #fff; }
+  .menu-list li ul {
+    border-left: 1px solid #dbdbdb;
+    margin: 0.75em;
+    padding-left: 0.75em; }
+
+.menu-label {
+  color: #7a7a7a;
+  font-size: 0.75em;
+  letter-spacing: 0.1em;
+  text-transform: uppercase; }
+  .menu-label:not(:first-child) {
+    margin-top: 1em; }
+  .menu-label:not(:last-child) {
+    margin-bottom: 1em; }
+
+.message {
+  background-color: whitesmoke;
+  border-radius: 4px;
+  font-size: 1rem; }
+  .message strong {
+    color: currentColor; }
+  .message a:not(.button):not(.tag):not(.dropdown-item) {
+    color: currentColor;
+    text-decoration: underline; }
+  .message.is-small, #documenter .docs-sidebar form.docs-search > input.message {
+    font-size: 0.75rem; }
+  .message.is-medium {
+    font-size: 1.25rem; }
+  .message.is-large {
+    font-size: 1.5rem; }
+  .message.is-white {
+    background-color: white; }
+    .message.is-white .message-header {
+      background-color: white;
+      color: #0a0a0a; }
+    .message.is-white .message-body {
+      border-color: white;
+      color: #4d4d4d; }
+  .message.is-black {
+    background-color: #fafafa; }
+    .message.is-black .message-header {
+      background-color: #0a0a0a;
+      color: white; }
+    .message.is-black .message-body {
+      border-color: #0a0a0a;
+      color: #090909; }
+  .message.is-light {
+    background-color: #fafafa; }
+    .message.is-light .message-header {
+      background-color: whitesmoke;
+      color: #363636; }
+    .message.is-light .message-body {
+      border-color: whitesmoke;
+      color: #505050; }
+  .message.is-dark, .content kbd.message {
+    background-color: #fafafa; }
+    .message.is-dark .message-header, .content kbd.message .message-header {
+      background-color: #363636;
+      color: whitesmoke; }
+    .message.is-dark .message-body, .content kbd.message .message-body {
+      border-color: #363636;
+      color: #2a2a2a; }
+  .message.is-primary, .docstring > section > a.message.docs-sourcelink {
+    background-color: #f6fbfd; }
+    .message.is-primary .message-header, .docstring > section > a.message.docs-sourcelink .message-header {
+      background-color: #4eb5de;
+      color: #fff; }
+    .message.is-primary .message-body, .docstring > section > a.message.docs-sourcelink .message-body {
+      border-color: #4eb5de;
+      color: #1f556a; }
+  .message.is-link {
+    background-color: #f7f9fd; }
+    .message.is-link .message-header {
+      background-color: #2e63b8;
+      color: #fff; }
+    .message.is-link .message-body {
+      border-color: #2e63b8;
+      color: #264981; }
+  .message.is-info {
+    background-color: #f6fbfe; }
+    .message.is-info .message-header {
+      background-color: #209cee;
+      color: #fff; }
+    .message.is-info .message-body {
+      border-color: #209cee;
+      color: #12537e; }
+  .message.is-success {
+    background-color: #f6fdf9; }
+    .message.is-success .message-header {
+      background-color: #22c35b;
+      color: #fff; }
+    .message.is-success .message-body {
+      border-color: #22c35b;
+      color: #0f361d; }
+  .message.is-warning {
+    background-color: #fffdf5; }
+    .message.is-warning .message-header {
+      background-color: #ffdd57;
+      color: rgba(0, 0, 0, 0.7); }
+    .message.is-warning .message-body {
+      border-color: #ffdd57;
+      color: #3b3108; }
+  .message.is-danger {
+    background-color: #fff5f5; }
+    .message.is-danger .message-header {
+      background-color: #da0b00;
+      color: #fff; }
+    .message.is-danger .message-body {
+      border-color: #da0b00;
+      color: #9b0c04; }
+
+.message-header {
+  align-items: center;
+  background-color: #222222;
+  border-radius: 4px 4px 0 0;
+  color: #fff;
+  display: flex;
+  font-weight: 700;
+  justify-content: space-between;
+  line-height: 1.25;
+  padding: 0.75em;
+  position: relative; }
+  .message-header .delete {
+    flex-grow: 0;
+    flex-shrink: 0;
+    margin-left: 0.75em; }
+  .message-header + .message-body {
+    border-width: 0;
+    border-top-left-radius: 0;
+    border-top-right-radius: 0; }
+
+.message-body {
+  border-color: #dbdbdb;
+  border-radius: 4px;
+  border-style: solid;
+  border-width: 0 0 0 4px;
+  color: #222222;
+  padding: 1em 1.25em; }
+  .message-body code,
+  .message-body pre {
+    background-color: white; }
+  .message-body pre code {
+    background-color: transparent; }
+
+.modal {
+  align-items: center;
+  display: none;
+  flex-direction: column;
+  justify-content: center;
+  overflow: hidden;
+  position: fixed;
+  z-index: 40; }
+  .modal.is-active {
+    display: flex; }
+
+.modal-background {
+  background-color: rgba(10, 10, 10, 0.86); }
+
+.modal-content,
+.modal-card {
+  margin: 0 20px;
+  max-height: calc(100vh - 160px);
+  overflow: auto;
+  position: relative;
+  width: 100%; }
+  @media screen and (min-width: 769px), print {
+    .modal-content,
+    .modal-card {
+      margin: 0 auto;
+      max-height: calc(100vh - 40px);
+      width: 640px; } }
+
+.modal-close {
+  background: none;
+  height: 40px;
+  position: fixed;
+  right: 20px;
+  top: 20px;
+  width: 40px; }
+
+.modal-card {
+  display: flex;
+  flex-direction: column;
+  max-height: calc(100vh - 40px);
+  overflow: hidden;
+  -ms-overflow-y: visible; }
+
+.modal-card-head,
+.modal-card-foot {
+  align-items: center;
+  background-color: whitesmoke;
+  display: flex;
+  flex-shrink: 0;
+  justify-content: flex-start;
+  padding: 20px;
+  position: relative; }
+
+.modal-card-head {
+  border-bottom: 1px solid #dbdbdb;
+  border-top-left-radius: 6px;
+  border-top-right-radius: 6px; }
+
+.modal-card-title {
+  color: #222222;
+  flex-grow: 1;
+  flex-shrink: 0;
+  font-size: 1.5rem;
+  line-height: 1; }
+
+.modal-card-foot {
+  border-bottom-left-radius: 6px;
+  border-bottom-right-radius: 6px;
+  border-top: 1px solid #dbdbdb; }
+  .modal-card-foot .button:not(:last-child) {
+    margin-right: 0.5em; }
+
+.modal-card-body {
+  -webkit-overflow-scrolling: touch;
+  background-color: white;
+  flex-grow: 1;
+  flex-shrink: 1;
+  overflow: auto;
+  padding: 20px; }
+
+.navbar {
+  background-color: white;
+  min-height: 3.25rem;
+  position: relative;
+  z-index: 30; }
+  .navbar.is-white {
+    background-color: white;
+    color: #0a0a0a; }
+    .navbar.is-white .navbar-brand > .navbar-item,
+    .navbar.is-white .navbar-brand .navbar-link {
+      color: #0a0a0a; }
+    .navbar.is-white .navbar-brand > a.navbar-item:focus, .navbar.is-white .navbar-brand > a.navbar-item:hover, .navbar.is-white .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-white .navbar-brand .navbar-link:focus,
+    .navbar.is-white .navbar-brand .navbar-link:hover,
+    .navbar.is-white .navbar-brand .navbar-link.is-active {
+      background-color: #f2f2f2;
+      color: #0a0a0a; }
+    .navbar.is-white .navbar-brand .navbar-link::after {
+      border-color: #0a0a0a; }
+    .navbar.is-white .navbar-burger {
+      color: #0a0a0a; }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-white .navbar-start > .navbar-item,
+      .navbar.is-white .navbar-start .navbar-link,
+      .navbar.is-white .navbar-end > .navbar-item,
+      .navbar.is-white .navbar-end .navbar-link {
+        color: #0a0a0a; }
+      .navbar.is-white .navbar-start > a.navbar-item:focus, .navbar.is-white .navbar-start > a.navbar-item:hover, .navbar.is-white .navbar-start > a.navbar-item.is-active,
+      .navbar.is-white .navbar-start .navbar-link:focus,
+      .navbar.is-white .navbar-start .navbar-link:hover,
+      .navbar.is-white .navbar-start .navbar-link.is-active,
+      .navbar.is-white .navbar-end > a.navbar-item:focus,
+      .navbar.is-white .navbar-end > a.navbar-item:hover,
+      .navbar.is-white .navbar-end > a.navbar-item.is-active,
+      .navbar.is-white .navbar-end .navbar-link:focus,
+      .navbar.is-white .navbar-end .navbar-link:hover,
+      .navbar.is-white .navbar-end .navbar-link.is-active {
+        background-color: #f2f2f2;
+        color: #0a0a0a; }
+      .navbar.is-white .navbar-start .navbar-link::after,
+      .navbar.is-white .navbar-end .navbar-link::after {
+        border-color: #0a0a0a; }
+      .navbar.is-white .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-white .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-white .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: #f2f2f2;
+        color: #0a0a0a; }
+      .navbar.is-white .navbar-dropdown a.navbar-item.is-active {
+        background-color: white;
+        color: #0a0a0a; } }
+  .navbar.is-black {
+    background-color: #0a0a0a;
+    color: white; }
+    .navbar.is-black .navbar-brand > .navbar-item,
+    .navbar.is-black .navbar-brand .navbar-link {
+      color: white; }
+    .navbar.is-black .navbar-brand > a.navbar-item:focus, .navbar.is-black .navbar-brand > a.navbar-item:hover, .navbar.is-black .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-black .navbar-brand .navbar-link:focus,
+    .navbar.is-black .navbar-brand .navbar-link:hover,
+    .navbar.is-black .navbar-brand .navbar-link.is-active {
+      background-color: black;
+      color: white; }
+    .navbar.is-black .navbar-brand .navbar-link::after {
+      border-color: white; }
+    .navbar.is-black .navbar-burger {
+      color: white; }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-black .navbar-start > .navbar-item,
+      .navbar.is-black .navbar-start .navbar-link,
+      .navbar.is-black .navbar-end > .navbar-item,
+      .navbar.is-black .navbar-end .navbar-link {
+        color: white; }
+      .navbar.is-black .navbar-start > a.navbar-item:focus, .navbar.is-black .navbar-start > a.navbar-item:hover, .navbar.is-black .navbar-start > a.navbar-item.is-active,
+      .navbar.is-black .navbar-start .navbar-link:focus,
+      .navbar.is-black .navbar-start .navbar-link:hover,
+      .navbar.is-black .navbar-start .navbar-link.is-active,
+      .navbar.is-black .navbar-end > a.navbar-item:focus,
+      .navbar.is-black .navbar-end > a.navbar-item:hover,
+      .navbar.is-black .navbar-end > a.navbar-item.is-active,
+      .navbar.is-black .navbar-end .navbar-link:focus,
+      .navbar.is-black .navbar-end .navbar-link:hover,
+      .navbar.is-black .navbar-end .navbar-link.is-active {
+        background-color: black;
+        color: white; }
+      .navbar.is-black .navbar-start .navbar-link::after,
+      .navbar.is-black .navbar-end .navbar-link::after {
+        border-color: white; }
+      .navbar.is-black .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-black .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-black .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: black;
+        color: white; }
+      .navbar.is-black .navbar-dropdown a.navbar-item.is-active {
+        background-color: #0a0a0a;
+        color: white; } }
+  .navbar.is-light {
+    background-color: whitesmoke;
+    color: #363636; }
+    .navbar.is-light .navbar-brand > .navbar-item,
+    .navbar.is-light .navbar-brand .navbar-link {
+      color: #363636; }
+    .navbar.is-light .navbar-brand > a.navbar-item:focus, .navbar.is-light .navbar-brand > a.navbar-item:hover, .navbar.is-light .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-light .navbar-brand .navbar-link:focus,
+    .navbar.is-light .navbar-brand .navbar-link:hover,
+    .navbar.is-light .navbar-brand .navbar-link.is-active {
+      background-color: #e8e8e8;
+      color: #363636; }
+    .navbar.is-light .navbar-brand .navbar-link::after {
+      border-color: #363636; }
+    .navbar.is-light .navbar-burger {
+      color: #363636; }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-light .navbar-start > .navbar-item,
+      .navbar.is-light .navbar-start .navbar-link,
+      .navbar.is-light .navbar-end > .navbar-item,
+      .navbar.is-light .navbar-end .navbar-link {
+        color: #363636; }
+      .navbar.is-light .navbar-start > a.navbar-item:focus, .navbar.is-light .navbar-start > a.navbar-item:hover, .navbar.is-light .navbar-start > a.navbar-item.is-active,
+      .navbar.is-light .navbar-start .navbar-link:focus,
+      .navbar.is-light .navbar-start .navbar-link:hover,
+      .navbar.is-light .navbar-start .navbar-link.is-active,
+      .navbar.is-light .navbar-end > a.navbar-item:focus,
+      .navbar.is-light .navbar-end > a.navbar-item:hover,
+      .navbar.is-light .navbar-end > a.navbar-item.is-active,
+      .navbar.is-light .navbar-end .navbar-link:focus,
+      .navbar.is-light .navbar-end .navbar-link:hover,
+      .navbar.is-light .navbar-end .navbar-link.is-active {
+        background-color: #e8e8e8;
+        color: #363636; }
+      .navbar.is-light .navbar-start .navbar-link::after,
+      .navbar.is-light .navbar-end .navbar-link::after {
+        border-color: #363636; }
+      .navbar.is-light .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-light .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-light .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: #e8e8e8;
+        color: #363636; }
+      .navbar.is-light .navbar-dropdown a.navbar-item.is-active {
+        background-color: whitesmoke;
+        color: #363636; } }
+  .navbar.is-dark, .content kbd.navbar {
+    background-color: #363636;
+    color: whitesmoke; }
+    .navbar.is-dark .navbar-brand > .navbar-item, .content kbd.navbar .navbar-brand > .navbar-item,
+    .navbar.is-dark .navbar-brand .navbar-link,
+    .content kbd.navbar .navbar-brand .navbar-link {
+      color: whitesmoke; }
+    .navbar.is-dark .navbar-brand > a.navbar-item:focus, .content kbd.navbar .navbar-brand > a.navbar-item:focus, .navbar.is-dark .navbar-brand > a.navbar-item:hover, .content kbd.navbar .navbar-brand > a.navbar-item:hover, .navbar.is-dark .navbar-brand > a.navbar-item.is-active, .content kbd.navbar .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-dark .navbar-brand .navbar-link:focus,
+    .content kbd.navbar .navbar-brand .navbar-link:focus,
+    .navbar.is-dark .navbar-brand .navbar-link:hover,
+    .content kbd.navbar .navbar-brand .navbar-link:hover,
+    .navbar.is-dark .navbar-brand .navbar-link.is-active,
+    .content kbd.navbar .navbar-brand .navbar-link.is-active {
+      background-color: #292929;
+      color: whitesmoke; }
+    .navbar.is-dark .navbar-brand .navbar-link::after, .content kbd.navbar .navbar-brand .navbar-link::after {
+      border-color: whitesmoke; }
+    .navbar.is-dark .navbar-burger, .content kbd.navbar .navbar-burger {
+      color: whitesmoke; }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-dark .navbar-start > .navbar-item, .content kbd.navbar .navbar-start > .navbar-item,
+      .navbar.is-dark .navbar-start .navbar-link,
+      .content kbd.navbar .navbar-start .navbar-link,
+      .navbar.is-dark .navbar-end > .navbar-item,
+      .content kbd.navbar .navbar-end > .navbar-item,
+      .navbar.is-dark .navbar-end .navbar-link,
+      .content kbd.navbar .navbar-end .navbar-link {
+        color: whitesmoke; }
+      .navbar.is-dark .navbar-start > a.navbar-item:focus, .content kbd.navbar .navbar-start > a.navbar-item:focus, .navbar.is-dark .navbar-start > a.navbar-item:hover, .content kbd.navbar .navbar-start > a.navbar-item:hover, .navbar.is-dark .navbar-start > a.navbar-item.is-active, .content kbd.navbar .navbar-start > a.navbar-item.is-active,
+      .navbar.is-dark .navbar-start .navbar-link:focus,
+      .content kbd.navbar .navbar-start .navbar-link:focus,
+      .navbar.is-dark .navbar-start .navbar-link:hover,
+      .content kbd.navbar .navbar-start .navbar-link:hover,
+      .navbar.is-dark .navbar-start .navbar-link.is-active,
+      .content kbd.navbar .navbar-start .navbar-link.is-active,
+      .navbar.is-dark .navbar-end > a.navbar-item:focus,
+      .content kbd.navbar .navbar-end > a.navbar-item:focus,
+      .navbar.is-dark .navbar-end > a.navbar-item:hover,
+      .content kbd.navbar .navbar-end > a.navbar-item:hover,
+      .navbar.is-dark .navbar-end > a.navbar-item.is-active,
+      .content kbd.navbar .navbar-end > a.navbar-item.is-active,
+      .navbar.is-dark .navbar-end .navbar-link:focus,
+      .content kbd.navbar .navbar-end .navbar-link:focus,
+      .navbar.is-dark .navbar-end .navbar-link:hover,
+      .content kbd.navbar .navbar-end .navbar-link:hover,
+      .navbar.is-dark .navbar-end .navbar-link.is-active,
+      .content kbd.navbar .navbar-end .navbar-link.is-active {
+        background-color: #292929;
+        color: whitesmoke; }
+      .navbar.is-dark .navbar-start .navbar-link::after, .content kbd.navbar .navbar-start .navbar-link::after,
+      .navbar.is-dark .navbar-end .navbar-link::after,
+      .content kbd.navbar .navbar-end .navbar-link::after {
+        border-color: whitesmoke; }
+      .navbar.is-dark .navbar-item.has-dropdown:focus .navbar-link, .content kbd.navbar .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-dark .navbar-item.has-dropdown:hover .navbar-link,
+      .content kbd.navbar .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-dark .navbar-item.has-dropdown.is-active .navbar-link,
+      .content kbd.navbar .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: #292929;
+        color: whitesmoke; }
+      .navbar.is-dark .navbar-dropdown a.navbar-item.is-active, .content kbd.navbar .navbar-dropdown a.navbar-item.is-active {
+        background-color: #363636;
+        color: whitesmoke; } }
+  .navbar.is-primary, .docstring > section > a.navbar.docs-sourcelink {
+    background-color: #4eb5de;
+    color: #fff; }
+    .navbar.is-primary .navbar-brand > .navbar-item, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > .navbar-item,
+    .navbar.is-primary .navbar-brand .navbar-link,
+    .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link {
+      color: #fff; }
+    .navbar.is-primary .navbar-brand > a.navbar-item:focus, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:focus, .navbar.is-primary .navbar-brand > a.navbar-item:hover, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:hover, .navbar.is-primary .navbar-brand > a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-primary .navbar-brand .navbar-link:focus,
+    .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:focus,
+    .navbar.is-primary .navbar-brand .navbar-link:hover,
+    .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:hover,
+    .navbar.is-primary .navbar-brand .navbar-link.is-active,
+    .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link.is-active {
+      background-color: #39acda;
+      color: #fff; }
+    .navbar.is-primary .navbar-brand .navbar-link::after, .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link::after {
+      border-color: #fff; }
+    .navbar.is-primary .navbar-burger, .docstring > section > a.navbar.docs-sourcelink .navbar-burger {
+      color: #fff; }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-primary .navbar-start > .navbar-item, .docstring > section > a.navbar.docs-sourcelink .navbar-start > .navbar-item,
+      .navbar.is-primary .navbar-start .navbar-link,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link,
+      .navbar.is-primary .navbar-end > .navbar-item,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-end > .navbar-item,
+      .navbar.is-primary .navbar-end .navbar-link,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link {
+        color: #fff; }
+      .navbar.is-primary .navbar-start > a.navbar-item:focus, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:focus, .navbar.is-primary .navbar-start > a.navbar-item:hover, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:hover, .navbar.is-primary .navbar-start > a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item.is-active,
+      .navbar.is-primary .navbar-start .navbar-link:focus,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:focus,
+      .navbar.is-primary .navbar-start .navbar-link:hover,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:hover,
+      .navbar.is-primary .navbar-start .navbar-link.is-active,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link.is-active,
+      .navbar.is-primary .navbar-end > a.navbar-item:focus,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:focus,
+      .navbar.is-primary .navbar-end > a.navbar-item:hover,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:hover,
+      .navbar.is-primary .navbar-end > a.navbar-item.is-active,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item.is-active,
+      .navbar.is-primary .navbar-end .navbar-link:focus,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:focus,
+      .navbar.is-primary .navbar-end .navbar-link:hover,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:hover,
+      .navbar.is-primary .navbar-end .navbar-link.is-active,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link.is-active {
+        background-color: #39acda;
+        color: #fff; }
+      .navbar.is-primary .navbar-start .navbar-link::after, .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link::after,
+      .navbar.is-primary .navbar-end .navbar-link::after,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link::after {
+        border-color: #fff; }
+      .navbar.is-primary .navbar-item.has-dropdown:focus .navbar-link, .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-primary .navbar-item.has-dropdown:hover .navbar-link,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-primary .navbar-item.has-dropdown.is-active .navbar-link,
+      .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: #39acda;
+        color: #fff; }
+      .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active {
+        background-color: #4eb5de;
+        color: #fff; } }
+  .navbar.is-link {
+    background-color: #2e63b8;
+    color: #fff; }
+    .navbar.is-link .navbar-brand > .navbar-item,
+    .navbar.is-link .navbar-brand .navbar-link {
+      color: #fff; }
+    .navbar.is-link .navbar-brand > a.navbar-item:focus, .navbar.is-link .navbar-brand > a.navbar-item:hover, .navbar.is-link .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-link .navbar-brand .navbar-link:focus,
+    .navbar.is-link .navbar-brand .navbar-link:hover,
+    .navbar.is-link .navbar-brand .navbar-link.is-active {
+      background-color: #2958a4;
+      color: #fff; }
+    .navbar.is-link .navbar-brand .navbar-link::after {
+      border-color: #fff; }
+    .navbar.is-link .navbar-burger {
+      color: #fff; }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-link .navbar-start > .navbar-item,
+      .navbar.is-link .navbar-start .navbar-link,
+      .navbar.is-link .navbar-end > .navbar-item,
+      .navbar.is-link .navbar-end .navbar-link {
+        color: #fff; }
+      .navbar.is-link .navbar-start > a.navbar-item:focus, .navbar.is-link .navbar-start > a.navbar-item:hover, .navbar.is-link .navbar-start > a.navbar-item.is-active,
+      .navbar.is-link .navbar-start .navbar-link:focus,
+      .navbar.is-link .navbar-start .navbar-link:hover,
+      .navbar.is-link .navbar-start .navbar-link.is-active,
+      .navbar.is-link .navbar-end > a.navbar-item:focus,
+      .navbar.is-link .navbar-end > a.navbar-item:hover,
+      .navbar.is-link .navbar-end > a.navbar-item.is-active,
+      .navbar.is-link .navbar-end .navbar-link:focus,
+      .navbar.is-link .navbar-end .navbar-link:hover,
+      .navbar.is-link .navbar-end .navbar-link.is-active {
+        background-color: #2958a4;
+        color: #fff; }
+      .navbar.is-link .navbar-start .navbar-link::after,
+      .navbar.is-link .navbar-end .navbar-link::after {
+        border-color: #fff; }
+      .navbar.is-link .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-link .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-link .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: #2958a4;
+        color: #fff; }
+      .navbar.is-link .navbar-dropdown a.navbar-item.is-active {
+        background-color: #2e63b8;
+        color: #fff; } }
+  .navbar.is-info {
+    background-color: #209cee;
+    color: #fff; }
+    .navbar.is-info .navbar-brand > .navbar-item,
+    .navbar.is-info .navbar-brand .navbar-link {
+      color: #fff; }
+    .navbar.is-info .navbar-brand > a.navbar-item:focus, .navbar.is-info .navbar-brand > a.navbar-item:hover, .navbar.is-info .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-info .navbar-brand .navbar-link:focus,
+    .navbar.is-info .navbar-brand .navbar-link:hover,
+    .navbar.is-info .navbar-brand .navbar-link.is-active {
+      background-color: #118fe4;
+      color: #fff; }
+    .navbar.is-info .navbar-brand .navbar-link::after {
+      border-color: #fff; }
+    .navbar.is-info .navbar-burger {
+      color: #fff; }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-info .navbar-start > .navbar-item,
+      .navbar.is-info .navbar-start .navbar-link,
+      .navbar.is-info .navbar-end > .navbar-item,
+      .navbar.is-info .navbar-end .navbar-link {
+        color: #fff; }
+      .navbar.is-info .navbar-start > a.navbar-item:focus, .navbar.is-info .navbar-start > a.navbar-item:hover, .navbar.is-info .navbar-start > a.navbar-item.is-active,
+      .navbar.is-info .navbar-start .navbar-link:focus,
+      .navbar.is-info .navbar-start .navbar-link:hover,
+      .navbar.is-info .navbar-start .navbar-link.is-active,
+      .navbar.is-info .navbar-end > a.navbar-item:focus,
+      .navbar.is-info .navbar-end > a.navbar-item:hover,
+      .navbar.is-info .navbar-end > a.navbar-item.is-active,
+      .navbar.is-info .navbar-end .navbar-link:focus,
+      .navbar.is-info .navbar-end .navbar-link:hover,
+      .navbar.is-info .navbar-end .navbar-link.is-active {
+        background-color: #118fe4;
+        color: #fff; }
+      .navbar.is-info .navbar-start .navbar-link::after,
+      .navbar.is-info .navbar-end .navbar-link::after {
+        border-color: #fff; }
+      .navbar.is-info .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-info .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-info .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: #118fe4;
+        color: #fff; }
+      .navbar.is-info .navbar-dropdown a.navbar-item.is-active {
+        background-color: #209cee;
+        color: #fff; } }
+  .navbar.is-success {
+    background-color: #22c35b;
+    color: #fff; }
+    .navbar.is-success .navbar-brand > .navbar-item,
+    .navbar.is-success .navbar-brand .navbar-link {
+      color: #fff; }
+    .navbar.is-success .navbar-brand > a.navbar-item:focus, .navbar.is-success .navbar-brand > a.navbar-item:hover, .navbar.is-success .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-success .navbar-brand .navbar-link:focus,
+    .navbar.is-success .navbar-brand .navbar-link:hover,
+    .navbar.is-success .navbar-brand .navbar-link.is-active {
+      background-color: #1ead51;
+      color: #fff; }
+    .navbar.is-success .navbar-brand .navbar-link::after {
+      border-color: #fff; }
+    .navbar.is-success .navbar-burger {
+      color: #fff; }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-success .navbar-start > .navbar-item,
+      .navbar.is-success .navbar-start .navbar-link,
+      .navbar.is-success .navbar-end > .navbar-item,
+      .navbar.is-success .navbar-end .navbar-link {
+        color: #fff; }
+      .navbar.is-success .navbar-start > a.navbar-item:focus, .navbar.is-success .navbar-start > a.navbar-item:hover, .navbar.is-success .navbar-start > a.navbar-item.is-active,
+      .navbar.is-success .navbar-start .navbar-link:focus,
+      .navbar.is-success .navbar-start .navbar-link:hover,
+      .navbar.is-success .navbar-start .navbar-link.is-active,
+      .navbar.is-success .navbar-end > a.navbar-item:focus,
+      .navbar.is-success .navbar-end > a.navbar-item:hover,
+      .navbar.is-success .navbar-end > a.navbar-item.is-active,
+      .navbar.is-success .navbar-end .navbar-link:focus,
+      .navbar.is-success .navbar-end .navbar-link:hover,
+      .navbar.is-success .navbar-end .navbar-link.is-active {
+        background-color: #1ead51;
+        color: #fff; }
+      .navbar.is-success .navbar-start .navbar-link::after,
+      .navbar.is-success .navbar-end .navbar-link::after {
+        border-color: #fff; }
+      .navbar.is-success .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-success .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-success .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: #1ead51;
+        color: #fff; }
+      .navbar.is-success .navbar-dropdown a.navbar-item.is-active {
+        background-color: #22c35b;
+        color: #fff; } }
+  .navbar.is-warning {
+    background-color: #ffdd57;
+    color: rgba(0, 0, 0, 0.7); }
+    .navbar.is-warning .navbar-brand > .navbar-item,
+    .navbar.is-warning .navbar-brand .navbar-link {
+      color: rgba(0, 0, 0, 0.7); }
+    .navbar.is-warning .navbar-brand > a.navbar-item:focus, .navbar.is-warning .navbar-brand > a.navbar-item:hover, .navbar.is-warning .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-warning .navbar-brand .navbar-link:focus,
+    .navbar.is-warning .navbar-brand .navbar-link:hover,
+    .navbar.is-warning .navbar-brand .navbar-link.is-active {
+      background-color: #ffd83d;
+      color: rgba(0, 0, 0, 0.7); }
+    .navbar.is-warning .navbar-brand .navbar-link::after {
+      border-color: rgba(0, 0, 0, 0.7); }
+    .navbar.is-warning .navbar-burger {
+      color: rgba(0, 0, 0, 0.7); }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-warning .navbar-start > .navbar-item,
+      .navbar.is-warning .navbar-start .navbar-link,
+      .navbar.is-warning .navbar-end > .navbar-item,
+      .navbar.is-warning .navbar-end .navbar-link {
+        color: rgba(0, 0, 0, 0.7); }
+      .navbar.is-warning .navbar-start > a.navbar-item:focus, .navbar.is-warning .navbar-start > a.navbar-item:hover, .navbar.is-warning .navbar-start > a.navbar-item.is-active,
+      .navbar.is-warning .navbar-start .navbar-link:focus,
+      .navbar.is-warning .navbar-start .navbar-link:hover,
+      .navbar.is-warning .navbar-start .navbar-link.is-active,
+      .navbar.is-warning .navbar-end > a.navbar-item:focus,
+      .navbar.is-warning .navbar-end > a.navbar-item:hover,
+      .navbar.is-warning .navbar-end > a.navbar-item.is-active,
+      .navbar.is-warning .navbar-end .navbar-link:focus,
+      .navbar.is-warning .navbar-end .navbar-link:hover,
+      .navbar.is-warning .navbar-end .navbar-link.is-active {
+        background-color: #ffd83d;
+        color: rgba(0, 0, 0, 0.7); }
+      .navbar.is-warning .navbar-start .navbar-link::after,
+      .navbar.is-warning .navbar-end .navbar-link::after {
+        border-color: rgba(0, 0, 0, 0.7); }
+      .navbar.is-warning .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-warning .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-warning .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: #ffd83d;
+        color: rgba(0, 0, 0, 0.7); }
+      .navbar.is-warning .navbar-dropdown a.navbar-item.is-active {
+        background-color: #ffdd57;
+        color: rgba(0, 0, 0, 0.7); } }
+  .navbar.is-danger {
+    background-color: #da0b00;
+    color: #fff; }
+    .navbar.is-danger .navbar-brand > .navbar-item,
+    .navbar.is-danger .navbar-brand .navbar-link {
+      color: #fff; }
+    .navbar.is-danger .navbar-brand > a.navbar-item:focus, .navbar.is-danger .navbar-brand > a.navbar-item:hover, .navbar.is-danger .navbar-brand > a.navbar-item.is-active,
+    .navbar.is-danger .navbar-brand .navbar-link:focus,
+    .navbar.is-danger .navbar-brand .navbar-link:hover,
+    .navbar.is-danger .navbar-brand .navbar-link.is-active {
+      background-color: #c10a00;
+      color: #fff; }
+    .navbar.is-danger .navbar-brand .navbar-link::after {
+      border-color: #fff; }
+    .navbar.is-danger .navbar-burger {
+      color: #fff; }
+    @media screen and (min-width: 1056px) {
+      .navbar.is-danger .navbar-start > .navbar-item,
+      .navbar.is-danger .navbar-start .navbar-link,
+      .navbar.is-danger .navbar-end > .navbar-item,
+      .navbar.is-danger .navbar-end .navbar-link {
+        color: #fff; }
+      .navbar.is-danger .navbar-start > a.navbar-item:focus, .navbar.is-danger .navbar-start > a.navbar-item:hover, .navbar.is-danger .navbar-start > a.navbar-item.is-active,
+      .navbar.is-danger .navbar-start .navbar-link:focus,
+      .navbar.is-danger .navbar-start .navbar-link:hover,
+      .navbar.is-danger .navbar-start .navbar-link.is-active,
+      .navbar.is-danger .navbar-end > a.navbar-item:focus,
+      .navbar.is-danger .navbar-end > a.navbar-item:hover,
+      .navbar.is-danger .navbar-end > a.navbar-item.is-active,
+      .navbar.is-danger .navbar-end .navbar-link:focus,
+      .navbar.is-danger .navbar-end .navbar-link:hover,
+      .navbar.is-danger .navbar-end .navbar-link.is-active {
+        background-color: #c10a00;
+        color: #fff; }
+      .navbar.is-danger .navbar-start .navbar-link::after,
+      .navbar.is-danger .navbar-end .navbar-link::after {
+        border-color: #fff; }
+      .navbar.is-danger .navbar-item.has-dropdown:focus .navbar-link,
+      .navbar.is-danger .navbar-item.has-dropdown:hover .navbar-link,
+      .navbar.is-danger .navbar-item.has-dropdown.is-active .navbar-link {
+        background-color: #c10a00;
+        color: #fff; }
+      .navbar.is-danger .navbar-dropdown a.navbar-item.is-active {
+        background-color: #da0b00;
+        color: #fff; } }
+  .navbar > .container {
+    align-items: stretch;
+    display: flex;
+    min-height: 3.25rem;
+    width: 100%; }
+  .navbar.has-shadow {
+    box-shadow: 0 2px 0 0 whitesmoke; }
+  .navbar.is-fixed-bottom, .navbar.is-fixed-top {
+    left: 0;
+    position: fixed;
+    right: 0;
+    z-index: 30; }
+  .navbar.is-fixed-bottom {
+    bottom: 0; }
+    .navbar.is-fixed-bottom.has-shadow {
+      box-shadow: 0 -2px 0 0 whitesmoke; }
+  .navbar.is-fixed-top {
+    top: 0; }
+
+html.has-navbar-fixed-top,
+body.has-navbar-fixed-top {
+  padding-top: 3.25rem; }
+
+html.has-navbar-fixed-bottom,
+body.has-navbar-fixed-bottom {
+  padding-bottom: 3.25rem; }
+
+.navbar-brand,
+.navbar-tabs {
+  align-items: stretch;
+  display: flex;
+  flex-shrink: 0;
+  min-height: 3.25rem; }
+
+.navbar-brand a.navbar-item:focus, .navbar-brand a.navbar-item:hover {
+  background-color: transparent; }
+
+.navbar-tabs {
+  -webkit-overflow-scrolling: touch;
+  max-width: 100vw;
+  overflow-x: auto;
+  overflow-y: hidden; }
+
+.navbar-burger {
+  color: #4a4a4a;
+  cursor: pointer;
+  display: block;
+  height: 3.25rem;
+  position: relative;
+  width: 3.25rem;
+  margin-left: auto; }
+  .navbar-burger span {
+    background-color: currentColor;
+    display: block;
+    height: 1px;
+    left: calc(50% - 8px);
+    position: absolute;
+    transform-origin: center;
+    transition-duration: 86ms;
+    transition-property: background-color, opacity, transform;
+    transition-timing-function: ease-out;
+    width: 16px; }
+    .navbar-burger span:nth-child(1) {
+      top: calc(50% - 6px); }
+    .navbar-burger span:nth-child(2) {
+      top: calc(50% - 1px); }
+    .navbar-burger span:nth-child(3) {
+      top: calc(50% + 4px); }
+  .navbar-burger:hover {
+    background-color: rgba(0, 0, 0, 0.05); }
+  .navbar-burger.is-active span:nth-child(1) {
+    transform: translateY(5px) rotate(45deg); }
+  .navbar-burger.is-active span:nth-child(2) {
+    opacity: 0; }
+  .navbar-burger.is-active span:nth-child(3) {
+    transform: translateY(-5px) rotate(-45deg); }
+
+.navbar-menu {
+  display: none; }
+
+.navbar-item,
+.navbar-link {
+  color: #4a4a4a;
+  display: block;
+  line-height: 1.5;
+  padding: 0.5rem 0.75rem;
+  position: relative; }
+  .navbar-item .icon:only-child,
+  .navbar-link .icon:only-child {
+    margin-left: -0.25rem;
+    margin-right: -0.25rem; }
+
+a.navbar-item,
+.navbar-link {
+  cursor: pointer; }
+  a.navbar-item:focus, a.navbar-item:focus-within, a.navbar-item:hover, a.navbar-item.is-active,
+  .navbar-link:focus,
+  .navbar-link:focus-within,
+  .navbar-link:hover,
+  .navbar-link.is-active {
+    background-color: #fafafa;
+    color: #2e63b8; }
+
+.navbar-item {
+  display: block;
+  flex-grow: 0;
+  flex-shrink: 0; }
+  .navbar-item img {
+    max-height: 1.75rem; }
+  .navbar-item.has-dropdown {
+    padding: 0; }
+  .navbar-item.is-expanded {
+    flex-grow: 1;
+    flex-shrink: 1; }
+  .navbar-item.is-tab {
+    border-bottom: 1px solid transparent;
+    min-height: 3.25rem;
+    padding-bottom: calc(0.5rem - 1px); }
+    .navbar-item.is-tab:focus, .navbar-item.is-tab:hover {
+      background-color: transparent;
+      border-bottom-color: #2e63b8; }
+    .navbar-item.is-tab.is-active {
+      background-color: transparent;
+      border-bottom-color: #2e63b8;
+      border-bottom-style: solid;
+      border-bottom-width: 3px;
+      color: #2e63b8;
+      padding-bottom: calc(0.5rem - 3px); }
+
+.navbar-content {
+  flex-grow: 1;
+  flex-shrink: 1; }
+
+.navbar-link:not(.is-arrowless) {
+  padding-right: 2.5em; }
+  .navbar-link:not(.is-arrowless)::after {
+    border-color: #2e63b8;
+    margin-top: -0.375em;
+    right: 1.125em; }
+
+.navbar-dropdown {
+  font-size: 0.875rem;
+  padding-bottom: 0.5rem;
+  padding-top: 0.5rem; }
+  .navbar-dropdown .navbar-item {
+    padding-left: 1.5rem;
+    padding-right: 1.5rem; }
+
+.navbar-divider {
+  background-color: whitesmoke;
+  border: none;
+  display: none;
+  height: 2px;
+  margin: 0.5rem 0; }
+
+@media screen and (max-width: 1055px) {
+  .navbar > .container {
+    display: block; }
+  .navbar-brand .navbar-item,
+  .navbar-tabs .navbar-item {
+    align-items: center;
+    display: flex; }
+  .navbar-link::after {
+    display: none; }
+  .navbar-menu {
+    background-color: white;
+    box-shadow: 0 8px 16px rgba(10, 10, 10, 0.1);
+    padding: 0.5rem 0; }
+    .navbar-menu.is-active {
+      display: block; }
+  .navbar.is-fixed-bottom-touch, .navbar.is-fixed-top-touch {
+    left: 0;
+    position: fixed;
+    right: 0;
+    z-index: 30; }
+  .navbar.is-fixed-bottom-touch {
+    bottom: 0; }
+    .navbar.is-fixed-bottom-touch.has-shadow {
+      box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); }
+  .navbar.is-fixed-top-touch {
+    top: 0; }
+  .navbar.is-fixed-top .navbar-menu, .navbar.is-fixed-top-touch .navbar-menu {
+    -webkit-overflow-scrolling: touch;
+    max-height: calc(100vh - 3.25rem);
+    overflow: auto; }
+  html.has-navbar-fixed-top-touch,
+  body.has-navbar-fixed-top-touch {
+    padding-top: 3.25rem; }
+  html.has-navbar-fixed-bottom-touch,
+  body.has-navbar-fixed-bottom-touch {
+    padding-bottom: 3.25rem; } }
+
+@media screen and (min-width: 1056px) {
+  .navbar,
+  .navbar-menu,
+  .navbar-start,
+  .navbar-end {
+    align-items: stretch;
+    display: flex; }
+  .navbar {
+    min-height: 3.25rem; }
+    .navbar.is-spaced {
+      padding: 1rem 2rem; }
+      .navbar.is-spaced .navbar-start,
+      .navbar.is-spaced .navbar-end {
+        align-items: center; }
+      .navbar.is-spaced a.navbar-item,
+      .navbar.is-spaced .navbar-link {
+        border-radius: 4px; }
+    .navbar.is-transparent a.navbar-item:focus, .navbar.is-transparent a.navbar-item:hover, .navbar.is-transparent a.navbar-item.is-active,
+    .navbar.is-transparent .navbar-link:focus,
+    .navbar.is-transparent .navbar-link:hover,
+    .navbar.is-transparent .navbar-link.is-active {
+      background-color: transparent !important; }
+    .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:hover .navbar-link {
+      background-color: transparent !important; }
+    .navbar.is-transparent .navbar-dropdown a.navbar-item:focus, .navbar.is-transparent .navbar-dropdown a.navbar-item:hover {
+      background-color: whitesmoke;
+      color: #0a0a0a; }
+    .navbar.is-transparent .navbar-dropdown a.navbar-item.is-active {
+      background-color: whitesmoke;
+      color: #2e63b8; }
+  .navbar-burger {
+    display: none; }
+  .navbar-item,
+  .navbar-link {
+    align-items: center;
+    display: flex; }
+  .navbar-item {
+    display: flex; }
+    .navbar-item.has-dropdown {
+      align-items: stretch; }
+    .navbar-item.has-dropdown-up .navbar-link::after {
+      transform: rotate(135deg) translate(0.25em, -0.25em); }
+    .navbar-item.has-dropdown-up .navbar-dropdown {
+      border-bottom: 2px solid #dbdbdb;
+      border-radius: 6px 6px 0 0;
+      border-top: none;
+      bottom: 100%;
+      box-shadow: 0 -8px 8px rgba(10, 10, 10, 0.1);
+      top: auto; }
+    .navbar-item.is-active .navbar-dropdown, .navbar-item.is-hoverable:focus .navbar-dropdown, .navbar-item.is-hoverable:focus-within .navbar-dropdown, .navbar-item.is-hoverable:hover .navbar-dropdown {
+      display: block; }
+      .navbar.is-spaced .navbar-item.is-active .navbar-dropdown, .navbar-item.is-active .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:focus .navbar-dropdown, .navbar-item.is-hoverable:focus .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:focus-within .navbar-dropdown, .navbar-item.is-hoverable:focus-within .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:hover .navbar-dropdown, .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed {
+        opacity: 1;
+        pointer-events: auto;
+        transform: translateY(0); }
+  .navbar-menu {
+    flex-grow: 1;
+    flex-shrink: 0; }
+  .navbar-start {
+    justify-content: flex-start;
+    margin-right: auto; }
+  .navbar-end {
+    justify-content: flex-end;
+    margin-left: auto; }
+  .navbar-dropdown {
+    background-color: white;
+    border-bottom-left-radius: 6px;
+    border-bottom-right-radius: 6px;
+    border-top: 2px solid #dbdbdb;
+    box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1);
+    display: none;
+    font-size: 0.875rem;
+    left: 0;
+    min-width: 100%;
+    position: absolute;
+    top: 100%;
+    z-index: 20; }
+    .navbar-dropdown .navbar-item {
+      padding: 0.375rem 1rem;
+      white-space: nowrap; }
+    .navbar-dropdown a.navbar-item {
+      padding-right: 3rem; }
+      .navbar-dropdown a.navbar-item:focus, .navbar-dropdown a.navbar-item:hover {
+        background-color: whitesmoke;
+        color: #0a0a0a; }
+      .navbar-dropdown a.navbar-item.is-active {
+        background-color: whitesmoke;
+        color: #2e63b8; }
+    .navbar.is-spaced .navbar-dropdown, .navbar-dropdown.is-boxed {
+      border-radius: 6px;
+      border-top: none;
+      box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1);
+      display: block;
+      opacity: 0;
+      pointer-events: none;
+      top: calc(100% + (-4px));
+      transform: translateY(-5px);
+      transition-duration: 86ms;
+      transition-property: opacity, transform; }
+    .navbar-dropdown.is-right {
+      left: auto;
+      right: 0; }
+  .navbar-divider {
+    display: block; }
+  .navbar > .container .navbar-brand,
+  .container > .navbar .navbar-brand {
+    margin-left: -.75rem; }
+  .navbar > .container .navbar-menu,
+  .container > .navbar .navbar-menu {
+    margin-right: -.75rem; }
+  .navbar.is-fixed-bottom-desktop, .navbar.is-fixed-top-desktop {
+    left: 0;
+    position: fixed;
+    right: 0;
+    z-index: 30; }
+  .navbar.is-fixed-bottom-desktop {
+    bottom: 0; }
+    .navbar.is-fixed-bottom-desktop.has-shadow {
+      box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); }
+  .navbar.is-fixed-top-desktop {
+    top: 0; }
+  html.has-navbar-fixed-top-desktop,
+  body.has-navbar-fixed-top-desktop {
+    padding-top: 3.25rem; }
+  html.has-navbar-fixed-bottom-desktop,
+  body.has-navbar-fixed-bottom-desktop {
+    padding-bottom: 3.25rem; }
+  html.has-spaced-navbar-fixed-top,
+  body.has-spaced-navbar-fixed-top {
+    padding-top: 5.25rem; }
+  html.has-spaced-navbar-fixed-bottom,
+  body.has-spaced-navbar-fixed-bottom {
+    padding-bottom: 5.25rem; }
+  a.navbar-item.is-active,
+  .navbar-link.is-active {
+    color: #0a0a0a; }
+  a.navbar-item.is-active:not(:focus):not(:hover),
+  .navbar-link.is-active:not(:focus):not(:hover) {
+    background-color: transparent; }
+  .navbar-item.has-dropdown:focus .navbar-link, .navbar-item.has-dropdown:hover .navbar-link, .navbar-item.has-dropdown.is-active .navbar-link {
+    background-color: #fafafa; } }
+
+.hero.is-fullheight-with-navbar {
+  min-height: calc(100vh - 3.25rem); }
+
+.pagination {
+  font-size: 1rem;
+  margin: -0.25rem; }
+  .pagination.is-small, #documenter .docs-sidebar form.docs-search > input.pagination {
+    font-size: 0.75rem; }
+  .pagination.is-medium {
+    font-size: 1.25rem; }
+  .pagination.is-large {
+    font-size: 1.5rem; }
+  .pagination.is-rounded .pagination-previous, #documenter .docs-sidebar form.docs-search > input.pagination .pagination-previous,
+  .pagination.is-rounded .pagination-next,
+  #documenter .docs-sidebar form.docs-search > input.pagination .pagination-next {
+    padding-left: 1em;
+    padding-right: 1em;
+    border-radius: 290486px; }
+  .pagination.is-rounded .pagination-link, #documenter .docs-sidebar form.docs-search > input.pagination .pagination-link {
+    border-radius: 290486px; }
+
+.pagination,
+.pagination-list {
+  align-items: center;
+  display: flex;
+  justify-content: center;
+  text-align: center; }
+
+.pagination-previous,
+.pagination-next,
+.pagination-link,
+.pagination-ellipsis {
+  font-size: 1em;
+  justify-content: center;
+  margin: 0.25rem;
+  padding-left: 0.5em;
+  padding-right: 0.5em;
+  text-align: center; }
+
+.pagination-previous,
+.pagination-next,
+.pagination-link {
+  border-color: #dbdbdb;
+  color: #363636;
+  min-width: 2.25em; }
+  .pagination-previous:hover,
+  .pagination-next:hover,
+  .pagination-link:hover {
+    border-color: #b5b5b5;
+    color: #363636; }
+  .pagination-previous:focus,
+  .pagination-next:focus,
+  .pagination-link:focus {
+    border-color: #2e63b8; }
+  .pagination-previous:active,
+  .pagination-next:active,
+  .pagination-link:active {
+    box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2); }
+  .pagination-previous[disabled],
+  .pagination-next[disabled],
+  .pagination-link[disabled] {
+    background-color: #dbdbdb;
+    border-color: #dbdbdb;
+    box-shadow: none;
+    color: #7a7a7a;
+    opacity: 0.5; }
+
+.pagination-previous,
+.pagination-next {
+  padding-left: 0.75em;
+  padding-right: 0.75em;
+  white-space: nowrap; }
+
+.pagination-link.is-current {
+  background-color: #2e63b8;
+  border-color: #2e63b8;
+  color: #fff; }
+
+.pagination-ellipsis {
+  color: #b5b5b5;
+  pointer-events: none; }
+
+.pagination-list {
+  flex-wrap: wrap; }
+
+@media screen and (max-width: 768px) {
+  .pagination {
+    flex-wrap: wrap; }
+  .pagination-previous,
+  .pagination-next {
+    flex-grow: 1;
+    flex-shrink: 1; }
+  .pagination-list li {
+    flex-grow: 1;
+    flex-shrink: 1; } }
+
+@media screen and (min-width: 769px), print {
+  .pagination-list {
+    flex-grow: 1;
+    flex-shrink: 1;
+    justify-content: flex-start;
+    order: 1; }
+  .pagination-previous {
+    order: 2; }
+  .pagination-next {
+    order: 3; }
+  .pagination {
+    justify-content: space-between; }
+    .pagination.is-centered .pagination-previous {
+      order: 1; }
+    .pagination.is-centered .pagination-list {
+      justify-content: center;
+      order: 2; }
+    .pagination.is-centered .pagination-next {
+      order: 3; }
+    .pagination.is-right .pagination-previous {
+      order: 1; }
+    .pagination.is-right .pagination-next {
+      order: 2; }
+    .pagination.is-right .pagination-list {
+      justify-content: flex-end;
+      order: 3; } }
+
+.panel {
+  font-size: 1rem; }
+  .panel:not(:last-child) {
+    margin-bottom: 1.5rem; }
+
+.panel-heading,
+.panel-tabs,
+.panel-block {
+  border-bottom: 1px solid #dbdbdb;
+  border-left: 1px solid #dbdbdb;
+  border-right: 1px solid #dbdbdb; }
+  .panel-heading:first-child,
+  .panel-tabs:first-child,
+  .panel-block:first-child {
+    border-top: 1px solid #dbdbdb; }
+
+.panel-heading {
+  background-color: whitesmoke;
+  border-radius: 4px 4px 0 0;
+  color: #222222;
+  font-size: 1.25em;
+  font-weight: 300;
+  line-height: 1.25;
+  padding: 0.5em 0.75em; }
+
+.panel-tabs {
+  align-items: flex-end;
+  display: flex;
+  font-size: 0.875em;
+  justify-content: center; }
+  .panel-tabs a {
+    border-bottom: 1px solid #dbdbdb;
+    margin-bottom: -1px;
+    padding: 0.5em; }
+    .panel-tabs a.is-active {
+      border-bottom-color: #4a4a4a;
+      color: #363636; }
+
+.panel-list a {
+  color: #222222; }
+  .panel-list a:hover {
+    color: #2e63b8; }
+
+.panel-block {
+  align-items: center;
+  color: #222222;
+  display: flex;
+  justify-content: flex-start;
+  padding: 0.5em 0.75em; }
+  .panel-block input[type="checkbox"] {
+    margin-right: 0.75em; }
+  .panel-block > .control {
+    flex-grow: 1;
+    flex-shrink: 1;
+    width: 100%; }
+  .panel-block.is-wrapped {
+    flex-wrap: wrap; }
+  .panel-block.is-active {
+    border-left-color: #2e63b8;
+    color: #363636; }
+    .panel-block.is-active .panel-icon {
+      color: #2e63b8; }
+
+a.panel-block,
+label.panel-block {
+  cursor: pointer; }
+  a.panel-block:hover,
+  label.panel-block:hover {
+    background-color: whitesmoke; }
+
+.panel-icon {
+  display: inline-block;
+  font-size: 14px;
+  height: 1em;
+  line-height: 1em;
+  text-align: center;
+  vertical-align: top;
+  width: 1em;
+  color: #7a7a7a;
+  margin-right: 0.75em; }
+  .panel-icon .fa {
+    font-size: inherit;
+    line-height: inherit; }
+
+.tabs {
+  -webkit-overflow-scrolling: touch;
+  align-items: stretch;
+  display: flex;
+  font-size: 1rem;
+  justify-content: space-between;
+  overflow: hidden;
+  overflow-x: auto;
+  white-space: nowrap; }
+  .tabs a {
+    align-items: center;
+    border-bottom-color: #dbdbdb;
+    border-bottom-style: solid;
+    border-bottom-width: 1px;
+    color: #222222;
+    display: flex;
+    justify-content: center;
+    margin-bottom: -1px;
+    padding: 0.5em 1em;
+    vertical-align: top; }
+    .tabs a:hover {
+      border-bottom-color: #222222;
+      color: #222222; }
+  .tabs li {
+    display: block; }
+    .tabs li.is-active a {
+      border-bottom-color: #2e63b8;
+      color: #2e63b8; }
+  .tabs ul {
+    align-items: center;
+    border-bottom-color: #dbdbdb;
+    border-bottom-style: solid;
+    border-bottom-width: 1px;
+    display: flex;
+    flex-grow: 1;
+    flex-shrink: 0;
+    justify-content: flex-start; }
+    .tabs ul.is-left {
+      padding-right: 0.75em; }
+    .tabs ul.is-center {
+      flex: none;
+      justify-content: center;
+      padding-left: 0.75em;
+      padding-right: 0.75em; }
+    .tabs ul.is-right {
+      justify-content: flex-end;
+      padding-left: 0.75em; }
+  .tabs .icon:first-child {
+    margin-right: 0.5em; }
+  .tabs .icon:last-child {
+    margin-left: 0.5em; }
+  .tabs.is-centered ul {
+    justify-content: center; }
+  .tabs.is-right ul {
+    justify-content: flex-end; }
+  .tabs.is-boxed a {
+    border: 1px solid transparent;
+    border-radius: 4px 4px 0 0; }
+    .tabs.is-boxed a:hover {
+      background-color: whitesmoke;
+      border-bottom-color: #dbdbdb; }
+  .tabs.is-boxed li.is-active a {
+    background-color: white;
+    border-color: #dbdbdb;
+    border-bottom-color: transparent !important; }
+  .tabs.is-fullwidth li {
+    flex-grow: 1;
+    flex-shrink: 0; }
+  .tabs.is-toggle a {
+    border-color: #dbdbdb;
+    border-style: solid;
+    border-width: 1px;
+    margin-bottom: 0;
+    position: relative; }
+    .tabs.is-toggle a:hover {
+      background-color: whitesmoke;
+      border-color: #b5b5b5;
+      z-index: 2; }
+  .tabs.is-toggle li + li {
+    margin-left: -1px; }
+  .tabs.is-toggle li:first-child a {
+    border-radius: 4px 0 0 4px; }
+  .tabs.is-toggle li:last-child a {
+    border-radius: 0 4px 4px 0; }
+  .tabs.is-toggle li.is-active a {
+    background-color: #2e63b8;
+    border-color: #2e63b8;
+    color: #fff;
+    z-index: 1; }
+  .tabs.is-toggle ul {
+    border-bottom: none; }
+  .tabs.is-toggle.is-toggle-rounded li:first-child a {
+    border-bottom-left-radius: 290486px;
+    border-top-left-radius: 290486px;
+    padding-left: 1.25em; }
+  .tabs.is-toggle.is-toggle-rounded li:last-child a {
+    border-bottom-right-radius: 290486px;
+    border-top-right-radius: 290486px;
+    padding-right: 1.25em; }
+  .tabs.is-small, #documenter .docs-sidebar form.docs-search > input.tabs {
+    font-size: 0.75rem; }
+  .tabs.is-medium {
+    font-size: 1.25rem; }
+  .tabs.is-large {
+    font-size: 1.5rem; }
+
+.column {
+  display: block;
+  flex-basis: 0;
+  flex-grow: 1;
+  flex-shrink: 1;
+  padding: 0.75rem; }
+  .columns.is-mobile > .column.is-narrow {
+    flex: none; }
+  .columns.is-mobile > .column.is-full {
+    flex: none;
+    width: 100%; }
+  .columns.is-mobile > .column.is-three-quarters {
+    flex: none;
+    width: 75%; }
+  .columns.is-mobile > .column.is-two-thirds {
+    flex: none;
+    width: 66.6666%; }
+  .columns.is-mobile > .column.is-half {
+    flex: none;
+    width: 50%; }
+  .columns.is-mobile > .column.is-one-third {
+    flex: none;
+    width: 33.3333%; }
+  .columns.is-mobile > .column.is-one-quarter {
+    flex: none;
+    width: 25%; }
+  .columns.is-mobile > .column.is-one-fifth {
+    flex: none;
+    width: 20%; }
+  .columns.is-mobile > .column.is-two-fifths {
+    flex: none;
+    width: 40%; }
+  .columns.is-mobile > .column.is-three-fifths {
+    flex: none;
+    width: 60%; }
+  .columns.is-mobile > .column.is-four-fifths {
+    flex: none;
+    width: 80%; }
+  .columns.is-mobile > .column.is-offset-three-quarters {
+    margin-left: 75%; }
+  .columns.is-mobile > .column.is-offset-two-thirds {
+    margin-left: 66.6666%; }
+  .columns.is-mobile > .column.is-offset-half {
+    margin-left: 50%; }
+  .columns.is-mobile > .column.is-offset-one-third {
+    margin-left: 33.3333%; }
+  .columns.is-mobile > .column.is-offset-one-quarter {
+    margin-left: 25%; }
+  .columns.is-mobile > .column.is-offset-one-fifth {
+    margin-left: 20%; }
+  .columns.is-mobile > .column.is-offset-two-fifths {
+    margin-left: 40%; }
+  .columns.is-mobile > .column.is-offset-three-fifths {
+    margin-left: 60%; }
+  .columns.is-mobile > .column.is-offset-four-fifths {
+    margin-left: 80%; }
+  .columns.is-mobile > .column.is-0 {
+    flex: none;
+    width: 0%; }
+  .columns.is-mobile > .column.is-offset-0 {
+    margin-left: 0%; }
+  .columns.is-mobile > .column.is-1 {
+    flex: none;
+    width: 8.33333%; }
+  .columns.is-mobile > .column.is-offset-1 {
+    margin-left: 8.33333%; }
+  .columns.is-mobile > .column.is-2 {
+    flex: none;
+    width: 16.66667%; }
+  .columns.is-mobile > .column.is-offset-2 {
+    margin-left: 16.66667%; }
+  .columns.is-mobile > .column.is-3 {
+    flex: none;
+    width: 25%; }
+  .columns.is-mobile > .column.is-offset-3 {
+    margin-left: 25%; }
+  .columns.is-mobile > .column.is-4 {
+    flex: none;
+    width: 33.33333%; }
+  .columns.is-mobile > .column.is-offset-4 {
+    margin-left: 33.33333%; }
+  .columns.is-mobile > .column.is-5 {
+    flex: none;
+    width: 41.66667%; }
+  .columns.is-mobile > .column.is-offset-5 {
+    margin-left: 41.66667%; }
+  .columns.is-mobile > .column.is-6 {
+    flex: none;
+    width: 50%; }
+  .columns.is-mobile > .column.is-offset-6 {
+    margin-left: 50%; }
+  .columns.is-mobile > .column.is-7 {
+    flex: none;
+    width: 58.33333%; }
+  .columns.is-mobile > .column.is-offset-7 {
+    margin-left: 58.33333%; }
+  .columns.is-mobile > .column.is-8 {
+    flex: none;
+    width: 66.66667%; }
+  .columns.is-mobile > .column.is-offset-8 {
+    margin-left: 66.66667%; }
+  .columns.is-mobile > .column.is-9 {
+    flex: none;
+    width: 75%; }
+  .columns.is-mobile > .column.is-offset-9 {
+    margin-left: 75%; }
+  .columns.is-mobile > .column.is-10 {
+    flex: none;
+    width: 83.33333%; }
+  .columns.is-mobile > .column.is-offset-10 {
+    margin-left: 83.33333%; }
+  .columns.is-mobile > .column.is-11 {
+    flex: none;
+    width: 91.66667%; }
+  .columns.is-mobile > .column.is-offset-11 {
+    margin-left: 91.66667%; }
+  .columns.is-mobile > .column.is-12 {
+    flex: none;
+    width: 100%; }
+  .columns.is-mobile > .column.is-offset-12 {
+    margin-left: 100%; }
+  @media screen and (max-width: 768px) {
+    .column.is-narrow-mobile {
+      flex: none; }
+    .column.is-full-mobile {
+      flex: none;
+      width: 100%; }
+    .column.is-three-quarters-mobile {
+      flex: none;
+      width: 75%; }
+    .column.is-two-thirds-mobile {
+      flex: none;
+      width: 66.6666%; }
+    .column.is-half-mobile {
+      flex: none;
+      width: 50%; }
+    .column.is-one-third-mobile {
+      flex: none;
+      width: 33.3333%; }
+    .column.is-one-quarter-mobile {
+      flex: none;
+      width: 25%; }
+    .column.is-one-fifth-mobile {
+      flex: none;
+      width: 20%; }
+    .column.is-two-fifths-mobile {
+      flex: none;
+      width: 40%; }
+    .column.is-three-fifths-mobile {
+      flex: none;
+      width: 60%; }
+    .column.is-four-fifths-mobile {
+      flex: none;
+      width: 80%; }
+    .column.is-offset-three-quarters-mobile {
+      margin-left: 75%; }
+    .column.is-offset-two-thirds-mobile {
+      margin-left: 66.6666%; }
+    .column.is-offset-half-mobile {
+      margin-left: 50%; }
+    .column.is-offset-one-third-mobile {
+      margin-left: 33.3333%; }
+    .column.is-offset-one-quarter-mobile {
+      margin-left: 25%; }
+    .column.is-offset-one-fifth-mobile {
+      margin-left: 20%; }
+    .column.is-offset-two-fifths-mobile {
+      margin-left: 40%; }
+    .column.is-offset-three-fifths-mobile {
+      margin-left: 60%; }
+    .column.is-offset-four-fifths-mobile {
+      margin-left: 80%; }
+    .column.is-0-mobile {
+      flex: none;
+      width: 0%; }
+    .column.is-offset-0-mobile {
+      margin-left: 0%; }
+    .column.is-1-mobile {
+      flex: none;
+      width: 8.33333%; }
+    .column.is-offset-1-mobile {
+      margin-left: 8.33333%; }
+    .column.is-2-mobile {
+      flex: none;
+      width: 16.66667%; }
+    .column.is-offset-2-mobile {
+      margin-left: 16.66667%; }
+    .column.is-3-mobile {
+      flex: none;
+      width: 25%; }
+    .column.is-offset-3-mobile {
+      margin-left: 25%; }
+    .column.is-4-mobile {
+      flex: none;
+      width: 33.33333%; }
+    .column.is-offset-4-mobile {
+      margin-left: 33.33333%; }
+    .column.is-5-mobile {
+      flex: none;
+      width: 41.66667%; }
+    .column.is-offset-5-mobile {
+      margin-left: 41.66667%; }
+    .column.is-6-mobile {
+      flex: none;
+      width: 50%; }
+    .column.is-offset-6-mobile {
+      margin-left: 50%; }
+    .column.is-7-mobile {
+      flex: none;
+      width: 58.33333%; }
+    .column.is-offset-7-mobile {
+      margin-left: 58.33333%; }
+    .column.is-8-mobile {
+      flex: none;
+      width: 66.66667%; }
+    .column.is-offset-8-mobile {
+      margin-left: 66.66667%; }
+    .column.is-9-mobile {
+      flex: none;
+      width: 75%; }
+    .column.is-offset-9-mobile {
+      margin-left: 75%; }
+    .column.is-10-mobile {
+      flex: none;
+      width: 83.33333%; }
+    .column.is-offset-10-mobile {
+      margin-left: 83.33333%; }
+    .column.is-11-mobile {
+      flex: none;
+      width: 91.66667%; }
+    .column.is-offset-11-mobile {
+      margin-left: 91.66667%; }
+    .column.is-12-mobile {
+      flex: none;
+      width: 100%; }
+    .column.is-offset-12-mobile {
+      margin-left: 100%; } }
+  @media screen and (min-width: 769px), print {
+    .column.is-narrow, .column.is-narrow-tablet {
+      flex: none; }
+    .column.is-full, .column.is-full-tablet {
+      flex: none;
+      width: 100%; }
+    .column.is-three-quarters, .column.is-three-quarters-tablet {
+      flex: none;
+      width: 75%; }
+    .column.is-two-thirds, .column.is-two-thirds-tablet {
+      flex: none;
+      width: 66.6666%; }
+    .column.is-half, .column.is-half-tablet {
+      flex: none;
+      width: 50%; }
+    .column.is-one-third, .column.is-one-third-tablet {
+      flex: none;
+      width: 33.3333%; }
+    .column.is-one-quarter, .column.is-one-quarter-tablet {
+      flex: none;
+      width: 25%; }
+    .column.is-one-fifth, .column.is-one-fifth-tablet {
+      flex: none;
+      width: 20%; }
+    .column.is-two-fifths, .column.is-two-fifths-tablet {
+      flex: none;
+      width: 40%; }
+    .column.is-three-fifths, .column.is-three-fifths-tablet {
+      flex: none;
+      width: 60%; }
+    .column.is-four-fifths, .column.is-four-fifths-tablet {
+      flex: none;
+      width: 80%; }
+    .column.is-offset-three-quarters, .column.is-offset-three-quarters-tablet {
+      margin-left: 75%; }
+    .column.is-offset-two-thirds, .column.is-offset-two-thirds-tablet {
+      margin-left: 66.6666%; }
+    .column.is-offset-half, .column.is-offset-half-tablet {
+      margin-left: 50%; }
+    .column.is-offset-one-third, .column.is-offset-one-third-tablet {
+      margin-left: 33.3333%; }
+    .column.is-offset-one-quarter, .column.is-offset-one-quarter-tablet {
+      margin-left: 25%; }
+    .column.is-offset-one-fifth, .column.is-offset-one-fifth-tablet {
+      margin-left: 20%; }
+    .column.is-offset-two-fifths, .column.is-offset-two-fifths-tablet {
+      margin-left: 40%; }
+    .column.is-offset-three-fifths, .column.is-offset-three-fifths-tablet {
+      margin-left: 60%; }
+    .column.is-offset-four-fifths, .column.is-offset-four-fifths-tablet {
+      margin-left: 80%; }
+    .column.is-0, .column.is-0-tablet {
+      flex: none;
+      width: 0%; }
+    .column.is-offset-0, .column.is-offset-0-tablet {
+      margin-left: 0%; }
+    .column.is-1, .column.is-1-tablet {
+      flex: none;
+      width: 8.33333%; }
+    .column.is-offset-1, .column.is-offset-1-tablet {
+      margin-left: 8.33333%; }
+    .column.is-2, .column.is-2-tablet {
+      flex: none;
+      width: 16.66667%; }
+    .column.is-offset-2, .column.is-offset-2-tablet {
+      margin-left: 16.66667%; }
+    .column.is-3, .column.is-3-tablet {
+      flex: none;
+      width: 25%; }
+    .column.is-offset-3, .column.is-offset-3-tablet {
+      margin-left: 25%; }
+    .column.is-4, .column.is-4-tablet {
+      flex: none;
+      width: 33.33333%; }
+    .column.is-offset-4, .column.is-offset-4-tablet {
+      margin-left: 33.33333%; }
+    .column.is-5, .column.is-5-tablet {
+      flex: none;
+      width: 41.66667%; }
+    .column.is-offset-5, .column.is-offset-5-tablet {
+      margin-left: 41.66667%; }
+    .column.is-6, .column.is-6-tablet {
+      flex: none;
+      width: 50%; }
+    .column.is-offset-6, .column.is-offset-6-tablet {
+      margin-left: 50%; }
+    .column.is-7, .column.is-7-tablet {
+      flex: none;
+      width: 58.33333%; }
+    .column.is-offset-7, .column.is-offset-7-tablet {
+      margin-left: 58.33333%; }
+    .column.is-8, .column.is-8-tablet {
+      flex: none;
+      width: 66.66667%; }
+    .column.is-offset-8, .column.is-offset-8-tablet {
+      margin-left: 66.66667%; }
+    .column.is-9, .column.is-9-tablet {
+      flex: none;
+      width: 75%; }
+    .column.is-offset-9, .column.is-offset-9-tablet {
+      margin-left: 75%; }
+    .column.is-10, .column.is-10-tablet {
+      flex: none;
+      width: 83.33333%; }
+    .column.is-offset-10, .column.is-offset-10-tablet {
+      margin-left: 83.33333%; }
+    .column.is-11, .column.is-11-tablet {
+      flex: none;
+      width: 91.66667%; }
+    .column.is-offset-11, .column.is-offset-11-tablet {
+      margin-left: 91.66667%; }
+    .column.is-12, .column.is-12-tablet {
+      flex: none;
+      width: 100%; }
+    .column.is-offset-12, .column.is-offset-12-tablet {
+      margin-left: 100%; } }
+  @media screen and (max-width: 1055px) {
+    .column.is-narrow-touch {
+      flex: none; }
+    .column.is-full-touch {
+      flex: none;
+      width: 100%; }
+    .column.is-three-quarters-touch {
+      flex: none;
+      width: 75%; }
+    .column.is-two-thirds-touch {
+      flex: none;
+      width: 66.6666%; }
+    .column.is-half-touch {
+      flex: none;
+      width: 50%; }
+    .column.is-one-third-touch {
+      flex: none;
+      width: 33.3333%; }
+    .column.is-one-quarter-touch {
+      flex: none;
+      width: 25%; }
+    .column.is-one-fifth-touch {
+      flex: none;
+      width: 20%; }
+    .column.is-two-fifths-touch {
+      flex: none;
+      width: 40%; }
+    .column.is-three-fifths-touch {
+      flex: none;
+      width: 60%; }
+    .column.is-four-fifths-touch {
+      flex: none;
+      width: 80%; }
+    .column.is-offset-three-quarters-touch {
+      margin-left: 75%; }
+    .column.is-offset-two-thirds-touch {
+      margin-left: 66.6666%; }
+    .column.is-offset-half-touch {
+      margin-left: 50%; }
+    .column.is-offset-one-third-touch {
+      margin-left: 33.3333%; }
+    .column.is-offset-one-quarter-touch {
+      margin-left: 25%; }
+    .column.is-offset-one-fifth-touch {
+      margin-left: 20%; }
+    .column.is-offset-two-fifths-touch {
+      margin-left: 40%; }
+    .column.is-offset-three-fifths-touch {
+      margin-left: 60%; }
+    .column.is-offset-four-fifths-touch {
+      margin-left: 80%; }
+    .column.is-0-touch {
+      flex: none;
+      width: 0%; }
+    .column.is-offset-0-touch {
+      margin-left: 0%; }
+    .column.is-1-touch {
+      flex: none;
+      width: 8.33333%; }
+    .column.is-offset-1-touch {
+      margin-left: 8.33333%; }
+    .column.is-2-touch {
+      flex: none;
+      width: 16.66667%; }
+    .column.is-offset-2-touch {
+      margin-left: 16.66667%; }
+    .column.is-3-touch {
+      flex: none;
+      width: 25%; }
+    .column.is-offset-3-touch {
+      margin-left: 25%; }
+    .column.is-4-touch {
+      flex: none;
+      width: 33.33333%; }
+    .column.is-offset-4-touch {
+      margin-left: 33.33333%; }
+    .column.is-5-touch {
+      flex: none;
+      width: 41.66667%; }
+    .column.is-offset-5-touch {
+      margin-left: 41.66667%; }
+    .column.is-6-touch {
+      flex: none;
+      width: 50%; }
+    .column.is-offset-6-touch {
+      margin-left: 50%; }
+    .column.is-7-touch {
+      flex: none;
+      width: 58.33333%; }
+    .column.is-offset-7-touch {
+      margin-left: 58.33333%; }
+    .column.is-8-touch {
+      flex: none;
+      width: 66.66667%; }
+    .column.is-offset-8-touch {
+      margin-left: 66.66667%; }
+    .column.is-9-touch {
+      flex: none;
+      width: 75%; }
+    .column.is-offset-9-touch {
+      margin-left: 75%; }
+    .column.is-10-touch {
+      flex: none;
+      width: 83.33333%; }
+    .column.is-offset-10-touch {
+      margin-left: 83.33333%; }
+    .column.is-11-touch {
+      flex: none;
+      width: 91.66667%; }
+    .column.is-offset-11-touch {
+      margin-left: 91.66667%; }
+    .column.is-12-touch {
+      flex: none;
+      width: 100%; }
+    .column.is-offset-12-touch {
+      margin-left: 100%; } }
+  @media screen and (min-width: 1056px) {
+    .column.is-narrow-desktop {
+      flex: none; }
+    .column.is-full-desktop {
+      flex: none;
+      width: 100%; }
+    .column.is-three-quarters-desktop {
+      flex: none;
+      width: 75%; }
+    .column.is-two-thirds-desktop {
+      flex: none;
+      width: 66.6666%; }
+    .column.is-half-desktop {
+      flex: none;
+      width: 50%; }
+    .column.is-one-third-desktop {
+      flex: none;
+      width: 33.3333%; }
+    .column.is-one-quarter-desktop {
+      flex: none;
+      width: 25%; }
+    .column.is-one-fifth-desktop {
+      flex: none;
+      width: 20%; }
+    .column.is-two-fifths-desktop {
+      flex: none;
+      width: 40%; }
+    .column.is-three-fifths-desktop {
+      flex: none;
+      width: 60%; }
+    .column.is-four-fifths-desktop {
+      flex: none;
+      width: 80%; }
+    .column.is-offset-three-quarters-desktop {
+      margin-left: 75%; }
+    .column.is-offset-two-thirds-desktop {
+      margin-left: 66.6666%; }
+    .column.is-offset-half-desktop {
+      margin-left: 50%; }
+    .column.is-offset-one-third-desktop {
+      margin-left: 33.3333%; }
+    .column.is-offset-one-quarter-desktop {
+      margin-left: 25%; }
+    .column.is-offset-one-fifth-desktop {
+      margin-left: 20%; }
+    .column.is-offset-two-fifths-desktop {
+      margin-left: 40%; }
+    .column.is-offset-three-fifths-desktop {
+      margin-left: 60%; }
+    .column.is-offset-four-fifths-desktop {
+      margin-left: 80%; }
+    .column.is-0-desktop {
+      flex: none;
+      width: 0%; }
+    .column.is-offset-0-desktop {
+      margin-left: 0%; }
+    .column.is-1-desktop {
+      flex: none;
+      width: 8.33333%; }
+    .column.is-offset-1-desktop {
+      margin-left: 8.33333%; }
+    .column.is-2-desktop {
+      flex: none;
+      width: 16.66667%; }
+    .column.is-offset-2-desktop {
+      margin-left: 16.66667%; }
+    .column.is-3-desktop {
+      flex: none;
+      width: 25%; }
+    .column.is-offset-3-desktop {
+      margin-left: 25%; }
+    .column.is-4-desktop {
+      flex: none;
+      width: 33.33333%; }
+    .column.is-offset-4-desktop {
+      margin-left: 33.33333%; }
+    .column.is-5-desktop {
+      flex: none;
+      width: 41.66667%; }
+    .column.is-offset-5-desktop {
+      margin-left: 41.66667%; }
+    .column.is-6-desktop {
+      flex: none;
+      width: 50%; }
+    .column.is-offset-6-desktop {
+      margin-left: 50%; }
+    .column.is-7-desktop {
+      flex: none;
+      width: 58.33333%; }
+    .column.is-offset-7-desktop {
+      margin-left: 58.33333%; }
+    .column.is-8-desktop {
+      flex: none;
+      width: 66.66667%; }
+    .column.is-offset-8-desktop {
+      margin-left: 66.66667%; }
+    .column.is-9-desktop {
+      flex: none;
+      width: 75%; }
+    .column.is-offset-9-desktop {
+      margin-left: 75%; }
+    .column.is-10-desktop {
+      flex: none;
+      width: 83.33333%; }
+    .column.is-offset-10-desktop {
+      margin-left: 83.33333%; }
+    .column.is-11-desktop {
+      flex: none;
+      width: 91.66667%; }
+    .column.is-offset-11-desktop {
+      margin-left: 91.66667%; }
+    .column.is-12-desktop {
+      flex: none;
+      width: 100%; }
+    .column.is-offset-12-desktop {
+      margin-left: 100%; } }
+  @media screen and (min-width: 1216px) {
+    .column.is-narrow-widescreen {
+      flex: none; }
+    .column.is-full-widescreen {
+      flex: none;
+      width: 100%; }
+    .column.is-three-quarters-widescreen {
+      flex: none;
+      width: 75%; }
+    .column.is-two-thirds-widescreen {
+      flex: none;
+      width: 66.6666%; }
+    .column.is-half-widescreen {
+      flex: none;
+      width: 50%; }
+    .column.is-one-third-widescreen {
+      flex: none;
+      width: 33.3333%; }
+    .column.is-one-quarter-widescreen {
+      flex: none;
+      width: 25%; }
+    .column.is-one-fifth-widescreen {
+      flex: none;
+      width: 20%; }
+    .column.is-two-fifths-widescreen {
+      flex: none;
+      width: 40%; }
+    .column.is-three-fifths-widescreen {
+      flex: none;
+      width: 60%; }
+    .column.is-four-fifths-widescreen {
+      flex: none;
+      width: 80%; }
+    .column.is-offset-three-quarters-widescreen {
+      margin-left: 75%; }
+    .column.is-offset-two-thirds-widescreen {
+      margin-left: 66.6666%; }
+    .column.is-offset-half-widescreen {
+      margin-left: 50%; }
+    .column.is-offset-one-third-widescreen {
+      margin-left: 33.3333%; }
+    .column.is-offset-one-quarter-widescreen {
+      margin-left: 25%; }
+    .column.is-offset-one-fifth-widescreen {
+      margin-left: 20%; }
+    .column.is-offset-two-fifths-widescreen {
+      margin-left: 40%; }
+    .column.is-offset-three-fifths-widescreen {
+      margin-left: 60%; }
+    .column.is-offset-four-fifths-widescreen {
+      margin-left: 80%; }
+    .column.is-0-widescreen {
+      flex: none;
+      width: 0%; }
+    .column.is-offset-0-widescreen {
+      margin-left: 0%; }
+    .column.is-1-widescreen {
+      flex: none;
+      width: 8.33333%; }
+    .column.is-offset-1-widescreen {
+      margin-left: 8.33333%; }
+    .column.is-2-widescreen {
+      flex: none;
+      width: 16.66667%; }
+    .column.is-offset-2-widescreen {
+      margin-left: 16.66667%; }
+    .column.is-3-widescreen {
+      flex: none;
+      width: 25%; }
+    .column.is-offset-3-widescreen {
+      margin-left: 25%; }
+    .column.is-4-widescreen {
+      flex: none;
+      width: 33.33333%; }
+    .column.is-offset-4-widescreen {
+      margin-left: 33.33333%; }
+    .column.is-5-widescreen {
+      flex: none;
+      width: 41.66667%; }
+    .column.is-offset-5-widescreen {
+      margin-left: 41.66667%; }
+    .column.is-6-widescreen {
+      flex: none;
+      width: 50%; }
+    .column.is-offset-6-widescreen {
+      margin-left: 50%; }
+    .column.is-7-widescreen {
+      flex: none;
+      width: 58.33333%; }
+    .column.is-offset-7-widescreen {
+      margin-left: 58.33333%; }
+    .column.is-8-widescreen {
+      flex: none;
+      width: 66.66667%; }
+    .column.is-offset-8-widescreen {
+      margin-left: 66.66667%; }
+    .column.is-9-widescreen {
+      flex: none;
+      width: 75%; }
+    .column.is-offset-9-widescreen {
+      margin-left: 75%; }
+    .column.is-10-widescreen {
+      flex: none;
+      width: 83.33333%; }
+    .column.is-offset-10-widescreen {
+      margin-left: 83.33333%; }
+    .column.is-11-widescreen {
+      flex: none;
+      width: 91.66667%; }
+    .column.is-offset-11-widescreen {
+      margin-left: 91.66667%; }
+    .column.is-12-widescreen {
+      flex: none;
+      width: 100%; }
+    .column.is-offset-12-widescreen {
+      margin-left: 100%; } }
+  @media screen and (min-width: 1408px) {
+    .column.is-narrow-fullhd {
+      flex: none; }
+    .column.is-full-fullhd {
+      flex: none;
+      width: 100%; }
+    .column.is-three-quarters-fullhd {
+      flex: none;
+      width: 75%; }
+    .column.is-two-thirds-fullhd {
+      flex: none;
+      width: 66.6666%; }
+    .column.is-half-fullhd {
+      flex: none;
+      width: 50%; }
+    .column.is-one-third-fullhd {
+      flex: none;
+      width: 33.3333%; }
+    .column.is-one-quarter-fullhd {
+      flex: none;
+      width: 25%; }
+    .column.is-one-fifth-fullhd {
+      flex: none;
+      width: 20%; }
+    .column.is-two-fifths-fullhd {
+      flex: none;
+      width: 40%; }
+    .column.is-three-fifths-fullhd {
+      flex: none;
+      width: 60%; }
+    .column.is-four-fifths-fullhd {
+      flex: none;
+      width: 80%; }
+    .column.is-offset-three-quarters-fullhd {
+      margin-left: 75%; }
+    .column.is-offset-two-thirds-fullhd {
+      margin-left: 66.6666%; }
+    .column.is-offset-half-fullhd {
+      margin-left: 50%; }
+    .column.is-offset-one-third-fullhd {
+      margin-left: 33.3333%; }
+    .column.is-offset-one-quarter-fullhd {
+      margin-left: 25%; }
+    .column.is-offset-one-fifth-fullhd {
+      margin-left: 20%; }
+    .column.is-offset-two-fifths-fullhd {
+      margin-left: 40%; }
+    .column.is-offset-three-fifths-fullhd {
+      margin-left: 60%; }
+    .column.is-offset-four-fifths-fullhd {
+      margin-left: 80%; }
+    .column.is-0-fullhd {
+      flex: none;
+      width: 0%; }
+    .column.is-offset-0-fullhd {
+      margin-left: 0%; }
+    .column.is-1-fullhd {
+      flex: none;
+      width: 8.33333%; }
+    .column.is-offset-1-fullhd {
+      margin-left: 8.33333%; }
+    .column.is-2-fullhd {
+      flex: none;
+      width: 16.66667%; }
+    .column.is-offset-2-fullhd {
+      margin-left: 16.66667%; }
+    .column.is-3-fullhd {
+      flex: none;
+      width: 25%; }
+    .column.is-offset-3-fullhd {
+      margin-left: 25%; }
+    .column.is-4-fullhd {
+      flex: none;
+      width: 33.33333%; }
+    .column.is-offset-4-fullhd {
+      margin-left: 33.33333%; }
+    .column.is-5-fullhd {
+      flex: none;
+      width: 41.66667%; }
+    .column.is-offset-5-fullhd {
+      margin-left: 41.66667%; }
+    .column.is-6-fullhd {
+      flex: none;
+      width: 50%; }
+    .column.is-offset-6-fullhd {
+      margin-left: 50%; }
+    .column.is-7-fullhd {
+      flex: none;
+      width: 58.33333%; }
+    .column.is-offset-7-fullhd {
+      margin-left: 58.33333%; }
+    .column.is-8-fullhd {
+      flex: none;
+      width: 66.66667%; }
+    .column.is-offset-8-fullhd {
+      margin-left: 66.66667%; }
+    .column.is-9-fullhd {
+      flex: none;
+      width: 75%; }
+    .column.is-offset-9-fullhd {
+      margin-left: 75%; }
+    .column.is-10-fullhd {
+      flex: none;
+      width: 83.33333%; }
+    .column.is-offset-10-fullhd {
+      margin-left: 83.33333%; }
+    .column.is-11-fullhd {
+      flex: none;
+      width: 91.66667%; }
+    .column.is-offset-11-fullhd {
+      margin-left: 91.66667%; }
+    .column.is-12-fullhd {
+      flex: none;
+      width: 100%; }
+    .column.is-offset-12-fullhd {
+      margin-left: 100%; } }
+
+.columns {
+  margin-left: -0.75rem;
+  margin-right: -0.75rem;
+  margin-top: -0.75rem; }
+  .columns:last-child {
+    margin-bottom: -0.75rem; }
+  .columns:not(:last-child) {
+    margin-bottom: calc(1.5rem - 0.75rem); }
+  .columns.is-centered {
+    justify-content: center; }
+  .columns.is-gapless {
+    margin-left: 0;
+    margin-right: 0;
+    margin-top: 0; }
+    .columns.is-gapless > .column {
+      margin: 0;
+      padding: 0 !important; }
+    .columns.is-gapless:not(:last-child) {
+      margin-bottom: 1.5rem; }
+    .columns.is-gapless:last-child {
+      margin-bottom: 0; }
+  .columns.is-mobile {
+    display: flex; }
+  .columns.is-multiline {
+    flex-wrap: wrap; }
+  .columns.is-vcentered {
+    align-items: center; }
+  @media screen and (min-width: 769px), print {
+    .columns:not(.is-desktop) {
+      display: flex; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-desktop {
+      display: flex; } }
+
+.columns.is-variable {
+  --columnGap: 0.75rem;
+  margin-left: calc(-1 * var(--columnGap));
+  margin-right: calc(-1 * var(--columnGap)); }
+  .columns.is-variable .column {
+    padding-left: var(--columnGap);
+    padding-right: var(--columnGap); }
+  .columns.is-variable.is-0 {
+    --columnGap: 0rem; }
+  @media screen and (max-width: 768px) {
+    .columns.is-variable.is-0-mobile {
+      --columnGap: 0rem; } }
+  @media screen and (min-width: 769px), print {
+    .columns.is-variable.is-0-tablet {
+      --columnGap: 0rem; } }
+  @media screen and (min-width: 769px) and (max-width: 1055px) {
+    .columns.is-variable.is-0-tablet-only {
+      --columnGap: 0rem; } }
+  @media screen and (max-width: 1055px) {
+    .columns.is-variable.is-0-touch {
+      --columnGap: 0rem; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-variable.is-0-desktop {
+      --columnGap: 0rem; } }
+  @media screen and (min-width: 1056px) and (max-width: 1215px) {
+    .columns.is-variable.is-0-desktop-only {
+      --columnGap: 0rem; } }
+  @media screen and (min-width: 1216px) {
+    .columns.is-variable.is-0-widescreen {
+      --columnGap: 0rem; } }
+  @media screen and (min-width: 1216px) and (max-width: 1407px) {
+    .columns.is-variable.is-0-widescreen-only {
+      --columnGap: 0rem; } }
+  @media screen and (min-width: 1408px) {
+    .columns.is-variable.is-0-fullhd {
+      --columnGap: 0rem; } }
+  .columns.is-variable.is-1 {
+    --columnGap: 0.25rem; }
+  @media screen and (max-width: 768px) {
+    .columns.is-variable.is-1-mobile {
+      --columnGap: 0.25rem; } }
+  @media screen and (min-width: 769px), print {
+    .columns.is-variable.is-1-tablet {
+      --columnGap: 0.25rem; } }
+  @media screen and (min-width: 769px) and (max-width: 1055px) {
+    .columns.is-variable.is-1-tablet-only {
+      --columnGap: 0.25rem; } }
+  @media screen and (max-width: 1055px) {
+    .columns.is-variable.is-1-touch {
+      --columnGap: 0.25rem; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-variable.is-1-desktop {
+      --columnGap: 0.25rem; } }
+  @media screen and (min-width: 1056px) and (max-width: 1215px) {
+    .columns.is-variable.is-1-desktop-only {
+      --columnGap: 0.25rem; } }
+  @media screen and (min-width: 1216px) {
+    .columns.is-variable.is-1-widescreen {
+      --columnGap: 0.25rem; } }
+  @media screen and (min-width: 1216px) and (max-width: 1407px) {
+    .columns.is-variable.is-1-widescreen-only {
+      --columnGap: 0.25rem; } }
+  @media screen and (min-width: 1408px) {
+    .columns.is-variable.is-1-fullhd {
+      --columnGap: 0.25rem; } }
+  .columns.is-variable.is-2 {
+    --columnGap: 0.5rem; }
+  @media screen and (max-width: 768px) {
+    .columns.is-variable.is-2-mobile {
+      --columnGap: 0.5rem; } }
+  @media screen and (min-width: 769px), print {
+    .columns.is-variable.is-2-tablet {
+      --columnGap: 0.5rem; } }
+  @media screen and (min-width: 769px) and (max-width: 1055px) {
+    .columns.is-variable.is-2-tablet-only {
+      --columnGap: 0.5rem; } }
+  @media screen and (max-width: 1055px) {
+    .columns.is-variable.is-2-touch {
+      --columnGap: 0.5rem; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-variable.is-2-desktop {
+      --columnGap: 0.5rem; } }
+  @media screen and (min-width: 1056px) and (max-width: 1215px) {
+    .columns.is-variable.is-2-desktop-only {
+      --columnGap: 0.5rem; } }
+  @media screen and (min-width: 1216px) {
+    .columns.is-variable.is-2-widescreen {
+      --columnGap: 0.5rem; } }
+  @media screen and (min-width: 1216px) and (max-width: 1407px) {
+    .columns.is-variable.is-2-widescreen-only {
+      --columnGap: 0.5rem; } }
+  @media screen and (min-width: 1408px) {
+    .columns.is-variable.is-2-fullhd {
+      --columnGap: 0.5rem; } }
+  .columns.is-variable.is-3 {
+    --columnGap: 0.75rem; }
+  @media screen and (max-width: 768px) {
+    .columns.is-variable.is-3-mobile {
+      --columnGap: 0.75rem; } }
+  @media screen and (min-width: 769px), print {
+    .columns.is-variable.is-3-tablet {
+      --columnGap: 0.75rem; } }
+  @media screen and (min-width: 769px) and (max-width: 1055px) {
+    .columns.is-variable.is-3-tablet-only {
+      --columnGap: 0.75rem; } }
+  @media screen and (max-width: 1055px) {
+    .columns.is-variable.is-3-touch {
+      --columnGap: 0.75rem; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-variable.is-3-desktop {
+      --columnGap: 0.75rem; } }
+  @media screen and (min-width: 1056px) and (max-width: 1215px) {
+    .columns.is-variable.is-3-desktop-only {
+      --columnGap: 0.75rem; } }
+  @media screen and (min-width: 1216px) {
+    .columns.is-variable.is-3-widescreen {
+      --columnGap: 0.75rem; } }
+  @media screen and (min-width: 1216px) and (max-width: 1407px) {
+    .columns.is-variable.is-3-widescreen-only {
+      --columnGap: 0.75rem; } }
+  @media screen and (min-width: 1408px) {
+    .columns.is-variable.is-3-fullhd {
+      --columnGap: 0.75rem; } }
+  .columns.is-variable.is-4 {
+    --columnGap: 1rem; }
+  @media screen and (max-width: 768px) {
+    .columns.is-variable.is-4-mobile {
+      --columnGap: 1rem; } }
+  @media screen and (min-width: 769px), print {
+    .columns.is-variable.is-4-tablet {
+      --columnGap: 1rem; } }
+  @media screen and (min-width: 769px) and (max-width: 1055px) {
+    .columns.is-variable.is-4-tablet-only {
+      --columnGap: 1rem; } }
+  @media screen and (max-width: 1055px) {
+    .columns.is-variable.is-4-touch {
+      --columnGap: 1rem; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-variable.is-4-desktop {
+      --columnGap: 1rem; } }
+  @media screen and (min-width: 1056px) and (max-width: 1215px) {
+    .columns.is-variable.is-4-desktop-only {
+      --columnGap: 1rem; } }
+  @media screen and (min-width: 1216px) {
+    .columns.is-variable.is-4-widescreen {
+      --columnGap: 1rem; } }
+  @media screen and (min-width: 1216px) and (max-width: 1407px) {
+    .columns.is-variable.is-4-widescreen-only {
+      --columnGap: 1rem; } }
+  @media screen and (min-width: 1408px) {
+    .columns.is-variable.is-4-fullhd {
+      --columnGap: 1rem; } }
+  .columns.is-variable.is-5 {
+    --columnGap: 1.25rem; }
+  @media screen and (max-width: 768px) {
+    .columns.is-variable.is-5-mobile {
+      --columnGap: 1.25rem; } }
+  @media screen and (min-width: 769px), print {
+    .columns.is-variable.is-5-tablet {
+      --columnGap: 1.25rem; } }
+  @media screen and (min-width: 769px) and (max-width: 1055px) {
+    .columns.is-variable.is-5-tablet-only {
+      --columnGap: 1.25rem; } }
+  @media screen and (max-width: 1055px) {
+    .columns.is-variable.is-5-touch {
+      --columnGap: 1.25rem; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-variable.is-5-desktop {
+      --columnGap: 1.25rem; } }
+  @media screen and (min-width: 1056px) and (max-width: 1215px) {
+    .columns.is-variable.is-5-desktop-only {
+      --columnGap: 1.25rem; } }
+  @media screen and (min-width: 1216px) {
+    .columns.is-variable.is-5-widescreen {
+      --columnGap: 1.25rem; } }
+  @media screen and (min-width: 1216px) and (max-width: 1407px) {
+    .columns.is-variable.is-5-widescreen-only {
+      --columnGap: 1.25rem; } }
+  @media screen and (min-width: 1408px) {
+    .columns.is-variable.is-5-fullhd {
+      --columnGap: 1.25rem; } }
+  .columns.is-variable.is-6 {
+    --columnGap: 1.5rem; }
+  @media screen and (max-width: 768px) {
+    .columns.is-variable.is-6-mobile {
+      --columnGap: 1.5rem; } }
+  @media screen and (min-width: 769px), print {
+    .columns.is-variable.is-6-tablet {
+      --columnGap: 1.5rem; } }
+  @media screen and (min-width: 769px) and (max-width: 1055px) {
+    .columns.is-variable.is-6-tablet-only {
+      --columnGap: 1.5rem; } }
+  @media screen and (max-width: 1055px) {
+    .columns.is-variable.is-6-touch {
+      --columnGap: 1.5rem; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-variable.is-6-desktop {
+      --columnGap: 1.5rem; } }
+  @media screen and (min-width: 1056px) and (max-width: 1215px) {
+    .columns.is-variable.is-6-desktop-only {
+      --columnGap: 1.5rem; } }
+  @media screen and (min-width: 1216px) {
+    .columns.is-variable.is-6-widescreen {
+      --columnGap: 1.5rem; } }
+  @media screen and (min-width: 1216px) and (max-width: 1407px) {
+    .columns.is-variable.is-6-widescreen-only {
+      --columnGap: 1.5rem; } }
+  @media screen and (min-width: 1408px) {
+    .columns.is-variable.is-6-fullhd {
+      --columnGap: 1.5rem; } }
+  .columns.is-variable.is-7 {
+    --columnGap: 1.75rem; }
+  @media screen and (max-width: 768px) {
+    .columns.is-variable.is-7-mobile {
+      --columnGap: 1.75rem; } }
+  @media screen and (min-width: 769px), print {
+    .columns.is-variable.is-7-tablet {
+      --columnGap: 1.75rem; } }
+  @media screen and (min-width: 769px) and (max-width: 1055px) {
+    .columns.is-variable.is-7-tablet-only {
+      --columnGap: 1.75rem; } }
+  @media screen and (max-width: 1055px) {
+    .columns.is-variable.is-7-touch {
+      --columnGap: 1.75rem; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-variable.is-7-desktop {
+      --columnGap: 1.75rem; } }
+  @media screen and (min-width: 1056px) and (max-width: 1215px) {
+    .columns.is-variable.is-7-desktop-only {
+      --columnGap: 1.75rem; } }
+  @media screen and (min-width: 1216px) {
+    .columns.is-variable.is-7-widescreen {
+      --columnGap: 1.75rem; } }
+  @media screen and (min-width: 1216px) and (max-width: 1407px) {
+    .columns.is-variable.is-7-widescreen-only {
+      --columnGap: 1.75rem; } }
+  @media screen and (min-width: 1408px) {
+    .columns.is-variable.is-7-fullhd {
+      --columnGap: 1.75rem; } }
+  .columns.is-variable.is-8 {
+    --columnGap: 2rem; }
+  @media screen and (max-width: 768px) {
+    .columns.is-variable.is-8-mobile {
+      --columnGap: 2rem; } }
+  @media screen and (min-width: 769px), print {
+    .columns.is-variable.is-8-tablet {
+      --columnGap: 2rem; } }
+  @media screen and (min-width: 769px) and (max-width: 1055px) {
+    .columns.is-variable.is-8-tablet-only {
+      --columnGap: 2rem; } }
+  @media screen and (max-width: 1055px) {
+    .columns.is-variable.is-8-touch {
+      --columnGap: 2rem; } }
+  @media screen and (min-width: 1056px) {
+    .columns.is-variable.is-8-desktop {
+      --columnGap: 2rem; } }
+  @media screen and (min-width: 1056px) and (max-width: 1215px) {
+    .columns.is-variable.is-8-desktop-only {
+      --columnGap: 2rem; } }
+  @media screen and (min-width: 1216px) {
+    .columns.is-variable.is-8-widescreen {
+      --columnGap: 2rem; } }
+  @media screen and (min-width: 1216px) and (max-width: 1407px) {
+    .columns.is-variable.is-8-widescreen-only {
+      --columnGap: 2rem; } }
+  @media screen and (min-width: 1408px) {
+    .columns.is-variable.is-8-fullhd {
+      --columnGap: 2rem; } }
+
+.tile {
+  align-items: stretch;
+  display: block;
+  flex-basis: 0;
+  flex-grow: 1;
+  flex-shrink: 1;
+  min-height: min-content; }
+  .tile.is-ancestor {
+    margin-left: -0.75rem;
+    margin-right: -0.75rem;
+    margin-top: -0.75rem; }
+    .tile.is-ancestor:last-child {
+      margin-bottom: -0.75rem; }
+    .tile.is-ancestor:not(:last-child) {
+      margin-bottom: 0.75rem; }
+  .tile.is-child {
+    margin: 0 !important; }
+  .tile.is-parent {
+    padding: 0.75rem; }
+  .tile.is-vertical {
+    flex-direction: column; }
+    .tile.is-vertical > .tile.is-child:not(:last-child) {
+      margin-bottom: 1.5rem !important; }
+  @media screen and (min-width: 769px), print {
+    .tile:not(.is-child) {
+      display: flex; }
+    .tile.is-1 {
+      flex: none;
+      width: 8.33333%; }
+    .tile.is-2 {
+      flex: none;
+      width: 16.66667%; }
+    .tile.is-3 {
+      flex: none;
+      width: 25%; }
+    .tile.is-4 {
+      flex: none;
+      width: 33.33333%; }
+    .tile.is-5 {
+      flex: none;
+      width: 41.66667%; }
+    .tile.is-6 {
+      flex: none;
+      width: 50%; }
+    .tile.is-7 {
+      flex: none;
+      width: 58.33333%; }
+    .tile.is-8 {
+      flex: none;
+      width: 66.66667%; }
+    .tile.is-9 {
+      flex: none;
+      width: 75%; }
+    .tile.is-10 {
+      flex: none;
+      width: 83.33333%; }
+    .tile.is-11 {
+      flex: none;
+      width: 91.66667%; }
+    .tile.is-12 {
+      flex: none;
+      width: 100%; } }
+
+.hero {
+  align-items: stretch;
+  display: flex;
+  flex-direction: column;
+  justify-content: space-between; }
+  .hero .navbar {
+    background: none; }
+  .hero .tabs ul {
+    border-bottom: none; }
+  .hero.is-white {
+    background-color: white;
+    color: #0a0a0a; }
+    .hero.is-white a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-white strong {
+      color: inherit; }
+    .hero.is-white .title {
+      color: #0a0a0a; }
+    .hero.is-white .subtitle {
+      color: rgba(10, 10, 10, 0.9); }
+      .hero.is-white .subtitle a:not(.button),
+      .hero.is-white .subtitle strong {
+        color: #0a0a0a; }
+    @media screen and (max-width: 1055px) {
+      .hero.is-white .navbar-menu {
+        background-color: white; } }
+    .hero.is-white .navbar-item,
+    .hero.is-white .navbar-link {
+      color: rgba(10, 10, 10, 0.7); }
+    .hero.is-white a.navbar-item:hover, .hero.is-white a.navbar-item.is-active,
+    .hero.is-white .navbar-link:hover,
+    .hero.is-white .navbar-link.is-active {
+      background-color: #f2f2f2;
+      color: #0a0a0a; }
+    .hero.is-white .tabs a {
+      color: #0a0a0a;
+      opacity: 0.9; }
+      .hero.is-white .tabs a:hover {
+        opacity: 1; }
+    .hero.is-white .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-white .tabs.is-boxed a, .hero.is-white .tabs.is-toggle a {
+      color: #0a0a0a; }
+      .hero.is-white .tabs.is-boxed a:hover, .hero.is-white .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-white .tabs.is-boxed li.is-active a, .hero.is-white .tabs.is-boxed li.is-active a:hover, .hero.is-white .tabs.is-toggle li.is-active a, .hero.is-white .tabs.is-toggle li.is-active a:hover {
+      background-color: #0a0a0a;
+      border-color: #0a0a0a;
+      color: white; }
+    .hero.is-white.is-bold {
+      background-image: linear-gradient(141deg, #e6e6e6 0%, white 71%, white 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-white.is-bold .navbar-menu {
+          background-image: linear-gradient(141deg, #e6e6e6 0%, white 71%, white 100%); } }
+  .hero.is-black {
+    background-color: #0a0a0a;
+    color: white; }
+    .hero.is-black a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-black strong {
+      color: inherit; }
+    .hero.is-black .title {
+      color: white; }
+    .hero.is-black .subtitle {
+      color: rgba(255, 255, 255, 0.9); }
+      .hero.is-black .subtitle a:not(.button),
+      .hero.is-black .subtitle strong {
+        color: white; }
+    @media screen and (max-width: 1055px) {
+      .hero.is-black .navbar-menu {
+        background-color: #0a0a0a; } }
+    .hero.is-black .navbar-item,
+    .hero.is-black .navbar-link {
+      color: rgba(255, 255, 255, 0.7); }
+    .hero.is-black a.navbar-item:hover, .hero.is-black a.navbar-item.is-active,
+    .hero.is-black .navbar-link:hover,
+    .hero.is-black .navbar-link.is-active {
+      background-color: black;
+      color: white; }
+    .hero.is-black .tabs a {
+      color: white;
+      opacity: 0.9; }
+      .hero.is-black .tabs a:hover {
+        opacity: 1; }
+    .hero.is-black .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-black .tabs.is-boxed a, .hero.is-black .tabs.is-toggle a {
+      color: white; }
+      .hero.is-black .tabs.is-boxed a:hover, .hero.is-black .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-black .tabs.is-boxed li.is-active a, .hero.is-black .tabs.is-boxed li.is-active a:hover, .hero.is-black .tabs.is-toggle li.is-active a, .hero.is-black .tabs.is-toggle li.is-active a:hover {
+      background-color: white;
+      border-color: white;
+      color: #0a0a0a; }
+    .hero.is-black.is-bold {
+      background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-black.is-bold .navbar-menu {
+          background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } }
+  .hero.is-light {
+    background-color: whitesmoke;
+    color: #363636; }
+    .hero.is-light a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-light strong {
+      color: inherit; }
+    .hero.is-light .title {
+      color: #363636; }
+    .hero.is-light .subtitle {
+      color: rgba(54, 54, 54, 0.9); }
+      .hero.is-light .subtitle a:not(.button),
+      .hero.is-light .subtitle strong {
+        color: #363636; }
+    @media screen and (max-width: 1055px) {
+      .hero.is-light .navbar-menu {
+        background-color: whitesmoke; } }
+    .hero.is-light .navbar-item,
+    .hero.is-light .navbar-link {
+      color: rgba(54, 54, 54, 0.7); }
+    .hero.is-light a.navbar-item:hover, .hero.is-light a.navbar-item.is-active,
+    .hero.is-light .navbar-link:hover,
+    .hero.is-light .navbar-link.is-active {
+      background-color: #e8e8e8;
+      color: #363636; }
+    .hero.is-light .tabs a {
+      color: #363636;
+      opacity: 0.9; }
+      .hero.is-light .tabs a:hover {
+        opacity: 1; }
+    .hero.is-light .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-light .tabs.is-boxed a, .hero.is-light .tabs.is-toggle a {
+      color: #363636; }
+      .hero.is-light .tabs.is-boxed a:hover, .hero.is-light .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-light .tabs.is-boxed li.is-active a, .hero.is-light .tabs.is-boxed li.is-active a:hover, .hero.is-light .tabs.is-toggle li.is-active a, .hero.is-light .tabs.is-toggle li.is-active a:hover {
+      background-color: #363636;
+      border-color: #363636;
+      color: whitesmoke; }
+    .hero.is-light.is-bold {
+      background-image: linear-gradient(141deg, #dfd8d9 0%, whitesmoke 71%, white 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-light.is-bold .navbar-menu {
+          background-image: linear-gradient(141deg, #dfd8d9 0%, whitesmoke 71%, white 100%); } }
+  .hero.is-dark, .content kbd.hero {
+    background-color: #363636;
+    color: whitesmoke; }
+    .hero.is-dark a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), .content kbd.hero a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-dark strong,
+    .content kbd.hero strong {
+      color: inherit; }
+    .hero.is-dark .title, .content kbd.hero .title {
+      color: whitesmoke; }
+    .hero.is-dark .subtitle, .content kbd.hero .subtitle {
+      color: rgba(245, 245, 245, 0.9); }
+      .hero.is-dark .subtitle a:not(.button), .content kbd.hero .subtitle a:not(.button),
+      .hero.is-dark .subtitle strong,
+      .content kbd.hero .subtitle strong {
+        color: whitesmoke; }
+    @media screen and (max-width: 1055px) {
+      .hero.is-dark .navbar-menu, .content kbd.hero .navbar-menu {
+        background-color: #363636; } }
+    .hero.is-dark .navbar-item, .content kbd.hero .navbar-item,
+    .hero.is-dark .navbar-link,
+    .content kbd.hero .navbar-link {
+      color: rgba(245, 245, 245, 0.7); }
+    .hero.is-dark a.navbar-item:hover, .content kbd.hero a.navbar-item:hover, .hero.is-dark a.navbar-item.is-active, .content kbd.hero a.navbar-item.is-active,
+    .hero.is-dark .navbar-link:hover,
+    .content kbd.hero .navbar-link:hover,
+    .hero.is-dark .navbar-link.is-active,
+    .content kbd.hero .navbar-link.is-active {
+      background-color: #292929;
+      color: whitesmoke; }
+    .hero.is-dark .tabs a, .content kbd.hero .tabs a {
+      color: whitesmoke;
+      opacity: 0.9; }
+      .hero.is-dark .tabs a:hover, .content kbd.hero .tabs a:hover {
+        opacity: 1; }
+    .hero.is-dark .tabs li.is-active a, .content kbd.hero .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-dark .tabs.is-boxed a, .content kbd.hero .tabs.is-boxed a, .hero.is-dark .tabs.is-toggle a, .content kbd.hero .tabs.is-toggle a {
+      color: whitesmoke; }
+      .hero.is-dark .tabs.is-boxed a:hover, .content kbd.hero .tabs.is-boxed a:hover, .hero.is-dark .tabs.is-toggle a:hover, .content kbd.hero .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-dark .tabs.is-boxed li.is-active a, .content kbd.hero .tabs.is-boxed li.is-active a, .hero.is-dark .tabs.is-boxed li.is-active a:hover, .content kbd.hero .tabs.is-boxed li.is-active a:hover, .hero.is-dark .tabs.is-toggle li.is-active a, .content kbd.hero .tabs.is-toggle li.is-active a, .hero.is-dark .tabs.is-toggle li.is-active a:hover, .content kbd.hero .tabs.is-toggle li.is-active a:hover {
+      background-color: whitesmoke;
+      border-color: whitesmoke;
+      color: #363636; }
+    .hero.is-dark.is-bold, .content kbd.hero.is-bold {
+      background-image: linear-gradient(141deg, #1f191a 0%, #363636 71%, #46403f 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-dark.is-bold .navbar-menu, .content kbd.hero.is-bold .navbar-menu {
+          background-image: linear-gradient(141deg, #1f191a 0%, #363636 71%, #46403f 100%); } }
+  .hero.is-primary, .docstring > section > a.hero.docs-sourcelink {
+    background-color: #4eb5de;
+    color: #fff; }
+    .hero.is-primary a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), .docstring > section > a.hero.docs-sourcelink a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-primary strong,
+    .docstring > section > a.hero.docs-sourcelink strong {
+      color: inherit; }
+    .hero.is-primary .title, .docstring > section > a.hero.docs-sourcelink .title {
+      color: #fff; }
+    .hero.is-primary .subtitle, .docstring > section > a.hero.docs-sourcelink .subtitle {
+      color: rgba(255, 255, 255, 0.9); }
+      .hero.is-primary .subtitle a:not(.button), .docstring > section > a.hero.docs-sourcelink .subtitle a:not(.button),
+      .hero.is-primary .subtitle strong,
+      .docstring > section > a.hero.docs-sourcelink .subtitle strong {
+        color: #fff; }
+    @media screen and (max-width: 1055px) {
+      .hero.is-primary .navbar-menu, .docstring > section > a.hero.docs-sourcelink .navbar-menu {
+        background-color: #4eb5de; } }
+    .hero.is-primary .navbar-item, .docstring > section > a.hero.docs-sourcelink .navbar-item,
+    .hero.is-primary .navbar-link,
+    .docstring > section > a.hero.docs-sourcelink .navbar-link {
+      color: rgba(255, 255, 255, 0.7); }
+    .hero.is-primary a.navbar-item:hover, .docstring > section > a.hero.docs-sourcelink a.navbar-item:hover, .hero.is-primary a.navbar-item.is-active, .docstring > section > a.hero.docs-sourcelink a.navbar-item.is-active,
+    .hero.is-primary .navbar-link:hover,
+    .docstring > section > a.hero.docs-sourcelink .navbar-link:hover,
+    .hero.is-primary .navbar-link.is-active,
+    .docstring > section > a.hero.docs-sourcelink .navbar-link.is-active {
+      background-color: #39acda;
+      color: #fff; }
+    .hero.is-primary .tabs a, .docstring > section > a.hero.docs-sourcelink .tabs a {
+      color: #fff;
+      opacity: 0.9; }
+      .hero.is-primary .tabs a:hover, .docstring > section > a.hero.docs-sourcelink .tabs a:hover {
+        opacity: 1; }
+    .hero.is-primary .tabs li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-primary .tabs.is-boxed a, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a, .hero.is-primary .tabs.is-toggle a, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a {
+      color: #fff; }
+      .hero.is-primary .tabs.is-boxed a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a:hover, .hero.is-primary .tabs.is-toggle a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-primary .tabs.is-boxed li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a, .hero.is-primary .tabs.is-boxed li.is-active a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a:hover, .hero.is-primary .tabs.is-toggle li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a, .hero.is-primary .tabs.is-toggle li.is-active a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a:hover {
+      background-color: #fff;
+      border-color: #fff;
+      color: #4eb5de; }
+    .hero.is-primary.is-bold, .docstring > section > a.hero.is-bold.docs-sourcelink {
+      background-image: linear-gradient(141deg, #1bc7de 0%, #4eb5de 71%, #5fa9e7 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-primary.is-bold .navbar-menu, .docstring > section > a.hero.is-bold.docs-sourcelink .navbar-menu {
+          background-image: linear-gradient(141deg, #1bc7de 0%, #4eb5de 71%, #5fa9e7 100%); } }
+  .hero.is-link {
+    background-color: #2e63b8;
+    color: #fff; }
+    .hero.is-link a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-link strong {
+      color: inherit; }
+    .hero.is-link .title {
+      color: #fff; }
+    .hero.is-link .subtitle {
+      color: rgba(255, 255, 255, 0.9); }
+      .hero.is-link .subtitle a:not(.button),
+      .hero.is-link .subtitle strong {
+        color: #fff; }
+    @media screen and (max-width: 1055px) {
+      .hero.is-link .navbar-menu {
+        background-color: #2e63b8; } }
+    .hero.is-link .navbar-item,
+    .hero.is-link .navbar-link {
+      color: rgba(255, 255, 255, 0.7); }
+    .hero.is-link a.navbar-item:hover, .hero.is-link a.navbar-item.is-active,
+    .hero.is-link .navbar-link:hover,
+    .hero.is-link .navbar-link.is-active {
+      background-color: #2958a4;
+      color: #fff; }
+    .hero.is-link .tabs a {
+      color: #fff;
+      opacity: 0.9; }
+      .hero.is-link .tabs a:hover {
+        opacity: 1; }
+    .hero.is-link .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-link .tabs.is-boxed a, .hero.is-link .tabs.is-toggle a {
+      color: #fff; }
+      .hero.is-link .tabs.is-boxed a:hover, .hero.is-link .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-link .tabs.is-boxed li.is-active a, .hero.is-link .tabs.is-boxed li.is-active a:hover, .hero.is-link .tabs.is-toggle li.is-active a, .hero.is-link .tabs.is-toggle li.is-active a:hover {
+      background-color: #fff;
+      border-color: #fff;
+      color: #2e63b8; }
+    .hero.is-link.is-bold {
+      background-image: linear-gradient(141deg, #1b6098 0%, #2e63b8 71%, #2d51d2 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-link.is-bold .navbar-menu {
+          background-image: linear-gradient(141deg, #1b6098 0%, #2e63b8 71%, #2d51d2 100%); } }
+  .hero.is-info {
+    background-color: #209cee;
+    color: #fff; }
+    .hero.is-info a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-info strong {
+      color: inherit; }
+    .hero.is-info .title {
+      color: #fff; }
+    .hero.is-info .subtitle {
+      color: rgba(255, 255, 255, 0.9); }
+      .hero.is-info .subtitle a:not(.button),
+      .hero.is-info .subtitle strong {
+        color: #fff; }
+    @media screen and (max-width: 1055px) {
+      .hero.is-info .navbar-menu {
+        background-color: #209cee; } }
+    .hero.is-info .navbar-item,
+    .hero.is-info .navbar-link {
+      color: rgba(255, 255, 255, 0.7); }
+    .hero.is-info a.navbar-item:hover, .hero.is-info a.navbar-item.is-active,
+    .hero.is-info .navbar-link:hover,
+    .hero.is-info .navbar-link.is-active {
+      background-color: #118fe4;
+      color: #fff; }
+    .hero.is-info .tabs a {
+      color: #fff;
+      opacity: 0.9; }
+      .hero.is-info .tabs a:hover {
+        opacity: 1; }
+    .hero.is-info .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-info .tabs.is-boxed a, .hero.is-info .tabs.is-toggle a {
+      color: #fff; }
+      .hero.is-info .tabs.is-boxed a:hover, .hero.is-info .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-info .tabs.is-boxed li.is-active a, .hero.is-info .tabs.is-boxed li.is-active a:hover, .hero.is-info .tabs.is-toggle li.is-active a, .hero.is-info .tabs.is-toggle li.is-active a:hover {
+      background-color: #fff;
+      border-color: #fff;
+      color: #209cee; }
+    .hero.is-info.is-bold {
+      background-image: linear-gradient(141deg, #04a6d7 0%, #209cee 71%, #3287f5 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-info.is-bold .navbar-menu {
+          background-image: linear-gradient(141deg, #04a6d7 0%, #209cee 71%, #3287f5 100%); } }
+  .hero.is-success {
+    background-color: #22c35b;
+    color: #fff; }
+    .hero.is-success a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-success strong {
+      color: inherit; }
+    .hero.is-success .title {
+      color: #fff; }
+    .hero.is-success .subtitle {
+      color: rgba(255, 255, 255, 0.9); }
+      .hero.is-success .subtitle a:not(.button),
+      .hero.is-success .subtitle strong {
+        color: #fff; }
+    @media screen and (max-width: 1055px) {
+      .hero.is-success .navbar-menu {
+        background-color: #22c35b; } }
+    .hero.is-success .navbar-item,
+    .hero.is-success .navbar-link {
+      color: rgba(255, 255, 255, 0.7); }
+    .hero.is-success a.navbar-item:hover, .hero.is-success a.navbar-item.is-active,
+    .hero.is-success .navbar-link:hover,
+    .hero.is-success .navbar-link.is-active {
+      background-color: #1ead51;
+      color: #fff; }
+    .hero.is-success .tabs a {
+      color: #fff;
+      opacity: 0.9; }
+      .hero.is-success .tabs a:hover {
+        opacity: 1; }
+    .hero.is-success .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-success .tabs.is-boxed a, .hero.is-success .tabs.is-toggle a {
+      color: #fff; }
+      .hero.is-success .tabs.is-boxed a:hover, .hero.is-success .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-success .tabs.is-boxed li.is-active a, .hero.is-success .tabs.is-boxed li.is-active a:hover, .hero.is-success .tabs.is-toggle li.is-active a, .hero.is-success .tabs.is-toggle li.is-active a:hover {
+      background-color: #fff;
+      border-color: #fff;
+      color: #22c35b; }
+    .hero.is-success.is-bold {
+      background-image: linear-gradient(141deg, #12a02c 0%, #22c35b 71%, #1fdf83 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-success.is-bold .navbar-menu {
+          background-image: linear-gradient(141deg, #12a02c 0%, #22c35b 71%, #1fdf83 100%); } }
+  .hero.is-warning {
+    background-color: #ffdd57;
+    color: rgba(0, 0, 0, 0.7); }
+    .hero.is-warning a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-warning strong {
+      color: inherit; }
+    .hero.is-warning .title {
+      color: rgba(0, 0, 0, 0.7); }
+    .hero.is-warning .subtitle {
+      color: rgba(0, 0, 0, 0.9); }
+      .hero.is-warning .subtitle a:not(.button),
+      .hero.is-warning .subtitle strong {
+        color: rgba(0, 0, 0, 0.7); }
+    @media screen and (max-width: 1055px) {
+      .hero.is-warning .navbar-menu {
+        background-color: #ffdd57; } }
+    .hero.is-warning .navbar-item,
+    .hero.is-warning .navbar-link {
+      color: rgba(0, 0, 0, 0.7); }
+    .hero.is-warning a.navbar-item:hover, .hero.is-warning a.navbar-item.is-active,
+    .hero.is-warning .navbar-link:hover,
+    .hero.is-warning .navbar-link.is-active {
+      background-color: #ffd83d;
+      color: rgba(0, 0, 0, 0.7); }
+    .hero.is-warning .tabs a {
+      color: rgba(0, 0, 0, 0.7);
+      opacity: 0.9; }
+      .hero.is-warning .tabs a:hover {
+        opacity: 1; }
+    .hero.is-warning .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-warning .tabs.is-boxed a, .hero.is-warning .tabs.is-toggle a {
+      color: rgba(0, 0, 0, 0.7); }
+      .hero.is-warning .tabs.is-boxed a:hover, .hero.is-warning .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-warning .tabs.is-boxed li.is-active a, .hero.is-warning .tabs.is-boxed li.is-active a:hover, .hero.is-warning .tabs.is-toggle li.is-active a, .hero.is-warning .tabs.is-toggle li.is-active a:hover {
+      background-color: rgba(0, 0, 0, 0.7);
+      border-color: rgba(0, 0, 0, 0.7);
+      color: #ffdd57; }
+    .hero.is-warning.is-bold {
+      background-image: linear-gradient(141deg, #ffaf24 0%, #ffdd57 71%, #fffa70 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-warning.is-bold .navbar-menu {
+          background-image: linear-gradient(141deg, #ffaf24 0%, #ffdd57 71%, #fffa70 100%); } }
+  .hero.is-danger {
+    background-color: #da0b00;
+    color: #fff; }
+    .hero.is-danger a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current),
+    .hero.is-danger strong {
+      color: inherit; }
+    .hero.is-danger .title {
+      color: #fff; }
+    .hero.is-danger .subtitle {
+      color: rgba(255, 255, 255, 0.9); }
+      .hero.is-danger .subtitle a:not(.button),
+      .hero.is-danger .subtitle strong {
+        color: #fff; }
+    @media screen and (max-width: 1055px) {
+      .hero.is-danger .navbar-menu {
+        background-color: #da0b00; } }
+    .hero.is-danger .navbar-item,
+    .hero.is-danger .navbar-link {
+      color: rgba(255, 255, 255, 0.7); }
+    .hero.is-danger a.navbar-item:hover, .hero.is-danger a.navbar-item.is-active,
+    .hero.is-danger .navbar-link:hover,
+    .hero.is-danger .navbar-link.is-active {
+      background-color: #c10a00;
+      color: #fff; }
+    .hero.is-danger .tabs a {
+      color: #fff;
+      opacity: 0.9; }
+      .hero.is-danger .tabs a:hover {
+        opacity: 1; }
+    .hero.is-danger .tabs li.is-active a {
+      opacity: 1; }
+    .hero.is-danger .tabs.is-boxed a, .hero.is-danger .tabs.is-toggle a {
+      color: #fff; }
+      .hero.is-danger .tabs.is-boxed a:hover, .hero.is-danger .tabs.is-toggle a:hover {
+        background-color: rgba(10, 10, 10, 0.1); }
+    .hero.is-danger .tabs.is-boxed li.is-active a, .hero.is-danger .tabs.is-boxed li.is-active a:hover, .hero.is-danger .tabs.is-toggle li.is-active a, .hero.is-danger .tabs.is-toggle li.is-active a:hover {
+      background-color: #fff;
+      border-color: #fff;
+      color: #da0b00; }
+    .hero.is-danger.is-bold {
+      background-image: linear-gradient(141deg, #a70013 0%, #da0b00 71%, #f43500 100%); }
+      @media screen and (max-width: 768px) {
+        .hero.is-danger.is-bold .navbar-menu {
+          background-image: linear-gradient(141deg, #a70013 0%, #da0b00 71%, #f43500 100%); } }
+  .hero.is-small .hero-body, #documenter .docs-sidebar form.docs-search > input.hero .hero-body {
+    padding-bottom: 1.5rem;
+    padding-top: 1.5rem; }
+  @media screen and (min-width: 769px), print {
+    .hero.is-medium .hero-body {
+      padding-bottom: 9rem;
+      padding-top: 9rem; } }
+  @media screen and (min-width: 769px), print {
+    .hero.is-large .hero-body {
+      padding-bottom: 18rem;
+      padding-top: 18rem; } }
+  .hero.is-halfheight .hero-body, .hero.is-fullheight .hero-body, .hero.is-fullheight-with-navbar .hero-body {
+    align-items: center;
+    display: flex; }
+    .hero.is-halfheight .hero-body > .container, .hero.is-fullheight .hero-body > .container, .hero.is-fullheight-with-navbar .hero-body > .container {
+      flex-grow: 1;
+      flex-shrink: 1; }
+  .hero.is-halfheight {
+    min-height: 50vh; }
+  .hero.is-fullheight {
+    min-height: 100vh; }
+
+.hero-video {
+  overflow: hidden; }
+  .hero-video video {
+    left: 50%;
+    min-height: 100%;
+    min-width: 100%;
+    position: absolute;
+    top: 50%;
+    transform: translate3d(-50%, -50%, 0); }
+  .hero-video.is-transparent {
+    opacity: 0.3; }
+  @media screen and (max-width: 768px) {
+    .hero-video {
+      display: none; } }
+
+.hero-buttons {
+  margin-top: 1.5rem; }
+  @media screen and (max-width: 768px) {
+    .hero-buttons .button {
+      display: flex; }
+      .hero-buttons .button:not(:last-child) {
+        margin-bottom: 0.75rem; } }
+  @media screen and (min-width: 769px), print {
+    .hero-buttons {
+      display: flex;
+      justify-content: center; }
+      .hero-buttons .button:not(:last-child) {
+        margin-right: 1.5rem; } }
+
+.hero-head,
+.hero-foot {
+  flex-grow: 0;
+  flex-shrink: 0; }
+
+.hero-body {
+  flex-grow: 1;
+  flex-shrink: 0;
+  padding: 3rem 1.5rem; }
+
+.section {
+  padding: 3rem 1.5rem; }
+  @media screen and (min-width: 1056px) {
+    .section.is-medium {
+      padding: 9rem 1.5rem; }
+    .section.is-large {
+      padding: 18rem 1.5rem; } }
+
+.footer {
+  background-color: #fafafa;
+  padding: 3rem 1.5rem 6rem; }
+
+h1 .docs-heading-anchor, h1 .docs-heading-anchor:hover, h1 .docs-heading-anchor:visited, h2 .docs-heading-anchor, h2 .docs-heading-anchor:hover, h2 .docs-heading-anchor:visited, h3 .docs-heading-anchor, h3 .docs-heading-anchor:hover, h3 .docs-heading-anchor:visited, h4 .docs-heading-anchor, h4 .docs-heading-anchor:hover, h4 .docs-heading-anchor:visited, h5 .docs-heading-anchor, h5 .docs-heading-anchor:hover, h5 .docs-heading-anchor:visited, h6 .docs-heading-anchor, h6 .docs-heading-anchor:hover, h6 .docs-heading-anchor:visited {
+  color: #222222; }
+
+h1 .docs-heading-anchor-permalink, h2 .docs-heading-anchor-permalink, h3 .docs-heading-anchor-permalink, h4 .docs-heading-anchor-permalink, h5 .docs-heading-anchor-permalink, h6 .docs-heading-anchor-permalink {
+  visibility: hidden;
+  vertical-align: middle;
+  margin-left: 0.5em;
+  font-size: 0.7rem; }
+  h1 .docs-heading-anchor-permalink::before, h2 .docs-heading-anchor-permalink::before, h3 .docs-heading-anchor-permalink::before, h4 .docs-heading-anchor-permalink::before, h5 .docs-heading-anchor-permalink::before, h6 .docs-heading-anchor-permalink::before {
+    font-family: "Font Awesome 5 Free";
+    font-weight: 900;
+    content: "\f0c1"; }
+
+h1:hover .docs-heading-anchor-permalink, h2:hover .docs-heading-anchor-permalink, h3:hover .docs-heading-anchor-permalink, h4:hover .docs-heading-anchor-permalink, h5:hover .docs-heading-anchor-permalink, h6:hover .docs-heading-anchor-permalink {
+  visibility: visible; }
+
+.docs-dark-only {
+  display: none !important; }
+
+.admonition {
+  background-color: #b5b5b5;
+  border-style: solid;
+  border-width: 1px;
+  border-color: #363636;
+  border-radius: 4px;
+  font-size: 1rem; }
+  .admonition strong {
+    color: currentColor; }
+  .admonition.is-small, #documenter .docs-sidebar form.docs-search > input.admonition {
+    font-size: 0.75rem; }
+  .admonition.is-medium {
+    font-size: 1.25rem; }
+  .admonition.is-large {
+    font-size: 1.5rem; }
+  .admonition.is-default {
+    background-color: #b5b5b5;
+    border-color: #363636; }
+    .admonition.is-default > .admonition-header {
+      background-color: #363636;
+      color: #fff; }
+    .admonition.is-default > .admonition-body {
+      color: #fff; }
+  .admonition.is-info {
+    background-color: #b8dffa;
+    border-color: #209cee; }
+    .admonition.is-info > .admonition-header {
+      background-color: #209cee;
+      color: #fff; }
+    .admonition.is-info > .admonition-body {
+      color: rgba(0, 0, 0, 0.7); }
+  .admonition.is-success {
+    background-color: #9beeb8;
+    border-color: #22c35b; }
+    .admonition.is-success > .admonition-header {
+      background-color: #22c35b;
+      color: #fff; }
+    .admonition.is-success > .admonition-body {
+      color: rgba(0, 0, 0, 0.7); }
+  .admonition.is-warning {
+    background-color: #fff3c5;
+    border-color: #ffdd57; }
+    .admonition.is-warning > .admonition-header {
+      background-color: #ffdd57;
+      color: rgba(0, 0, 0, 0.7); }
+    .admonition.is-warning > .admonition-body {
+      color: rgba(0, 0, 0, 0.7); }
+  .admonition.is-danger {
+    background-color: #ff857e;
+    border-color: #da0b00; }
+    .admonition.is-danger > .admonition-header {
+      background-color: #da0b00;
+      color: #fff; }
+    .admonition.is-danger > .admonition-body {
+      color: #fff; }
+  .admonition.is-compat {
+    background-color: #99e6f0;
+    border-color: #1db5c9; }
+    .admonition.is-compat > .admonition-header {
+      background-color: #1db5c9;
+      color: #fff; }
+    .admonition.is-compat > .admonition-body {
+      color: rgba(0, 0, 0, 0.7); }
+
+.admonition-header {
+  color: #fff;
+  background-color: #363636;
+  align-items: center;
+  font-weight: 700;
+  justify-content: space-between;
+  line-height: 1.25;
+  padding: 0.75em;
+  position: relative; }
+  .admonition-header:before {
+    font-family: "Font Awesome 5 Free";
+    font-weight: 900;
+    margin-right: 0.75em;
+    content: "\f06a"; }
+
+.admonition-body {
+  color: #222222;
+  padding: 1em 1.25em; }
+  .admonition-body pre {
+    background-color: whitesmoke; }
+  .admonition-body code {
+    background-color: rgba(0, 0, 0, 0.05); }
+
+.docstring {
+  margin-bottom: 1em;
+  background-color: transparent;
+  border: 1px solid #dbdbdb;
+  box-shadow: 2px 2px 3px rgba(10, 10, 10, 0.1);
+  max-width: 100%; }
+  .docstring > header {
+    display: flex;
+    flex-grow: 1;
+    align-items: stretch;
+    padding: 0.75rem;
+    background-color: whitesmoke;
+    box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1);
+    box-shadow: none;
+    border-bottom: 1px solid #dbdbdb; }
+    .docstring > header code {
+      background-color: transparent; }
+    .docstring > header .docstring-binding {
+      margin-right: 0.3em; }
+    .docstring > header .docstring-category {
+      margin-left: 0.3em; }
+  .docstring > section {
+    position: relative;
+    padding: 1rem 1.25rem;
+    border-bottom: 1px solid #dbdbdb; }
+    .docstring > section:last-child {
+      border-bottom: none; }
+    .docstring > section > a.docs-sourcelink {
+      transition: opacity 0.3s;
+      opacity: 0;
+      position: absolute;
+      right: 0.625rem;
+      bottom: 0.5rem; }
+  .docstring:hover > section > a.docs-sourcelink {
+    opacity: 0.2; }
+  .docstring > section:hover a.docs-sourcelink {
+    opacity: 1; }
+
+.documenter-example-output {
+  background-color: white; }
+
+.content pre {
+  border: 1px solid #dbdbdb; }
+
+.content code {
+  font-weight: inherit; }
+
+.content a code {
+  color: #2e63b8; }
+
+.content h1 code, .content h2 code, .content h3 code, .content h4 code, .content h5 code, .content h6 code {
+  color: #222222; }
+
+.content table {
+  display: block;
+  width: initial;
+  max-width: 100%;
+  overflow-x: auto; }
+
+.content blockquote > ul:first-child, .content blockquote > ol:first-child, .content .admonition-body > ul:first-child, .content .admonition-body > ol:first-child {
+  margin-top: 0; }
+
+.breadcrumb a.is-disabled {
+  cursor: default;
+  pointer-events: none; }
+  .breadcrumb a.is-disabled, .breadcrumb a.is-disabled:hover {
+    color: #222222; }
+
+.hljs {
+  background: initial !important;
+  padding: initial !important; }
+
+.katex .katex-mathml {
+  top: 0;
+  right: 0; }
+
+.katex-display, mjx-container, .MathJax_Display {
+  margin: 0.5em 0 !important; }
+
+html {
+  -moz-osx-font-smoothing: auto;
+  -webkit-font-smoothing: auto; }
+
+/* This file contain the overall layout.
+ *
+ * The main container is <div> that is identified by id #documenter.
+ */
+#documenter .docs-main > article {
+  overflow-wrap: break-word; }
+  #documenter .docs-main > article .math-container {
+    overflow-x: auto; }
+
+@media screen and (min-width: 1056px) {
+  #documenter .docs-main {
+    max-width: 52rem;
+    margin-left: 20rem;
+    padding-right: 1rem; } }
+
+@media screen and (max-width: 1055px) {
+  #documenter .docs-main {
+    width: 100%; }
+    #documenter .docs-main > article {
+      max-width: 52rem;
+      margin-left: auto;
+      margin-right: auto;
+      margin-bottom: 1rem;
+      padding: 0 1rem; }
+    #documenter .docs-main > header, #documenter .docs-main > nav {
+      max-width: 100%;
+      width: 100%;
+      margin: 0; } }
+
+#documenter .docs-main header.docs-navbar {
+  background-color: white;
+  border-bottom: 1px solid #dbdbdb;
+  z-index: 2;
+  min-height: 4rem;
+  margin-bottom: 1rem;
+  display: flex; }
+  #documenter .docs-main header.docs-navbar .breadcrumb {
+    flex-grow: 1; }
+  #documenter .docs-main header.docs-navbar .docs-right {
+    display: flex;
+    white-space: nowrap; }
+    #documenter .docs-main header.docs-navbar .docs-right .docs-icon, #documenter .docs-main header.docs-navbar .docs-right .docs-label, #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button {
+      display: inline-block; }
+    #documenter .docs-main header.docs-navbar .docs-right .docs-label {
+      padding: 0;
+      margin-left: 0.3em; }
+    #documenter .docs-main header.docs-navbar .docs-right .docs-settings-button {
+      margin: auto 0 auto 1rem; }
+    #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button {
+      font-size: 1.5rem;
+      margin: auto 0 auto 1rem; }
+  #documenter .docs-main header.docs-navbar > * {
+    margin: auto 0; }
+  @media screen and (max-width: 1055px) {
+    #documenter .docs-main header.docs-navbar {
+      position: sticky;
+      top: 0;
+      padding: 0 1rem;
+      /* For Headroom.js */
+      transition-property: top, box-shadow;
+      -webkit-transition-property: top, box-shadow;
+      /* Safari */
+      transition-duration: 0.3s;
+      -webkit-transition-duration: 0.3s;
+      /* Safari */ }
+      #documenter .docs-main header.docs-navbar.headroom--not-top {
+        box-shadow: 0.2rem 0rem 0.4rem #bbb;
+        transition-duration: 0.7s;
+        -webkit-transition-duration: 0.7s;
+        /* Safari */ }
+      #documenter .docs-main header.docs-navbar.headroom--unpinned.headroom--not-top.headroom--not-bottom {
+        top: -4.5rem;
+        transition-duration: 0.7s;
+        -webkit-transition-duration: 0.7s;
+        /* Safari */ } }
+
+#documenter .docs-main section.footnotes {
+  border-top: 1px solid #dbdbdb; }
+  #documenter .docs-main section.footnotes li .tag:first-child, #documenter .docs-main section.footnotes li .docstring > section > a.docs-sourcelink:first-child, #documenter .docs-main section.footnotes li .content kbd:first-child, .content #documenter .docs-main section.footnotes li kbd:first-child {
+    margin-right: 1em;
+    margin-bottom: 0.4em; }
+
+#documenter .docs-main .docs-footer {
+  display: flex;
+  flex-wrap: wrap;
+  margin-left: 0;
+  margin-right: 0;
+  border-top: 1px solid #dbdbdb;
+  padding-top: 1rem;
+  padding-bottom: 1rem; }
+  @media screen and (max-width: 1055px) {
+    #documenter .docs-main .docs-footer {
+      padding-left: 1rem;
+      padding-right: 1rem; } }
+  #documenter .docs-main .docs-footer .docs-footer-nextpage, #documenter .docs-main .docs-footer .docs-footer-prevpage {
+    flex-grow: 1; }
+  #documenter .docs-main .docs-footer .docs-footer-nextpage {
+    text-align: right; }
+  #documenter .docs-main .docs-footer .flexbox-break {
+    flex-basis: 100%;
+    height: 0; }
+  #documenter .docs-main .docs-footer .footer-message {
+    font-size: 0.8em;
+    margin: 0.5em auto 0 auto;
+    text-align: center; }
+
+#documenter .docs-sidebar {
+  display: flex;
+  flex-direction: column;
+  color: #0a0a0a;
+  background-color: whitesmoke;
+  border-right: 1px solid #dbdbdb;
+  padding: 0;
+  flex: 0 0 18rem;
+  z-index: 5;
+  font-size: 1rem;
+  position: fixed;
+  left: -18rem;
+  width: 18rem;
+  height: 100%;
+  transition: left 0.3s;
+  /* Setting up a nicer theme style for the scrollbar */ }
+  #documenter .docs-sidebar.visible {
+    left: 0;
+    box-shadow: 0.4rem 0rem 0.8rem #bbb; }
+    @media screen and (min-width: 1056px) {
+      #documenter .docs-sidebar.visible {
+        box-shadow: none; } }
+  @media screen and (min-width: 1056px) {
+    #documenter .docs-sidebar {
+      left: 0;
+      top: 0; } }
+  #documenter .docs-sidebar .docs-logo {
+    margin-top: 1rem;
+    padding: 0 1rem; }
+    #documenter .docs-sidebar .docs-logo > img {
+      max-height: 6rem;
+      margin: auto; }
+  #documenter .docs-sidebar .docs-package-name {
+    flex-shrink: 0;
+    font-size: 1.5rem;
+    font-weight: 700;
+    text-align: center;
+    white-space: nowrap;
+    overflow: hidden;
+    padding: 0.5rem 0; }
+    #documenter .docs-sidebar .docs-package-name .docs-autofit {
+      max-width: 16.2rem; }
+  #documenter .docs-sidebar .docs-version-selector {
+    border-top: 1px solid #dbdbdb;
+    display: none;
+    padding: 0.5rem; }
+    #documenter .docs-sidebar .docs-version-selector.visible {
+      display: flex; }
+  #documenter .docs-sidebar ul.docs-menu {
+    flex-grow: 1;
+    user-select: none;
+    border-top: 1px solid #dbdbdb;
+    padding-bottom: 1.5rem;
+    /* Managing collapsible submenus */ }
+    #documenter .docs-sidebar ul.docs-menu > li > .tocitem {
+      font-weight: bold; }
+    #documenter .docs-sidebar ul.docs-menu > li li {
+      font-size: 0.95rem;
+      margin-left: 1em;
+      border-left: 1px solid #dbdbdb; }
+    #documenter .docs-sidebar ul.docs-menu input.collapse-toggle {
+      display: none; }
+    #documenter .docs-sidebar ul.docs-menu ul.collapsed {
+      display: none; }
+    #documenter .docs-sidebar ul.docs-menu input:checked ~ ul.collapsed {
+      display: block; }
+    #documenter .docs-sidebar ul.docs-menu label.tocitem {
+      display: flex; }
+      #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-label {
+        flex-grow: 2; }
+      #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron {
+        display: inline-block;
+        font-style: normal;
+        font-variant: normal;
+        text-rendering: auto;
+        line-height: 1;
+        font-size: 0.75rem;
+        margin-left: 1rem;
+        margin-top: auto;
+        margin-bottom: auto; }
+        #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron::before {
+          font-family: "Font Awesome 5 Free";
+          font-weight: 900;
+          content: "\f054"; }
+    #documenter .docs-sidebar ul.docs-menu input:checked ~ label.tocitem .docs-chevron::before {
+      content: "\f078"; }
+    #documenter .docs-sidebar ul.docs-menu .tocitem {
+      display: block;
+      padding: 0.5rem 0.5rem; }
+      #documenter .docs-sidebar ul.docs-menu .tocitem, #documenter .docs-sidebar ul.docs-menu .tocitem:hover {
+        color: #0a0a0a;
+        background: whitesmoke; }
+    #documenter .docs-sidebar ul.docs-menu a.tocitem:hover, #documenter .docs-sidebar ul.docs-menu label.tocitem:hover {
+      color: #0a0a0a;
+      background-color: #ebebeb; }
+    #documenter .docs-sidebar ul.docs-menu li.is-active {
+      border-top: 1px solid #dbdbdb;
+      border-bottom: 1px solid #dbdbdb;
+      background-color: white; }
+      #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem, #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem:hover {
+        background-color: white;
+        color: #0a0a0a; }
+      #documenter .docs-sidebar ul.docs-menu li.is-active ul.internal .tocitem:hover {
+        background-color: #ebebeb;
+        color: #0a0a0a; }
+    #documenter .docs-sidebar ul.docs-menu > li.is-active:first-child {
+      border-top: none; }
+    #documenter .docs-sidebar ul.docs-menu ul.internal {
+      margin: 0 0.5rem 0.5rem;
+      border-top: 1px solid #dbdbdb; }
+      #documenter .docs-sidebar ul.docs-menu ul.internal li {
+        font-size: 0.85rem;
+        border-left: none;
+        margin-left: 0;
+        margin-top: 0.5rem; }
+      #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem {
+        width: 100%;
+        padding: 0; }
+        #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem::before {
+          content: "⚬";
+          margin-right: 0.4em; }
+  #documenter .docs-sidebar form.docs-search {
+    margin: auto;
+    margin-top: 0.5rem;
+    margin-bottom: 0.5rem; }
+    #documenter .docs-sidebar form.docs-search > input {
+      width: 14.4rem; }
+  @media screen and (min-width: 1056px) {
+    #documenter .docs-sidebar ul.docs-menu {
+      overflow-y: auto;
+      -webkit-overflow-scroll: touch; }
+      #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar {
+        width: .3rem;
+        background: none; }
+      #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb {
+        border-radius: 5px 0px 0px 5px;
+        background: #e0e0e0; }
+      #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb:hover {
+        background: #cccccc; } }
+  @media screen and (max-width: 1055px) {
+    #documenter .docs-sidebar {
+      overflow-y: auto;
+      -webkit-overflow-scroll: touch; }
+      #documenter .docs-sidebar::-webkit-scrollbar {
+        width: .3rem;
+        background: none; }
+      #documenter .docs-sidebar::-webkit-scrollbar-thumb {
+        border-radius: 5px 0px 0px 5px;
+        background: #e0e0e0; }
+      #documenter .docs-sidebar::-webkit-scrollbar-thumb:hover {
+        background: #cccccc; } }
+
+#documenter .docs-main #documenter-search-info {
+  margin-bottom: 1rem; }
+
+#documenter .docs-main #documenter-search-results {
+  list-style-type: circle;
+  list-style-position: outside; }
+  #documenter .docs-main #documenter-search-results li {
+    margin-left: 2rem; }
+  #documenter .docs-main #documenter-search-results .docs-highlight {
+    background-color: yellow; }
+
+/*
+
+Original highlight.js style (c) Ivan Sagalaev <maniac@softwaremaniacs.org>
+
+*/
+.hljs {
+  display: block;
+  overflow-x: auto;
+  padding: 0.5em;
+  background: #F0F0F0; }
+
+/* Base color: saturation 0; */
+.hljs,
+.hljs-subst {
+  color: #444; }
+
+.hljs-comment {
+  color: #888888; }
+
+.hljs-keyword,
+.hljs-attribute,
+.hljs-selector-tag,
+.hljs-meta-keyword,
+.hljs-doctag,
+.hljs-name {
+  font-weight: bold; }
+
+/* User color: hue: 0 */
+.hljs-type,
+.hljs-string,
+.hljs-number,
+.hljs-selector-id,
+.hljs-selector-class,
+.hljs-quote,
+.hljs-template-tag,
+.hljs-deletion {
+  color: #880000; }
+
+.hljs-title,
+.hljs-section {
+  color: #880000;
+  font-weight: bold; }
+
+.hljs-regexp,
+.hljs-symbol,
+.hljs-variable,
+.hljs-template-variable,
+.hljs-link,
+.hljs-selector-attr,
+.hljs-selector-pseudo {
+  color: #BC6060; }
+
+/* Language color: hue: 90; */
+.hljs-literal {
+  color: #78A960; }
+
+.hljs-built_in,
+.hljs-bullet,
+.hljs-code,
+.hljs-addition {
+  color: #397300; }
+
+/* Meta color: hue: 200 */
+.hljs-meta {
+  color: #1f7199; }
+
+.hljs-meta-string {
+  color: #4d99bf; }
+
+/* Misc effects */
+.hljs-emphasis {
+  font-style: italic; }
+
+.hljs-strong {
+  font-weight: bold; }
diff --git a/docs/assets/themeswap.js b/docs/assets/themeswap.js
index 9f5eebe..c58e993 100644
--- a/docs/assets/themeswap.js
+++ b/docs/assets/themeswap.js
@@ -1,17 +1,20 @@
 // Small function to quickly swap out themes. Gets put into the <head> tag..
 function set_theme_from_local_storage() {
-  // Initialize the theme to null, which means default
+  // Intialize the theme to null, which means default
   var theme = null;
   // If the browser supports the localstorage and is not disabled then try to get the
   // documenter theme
-  if (window.localStorage != null) {
+  if(window.localStorage != null) {
     // Get the user-picked theme from localStorage. May be `null`, which means the default
     // theme.
-    theme = window.localStorage.getItem("documenter-theme");
+    theme =  window.localStorage.getItem("documenter-theme");
   }
+  // Check if the browser supports user color preference
+  var darkPreference = false;
   // Check if the users preference is for dark color scheme
-  var darkPreference =
-    window.matchMedia("(prefers-color-scheme: dark)").matches === true;
+  if(window.matchMedia('(prefers-color-scheme: dark)').matches === true) {
+    darkPreference = true;
+  }
   // Initialize a few variables for the loop:
   //
   //  - active: will contain the index of the theme that should be active. Note that there
@@ -21,64 +24,43 @@ function set_theme_from_local_storage() {
   //
   //  - disabled: style sheets that should be disabled (i.e. all the theme style sheets
   //    that are not the currently active theme)
-  var active = null;
-  var disabled = [];
-  var primaryLightTheme = null;
-  var primaryDarkTheme = null;
+  var active = null; var disabled = []; var darkTheme = null;
   for (var i = 0; i < document.styleSheets.length; i++) {
     var ss = document.styleSheets[i];
     // The <link> tag of each style sheet is expected to have a data-theme-name attribute
     // which must contain the name of the theme. The names in localStorage much match this.
     var themename = ss.ownerNode.getAttribute("data-theme-name");
     // attribute not set => non-theme stylesheet => ignore
-    if (themename === null) continue;
+    if(themename === null) continue;
     // To distinguish the default (primary) theme, it needs to have the data-theme-primary
     // attribute set.
-    if (ss.ownerNode.getAttribute("data-theme-primary") !== null) {
-      primaryLightTheme = themename;
-    }
-    // Check if the theme is primary dark theme so that we could store its name in darkTheme
-    if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) {
-      primaryDarkTheme = themename;
-    }
+    var isprimary = (ss.ownerNode.getAttribute("data-theme-primary") !== null);
+    // Check if the theme is primary dark theme
+    var isDarkTheme = (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null);
+    // If ss is for dark theme then set the value of darkTheme to the name of the theme
+    if(isDarkTheme) darkTheme = themename;
     // If we find a matching theme (and it's not the default), we'll set active to non-null
-    if (themename === theme) active = i;
+    if(themename === theme) active = i;
     // Store the style sheets of inactive themes so that we could disable them
-    if (themename !== theme) disabled.push(ss);
+    if(themename !== theme) disabled.push(ss);
   }
-  var activeTheme = null;
-  if (active !== null) {
+  if(active !== null) {
     // If we did find an active theme, we'll (1) add the theme--$(theme) class to <html>
-    document.getElementsByTagName("html")[0].className = "theme--" + theme;
-    activeTheme = theme;
-  } else {
-    // If we did _not_ find an active theme, then we need to fall back to the primary theme
-    // which can either be dark or light, depending on the user's OS preference.
-    var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme;
-    // In case it somehow happens that the relevant primary theme was not found in the
-    // preceding loop, we abort without doing anything.
-    if (activeTheme === null) {
-      console.error("Unable to determine primary theme.");
-      return;
-    }
-    // When switching to the primary light theme, then we must not have a class name
-    // for the <html> tag. That's only for non-primary or the primary dark theme.
-    if (darkPreference) {
-      document.getElementsByTagName("html")[0].className =
-        "theme--" + activeTheme;
-    } else {
-      document.getElementsByTagName("html")[0].className = "";
-    }
+    document.getElementsByTagName('html')[0].className = "theme--" + theme;
+    // and (2) disable all the other theme stylesheets
+    disabled.forEach(function(ss){
+      ss.disabled = true;
+    });
   }
-  for (var i = 0; i < document.styleSheets.length; i++) {
-    var ss = document.styleSheets[i];
-    // The <link> tag of each style sheet is expected to have a data-theme-name attribute
-    // which must contain the name of the theme. The names in localStorage much match this.
-    var themename = ss.ownerNode.getAttribute("data-theme-name");
-    // attribute not set => non-theme stylesheet => ignore
-    if (themename === null) continue;
-    // we'll disable all the stylesheets, except for the active one
-    ss.disabled = !(themename == activeTheme);
+  else if(darkTheme !== null && darkPreference === true) {
+    // If we did find an active theme, we'll (1) add the theme--$(theme) class to <html>
+    document.getElementsByTagName('html')[0].className = "theme--" + darkTheme;
+    // and (2) disable all the other theme stylesheets
+    disabled.forEach(function(ss){
+      if (ss.ownerNode.getAttribute("data-theme-name") !== darkTheme) {
+        ss.disabled = true;
+      }
+    });
   }
 }
 set_theme_from_local_storage();
diff --git a/docs/assets/warner.js b/docs/assets/warner.js
deleted file mode 100644
index 3f6f5d0..0000000
--- a/docs/assets/warner.js
+++ /dev/null
@@ -1,52 +0,0 @@
-function maybeAddWarning() {
-  // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE
-  // in siteinfo.js.
-  // If either of these are undefined something went horribly wrong, so we abort.
-  if (
-    window.DOCUMENTER_NEWEST === undefined ||
-    window.DOCUMENTER_CURRENT_VERSION === undefined ||
-    window.DOCUMENTER_STABLE === undefined
-  ) {
-    return;
-  }
-
-  // Current version is not a version number, so we can't tell if it's the newest version. Abort.
-  if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) {
-    return;
-  }
-
-  // Current version is newest version, so no need to add a warning.
-  if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) {
-    return;
-  }
-
-  // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs.
-  if (document.body.querySelector('meta[name="robots"]') === null) {
-    const meta = document.createElement("meta");
-    meta.name = "robots";
-    meta.content = "noindex";
-
-    document.getElementsByTagName("head")[0].appendChild(meta);
-  }
-
-  const div = document.createElement("div");
-  div.classList.add("outdated-warning-overlay");
-  const closer = document.createElement("button");
-  closer.classList.add("outdated-warning-closer", "delete");
-  closer.addEventListener("click", function () {
-    document.body.removeChild(div);
-  });
-  const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE;
-  div.innerHTML =
-    'This documentation is not for the latest stable release, but for either the development version or an older release.<br><a href="' +
-    href +
-    '">Click here to go to the documentation for the latest stable release.</a>';
-  div.appendChild(closer);
-  document.body.appendChild(div);
-}
-
-if (document.readyState === "loading") {
-  document.addEventListener("DOMContentLoaded", maybeAddWarning);
-} else {
-  maybeAddWarning();
-}
diff --git a/docs/convergence/index.html b/docs/convergence/index.html
new file mode 100644
index 0000000..29e4a43
--- /dev/null
+++ b/docs/convergence/index.html
@@ -0,0 +1,2 @@
+<!DOCTYPE html>
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+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Developers · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img src="../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Introduction</a></li><li><a class="tocitem" href="../user-guide/">User Guide</a></li><li><a class="tocitem" href="../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../methods/">Methods</a></li><li class="is-active"><a class="tocitem" href>Developers</a><ul class="internal"><li><a class="tocitem" href="#Simulation-Workflow"><span>Simulation Workflow</span></a></li><li><a class="tocitem" href="#How-to-Contribute"><span>How to Contribute</span></a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Developers</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Developers</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/dev.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Developers"><a class="docs-heading-anchor" href="#Developers">Developers</a><a id="Developers-1"></a><a class="docs-heading-anchor-permalink" href="#Developers" title="Permalink"></a></h1><h2 id="Simulation-Workflow"><a class="docs-heading-anchor" href="#Simulation-Workflow">Simulation Workflow</a><a id="Simulation-Workflow-1"></a><a class="docs-heading-anchor-permalink" href="#Simulation-Workflow" title="Permalink"></a></h2><p><img src="../assets/flowchart.png" alt="Flowchart of the simulation workflow"/></p><h2 id="How-to-Contribute"><a class="docs-heading-anchor" href="#How-to-Contribute">How to Contribute</a><a id="How-to-Contribute-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-Contribute" title="Permalink"></a></h2><p>Contributions are welcome! 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-<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>The Anderson Impurity Model · MPSDynamics.jl</title><meta name="title" content="The Anderson Impurity Model · MPSDynamics.jl"/><meta property="og:title" content="The Anderson Impurity Model · MPSDynamics.jl"/><meta property="twitter:title" content="The Anderson Impurity Model · MPSDynamics.jl"/><meta name="description" content="Documentation for MPSDynamics.jl."/><meta property="og:description" content="Documentation for MPSDynamics.jl."/><meta property="twitter:description" content="Documentation for MPSDynamics.jl."/><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link 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href="../../">MPSDynamics.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">Introduction</a></li><li><a class="tocitem" href="../../user-guide/">User Guide</a></li><li><a class="tocitem" href="../../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../timedep/">Time-dependent Hamiltonian</a></li><li class="is-active"><a class="tocitem" href>The Anderson Impurity Model</a><ul class="internal"><li><a class="tocitem" href="#Basics-of-the-fermionic-chain-mapping"><span>Basics of the fermionic chain mapping</span></a></li><li><a class="tocitem" href="#Code-implementation"><span>Code implementation</span></a></li><li><a class="tocitem" href="#Bibliography"><span>Bibliography</span></a></li></ul></li><li><a class="tocitem" href="../bath-observables/">Inspecting the bath by undoing the chain mapping</a></li><li><a class="tocitem" href="../protontransfer/">Proton Transfer Model</a></li></ul></li><li><a class="tocitem" href="../../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../../methods/">Methods</a></li><li><a class="tocitem" href="../../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>The Anderson Impurity Model</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The Anderson Impurity Model</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-Anderson-Impurity-Model"><a class="docs-heading-anchor" href="#The-Anderson-Impurity-Model">The Anderson Impurity Model</a><a id="The-Anderson-Impurity-Model-1"></a><a class="docs-heading-anchor-permalink" href="#The-Anderson-Impurity-Model" title="Permalink"></a></h1><p>Here we give some context on the examples provided in <code>MPSDynamics/examples/anderson_model_double.jl</code> and <code>MPSDynamics/examples/anderson_model_interleaved.jl</code>. In these two examples, we use the fermionic chain mapping proposed in <sup class="footnote-reference"><a id="citeref-khon_efficient_2021" href="#footnote-khon_efficient_2021">[khon_efficient_2021]</a></sup> to show how to perform tensor network simulations of the <a href="https://en.wikipedia.org/wiki/Anderson_impurity_model">Single Impurity Anderson Model</a> (SIAM) with the <code>MPSDynamics.jl</code> library. </p><h2 id="Basics-of-the-fermionic-chain-mapping"><a class="docs-heading-anchor" href="#Basics-of-the-fermionic-chain-mapping">Basics of the fermionic chain mapping</a><a id="Basics-of-the-fermionic-chain-mapping-1"></a><a class="docs-heading-anchor-permalink" href="#Basics-of-the-fermionic-chain-mapping" title="Permalink"></a></h2><p>Before giving the code implementation, a short recap on the problem statement and on the fermionic mapping used to solved it. The SIAM Hamiltonian is defined as:</p><p class="math-container">\[    \hat H^\text{SIAM}  = \hat H_\text{loc} + \hat H_\text{hyb} + \hat H_\text{cond} = \overbrace{\epsilon_d \hat d^\dagger \hat d}^{\hat H_\text{loc}} + \underbrace{\sum_{k} V_k \Big( \hat d^\dagger \hat c_k + \hat c_k^\dagger \hat d \Big)}_{\hat H_\text{hyb}} + \underbrace{\sum_k \epsilon_k \hat c_k^\dagger \hat c_k}_{\hat H_\text{cond}}.\]</p><p>This Hamiltonian represents a located impurity (<span>$\hat H_\text{loc}$</span>), conduction electrons (<span>$\hat H_\text{cond}$</span>) and a hybridization term between the impurity and the conduction electrons (<span>$\hat H_\text{hyb}$</span>). All of the operators obey to the usual fermionic anti-commutation relations: <span>$\{\hat c_i, \hat c_j^\dagger \} = \delta_{ij}$</span>, <span>$\{\hat c_i, \hat c_j \} =\{\hat c_i^\dagger, \hat c_j^\dagger \} =0$</span> <span>$\forall i,j$</span>. The chain mapping is based on a thermofield-like transformation  <sup class="footnote-reference"><a id="citeref-devega_thermo_2015" href="#footnote-devega_thermo_2015">[devega_thermo_2015]</a></sup>, performed with fermions: ancillary fermionic operators <span>$\hat c_{2k}$</span> are defined, one for each of the original fermionic modes <span>$\hat c_{1k}$</span>. A Bogoliubov transformation is then applied, so that two new fermionic modes <span>$\hat f_{1k}$</span> and <span>$\hat f_{2k}$</span> are defined as a linear combination of <span>$\hat c_{1k}$</span> and <span>$\hat c_{2k}$</span>. Two chains are defined: the chain labelled <span>$1$</span> for the empty modes:</p><p class="math-container">\[\hat f_{1k}=e^{-iG}  \hat c_k e^{iG}= \cosh(\theta_k) \hat c_{1k} -\sinh(\theta_k)  \hat c_{2k}^\dagger \\
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>The Anderson Impurity Model · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Introduction</a></li><li><a class="tocitem" href="../../user-guide/">User Guide</a></li><li><a class="tocitem" href="../../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../timedep/">Time-dependent Hamiltonian</a></li><li class="is-active"><a class="tocitem" href>The Anderson Impurity Model</a><ul class="internal"><li><a class="tocitem" href="#Basics-of-the-fermionic-chain-mapping"><span>Basics of the fermionic chain mapping</span></a></li><li><a class="tocitem" href="#Code-implementation"><span>Code implementation</span></a></li></ul></li><li><a class="tocitem" href="../bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="../../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../../methods/">Methods</a></li><li><a class="tocitem" href="../../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>The Anderson Impurity Model</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The Anderson Impurity Model</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/examples/anderson-model.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="The-Anderson-Impurity-Model"><a class="docs-heading-anchor" href="#The-Anderson-Impurity-Model">The Anderson Impurity Model</a><a id="The-Anderson-Impurity-Model-1"></a><a class="docs-heading-anchor-permalink" href="#The-Anderson-Impurity-Model" title="Permalink"></a></h1><p>Here we give some context on the examples provided in <code>MPSDynamics/examples/anderson_model_double.jl</code> and <code>MPSDynamics/examples/anderson_model_interleaved.jl</code>. In these two examples, we use the fermionic chain mapping proposed in <sup class="footnote-reference"><a id="citeref-khon_efficient_2021" href="#footnote-khon_efficient_2021">[khon_efficient_2021]</a></sup> to show how to perform tensor network simulations of the Single Impurity Anderson Model (SIAM) with the <code>MPSDynamics.jl</code> library. </p><h2 id="Basics-of-the-fermionic-chain-mapping"><a class="docs-heading-anchor" href="#Basics-of-the-fermionic-chain-mapping">Basics of the fermionic chain mapping</a><a id="Basics-of-the-fermionic-chain-mapping-1"></a><a class="docs-heading-anchor-permalink" href="#Basics-of-the-fermionic-chain-mapping" title="Permalink"></a></h2><p>Before giving the code implementation, a short recap on the problem statement and on the fermionic mapping used to solved it. The SIAM Hamiltonian is defined as:</p><p class="math-container">\[    \hat H^\text{SIAM}  = \hat H_\text{loc} + \hat H_\text{hyb} + \hat H_\text{cond} = \overbrace{\epsilon_d \hat d^\dagger \hat d}^{\hat H_\text{loc}} + \underbrace{\sum_{k} V_k \Big( \hat d^\dagger \hat c_k + \hat c_k^\dagger \hat d \Big)}_{H_\text{hyb}} + \underbrace{\sum_k \epsilon_k \hat c_k^\dagger \hat c_k}_{H_I^\text{chain}}.\]</p><p>All of the operators obey to the usual fermionic anti-commutation relations: <span>$\{\hat c_i, \hat c_j^\dagger \} = \delta_{ij}$</span>, <span>$\{\hat c_i, \hat c_j \} =\{\hat c_i^\dagger, \hat c_j^\dagger \} =0$</span> <span>$\forall i,j$</span>. The chain mapping is based on a thermofield-like transformation  <sup class="footnote-reference"><a id="citeref-devega_thermo_2015" href="#footnote-devega_thermo_2015">[devega_thermo_2015]</a></sup>, performed with fermions: ancillary fermionic operators <span>$\hat c_{2k}$</span> are defined, one for each of the original fermionic modes <span>$\hat c_{1k}$</span>. A Bogoliubov transformation is then applied, so that two new fermionic modes <span>$\hat f_{1k}$</span> and <span>$\hat f_{2k}$</span> are defined as a linear combination of <span>$\hat c_{1k}$</span> and <span>$\hat c_{2k}$</span>. Two chains are defined: the chain labelled <span>$1$</span> for the empty modes:</p><p class="math-container">\[\hat f_{1k}=e^{-iG}  \hat c_k e^{iG}= \cosh(\theta_k) \hat c_{1k} -\sinh(\theta_k)  \hat c_{2k}^\dagger \\
 \hat f_{2k}=e^{-iG} \hat c_k e^{iG}= \cosh(\theta_k)  \hat c_{1k} +\sinh(\theta_k)  \hat c_{2k}^\dagger.\]</p><p>We remark that this is the same Bogoliubov transformation used in the thermofield<sup class="footnote-reference"><a id="citeref-devega_thermo_2015" href="#footnote-devega_thermo_2015">[devega_thermo_2015]</a></sup> for the bosonic case: the only thing that changes is a plus sign, that takes into account the fermionic anti-commutation relations. With these new fermionic operators we obtain the thermofield-transformed Hamiltonian, where the system interacts with two environments at zero temperature, allowing for pure state simulations (and thus the employement of MPS).</p><p>The thermofield-transformed Hamiltonian is then mapped on two chains, defined and constructed using the TEDOPA chain mapping: the chain labelled <span>$1$</span> is for the empty modes, the chain labelled <span>$2$</span> for the filled modes. The following relations are used to define the functions equivalent to the spectral density of the bosonic case, one for each chain:</p><p class="math-container">\[\begin{aligned}
    &amp;V_{1k} = V_{k} \sin \theta_k = \sqrt{\frac{1}{e^{\beta \epsilon_k}+1}} \\
    &amp;V_{2k} = V_{k} \cos \theta_k = \sqrt{\frac{1}{e^{-\beta \epsilon_k}+1}}, 
-\end{aligned}\]</p><p>where we choose the spectral function that characterizes the fermionic bath to be: <span>$V_k= \sqrt{1-k^2}$</span>, and we define the dispersion relation as: <span>$e_k = k$</span>, that is, a linear dispersion relation with propagation speed equal to <span>$1$</span>. This latter choice corresponds to a model of metals (gapless energy spectrum). We select a filled state as the initial state of the defect. Using the mapping proposed, the chain Hamiltonian becomes:</p><p class="math-container">\[    \begin{aligned}
+\end{aligned}\]</p><p>where we choose the spectral function that characterizes the fermionic bath to be: <span>$V_k= \sqrt{1-k^2}$</span>, and we define the dispersion relation as: <span>$e_k = k$</span>, that is, a linear dispersion relation with propagation speed equal to <span>$1$</span>. This latter choice corresponds to a model of metals (gapless energy spectrum). We select a filled state as the initial state of the defect. Using the mapping proposed in [1], the chain Hamiltonian becomes:</p><p class="math-container">\[    \begin{aligned}
     \hat H^\text{chain}  = \hat H_\text{loc} &amp;+ \sum_{i = \{1,2\}}\bigg[ J_{i,0} \Big(\hat d^\dagger \hat a_{i,0} + \hat d \hat a_{i,0}^\dagger \Big) + \\ &amp;+ \sum_{n=1}^\infty  \Big( J_{i,n} \hat a_{i,n}^\dagger \hat a_{i,n-1} +  J_{i,n} \hat  a_{i,n-1}^\dagger \hat a_{i,n} \Big) + \sum_{n=0}^\infty E_{i,n} \hat  a_{i,n}^\dagger \hat a_{i,n} \bigg],
     \end{aligned}\]</p><p>where the <span>$J_{i,n}$</span> coefficients are the couplings between the chain sites and the <span>$E_{i,n}$</span> coefficients are the energies associated to each chain site. Clearly, the interactions are between nearest neighbors. This, combined with the fact that the fermions in our model are spinless, enables a straightforward mapping into fermionic operators of the bosonic creation and annihilation operators, that on their part obey to the bosonic commutation relations: <span>$[\hat b_i, \hat b_j^\dagger] = \delta_{ij}$</span>, <span>$[\hat b_i, \hat b_j] =[\hat b_i^\dagger, \hat b_j^\dagger] =0$</span> <span>$\forall i,j$</span>. The mapping derived from Jordan-Wigner transformations for spinless fermions is:</p><p class="math-container">\[    \hat a_{i}^\dagger \hat a_{i+1} + \hat a_{i+1}^\dagger \hat a_{i} = \hat b_{i}^\dagger \hat b_{i+1} + \hat b_{i+1}^\dagger \hat b_{i}.  \]</p><h3 id="Double-chain-MPO"><a class="docs-heading-anchor" href="#Double-chain-MPO">Double chain MPO</a><a id="Double-chain-MPO-1"></a><a class="docs-heading-anchor-permalink" href="#Double-chain-MPO" title="Permalink"></a></h3><p><img src="../doublechain.png" alt="image"/></p><p>We can represent the Hamiltonian <span>$\hat H^\text{chain}$</span> as a MPOs bond dimension is: <span>$\chi = 4$</span>. The MPO has the structure of a double chain, with an impurty site at the center, and therefore can be seen as the product of the following matrices:</p><p class="math-container">\[    H = W_{1N} \cdot...\cdot W_{1 0} \cdot W_d \cdot W_{20} \cdot ... \cdot W_{2 N},  \]</p><p>where the matrices are defined as:</p><p class="math-container">\[\begin{aligned}
 &amp; W_{1N} = 
@@ -34,7 +34,7 @@
 \begin{bmatrix}
  E_{2,N} \hat b_{2,N}^\dagger \hat b_{2,N} \\ J_{2,N} \hat b_{2,N} \\ J_{2,N} \hat b_{2,N}^\dagger \\ \hat{\mathbb I}
 \end{bmatrix}.
-\end{aligned}\]</p><h3 id="Interleaved-chain-MPO"><a class="docs-heading-anchor" href="#Interleaved-chain-MPO">Interleaved chain MPO</a><a id="Interleaved-chain-MPO-1"></a><a class="docs-heading-anchor-permalink" href="#Interleaved-chain-MPO" title="Permalink"></a></h3><p><img src="../foldedchain.png" alt="image"/></p><p>The drawback of the double chain representation is that the particle-hole pairs are spatially separated in the MPS, creating correlations and therefore leading to a dramatic increase in the bond dimensions. This is why Kohn and Santoro propose an interleaved geometry, the advantages of which are thoroughly explained in <sup class="footnote-reference"><a id="citeref-khon_eff_2022" href="#footnote-khon_eff_2022">[khon_eff_2022]</a></sup>. Exploiting the interleaved representation, the interaction comes to be between next-nearest neighbors: a string operator appears in the Jordan-Wigner transformation from bosons to fermions:</p><p class="math-container">\[    \hat a_{i}^\dagger \hat a_{i+2} + \hat a_{i+2}^\dagger \hat a_{i} = \hat b_{i}^\dagger \hat F_{i+1} \hat b_{i+2} + \hat b_{i} \hat F_{i+1} \hat b_{i+2}^\dagger,\]</p><p>where the string operator <span>$\hat F_i$</span> is defined as: <span>$\hat F_i = (-1)^{\hat n_i} = \hat{\mathbb I} -2 \hat n_i = \hat{\mathbb I}-2 \hat b_i^\dagger \hat b_i$</span>.  It is possible to find the analytical form also for MPOs with long range interaction. In the case of next-nearest neighbors interactions between spinless fermions, in the interleaved geometry of the figure above, the MPO representation will require a bond dimension <span>$\chi=6$</span>. We explicitly write it as:</p><p class="math-container">\[    H = W_{d} \cdot W_{2 0} \cdot W_{1 0} \cdot...\cdot W_{2N} \cdot W_{1 N},  \]</p><p>where the matrices are defined as: </p><p class="math-container">\[\begin{aligned}
+\end{aligned}\]</p><h3 id="Interleaved-chain-MPO"><a class="docs-heading-anchor" href="#Interleaved-chain-MPO">Interleaved chain MPO</a><a id="Interleaved-chain-MPO-1"></a><a class="docs-heading-anchor-permalink" href="#Interleaved-chain-MPO" title="Permalink"></a></h3><p><img src="../foldedchain.png" alt="image"/></p><p>The drawback of the double chain representation is that the particle-hole pairs are spatially separated in the MPS, creating correlations and therefore leading to a dramatic increase in the bond dimensions. This is why Kohn and Santoro propose an interleaved geometry, the advantages of which are thoroughly explained in <sup class="footnote-reference"><a id="citeref-khon_eff_2022" href="#footnote-khon_eff_2022">[khon_eff_2022]</a></sup>. Exploiting the interleaved representation, the interaction comes to be between next-nearest neighbors: a string operator appears in the Jordan-Wigner transformation from bosons to fermions:</p><p class="math-container">\[    \hat a_{i}^\dagger \hat a_{i+2} + \hat a_{i+2}^\dagger \hat a_{i} = \hat b_{i}^\dagger \hat F_{i+1} \hat b_{i+2} + \hat b_{i} \hat F_{i+1} \hat b_{i+2}^\dagger,\]</p><p>where the string operator <span>$\hat F_i$</span> is defined as: <span>$\hat F_i = (-1)^{\hat n_i} = \hat{\mathbb I} -2 \hat n_i = \hat{\mathbb I}-2 \hat b_i^\dagger \hat b_i$</span>.  It is possible to find the analytical form also for MPOs with long range interaction \cite{mpo}. In the case of next-nearest neighbors interactions between spinless fermions, in the interleaved geometry of Fig. \ref{subfig:folded}, the MPO representation will require a bond dimension <span>$\chi=6$</span>. We explicitly write it as:</p><p class="math-container">\[    H = W_{d} \cdot W_{2 0} \cdot W_{1 0} \cdot...\cdot W_{2N} \cdot W_{1 N},  \]</p><p>where the matrices are defined as: </p><p class="math-container">\[\begin{aligned}
 &amp; W_d = 
 \begin{bmatrix}
  \hat{\mathbb I} &amp; \hat d &amp; \hat d^\dagger &amp; 0 &amp; 0 &amp; E_{d} \hat d^\dagger \hat d 
@@ -69,45 +69,45 @@
 \begin{bmatrix}
  E_{1,N} \hat b_{1,N}^\dagger \hat b_{1,N} \\ 0 \\0 \\ J_{1,N} \hat b_{1,N}^\dagger \\ J_{1,N} \hat b_{1,N} \\ \hat{\mathbb I}
 \end{bmatrix} .
-\end{aligned}\]</p><h2 id="Code-implementation"><a class="docs-heading-anchor" href="#Code-implementation">Code implementation</a><a id="Code-implementation-1"></a><a class="docs-heading-anchor-permalink" href="#Code-implementation" title="Permalink"></a></h2><p>The fermionic mapping can be strightforwardly implemented with the methods of <code>MPSDynamics.jl</code>. We provide two examples: </p><ul><li><code>examples/anderson_model_double</code>, simulating the SIAM with the double-chain geometry</li><li><code>examples/anderson_model_interleaved</code>, simulating the SIAM with the interleaved-chain geometry</li></ul><p>They only differ on the way the Hamiltonian MPO is defined. We now briefly review the important bits of the code.</p><p>Both of the examples start with the definition of the physical parameters,</p><pre><code class="language-julia hljs">N = 40      # number of chain sites
+\end{aligned}\]</p><h2 id="Code-implementation"><a class="docs-heading-anchor" href="#Code-implementation">Code implementation</a><a id="Code-implementation-1"></a><a class="docs-heading-anchor-permalink" href="#Code-implementation" title="Permalink"></a></h2><p>The fermionic mapping can be strightforwardly implemented with the methods of <code>MPSDyanmics.jl</code>. We provide two examples: </p><ul><li><code>examples/anderson_model_double</code>, simulating the SIAM with the double-chain geometry</li><li><code>examples/anderson_model_interleaved</code>, simulating the SIAM with the interleaved-chain geometry</li></ul><p>They only differ on the way the Hamiltonian MPO is defined. We now briefly review the important bits of the code.</p><p>Both of the examples start with the definition of the physical parameters,</p><pre><code class="language-julia">N = 40      # number of chain sites
 β = 10.0     # inverse temperature
 μ = 0.      # chemical potential
 Ed = 0.3    # energy of the impurity
-ϵd = Ed - μ # energy of the impurity minus the chemical potential</code></pre><p>The <a href="../../methods/#MPSDynamics.chaincoeffs_fermionic-Tuple{Any, Any, Any}"><code>MPSDynamics.chaincoeffs_fermionic</code></a> function is needed to compute the chain coefficients. It requires as inputs the number of modes of each chain <code>N</code>, the inverse temperature <code>β</code>, a label to specify if the chain modes are empty (label is <code>1.0</code>) or filled (label is <code>2.0</code>), and both the dispersion relation <span>$\epsilon_k$</span> and the fermionic spectral density funciton <span>$V_k$</span>.</p><pre><code class="language-julia hljs">function ϵ(x)
+ϵd = Ed - μ # energy of the impurity minus the chemical potential</code></pre><p>The <code>chaincoeffs_fermionic</code> function is needed to compute the chain coefficients. It requires as inputs the number of modes of each chain <code>N</code>, the inverse temperature <code>\beta</code>, a label to specify if the chain modes are empty (label is <code>1.0</code>) or filled (label is <code>2.0</code>), and both the dispersion relation <span>$\epsilon_k$</span> and the fermionic spectral density funciton <span>$V_k$</span>.</p><pre><code class="language-julia">function ϵ(x)
     return x
 end
 
-function V(x)
+function J(x)
     return sqrt(1 - x^2) # semi-circular density of states
 end
 
-chainparams1 = chaincoeffs_fermionic(N, β, 1.0; ϵ, V, save=false) # empty
-chainparams2 = chaincoeffs_fermionic(N, β, 2.0; ϵ, V, save=false) # filled</code></pre><p>We then specify the simulation parameters</p><pre><code class="language-julia hljs">dt = 0.25           # time step
+chainparams1 = chaincoeffs_fermionic(N, β, 1.0; ϵ, J, save=false) # empty
+chainparams2 = chaincoeffs_fermionic(N, β, 2.0; ϵ, J, save=false) # filled</code></pre><p>We then specify the simulation parameters</p><pre><code class="language-julia">dt = 0.25           # time step
 T = 15.0            # simulation time
 method = :DTDVP     # time-evolution method
 Dmax = 150          # MPS max bond dimension
 prec = 0.0001       # precision for the adaptive TDVP
-</code></pre><p>and with this we are ready to construct the Hamiltonian MPO and specify the initial state, which will obviously differ depending on the chosen geometry.</p><h4 id="Double-chain-geometry"><a class="docs-heading-anchor" href="#Double-chain-geometry">Double chain geometry</a><a id="Double-chain-geometry-1"></a><a class="docs-heading-anchor-permalink" href="#Double-chain-geometry" title="Permalink"></a></h4><p>The Hamiltonian is defined using the <a href="../../methods/#MPSDynamics.tightbinding_mpo-NTuple{4, Any}"><code>MPSDynamics.tightbinding_mpo</code></a> function, which takes as an input the number of modes of each chain <code>N</code>, the defect&#39;s energy <code>ϵd</code>, and the chain coefficients of the first <code>chainparams1</code> and second <code>chainparams2</code> chain. The MPS for the initial state is a factorized state made of: N filled states, a filled impurity, and N empty states. </p><pre><code class="language-julia hljs">H = tightbinding_mpo(N, ϵd, chainparams1, chainparams2)
+</code></pre><p>and with this we are ready to construct the Hamiltonian MPO and specify the initial state, which will obviously differ depending on the chosen geometry.</p><h4 id="Double-chain-geometry"><a class="docs-heading-anchor" href="#Double-chain-geometry">Double chain geometry</a><a id="Double-chain-geometry-1"></a><a class="docs-heading-anchor-permalink" href="#Double-chain-geometry" title="Permalink"></a></h4><p>The Hamiltonian is defined using the <code>tightbinding_mpo</code> function, which takes as an input the number of modes of each chain <code>N</code>, the defect&#39;s energy <code>\epsilon_d</code>, and the chain coefficients of the first <code>chainparams1</code> and second <code>chainparams2</code> chain. The MPS for the initial state is a factorized state made of: N filled states, a filled impurity, and N empty states. </p><pre><code class="language-julia">H = tightbinding_mpo(N, ϵd, chainparams1, chainparams2)
 
 ψ =  unitcol(2,2) # (0,1) filled impurity state
-A = productstatemps(physdims(H), state=[fill(unitcol(2,2), N)..., ψ, fill(unitcol(1,2), N)...]) # MPS</code></pre><p>To avoid the <code>DTDVP</code> algorithm from getting stuck in a local minimum, it is better to embed the MPS in a manifold of bond dimension 2 (or more):</p><pre><code class="language-julia hljs">mpsembed!(A, 2) # to embed the MPS in a manifold of bond dimension 2</code></pre><p>We can now define the observables for the two chains and for the impurity</p><pre><code class="language-julia hljs">ob1 = OneSiteObservable(&quot;chain1_filled_occup&quot;, numb(2), (1,N))
+A = productstatemps(physdims(H), state=[fill(unitcol(2,2), N)..., ψ, fill(unitcol(1,2), N)...]) # MPS</code></pre><p>To avoid the <code>DTDVP</code> algorithm from getting stuck in a local minimum, it is better to embed the MPS in a manifold of bond dimension 2 (or more):</p><pre><code class="language-julia">mpsembed!(A, 2) # to embed the MPS in a manifold of bond dimension 2</code></pre><p>We can now define the observables for the two chains and for the impurity</p><pre><code class="language-julia">ob1 = OneSiteObservable(&quot;chain1_filled_occup&quot;, numb(2), (1,N))
 ob2 = OneSiteObservable(&quot;chain2_empty_occup&quot;, numb(2), (N+2, 2N+1))
-ob3 = OneSiteObservable(&quot;system_occup&quot;, numb(2), N+1)</code></pre><p>and run the simulation</p><pre><code class="language-julia hljs">A, dat = runsim(dt, T, A, H;
+ob3 = OneSiteObservable(&quot;system_occup&quot;, numb(2), N+1)</code></pre><p>and run the simulation</p><pre><code class="language-julia">A, dat = runsim(dt, T, A, H;
                 name = &quot;Anderson impurity problem (folded chain)&quot;,
                 method = method,
                 obs = [ob1, ob2, ob3], 
                 convobs = [ob1],
-                params = @LogParams(N, ϵd, β),
+                params = @LogParams(N, ϵd, β, c1, c2),
                 convparams = [prec],   
                 Dlim = Dmax,          
                 savebonddims = true,   # we want to save the bond dimension
                 verbose = false,
                 save = false,
                 plot = true,
-                );</code></pre><p>With very minimal post-processing of the data</p><pre><code class="language-julia hljs"># Reshaping the vector to a column matrix and horizontal concatenation
+                );</code></pre><p>With very minimal post-processing of the data</p><pre><code class="language-julia"># Reshaping the vector to a column matrix and horizontal concatenation
 system_occup_col = reshape(dat[&quot;data/system_occup&quot;], :, 1)
 occ = hcat(dat[&quot;data/chain1_filled_occup&quot;]&#39;, system_occup_col)
-occ = vcat(occ&#39;, dat[&quot;data/chain2_empty_occup&quot;])</code></pre><p>we plot the results:</p><pre><code class="language-julia hljs"># Plot the system occupation    
+occ = vcat(occ&#39;, dat[&quot;data/chain2_empty_occup&quot;])</code></pre><p>we plot the results:</p><pre><code class="language-julia"># Plot the system occupation    
 p1 = plot(
     dat[&quot;data/times&quot;],
     dat[&quot;data/system_occup&quot;],
@@ -162,7 +162,7 @@
 
 # Display the plots
 plot(p2, p3, p4, p5, p1, layout = (3, 2), size = (1400, 1200))
-</code></pre><h4 id="Interleaved-chain-geometry"><a class="docs-heading-anchor" href="#Interleaved-chain-geometry">Interleaved chain geometry</a><a id="Interleaved-chain-geometry-1"></a><a class="docs-heading-anchor-permalink" href="#Interleaved-chain-geometry" title="Permalink"></a></h4><p>The Hamiltonian is defined using the <a href="../../methods/#MPSDynamics.interleaved_tightbinding_mpo-NTuple{4, Any}"><code>MPSDynamics.interleaved_tightbinding_mpo</code></a> function, which takes as an input the number of modes of each chain <code>N</code>, the defect&#39;s energy <code>ϵd</code>, and the chain coefficients of the first <code>chainparams1</code> and second <code>chainparams2</code> chain. The MPS for the initial state is a factorized state (bond dimension 1) made of: a filled impurity, and 2N alternate filled-empty states. </p><pre><code class="language-julia hljs">H = interleaved_tightbinding_mpo(N, ϵd, chainparams1, chainparams2)
+</code></pre><h4 id="Interleaved-chain-geometry"><a class="docs-heading-anchor" href="#Interleaved-chain-geometry">Interleaved chain geometry</a><a id="Interleaved-chain-geometry-1"></a><a class="docs-heading-anchor-permalink" href="#Interleaved-chain-geometry" title="Permalink"></a></h4><p>The Hamiltonian is defined using the <code>interleaved_tightbinding_mpo</code> function, which takes as an input the number of modes of each chain <code>N</code>, the defect&#39;s energy <code>\epsilon_d</code>, and the chain coefficients of the first <code>chainparams1</code> and second <code>chainparams2</code> chain. The MPS for the initial state is a factorized state (bond dimension 1) made of: a filled impurity, and 2N alternate filled-empty states. </p><pre><code class="language-julia">H = interleaved_tightbinding_mpo(N, ϵd, chainparams1, chainparams2)
 
 ψ =  unitcol(2,2) # (0,1) filled impurity state
 Tot = Any[]
@@ -172,20 +172,20 @@
     push!(Tot, unitcol(1,2))
 end
 
-A = productstatemps(physdims(H), state=Tot) # MPS</code></pre><p>To avoid the <code>DTDVP</code> algorithm from getting stuck in a local minimum, it is better to embed the MPS in a manifold of bond dimension 2 (or more):</p><pre><code class="language-julia hljs">mpsembed!(A, 2) # to embed the MPS in a manifold of bond dimension 2</code></pre><p>We can finally define the observables for the interleaved chain</p><pre><code class="language-julia hljs">ob1 = OneSiteObservable(&quot;system_occup&quot;, numb(2), 1)
-ob2 = OneSiteObservable(&quot;folded_chain_occup&quot;, numb(2), (2,2N+1))</code></pre><p>and run the simulation</p><pre><code class="language-julia hljs">A, dat = runsim(dt, T, A, H;
+A = productstatemps(physdims(H), state=Tot) # MPS</code></pre><p>To avoid the <code>DTDVP</code> algorithm from getting stuck in a local minimum, it is better to embed the MPS in a manifold of bond dimension 2 (or more):</p><pre><code class="language-julia">mpsembed!(A, 2) # to embed the MPS in a manifold of bond dimension 2</code></pre><p>We can finally define the observables for the interleaved chain</p><pre><code class="language-julia">ob1 = OneSiteObservable(&quot;system_occup&quot;, numb(2), 1)
+ob2 = OneSiteObservable(&quot;folded_chain_occup&quot;, numb(2), (2,2N+1))</code></pre><p>and run the simulation</p><pre><code class="language-julia">A, dat = runsim(dt, T, A, H;
                 name = &quot;Anderson impurity problem (folded chain)&quot;,
                 method = method,
                 obs = [ob1, ob2], 
                 convobs = [ob1],
-                params = @LogParams(N, ϵd, β),
+                params = @LogParams(N, ϵd, β, c1, c2),
                 convparams = [prec],   
                 Dlim = Dmax,          
                 savebonddims = true,   # we want to save the bond dimension
                 verbose = false,
                 save = false,
                 plot = true,
-                );</code></pre><p>To show the data in a clear way, we do a bit of post-processing of the data to <em>unfold</em> the chain, and move back to the double chain representation:</p><pre><code class="language-julia hljs">unfolded_occ = Vector{Vector{Float64}}()  # Assuming the elements are of type Float64
+                );</code></pre><p>To show the data in a clear way, we do a bit of post-processing of the data to <em>unfold</em> the chain, and move back to the double chain representation:</p><pre><code class="language-julia">unfolded_occ = Vector{Vector{Float64}}()  # Assuming the elements are of type Float64
 unfolded_bonds = Vector{Vector{Float64}}()  # Adjust the type based on actual data
 
 # Populate unfolded_occ by iterating in the specific order mentioned
@@ -211,7 +211,7 @@
 end
 
 unfolded_bonds_matrix = hcat(unfolded_bonds...)&#39;
-unfolded_occ_matrix = hcat(unfolded_occ...)&#39;</code></pre><p>We conclude by plotting the data</p><pre><code class="language-julia hljs"># Plot the system occupation    
+unfolded_occ_matrix = hcat(unfolded_occ...)&#39;</code></pre><p>We conclude by plotting the data</p><pre><code class="language-julia"># Plot the system occupation    
 p1 = plot(
     dat[&quot;data/times&quot;],
     dat[&quot;data/system_occup&quot;],
@@ -266,4 +266,4 @@
 end
 
 # Display the plots
-plot(p2, p3, p4, p5, p1, layout = (3, 2), size = (1400, 1200))</code></pre><p>________________</p><h2 id="Bibliography"><a class="docs-heading-anchor" href="#Bibliography">Bibliography</a><a id="Bibliography-1"></a><a class="docs-heading-anchor-permalink" href="#Bibliography" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-khon_efficient_2021"><a class="tag is-link" href="#citeref-khon_efficient_2021">khon_efficient_2021</a><blockquote><p>Kohn, L.; Santoro, G. E. Efficient mapping for anderson impurity problems with matrix product states. Phys. Rev. B 2021, 104 (1), 014303. https://doi.org/10.1103/PhysRevB.104.014303.</p></blockquote></li><li class="footnote" id="footnote-devega_thermo_2015"><a class="tag is-link" href="#citeref-devega_thermo_2015">devega_thermo_2015</a><blockquote><p>de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.</p></blockquote></li><li class="footnote" id="footnote-khon_eff_2022"><a class="tag is-link" href="#citeref-khon_eff_2022">khon_eff_2022</a><blockquote><p>Kohn L.; Santoro G. E. J. Stat. Mech. (2022) 063102 DOI 10.1088/1742-5468/ac729b.</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../timedep/">« Time-dependent Hamiltonian</a><a class="docs-footer-nextpage" href="../bath-observables/">Inspecting the bath by undoing the chain mapping »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+plot(p2, p3, p4, p5, p1, layout = (3, 2), size = (1400, 1200))</code></pre><p>________________</p><h3 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h3><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-khon_efficient_2021"><a class="tag is-link" href="#citeref-khon_efficient_2021">khon_efficient_2021</a><blockquote><p>Kohn, L.; Santoro, G. E. Efficient mapping for anderson impurity problems with matrix product states. Phys. Rev. B 2021, 104 (1), 014303. https://doi.org/10.1103/PhysRevB.104.014303.</p></blockquote></li><li class="footnote" id="footnote-devega_thermo_2015"><a class="tag is-link" href="#citeref-devega_thermo_2015">devega_thermo_2015</a><blockquote><p>de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.</p></blockquote></li><li class="footnote" id="footnote-khon_eff_2022"><a class="tag is-link" href="#citeref-khon_eff_2022">khon_eff_2022</a><blockquote><p>Kohn L.; Santoro G. E. J. Stat. Mech. (2022) 063102 DOI 10.1088/1742-5468/ac729b.</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../timedep/">« Time-dependent Hamiltonian</a><a class="docs-footer-nextpage" href="../bath-observables/">Inspecting the bath by undoing the chain mapping »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/docs/examples/bath-observables/index.html b/docs/examples/bath-observables/index.html
index 42cf843..e8251d7 100644
--- a/docs/examples/bath-observables/index.html
+++ b/docs/examples/bath-observables/index.html
@@ -1,27 +1,27 @@
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the bath by undoing the chain mapping</a><ul class="internal"><li><a class="tocitem" href="#The-code"><span>The code</span></a></li><li><a class="tocitem" href="#Bibliography"><span>Bibliography</span></a></li></ul></li><li><a class="tocitem" href="../protontransfer/">Proton Transfer Model</a></li></ul></li><li><a class="tocitem" href="../../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../../methods/">Methods</a></li><li><a class="tocitem" href="../../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Inspecting the bath by undoing the chain mapping</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Inspecting the bath by undoing the chain mapping</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Inspecting-the-bath-by-undoing-the-chain-mapping"><a class="docs-heading-anchor" href="#Inspecting-the-bath-by-undoing-the-chain-mapping">Inspecting the bath by undoing the chain mapping</a><a id="Inspecting-the-bath-by-undoing-the-chain-mapping-1"></a><a class="docs-heading-anchor-permalink" href="#Inspecting-the-bath-by-undoing-the-chain-mapping" title="Permalink"></a></h1><p>Here we give some context on the example script provided in <code>MPSDynamics/example/bath-observables.jl</code>. This example demonstrates the setup and execution of a simulation for a two-level system coupled to an Ohmic bath at finite temperature, where we exploit the access to the chain observables to:</p><ul><li>undo the chain mapping<sup class="footnote-reference"><a id="citeref-chin_exact_2010" href="#footnote-chin_exact_2010">[chin_exact_2010]</a></sup>, thus obtaining their representation in the extended bath of T-TEDOPA <sup class="footnote-reference"><a id="citeref-tamascelli_efficient_2019" href="#footnote-tamascelli_efficient_2019">[tamascelli_efficient_2019]</a></sup>, characterized by <span>$J(\omega, \beta)$</span></li><li>inverting the thermofield transformation<sup class="footnote-reference"><a id="citeref-devega_thermo_2015" href="#footnote-devega_thermo_2015">[devega_thermo_2015]</a></sup>, thus obtaining the representation of the physical frequencies in the original environment, characterized by <span>$J(\omega)$</span></li></ul><p>T-TEDOPA allows to substitute a thermally occupied bath by an extended one, in the pure state of the vacuum, extending the bath of frequencies to negative values: the <em>creation</em> from the system of a mode of <em>negative frequency</em> in the extended bath of frequencies corresponds to the <em>absorption</em> of energy for the system from the thermally occupied modes of the environment. However, by exploiting the thermofield transformation, it is possible to recover the picture of the occupations of the steady states in the physical bath of frequencies<sup class="footnote-reference"><a id="citeref-riva_thermal_2023" href="#footnote-riva_thermal_2023">[riva_thermal_2023]</a></sup>. </p><h2 id="The-code"><a class="docs-heading-anchor" href="#The-code">The code</a><a id="The-code-1"></a><a class="docs-heading-anchor-permalink" href="#The-code" title="Permalink"></a></h2><p>We start by defining the parameters of the simulation:</p><pre><code class="language-julia hljs">d = 10      # number of Fock states of the chain modes
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Inspecting the bath by undoing the chain mapping · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Introduction</a></li><li><a class="tocitem" href="../../user-guide/">User Guide</a></li><li><a class="tocitem" href="../../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../anderson-model/">The Anderson Impurity Model</a></li><li class="is-active"><a class="tocitem" href>Inspecting the bath by undoing the chain mapping</a><ul class="internal"><li><a class="tocitem" href="#The-code"><span>The code</span></a></li><li><a class="tocitem" href="#References"><span>References</span></a></li></ul></li></ul></li><li><a class="tocitem" href="../../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../../methods/">Methods</a></li><li><a class="tocitem" href="../../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Inspecting the bath by undoing the chain mapping</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Inspecting the bath by undoing the chain mapping</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/examples/bath-observables.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Inspecting-the-bath-by-undoing-the-chain-mapping"><a class="docs-heading-anchor" href="#Inspecting-the-bath-by-undoing-the-chain-mapping">Inspecting the bath by undoing the chain mapping</a><a id="Inspecting-the-bath-by-undoing-the-chain-mapping-1"></a><a class="docs-heading-anchor-permalink" href="#Inspecting-the-bath-by-undoing-the-chain-mapping" title="Permalink"></a></h1><p>Here we give some context on the example script provided in <code>MPSDynamics/example/bath-observables.jl</code>. This example demonstrates the setup and execution of a simulation for a two-level system coupled to an Ohmic bath at finite temperature, where we exploit the access to the chain observables to:</p><ul><li>undo the chain mapping<sup class="footnote-reference"><a id="citeref-chin_exact_2010" href="#footnote-chin_exact_2010">[chin_exact_2010]</a></sup>, thus obtaining their representation in the extended bath of T-TEDOPA <sup class="footnote-reference"><a id="citeref-tamascelli_efficient_2019" href="#footnote-tamascelli_efficient_2019">[tamascelli_efficient_2019]</a></sup>, characterized by <span>$J(\omega, \beta)$</span></li><li>inverting the thermofield transformation<sup class="footnote-reference"><a id="citeref-devega_thermo_2015" href="#footnote-devega_thermo_2015">[devega_thermo_2015]</a></sup>, thus obtaining the representation of the physical frequencies in the original environment, characterized by <span>$J(\omega)$</span></li></ul><p>T-TEDOPA allows to substitute a thermally occupied bath by an extended one, in the pure state of the vacuum, extending the bath of frequencies to negative values: the <em>creation</em> from the system of a mode of <em>negative frequency</em> in the extended bath of frequencies corresponds to the <em>absorption</em> of energy for the system from the thermally occupied modes of the environment. However, by exploiting the thermofield transformation, it is possible to recover the picture of the occupations of the steady states in the physical bath of frequencies<sup class="footnote-reference"><a id="citeref-riva_thermal_2023" href="#footnote-riva_thermal_2023">[riva_thermal_2023]</a></sup>. </p><h2 id="The-code"><a class="docs-heading-anchor" href="#The-code">The code</a><a id="The-code-1"></a><a class="docs-heading-anchor-permalink" href="#The-code" title="Permalink"></a></h2><p>We start by defining the parameters of the simulation:</p><pre><code class="language-julia">d = 10      # number of Fock states of the chain modes
 N = 60      # length of the chain
 α = 0.01    # coupling strength
 ω0 = 0.2  # TLS gap
 s = 1       # ohmicity
 ωc = 1.  # Cut-off of the spectral density J(ω)
-β = 20    # Thermalized environment</code></pre><p>We set the specifics of the simulation:</p><pre><code class="language-julia hljs">method = :TDVP1         # time-evolution method
+β = 20    # Thermalized environment</code></pre><p>We set the specifics of the simulation:</p><pre><code class="language-julia">method = :TDVP1         # time-evolution method
 conv = 3                # bond dimension for the TDVP1
 dt = 0.5                # time step
-tfinal = 60.0           # simulation time</code></pre><p>And then compute the chain coefficients, i.e. on-site energies <span>$\epsilon_i$</span>, hopping energies <span>$t_i$</span>, and system-chain coupling <span>$c_0$</span>, that define the chain representation of the pure-dephasing model:</p><pre><code class="language-julia hljs">cpars = chaincoeffs_finiteT(N, β; α=α, s=s, J=nothing, ωc=ωc, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=false)</code></pre><p>with this, we can compute the corresponding MPO, and the initial state in MPS form (<span>$1/\sqrt{2}(|0\rangle + |1\rangle)$</span>):</p><pre><code class="language-julia hljs">H = puredephasingmpo(ω0, d, N, cpars)
+tfinal = 60.0           # simulation time</code></pre><p>And then compute the chain coefficients, i.e. on-site energies <span>$\epsilon_i$</span>, hopping energies <span>$t_i$</span>, and system-chain coupling <span>$c_0$</span>, that define the chain representation of the pure-dephasing model:</p><pre><code class="language-julia">cpars = chaincoeffs_finiteT(N, β; α=α, s=s, J=nothing, ωc=ωc, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=false)</code></pre><p>with this, we can compute the corresponding MPO, and the initial state in MPS form (<span>$1/\sqrt{2}(|0\rangle + |1\rangle)$</span>):</p><pre><code class="language-julia">H = puredephasingmpo(ω0, d, N, cpars)
 
 ψ = zeros(2)
 ψ[1] = 1/sqrt(2)
 ψ[2] = 1/sqrt(2)
 
-A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...])</code></pre><p>We can now define the observables we are interested in computing. Importantly, we not only compute observables related to the system, but also to the chain modes, so that we can inspect the environment.</p><pre><code class="language-julia hljs">ob1 = OneSiteObservable(&quot;sz&quot;, sz, 1)
+A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...])</code></pre><p>We can now define the observables we are interested in computing. Importantly, we not only compute observables related to the system, but also to the chain modes, so that we can inspect the environment.</p><pre><code class="language-julia">ob1 = OneSiteObservable(&quot;sz&quot;, sz, 1)
 ob2 = OneSiteObservable(&quot;sx&quot;, sx, 1)
 ob3 = OneSiteObservable(&quot;chain_mode_occupation&quot;, numb(d), (2,N+1))
 ob4 = OneSiteObservable(&quot;c&quot;, crea(d), collect(2:N+1))
 ob5 = OneSiteObservable(&quot;cdag&quot;, crea(d), collect(2:N+1))
 ob6 = TwoSiteObservable(&quot;cdagc&quot;, crea(d), anih(d), collect(2:N+1), collect(2:N+1))
 ob7 = TwoSiteObservable(&quot;cdagcdag&quot;, crea(d), crea(d), collect(2:N+1), collect(2:N+1))
-ob8 = TwoSiteObservable(&quot;cc&quot;, anih(d), anih(d), collect(2:N+1), collect(2:N+1))</code></pre><p>with this we run the simulation:</p><pre><code class="language-julia hljs">A, dat = runsim(dt, tfinal, A, H, prec=1E-4;
+ob8 = TwoSiteObservable(&quot;cc&quot;, anih(d), anih(d), collect(2:N+1), collect(2:N+1))</code></pre><p>with this we run the simulation:</p><pre><code class="language-julia">A, dat = runsim(dt, tfinal, A, H, prec=1E-4;
                 name = &quot;Bath observables in the pure dephasing model&quot;,
                 method = method,
                 obs = [ob1, ob2, ob3, ob4, ob5, ob6, ob7, ob8],
@@ -32,18 +32,18 @@
                 verbose = false,
                 save = false,
                 plot = true,
-                );</code></pre><p>After the simulation, we need to post-process the data a bit, in order to map the correlations from the chain representation to the original environment. We start by defining the matrices associated to the mean values of <span>$c$</span> and <span>$c^\dagger$</span>:</p><pre><code class="language-julia hljs">T = length(dat[&quot;data/times&quot;])
+                );</code></pre><p>After the simulation, we need to post-process the data a bit, in order to map the correlations from the chain representation to the original environment. We start by defining the matrices associated to the mean values of <span>$c$</span> and <span>$c^\dagger$</span>:</p><pre><code class="language-julia">T = length(dat[&quot;data/times&quot;])
 
 constr = Array{ComplexF64}(undef, N, N, T)
 destr = Array{ComplexF64}(undef, N, N, T)
 for t in 1:T
     constr[:,:,t] = diagm(0 =&gt; dat[&quot;data/cdag&quot;][:,t])
     destr[:,:,t] = diagm(0 =&gt; dat[&quot;data/c&quot;][:,t])
-end</code></pre><p>During a numerical simulation we work with chain of finite length <span>$N$</span>. This truncation on chain modes, introduces a sampling on the modes in the original star-like environment. To recover the frequency modes that are implicitly sampled, one has to diagonalize the tri-diagonal <span>$N\times N$</span> matrix <span>$H^\text{chain}$</span>, where the diagonal is formed by the <span>$e_n$</span> coefficients, that is the chain&#39;s frequencies, and the upper and lower diagonals by the <span>$N-1$</span> hopping coefficients <span>$t_n$</span>. The unitary matrix that defines the change of basis from the star-like to the chain-like environment is <span>$U_n$</span>, constituted by the eigenvectors of <span>$H^\text{chain}$</span>. In the code, we use the <a href="../../methods/#MPSDynamics.eigenchain-Tuple{Any}"><code>MPSDynamics.eigenchain</code></a> function:</p><pre><code class="language-julia hljs">omeg = eigenchain(cpars, nummodes=N).values</code></pre><p>At each time step of the simulation, a number of one-site and two-sites observables where evaluated on the chain. To obtain their value in the extended bath of T-tedopa, characterized by <span>$J(\omega,\beta)$</span>, the unitary transformation that maps the extended bath Hamiltonian into the chain representation has to be reversed. For instance, when measuring the single site <span>$\hat n^c_i=\hat c_i^\dagger \hat c_i$</span> occupation number, we are not measuring the occupation number of the bosonic mode associated to the <span>$\omega_i$</span> frequency, but the occupation number of the <span>$i-$</span>th chain mode. Therefore to calculate the number of modes of the environment associated to a specific frequency <span>$\omega_i$</span>, the mapping must be reversed, to obtain the diagonal representation of the bosonic number operator:</p><p class="math-container">\[\begin{aligned}
+end</code></pre><p>During a numerical simulation we work with chain of finite length <span>$N$</span>. This truncation on chain modes, introduces a sampling on the modes in the original star-like environment. To recover the frequency modes that are implicitly sampled, one has to diagonalize the tri-diagonal <span>$N\times N$</span> matrix <span>$H^\text{chain}$</span>, where the diagonal is formed by the <span>$e_n$</span> coefficients, that is the chain&#39;s frequencies, and the upper and lower diagonals by the <span>$N-1$</span> hopping coefficients <span>$t_n$</span>. The unitary matrix that defines the change of basis from the star-like to the chain-like environment is <span>$U_n$</span>, constituted by the eigenvectors of <span>$H^\text{chain}$</span>. In the code, we use the <code>eigenchain</code> function:</p><pre><code class="language-julia">omeg = eigenchain(cpars, nummodes=N).values</code></pre><p>At each time step of the simulation, a number of one-site and two-sites observables where evaluated on the chain. To obtain their value in the extended bath of T-tedopa, characterized by <span>$J(\omega,\beta)$</span>, the unitary transformation that maps the extended bath Hamiltonian into the chain representation has to be reversed. For instance, when measuring the single site <span>$\hat n^c_i=\hat c_i^\dagger \hat c_i$</span> occupation number, we are not measuring the occupation number of the bosonic mode associated to the <span>$\omega_i$</span> frequency, but the occupation number of the <span>$i-$</span>th chain mode. Therefore to calculate the number of modes of the environment associated to a specific frequency <span>$\omega_i$</span>, the mapping must be reversed, to obtain the diagonal representation of the bosonic number operator:</p><p class="math-container">\[\begin{aligned}
 \hat n^b_{i} = \hat b_i^\dagger \hat b_i = \sum_{k,l} U_{ik}^* \hat c_k^\dagger \hat c_l U_{li}.  
-\end{aligned}\]</p><p>This is done in the code using the <a href="../../methods/#MPSDynamics.measuremodes"><code>MPSDynamics.measuremodes</code></a> function, which outputs the vector of the diagonal elements of the operators, in the following way:</p><pre><code class="language-julia hljs">bath_occup = mapslices(X -&gt; measuremodes(X, cpars[1], cpars[2]), dat[&quot;data/cdagc&quot;], dims = [1,2])
+\end{aligned}\]</p><p>This is done in the code using the <code>measuremodes(X, cpars[1], cpars[2])</code> function, which outputs the vector of the diagonal elements of the operators, in the following way:</p><pre><code class="language-julia">bath_occup = mapslices(X -&gt; measuremodes(X, cpars[1], cpars[2]), dat[&quot;data/cdagc&quot;], dims = [1,2])
 cdag_average = mapslices(X -&gt; measuremodes(X, cpars[1], cpars[2]), constr, dims = [1,2])
-c_average = mapslices(X -&gt; measuremodes(X, cpars[1], cpars[2]), destr, dims = [1,2])</code></pre><p>To compute the correlators, we need the full matrix in the original basis. We therefore use the <code>measurecorrs</code> function:</p><pre><code class="language-julia hljs">cc_average = mapslices(X -&gt; measurecorrs(X, cpars[1], cpars[2]), dat[&quot;data/cc&quot;], dims = [1,2])
+c_average = mapslices(X -&gt; measuremodes(X, cpars[1], cpars[2]), destr, dims = [1,2])</code></pre><p>To compute the correlators, we need the full matrix in the original basis. We therefore use the <code>measurecorrs</code> function:</p><pre><code class="language-julia">cc_average = mapslices(X -&gt; measurecorrs(X, cpars[1], cpars[2]), dat[&quot;data/cc&quot;], dims = [1,2])
 cdagcdag_average = mapslices(X -&gt; measurecorrs(X, cpars[1], cpars[2]), dat[&quot;data/cdagcdag&quot;], dims = [1,2])
 
 correlations_c = [
@@ -55,10 +55,10 @@
     for i in 1:size(cdagcdag_average, 1), j in 1:size(cdagcdag_average, 2), t in 1:size(cdagcdag_average,3)
 ]</code></pre><p>It is possible to invert the thermofield transformation (details in <sup class="footnote-reference"><a id="citeref-riva_thermal_2023" href="#footnote-riva_thermal_2023">[riva_thermal_2023]</a></sup>). The expression of the mean value of the number operator for the physical modes can be expressed as a function of mean values in the extended bath, which we denote <span>$\langle \hat a_{2k}^\dagger \hat a_{2k} \rangle$</span>:</p><p class="math-container">\[\begin{aligned}
     \langle \hat b_k^\dagger \hat b_k \rangle = \cosh{\theta_k}\sinh{\theta_k} (\langle \hat a_{2k}\hat a_{1k}\rangle + \langle \hat a_{1k}^\dagger\hat a_{2k}^\dagger\rangle ) + \sinh^2{\theta_k} (1+ \langle \hat a_{2k}^\dagger \hat a_{2k} \rangle ) ++ \cosh^2{\theta_k} \langle \hat a_{1k}^\dagger \hat a_{1k} \rangle
-\end{aligned}\]</p><p>We remark that in the thermofield case, a negative frequency <span>$\omega_{2k}$</span> is associated to each positive frequency <span>$\omega_{1k}$</span>. The sampling is therefore symmetric around zero. This marks a difference with T-TEDOPA, where the sampling of frequencies was obtained through the thermalized measure <span>$d\mu(\beta) = \sqrt{J(\omega, \beta)}d\omega$</span>, and was not symmetric. To recover the results for the physical bath of frequencies starting from the results of our simulations, that were conducted using the T-TEDOPA chain mapping, we need to do an extrapolation for all of the mean values, in order to have their values for each <span>$\omega$</span> at <span>$-\omega$</span> as well. This is done in the code with the <a href="../../methods/#MPSDynamics.physical_occup-NTuple{6, Any}"><code>MPSDynamics.physical_occup</code></a> function:</p><pre><code class="language-julia hljs">bath_occup_phys = physical_occup(cdagcdag_average[:,:,T], cc_average[:,:,T], omeg, bath_occup[:,:,T], β, N)</code></pre><p>Finally, in the pure dephasing case, it is also possible to obtain the analytical prediction of the time evolution of the occupations of the bath&#39;s modes, so that we can compare our numerical results with the analytical ones, exploiting the Heisenberg time evolution relation:</p><p class="math-container">\[\begin{aligned}
+\end{aligned}\]</p><p>We remark that in the thermofield case, a negative frequency <span>$\omega_{2k}$</span> is associated to each positive frequency <span>$\omega_{1k}$</span>. The sampling is therefore symmetric around zero. This marks a difference with T-TEDOPA, where the sampling of frequencies was obtained through the thermalized measure <span>$d\mu(\beta) = \sqrt{J(\omega, \beta)}d\omega$</span>, and was not symmetric. To recover the results for the physical bath of frequencies starting from the results of our simulations, that were conducted using the T-TEDOPA chain mapping, we need to do an extrapolation for all of the mean values, in order to have their values for each <span>$\omega$</span> at <span>$-\omega$</span> as well. This is done in the code with the <code>physical_occup</code> function:</p><pre><code class="language-julia">bath_occup_phys = physical_occup(cdagcdag_average[:,:,T], cc_average[:,:,T], omeg, bath_occup[:,:,T], β, N)</code></pre><p>Finally, in the pure dephasing case, it is also possible to obtain the analytical prediction of the time evolution of the occupations of the bath&#39;s modes, so that we can compare our numerical results with the analytical ones, exploiting the Heisenberg time evolution relation:</p><p class="math-container">\[\begin{aligned}
 \frac{d \langle \hat b_\omega \rangle}{dt} = -i \langle[ \hat b_\omega, \hat H] \rangle = - i \omega \langle\hat b_\omega \rangle - i \frac{\langle \hat \sigma_x \rangle}{2} \sqrt{J(\omega, \beta)},     \\
 \frac{d \langle \hat n_\omega \rangle}{dt} = -i \langle[\hat b_\omega^\dagger \hat b_\omega, \hat H] \rangle= 2 \frac{|J(\omega,\beta)|}{\omega} \sin(\omega t).
-\end{aligned}\]</p><p>To this end, it is convenient to choose one of the eigenstates of <span>$\hat \sigma_z$</span> as the initial state, so that <span>$\langle \hat \sigma_x \rangle = \pm 1$</span>. By solving these differential equations, one obtains the time evolved theoretical behavior of the bath. We define the function for the comparison with analytical predictions:</p><pre><code class="language-julia hljs">Johmic(ω,s) = (2*α*ω^s)/(ωc^(s-1))
+\end{aligned}\]</p><p>To this end, it is convenient to choose one of the eigenstates of <span>$\hat \sigma_z$</span> as the initial state, so that <span>$\langle \hat \sigma_x \rangle = \pm 1$</span>. By solving these differential equations, one obtains the time evolved theoretical behavior of the bath. We define the function for the comparison with analytical predictions:</p><pre><code class="language-julia">Johmic(ω,s) = (2*α*ω^s)/(ωc^(s-1))
 
 time_analytical = LinRange(0.0, tfinal, Int(tfinal))
 
@@ -78,7 +78,7 @@
     end
 end
 
-bath_occup_analytical(ω, t) = abs(Jtherm(ω))/(ω^2)*2*(1-cos(ω*t)) </code></pre><p>We conclude the example by plotting.</p><pre><code class="language-julia hljs">ρ12 = abs.(dat[&quot;data/Reduced ρ&quot;][1,2,:])
+bath_occup_analytical(ω, t) = abs(Jtherm(ω))/(ω^2)*2*(1-cos(ω*t)) </code></pre><p>We conclude the example by plotting.</p><pre><code class="language-julia">ρ12 = abs.(dat[&quot;data/Reduced ρ&quot;][1,2,:])
 
 p1 = plot(time_analytical, t-&gt;Decoherence_ohmic(t), label=&quot;Analytics&quot;, title=L&quot;Pure Dephasing, Ohmic $s=%$s$, $\beta = %$β ~\mathrm{K}$&quot;, linecolor=:black, xlabel=&quot;Time (arb. units)&quot;, ylabel=L&quot;Coherence $|\rho_{12}(t)|$&quot;, linewidth=4, titlefontsize=16, legend=:best, legendfontsize=16, xguidefontsize=16, yguidefontsize=16, tickfontsize=10)
 p1 = plot!(dat[&quot;data/times&quot;], ρ12, lw=4, ls=:dash, label=&quot;Numerics&quot;)
@@ -102,4 +102,4 @@
             xlabel=L&quot;\omega&quot;, ylabel=L&quot;\langle n^b_\omega \rangle&quot;,
             title=&quot;Mode occupation in the physical bath&quot;)
 
-plot(p1, p2, p3, p4, layout = (2, 2), size = (1400, 1200))</code></pre><p>___________________</p><h2 id="Bibliography"><a class="docs-heading-anchor" href="#Bibliography">Bibliography</a><a id="Bibliography-1"></a><a class="docs-heading-anchor-permalink" href="#Bibliography" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-chin_exact_2010"><a class="tag is-link" href="#citeref-chin_exact_2010">chin_exact_2010</a><blockquote><p>Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. Journal of Mathematical Physics 2010, 51 (9), 092109. https://doi.org/10.1063/1.3490188.</p></blockquote></li><li class="footnote" id="footnote-tamascelli_efficient_2019"><a class="tag is-link" href="#citeref-tamascelli_efficient_2019">tamascelli_efficient_2019</a><blockquote><p>Tamascelli, D.; Smirne, A.; Lim, J.; Huelga, S. F.; Plenio, M. B. Efficient Simulation of Finite-Temperature Open Quantum Systems. Phys. Rev. Lett. 2019, 123 (9), 090402. https://doi.org/10.1103/PhysRevLett.123.090402.</p></blockquote></li><li class="footnote" id="footnote-devega_thermo_2015"><a class="tag is-link" href="#citeref-devega_thermo_2015">devega_thermo_2015</a><blockquote><p>de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.</p></blockquote></li><li class="footnote" id="footnote-riva_thermal_2023"><a class="tag is-link" href="#citeref-riva_thermal_2023">riva_thermal_2023</a><blockquote><p>Riva, A.; Tamascelli, D.; Dunnett, A. J.; Chin, A. W. Thermal cycle and polaron formation in structured bosonic environments. Phys. Rev. B 2023, 108, 195138, https://doi.org/10.1103/PhysRevB.108.195138.</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../anderson-model/">« The Anderson Impurity Model</a><a class="docs-footer-nextpage" href="../protontransfer/">Proton Transfer Model »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+plot(p1, p2, p3, p4, layout = (2, 2), size = (1400, 1200))</code></pre><p>___________________</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-chin_exact_2010"><a class="tag is-link" href="#citeref-chin_exact_2010">chin_exact_2010</a><blockquote><p>Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. Journal of Mathematical Physics 2010, 51 (9), 092109. https://doi.org/10.1063/1.3490188.</p></blockquote></li><li class="footnote" id="footnote-tamascelli_efficient_2019"><a class="tag is-link" href="#citeref-tamascelli_efficient_2019">tamascelli_efficient_2019</a><blockquote><p>Tamascelli, D.; Smirne, A.; Lim, J.; Huelga, S. F.; Plenio, M. B. Efficient Simulation of Finite-Temperature Open Quantum Systems. Phys. Rev. Lett. 2019, 123 (9), 090402. https://doi.org/10.1103/PhysRevLett.123.090402.</p></blockquote></li><li class="footnote" id="footnote-devega_thermo_2015"><a class="tag is-link" href="#citeref-devega_thermo_2015">devega_thermo_2015</a><blockquote><p>de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.</p></blockquote></li><li class="footnote" id="footnote-riva_thermal_2023"><a class="tag is-link" href="#citeref-riva_thermal_2023">riva_thermal_2023</a><blockquote><p>Riva, A.; Tamascelli, D.; Dunnett, A. J.; Chin, A. W. Thermal cycle and polaron formation in structured bosonic environments. Phys. Rev. B 2023, 108, 195138, https://doi.org/10.1103/PhysRevB.108.195138.</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../anderson-model/">« The Anderson Impurity Model</a><a class="docs-footer-nextpage" href="../../theory/">Theoretical Background »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-disabled">Examples</a></li><li class="is-active"><a href>Proton Transfer Model</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Proton Transfer Model</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Proton-Transfer-Model"><a class="docs-heading-anchor" href="#Proton-Transfer-Model">Proton Transfer Model</a><a id="Proton-Transfer-Model-1"></a><a class="docs-heading-anchor-permalink" href="#Proton-Transfer-Model" title="Permalink"></a></h1><h2 id="Context"><a class="docs-heading-anchor" href="#Context">Context</a><a id="Context-1"></a><a class="docs-heading-anchor-permalink" href="#Context" title="Permalink"></a></h2><p>The MPS formalism can also be used for physical chemistry problems. One development done with the <code>MPSDynamics.jl</code> package is the introduction of a reaction coordinate tensor, allowing the system to be described in space <sup class="footnote-reference"><a id="citeref-lede_ESIPT_2024" href="#footnote-lede_ESIPT_2024">[lede_ESIPT_2024]</a></sup>. It can model an electronic system with discretized states being described along a reaction coordinate. The introduction of a reaction coordinate allows to recover the well-known doublewell viewpoint and wavepacket dynamics can be analyzed with the reduced density matrix. </p><p>Here is an illustrative example of two electronic configurations undergoing a tautomerization : the enol named <span>$|e\rangle$</span> and the keto named <span>$|k\rangle$</span> (<span>$\hbar = 1$</span>) :</p><p class="math-container">\[H_S = \omega^0_{e} |e\rangle \langle e| + \omega^0_{k} |k\rangle \langle k| + \Delta (|e\rangle \langle k| + |k\rangle \langle e|) \]</p><p>This description requires the tensor of the system to be linked to another tensor representing an harmonic oscillator. This reaction coordinate oscillator is expressed as RC and represents the reaction path of the proton.</p><p class="math-container">\[H_{RC} + H_{int}^{S-RC} = \omega_{RC} (d^{\dagger}d + \frac{1}{2}) + g_{e} |e\rangle \langle e|( d + d^{\dagger})+ g_{k} |k \rangle \langle k|( d + d^{\dagger})\]</p><p>The coefficients <span>$g$</span> are set up in order to recover the doublewell formalism. A bath is introduced in the description and can represent intramolecular vibrations, damping the dynamics.</p><p class="math-container">\[H_B + H_{int}^{RC-B} = \int_{-∞}^{+∞} \mathrm{d}k \omega_k b_k^\dagger b_k - (d + d^{\dagger})\int_0^∞ \mathrm{d}\omega \sqrt{J(\omega)}(b_\omega^\dagger+b_ω) + \lambda_\text{reorg}(d + d^{\dagger})^2\]</p><p>with the reorganization energy taken into account that reads</p><p class="math-container">\[\lambda_\text{reorg} = \int \frac{J(\omega)}{\omega}\mathrm{d}\omega\]</p><p>.</p><h2 id="The-code"><a class="docs-heading-anchor" href="#The-code">The code</a><a id="The-code-1"></a><a class="docs-heading-anchor-permalink" href="#The-code" title="Permalink"></a></h2><p>First we load the <code>MPSdynamics.jl</code> package to be able to perform the simulation, the <code>Plots.jl</code> one to plot the results, and the <code>LaTeXStrings.jl</code> one to be able to use <span>$\LaTeX$</span> in the plots. <code>ColorSchemes.jl</code> is uploaded for colors in plots, <code>PolyChaos.jl</code> is needed to calculate the Hermite Polynomials that are necessary for the translation of the reduced density matrix from the Fock space dimension to the <span>$x$</span> space dimension. Eventually, LinearAlgebra is loaded for the <code>LinearAlgebra.diag</code> function for post-processing. </p><pre><code class="language-julia hljs">using MPSDynamics, Plots, LaTeXStrings, ColorSchemes, PolyChaos, LinearAlgebra</code></pre><p>We then define variables for the physical parameters of the simulation. The argument <code>isolated</code> can be set up to true to simulate only the electronic system with the RC oscillator. <code>isolated=false</code> adds an environment with the usual parameters <code>d</code> <code>N</code> and <code>α</code> that can be changed</p><pre><code class="language-julia hljs">#----------------------------
-# Physical parameters
-#----------------------------
-# Enol / Keto
-
-ω0e= 0.8  # Enol energy at x=0
-ω0k= 0.8  # Keto energy at x=0
-
-x0e = -0.25 # Enol Equilibrium displacement 
-x0k = 0.25 # Keto Equilibrium displacement 
-
-Δ = 0.05 # Direct coupling between enol and keto
-
-m=1.83E3 # Mass of the reaction coordinate particle. 1.83E3 is the proton mass in atomic units. 
-ħ=1 # Atomic Units convention
-
-dFockRC = 25 # Fock space of the RC tensor
-
-ωRC = 0.0347 # Frequency of the RC tensor
-
-γ = sqrt(m*ωRC/2)*x0e # γ = displacement RC ; γ = \sqrt{m\omega/2} x_disp
-
-cparsRC = [ωRC,ωRC*sqrt(m*ωRC/2)] # Energy and g RC coupling parameter
-
-isolated = true # isolated = true : no environment ; isolated = false : system coupled to a bosonic environment
-# Creates the chain depending on the isolated condition. The parameters for isolated = false can be modified as desired.
-if isolated
-    d=1; N=2; α = 0.0  # number of Fock states of the chain modes ; length of the chain ; coupling strength
-else
-    d=4; N=40; α = 0.008 # number of Fock states of the chain modes ; length of the chain ; coupling strength
-end
-
-s = 1 # ohmicity
-
-ωc = 2*ωRC # Cut-off of the spectral density J(ω)
-
-λreorg = (2*α*ωc)/s # Reorganisation Energy taken into account in the Hamiltonian
-
-cpars = chaincoeffs_ohmic(N, α, s; ωc=ωc) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0</code></pre><p>We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is <span>$\mathcal{O}(dt^3)$</span>. In this example we present only the regular one-site method with the keyword <code>:TDVP1</code> where all the virtual bonds of the MPS have the same bond dimension <span>$D$</span>.</p><p>Logically the constant bond dimension of the MPS for TDVP1 is the respective convergence parameter.</p><pre><code class="language-julia hljs">#-----------------------
-# Simulation parameters
-#-----------------------
-
-dt = 10.0 # time step
-
-tfinal = 2000.0 # simulation time
-
-numsteps = length(collect(0:dt:tfinal))-1
-
-method = :TDVP1 # time-evolution method
-
-D = 5 # MPS bond dimension</code></pre><p>This initial state is a product state between the system, the RC oscillator and the chain. While the chain is initially in the vacuum, the RC oscillator is initialised coherently representing a displaced state. The electronic system is diagonalized at this chosen displacement value in order to mainly initialize it in the adiabatic LOW state</p><p>The energy of the starting system is printed thanks to the function <a href="../../methods/#MPSDynamics.measurempo-Tuple{Vector, Vector, Tuple{Int64, Int64}}"><code>MPSDynamics.measurempo</code></a>.</p><pre><code class="language-julia hljs">#---------------------------
-# MPO and initial state MPS
-#---------------------------
-
-# Initial electronic system in the left well, at mean displacement . It results to a superposition of 1 and 2
-# This state is obtained with the diagonalization of the diabatic Hamiltonian at x=γ
-ψ = zeros(2)
-X2 = zeros(2)
-
-Ae = ω0e  - 0.5*m*ωRC^2*(x0e)^2
-Ak = ω0k  - 0.5*m*ωRC^2*(x0k)^2
-xini = γ/(sqrt(m*ωRC/2))
-
-Adumb = (ω0e-ω0k+0.5*m*ωRC^2*((xini-x0e)^2-(xini-x0k)^2))
-Bdumb = sqrt((Adumb)^2 + 4*Δ^2)
-X2[1] = (Adumb-Bdumb)/(2*Δ)
-X2[2] = 1
-
-mod_X2 = sqrt(sum(X2[i]^2 for i=1:2))
-
-ψ[1] = X2[1] / mod_X2
-ψ[2] = X2[2] / mod_X2
-
-#### Coherent state . Initialise the RC oscillator at displacement γ thanks to the Taylor expression of the displacement operator
-
-coherent_RC = [fill(unitcol(1,dFockRC), 1)...]
-chainpop_RCini =zeros(dFockRC,2)
-
-chainpop_RCini[1,2]=1
-chainpop_RCini[1,1]=exp(-(γ^2)/2)
-
-for j=2:dFockRC            
-    chainpop_RCini[j,1]=chainpop_RCini[j-1,1]*(γ/sqrt(j-1))
-end
-
-coherent_RC[1] = hcat(chainpop_RCini[:,1]...) #Manipulation to get good format for MPS, does not change value
-
-H = protontransfermpo(ω0e, ω0k, x0e, x0k, Δ, dFockRC, d, N, cpars, cparsRC, λreorg)
-
-A = productstatemps(physdims(H), state=[ψ, coherent_RC..., fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Displaced RC&gt;|Vacuum&gt;
-
-Eini = measurempo(A,H)
-print(&quot;\n Energy =&quot;,Eini)</code></pre><p>We then choose the observables that will be stored in the data and the <a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> arguments. The main analyzed data will be here the reduced density matrix.</p><pre><code class="language-julia hljs">#---------------------------
-# Definition of observables
-#---------------------------
-
-ob1 = OneSiteObservable(&quot;sz&quot;, sz, 1)
-#-------------
-ob2 = OneSiteObservable(&quot;RC displacement&quot;, MPSDynamics.disp(dFockRC,ωRC,m), 2)</code></pre><p><a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> is called to perform the dynamics. The argument <code>Nrho</code> is set up to 2 to specifiy the MPS site of the reduced density matrix. </p><pre><code class="language-julia hljs">#-------------
-# Simulation
-#------------
-
-A, dat = runsim(dt, tfinal, A, H;
-                name = &quot;proton transfer and tunneling at zero temperature&quot;,
-                method = method,
-                obs = [ob1, ob2],
-                convobs = [ob1, ob2],
-                params = @LogParams(N, d, α, s),
-                convparams = D,
-                Nrho = 2, # Need to precise that the reduced density matrix is
-                # calculated for the 2nd tensor. The value by default is 1. 
-                reduceddensity= true,
-                verbose = false,
-                save = true,
-                plot = false,
-                );</code></pre><p>Plot parameters are chosen. Among them, <code>xlist_rho</code> determines the resolution of the reduced density matrix in space. </p><pre><code class="language-julia hljs">#-----------------------
-# Plot parameters
-#-----------------------
-
-xlist_rho =collect(-1.2:0.05:1.2) # x definition for the reduced density matrix expressed in space
-# other values of xlist_rho can be chosen to gain numerical time
-
-palette = ColorSchemes.okabe_ito # color palette for plots</code></pre><p>We begin by representing the doublewell energy landscape described by the parameters.</p><pre><code class="language-julia hljs">#-------------
-# Plots
-#------------
-
-#### Plot the doublewell in the adiabatic basis. It is obtained with the diagonalization of the diabatic space Hamiltonian #####
-
-function fp(ɸ)
-    H11 = Ae + 0.5*m*ωRC^2*(ɸ-x0e)^2
-    H22 = Ak + 0.5*m*ωRC^2*(ɸ-x0k)^2 
-    H12 = Δ
-    λp = ((H11+H22)/2)+0.5*sqrt((H11-H22)^2+4*H12^2)
-    return λp
-end
-
-function fm(ɸ)
-    H11 = Ae + 0.5*m*ωRC^2*(ɸ-x0e)^2
-    H22 = Ak + 0.5*m*ωRC^2*(ɸ-x0k)^2
-    H12 = Δ
-    λm = ((H11+H22)/2)-0.5*sqrt((H11-H22)^2+4*H12^2)
-    return λm
-end
-
-function gaussian(x)
-    return Eini .+((1/0.06*sqrt(2*pi)).*exp.(-(x.+(0.25)).^2 ./(2*0.06^2))) .*0.001
-end
-
-ɸ = [i for i in -0.75:0.001:0.75]
-ɸwp = [i for i in -0.45:0.001:-0.05]
-xmin = -0.75 ; xmax = 0.75 ; ymin = 0.7 ; ymax = 0.9 
-
-plt=plot(ɸ,fm, label=&quot;LOW&quot;,xlabel=&quot;Reaction Coordinate (arb. units)&quot;,ylabel=L&quot;$\omega$ (arb. units)&quot;,linewidth=4,thickness_scaling = 1, bg_legend=:transparent, legendfont=16, legendfontsize=16, xguidefontsize=16, yguidefontsize=16, tickfontsize=(12), xlims=(xmin,xmax),ylims=(ymin,ymax),xticks=(xmin:0.25:xmax),yticks=(ymin:0.05:ymax),legend=(0.6,0.6), grid=false,linecolor =palette[8])
-
-plt=plot!(ɸ,fp,label=&quot;UP&quot;,lc =palette[3], linewidth=3,thickness_scaling = 1)
-
-plt = hline!([Eini], lw=2,  lc=:black, ls=:dash, label=L&quot;\omega_{system}&quot;)
-
-plt=plot!(ɸwp,gaussian,linecolor=:black,linewidth=3,  label=&quot;&quot;, fillrange = (Eini , gaussian(ɸwp)),c = :black)
-
-display(plt)</code></pre><p>The stored observables can be showed as usual.</p><pre><code class="language-julia hljs">##### Results #####
-
-display(plot(dat[&quot;data/times&quot;], dat[&quot;data/RC displacement&quot;],label=&quot;&lt;X&gt; (arb. units)&quot;, linecolor =:black, xlabel=&quot;Time (arb. units)&quot;,ylabel=&quot;&lt;X&gt;&quot;, title=&quot;&quot;, linewidth=4, legend=:none, tickfontsize=10,xlims=(0,2000)))</code></pre><p>Eventually, the reduced density matrix is analyzed. Initially expressed in the Fock dimension, the eigenvectors of the RC oscillator that are constructed here allows us to translate it into the reaction coordinate space dimension.</p><pre><code class="language-julia hljs">#### Reduced Density Matrix in space#####
-
-# Loads the Hermite coefficients to calculate the eigenstates of the harmonic oscillator. The coefficient 2 comes from the definition of the physics field. 
-αherm, βherm = 2.0 .*rm_hermite(dFockRC)
-
-#Fct building the nth harmonic eigenstate at coordinate x
-# sqrt(m*ωRC/ħ)*sqrt(2)*x in the Hermite function to recover the form of the physical Hermite polynomial in the atomic units
-function Ψ(n,x)
-return Acoef(n)*(m*ωRC/(ħ*π))^(1/4)*exp(-m*ωRC*(x)^2/(2*ħ))*PolyChaos.evaluate(n,(sqrt(m*ωRC/ħ)*sqrt(2)*x),αherm,βherm)
-end
-
-
-#Function to compute the pre factor of Ψ(n,x)
-  function Acoef(n)
-    if n == 0
-        return 1.0
-    else
-        return (1/sqrt(n*2))*2^(1/2)*Acoef(n-1)
-    end
-end
-
-#### Calculation of functions to express the reduced densty matrix in space. These functions build the matrix of oscillator eigenstates
-# Ψ(i,x) Ψ(j,x) ∀i,j ∈ dFockRC^2 and ∀ x ∈ xlist_rho
-Pmatrix = zeros(length(xlist_rho),dFockRC)
-Pmatrixnormed = zeros(length(xlist_rho),dFockRC)
-normalization = zeros(dFockRC)
-
-for n=1:dFockRC
-    for x=1:length(xlist_rho)
-        Pmatrix[x,n] = Ψ(n-1,xlist_rho[x])
-    end
-end
-
-for n=1:dFockRC
-    normalization[n] = sqrt(sum(Pmatrix[k,n]^2 for k=1:length(xlist_rho)))
-end
-
-
-for n=1:dFockRC
-    Pmatrixnormed[:,n] = Pmatrix[:,n]/normalization[n]
-end
-
-### Converts the reduced density matrix expressed in Fock states in a reduced density matrix expressed in space.
-ρx=zeros(ComplexF64,length(xlist_rho),length(xlist_rho),length(dat[&quot;data/times&quot;]))
-for t=1:length(dat[&quot;data/times&quot;])
-    print(&quot; \n t = &quot;, dat[&quot;data/times&quot;][t])
-    for x=1:length(xlist_rho)
-        for y=1:length(xlist_rho)
-            for n=1:dFockRC
-                for m=1:dFockRC
-                    ρx[x,y,t] +=  Pmatrixnormed[x,n]*Pmatrixnormed[y,m]*dat[&quot;data/Reduced ρ&quot;][n,m,t]
-                end
-            end
-        end
-    end
-end</code></pre><p>The reduced density matrixi in space can then be illustrated over time. The diagonal elements represent the population while off-diagonal elements are coherences.</p><pre><code class="language-julia hljs">#Creates a GIF of the diagonal elements of the reduced density over time, ie the population dynamics. Here, the reduced density matrix is expressed in space.
-
-anim = @animate for t=1:length(dat[&quot;data/times&quot;])
-    k=(dat[&quot;data/times&quot;][t])
-    print(&quot;\n k = &quot;, k)
-    plot(xlist_rho, abs.(diag(ρx[:,:,t])), title=&quot;Time = $k (arb. units)&quot;, xlabel=L&quot;Diag( $||\rho_{RC}(x,x)||$ )&quot;, ylabel=&quot;Amplitude&quot;, xlims=(-0.5, 0.5), ylims=(0,0.25),legend=:none)
-end
-display(gif(anim, &quot;gif_population.gif&quot;, fps = 2.5))
-</code></pre><pre><code class="language-julia hljs">#Creates a GIF of the reduced density expressed over time. Here, the reduced density matrix is expressed in space. Diagonal elements are populations 
-#whereas anti-diagonal elements  represent coherences
-
-anim = @animate for t=1:length(dat[&quot;data/times&quot;])
-    k=(dat[&quot;data/times&quot;][t])
-    print(&quot;\n k = &quot;, k)
-    (plot(heatmap(xlist_rho, xlist_rho, abs.(ρx[:,:,t]), c=:Blues ), title=&quot;Time = $k (arb. units)&quot;, xlabel=L&quot;$x$ (arb. units)&quot;,ylabel=L&quot;$x$ (arb. units)&quot;, tickfontsize=(12),colorbar_title = L&quot;||\rho_{RC}(x,x)||&quot;, legend=:none, xlims=(-0.5, 0.5), ylims=(-0.5,0.5), clims=(0,0.18),aspect_ratio=:equal))
-end
-display(gif(anim, &quot;gif_reducedrho.gif&quot;, fps = 2.5))</code></pre><p>________________</p><h2 id="Bibliography"><a class="docs-heading-anchor" href="#Bibliography">Bibliography</a><a id="Bibliography-1"></a><a class="docs-heading-anchor-permalink" href="#Bibliography" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-lede_ESIPT_2024"><a class="tag is-link" href="#citeref-lede_ESIPT_2024">lede_ESIPT_2024</a><blockquote><p>Le Dé, B.; Huppert, S.; Spezia, R.; Chin, A.W Extending Non-Perturbative Simulation Techniques for Open-Quantum Systems to Excited-State Proton Transfer and Ultrafast Non-Adiabatic Dynamics https://arxiv.org/abs/2405.08693</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../bath-observables/">« Inspecting the bath by undoing the chain mapping</a><a class="docs-footer-nextpage" href="../../theory/">Theoretical Background »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/docs/examples/puredephasing/index.html b/docs/examples/puredephasing/index.html
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--- a/docs/examples/puredephasing/index.html
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href="../../">MPSDynamics.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">Introduction</a></li><li><a class="tocitem" href="../../user-guide/">User Guide</a></li><li><a class="tocitem" href="../../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../sbm/">The Spin-Boson Model</a></li><li class="is-active"><a class="tocitem" href>Pure-Dephasing</a><ul class="internal"><li><a class="tocitem" href="#Context"><span>Context</span></a></li><li><a class="tocitem" href="#The-code"><span>The code</span></a></li><li><a class="tocitem" href="#Bibliography"><span>Bibliography</span></a></li></ul></li><li><a class="tocitem" href="../timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" 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class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Pure-Dephasing</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Pure-Dephasing</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Pure-Dephasing"><a class="docs-heading-anchor" href="#Pure-Dephasing">Pure-Dephasing</a><a id="Pure-Dephasing-1"></a><a class="docs-heading-anchor-permalink" href="#Pure-Dephasing" title="Permalink"></a></h1><h2 id="Context"><a class="docs-heading-anchor" href="#Context">Context</a><a id="Context-1"></a><a class="docs-heading-anchor-permalink" href="#Context" title="Permalink"></a></h2><p>The Pure-Dephasing Model describes a two-level system interacting linearly with an environment characterised by a spectral density (SD) <span>$J(\omega)$</span>. The coupling only acts on diagonal terms through the <span>$\sigma_z$</span> operator. The Hamiltonian reads (<span>$\hbar = 1$</span>)</p><p class="math-container">\[        \hat{H} = \frac{\omega_0}{2}\hat{\sigma}_z + \int_0^{+\infty}\omega \hat{a}^\dagger_\omega\hat{a}_\omega \mathrm{d}\omega + \frac{\hat{\sigma}_z}{2}\int_0^{+\infty}\sqrt{J(\omega)}(\hat{a}_\omega + \hat{a}^\dagger_\omega)\mathrm{d}\omega\]</p><p>Although this interaction will not change the population of the two-level system, the coherences between the two states will vary due to the environment. Introducing the two-level system reduced density matrix <span>$\rho_{ij}(t)$</span> with <span>$i,j \in (0,1)$</span>, the diagonal terms <span>$\rho_{ii}(t)$</span> are the populations of the states and the anti-diagonal terms <span>$\rho_{ij}(t)$</span> with <span>$i \neq j$</span> are the coherences between the two states. The effect of the <span>$\sigma_z$</span> bath interaction is to decouple the two states <span>$|0\rangle$</span> and <span>$|1\rangle$</span>. </p><p>The density matrix can be calculated within the MPS formalism with the MPO <span>$|\psi\rangle \langle \psi|$</span>. Tracing out the environment leads to the reduced density matrix. This can be done with the function <a href="../../methods/#MPSDynamics.rhoreduced_1site"><code>MPSDynamics.rhoreduced_1site</code></a>. </p><p>An analytical formula can be found for the decoherence function <span>$\Gamma(t)$</span>, taking into account the SD as well as the temperature of the environment <sup class="footnote-reference"><a id="citeref-breuer" href="#footnote-breuer">[breuer]</a></sup></p><p class="math-container">\[    \Gamma(t) = - \int_0^{\omega_c} \mathrm{d} \omega J(\omega)\frac{(1 - \cos(\omega t))}{\omega^2} \coth(β\omega/2) ,\]</p><p>with <span>$\beta = (k_B T)^{-1}$</span>. For the case where <span>$\beta \longrightarrow \infty$</span>, the integral reads</p><p class="math-container">\[    \Gamma(t) = - \int_0^{\omega_c} \mathrm{d} \omega J(\omega)\frac{(1 - \cos(\omega t))}{\omega^2} ,\]</p><p>The time-dependent anti-diagonal terms of the reduced density matrix are then expressed as</p><p class="math-container">\[    \rho_{12}(t) = \rho_{21}(t)^* =\rho_{12}(0) \exp(\Gamma(t)) \]</p><p>Setting up the initial two-level system as a cat state <span>$|\psi\rangle_S(0) = \frac{|0\rangle \pm |1\rangle}{\sqrt{2}}$</span>, this leads to <span>$\rho_{12}(0)=\frac{1}{2}$</span>.</p><p>Here we break out and comment the script in <code>MPSDynamics/examples/puredephasing.jl</code> and <code>MPSDynamics/examples/puredephasing_temperature.jl</code> to show how to simulate this model with an Ohmic SD (hard cut-off) using the T-TEDOPA method as implemented in <code>MPSDynamics.jl</code>.</p><p>The T-TEDOPA method relies on a truncated chain mapping that transform the initial Hamiltonian into</p><p class="math-container">\[        \hat{H} = \frac{\omega_0}{2} \hat{\sigma}_z +  c_0 \frac{\hat{\sigma}_z}{2}(\hat{b}_0^\dagger + \hat{b}_0) + \sum_{i=0}^{N-1} t_i (\hat{b}_{i+1}^\dagger \hat{b}_i + \mathrm{h.c.}) + \sum_{i=0}^{N-1} \epsilon_i \hat{b}_i^\dagger \hat{b}_i\]</p><h2 id="The-code"><a class="docs-heading-anchor" href="#The-code">The code</a><a id="The-code-1"></a><a class="docs-heading-anchor-permalink" href="#The-code" title="Permalink"></a></h2><p>First, we load the <code>MPSdynamics.jl</code> package to be able to perform the simulation, the <code>Plots.jl</code> one to plot the results, the <code>LaTeXStrings.jl</code> one to be able to use <span>$\LaTeX$</span> in the plots and eventually <code>QuadGK.jl</code> to perform the analytical integral calculations. A notation is introduced for <code>+∞</code>.</p><pre><code class="language-julia hljs">using MPSDynamics, Plots, LaTeXStrings, QuadGK
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Pure-Dephasing · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Introduction</a></li><li><a class="tocitem" href="../../user-guide/">User Guide</a></li><li><a class="tocitem" href="../../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../sbm/">The Spin-Boson Model</a></li><li class="is-active"><a class="tocitem" href>Pure-Dephasing</a><ul class="internal"><li><a class="tocitem" href="#Context"><span>Context</span></a></li><li><a class="tocitem" href="#The-code"><span>The code</span></a></li><li><a class="tocitem" href="#Bibliography"><span>Bibliography</span></a></li></ul></li><li><a class="tocitem" href="../timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="../../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../../methods/">Methods</a></li><li><a class="tocitem" href="../../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Pure-Dephasing</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Pure-Dephasing</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/examples/puredephasing.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Pure-Dephasing"><a class="docs-heading-anchor" href="#Pure-Dephasing">Pure-Dephasing</a><a id="Pure-Dephasing-1"></a><a class="docs-heading-anchor-permalink" href="#Pure-Dephasing" title="Permalink"></a></h1><h2 id="Context"><a class="docs-heading-anchor" href="#Context">Context</a><a id="Context-1"></a><a class="docs-heading-anchor-permalink" href="#Context" title="Permalink"></a></h2><p>The Pure-Dephasing Model describes a two-level system interacting linearly with an environment characterised by a spectral density (SD) <span>$J(\omega)$</span>. The coupling only acts on diagonal terms through the <span>$\sigma_z$</span> operator. The Hamiltonian reads</p><p class="math-container">\[        \hat{H} = \frac{\omega_0}{2}\hat{\sigma}_z + \int_0^{+\infty}\omega \hat{a}^\dagger_\omega\hat{a}_\omega \mathrm{d}\omega + \frac{\hat{\sigma}_z}{2}\int_0^{+\infty}\sqrt{J(\omega)}(\hat{a}_\omega + \hat{a}^\dagger_\omega)\mathrm{d}\omega\]</p><p>Although this interaction will not change the population of the two-level system, the coherences between the two states will vary due to the environment. Introducing the two-level system reduced density matrix <span>$\rho_{ij}(t)$</span> with <span>$i,j \in (0,1)$</span>, the diagonal terms <span>$\rho_{ii}(t)$</span> are the populations of the states and the anti-diagonal terms <span>$\rho_{ij}(t)$</span> with <span>$i \neq j$</span> are the coherences between the two states. The effect of the <span>$\sigma_z$</span> bath interaction is to decouple the two states <span>$|0\rangle$</span> and <span>$|1\rangle$</span>. </p><p>The density matrix can be calculated within the MPS formalism with the MPO <span>$|\psi\rangle \langle \psi|$</span>. Tracing out the environment leads to the reduced density matrix. This can be done with the function <a href="examples/@ref"><code>MPSDynamics.rhoreduced_1site</code></a>. </p><p>An analytical formula can be found for the decoherence function <span>$\Gamma(t)$</span>, taking into account the SD as well as the temperature of the environment <sup class="footnote-reference"><a id="citeref-breuer" href="#footnote-breuer">[breuer]</a></sup></p><p class="math-container">\[    \Gamma(t) = - \int_0^{\omega_c} \mathrm{d} \omega J(\omega)\frac{(1 - \cos(\omega t))}{\omega^2} \coth(β\omega/2) ,\]</p><p>with <span>$\beta = (k_B T)^{-1}$</span>. For the case where <span>$\beta \longrightarrow \infty$</span>, the integral reads</p><p class="math-container">\[    \Gamma(t) = - \int_0^{\omega_c} \mathrm{d} \omega J(\omega)\frac{(1 - \cos(\omega t))}{\omega^2} ,\]</p><p>The time-dependent anti-diagonal terms of the reduced density matrix are then expressed as</p><p class="math-container">\[    \rho_{12}(t) = \rho_{21}(t)^* =\rho_{12}(0) \exp(\Gamma(t)) \]</p><p>Setting up the initial two-level system as a cat state <span>$|\psi\rangle_S(0) = \frac{|0\rangle \pm |1\rangle}{\sqrt{2}}$</span>, this leads to <span>$\rho_{12}(0)=\frac{1}{2}$</span>.</p><p>Here we break out and comment the script in <code>MPSDynamics/examples/puredephasing.jl</code> and <code>MPSDynamics/examples/puredephasing_temperature.jl</code> to show how to simulate this model with an Ohmic SD (hard cut-off) using the T-TEDOPA method as implemented in <code>MPSDynamics.jl</code>.</p><p>The T-TEDOPA method relies on a truncated chain mapping that transform the initial Hamiltonian into</p><p class="math-container">\[        \hat{H} = \frac{\omega_0}{2} \hat{\sigma}_z +  c_0 \frac{\hat{\sigma}_z}{2}(\hat{b}_0^\dagger + \hat{b}_0) + \sum_{i=0}^{N-1} t_i (\hat{b}_{i+1}^\dagger \hat{b}_i + \mathrm{h.c.}) + \sum_{i=0}^{N-1} \epsilon_i \hat{b}_i^\dagger \hat{b}_i\]</p><h2 id="The-code"><a class="docs-heading-anchor" href="#The-code">The code</a><a id="The-code-1"></a><a class="docs-heading-anchor-permalink" href="#The-code" title="Permalink"></a></h2><p>First, we load the <code>MPSdynamics.jl</code> package to be able to perform the simulation, the <code>Plots.jl</code> one to plot the results, the <code>LaTeXStrings.jl</code> one to be able to use <span>$\LaTeX$</span> in the plots and eventually <code>QuadGK.jl</code> to perform the analytical integral calculations. A notation is introduced for <code>+∞</code>.</p><pre><code class="language-julia">using MPSDynamics, Plots, LaTeXStrings, QuadGK
 
-const ∞  = Inf</code></pre><p>We then define variables for the physical parameters of the simulation. Among these, two are convergence parameters:</p><ul><li><code>d</code> is the number of states we retain for the truncated harmonic oscillators representation of environmental modes</li><li><code>N</code> is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results</li></ul><p>The value of <code>β</code> determines whether the environment is thermalized or not. The example as it is is the zero-temperature case. For the finite-temperature case, &#39;β = ∞&#39; has be commented instead of the line above that precises a <code>β</code> value.</p><pre><code class="language-julia hljs">#----------------------------
+const ∞  = Inf</code></pre><p>We then define variables for the physical parameters of the simulation. Among these, two are convergence parameters:</p><ul><li><code>d</code> is the number of states we retain for the truncated harmonic oscillators representation of environmental modes</li><li><code>N</code> is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results</li></ul><p>The value of <code>β</code> determines whether the environment is thermalized or not. The example as it is is the zero-temperature case. For the finite-temperature case, &#39;β = ∞&#39; has be commented instead of the line above that precises a <code>β</code> value.</p><pre><code class="language-julia">#----------------------------
 # Physical parameters
 #----------------------------
 
@@ -19,7 +19,7 @@
 
 #β = 100 # Thermalized environment
 
-β = ∞ # Case zero temperature T=0, β → ∞</code></pre><p>The chain coefficient have then to be calculated. This part differs for the zero-temperature case and the thermalized bath. For <span>$β = ∞$</span>, the chain coeffficients can be calculated with the function <a href="../../methods/#MPSDynamics.chaincoeffs_ohmic-Tuple{Any, Any, Any}"><code>MPSDynamics.chaincoeffs_ohmic</code></a> whereas the function <a href="../../methods/#MPSDynamics.chaincoeffs_finiteT"><code>MPSDynamics.chaincoeffs_finiteT</code></a> is used when <span>$T \neq 0 ~ \text{K}$</span>:</p><pre><code class="language-julia hljs">if β == ∞
+β = ∞ # Case zero temperature T=0, β → ∞</code></pre><p>The chain coefficient have then to be calculated. This part differs for the zero-temperature case and the thermalized bath. For <span>$β = ∞$</span>, the chain coeffficients can be calculated with the function <a href="../../methods/#MPSDynamics.chaincoeffs_ohmic-Tuple{Any, Any, Any}"><code>MPSDynamics.chaincoeffs_ohmic</code></a> whereas the function <a href="../../methods/#MPSDynamics.chaincoeffs_finiteT"><code>MPSDynamics.chaincoeffs_finiteT</code></a> is used when <span>$T \neq 0 ~ \text{K}$</span>:</p><pre><code class="language-julia">if β == ∞
     cpars = chaincoeffs_ohmic(N, α, s; ωc=ωc)  # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0
 else
     cpars = chaincoeffs_finiteT(N, β; α=α, s=s, J=nothing, ωc=ωc, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=false)  # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0
@@ -28,7 +28,7 @@
     dir_chaincoeff = abspath(joinpath(curdir, &quot;../ChainOhmT/ohmicT&quot;))
     cpars  = readchaincoeffs(&quot;$dir_chaincoeff/chaincoeffs.h5&quot;,N,α,s,β) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0
     =#
-end</code></pre><p>An option is provided if the coefficient are saved to be reused.</p><p>We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is <span>$\mathcal{O}(dt^3)$</span>. In this example we present the regular one-site method with the keyword <code>:TDVP1</code> where all the virtual bonds of the MPS have the same bond dimension <span>$D$</span>. It is worh pointing out that DTDVP performs poorly in this specific example due to the exact solution of the system.</p><p>Logically the constant bond dimension of the MPS for TDVP1 is the convergence parameter.</p><pre><code class="language-julia hljs">#-----------------------
+end</code></pre><p>An option is provided if the coefficient are saved to be reused.</p><p>We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is <span>$\mathcal{O}(dt^3)$</span>. In this example we present two one-site implementation of TDVP that both preserves the unitarity of the evolution:</p><ul><li>the regular one-site method with the keyword <code>:TDVP1</code> where all the virtual bonds of the MPS have the same bond dimension <span>$D$</span></li><li>the adaptive method with the keyword <code>:DTDVP</code> where the bond dimension is locally increased at each time step if the TDVP projection error crosses a threshold value</li></ul><p>Logically the constant bond dimension of the MPS for TDVP1 and the threshold of the projection error for DTDVP are their respective convergence parameter.</p><pre><code class="language-julia">#-----------------------
 # Simulation parameters
 #-----------------------
 
@@ -38,7 +38,9 @@
 
 method = :TDVP1 # time-evolution method
 
-D = 2 # MPS bond dimension</code></pre><p>Using <code>MPSDynamics.jl</code> built-in methods we define the Pure Dephasing model MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the &#39;local state&#39; of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.</p><pre><code class="language-julia hljs">#---------------------------
+#method = :DTDVP # time-evolution method
+
+D = 2 # MPS bond dimension</code></pre><p>Using <code>MPSDynamics.jl</code> built-in methods we define the Pure Dephasing model MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the &#39;local state&#39; of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.</p><pre><code class="language-julia">#---------------------------
 # MPO and initial state MPS
 #---------------------------
 
@@ -49,7 +51,7 @@
 ψ[1] = 1/sqrt(2)
 ψ[2] = 1/sqrt(2)
 
-A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre><p>We then chose the observables that will be stored in the data and the <a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> arguments. This example relies on the storage of the reduced density matrix (called <code>reduceddensity</code> in runsim).</p><pre><code class="language-julia hljs">
+A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre><p>We then chose the observables that will be stored in the data and the <a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> arguments. This example relies on the storage of the reduced density matrix (called <code>reduceddensity</code> in runsim).</p><pre><code class="language-julia">
 #---------------------------
 # Definition of observables
 #---------------------------
@@ -72,7 +74,7 @@
                 verbose = false,
                 save = true,
                 plot = true,
-                );</code></pre><p>After having performed the dynamics, the analytical decoherence function is numerically calculated with the help of the quadgk function (from the <code>QuadGK.jl</code> package). It reads</p><pre><code class="language-julia hljs">#----------
+                );</code></pre><p>After having performed the dynamics, the analytical decoherence function is numerically calculated with the help of the quadgk function (from the <code>QuadGK.jl</code> package). It reads</p><pre><code class="language-julia">#----------
 # Analytical results at specified temperature 
 # (see The Theory of Open Quantum System, H.-P. Breuer &amp; F. Petruccione 2002, Chapter 4)
 #----------
@@ -83,7 +85,7 @@
 
 Γohmic(t) = - quadgk(x -&gt; Johmic(x,s)*(1 - cos(x*t))*coth(β*x/2)/x^2, 0, ωc)[1]
 
-Decoherence_ohmic(t) = 0.5*exp(Γohmic(t))</code></pre><p>Eventually, the stored reduced density matrix is compared against the analytical formula</p><pre><code class="language-julia hljs">#-------------
+Decoherence_ohmic(t) = 0.5*exp(Γohmic(t))</code></pre><p>Eventually, the stored reduced density matrix is compared against the analytical formula</p><pre><code class="language-julia">#-------------
 # Plots
 #------------
 
@@ -91,4 +93,4 @@
 
 plot(time_analytical, t-&gt;Decoherence_ohmic(t), label=&quot;Analytics&quot;, title=L&quot;Pure Dephasing, Ohmic $s=%$s$, $\beta = %$β ~\mathrm{K}$&quot;, linecolor=:black, xlabel=&quot;Time (arb. units)&quot;, ylabel=L&quot;Coherence $|\rho_{12}(t)|$&quot;, linewidth=4, titlefontsize=16, legend=:best, legendfontsize=16, xguidefontsize=16, yguidefontsize=16, tickfontsize=10)
 
-plot!(dat[&quot;data/times&quot;], ρ12, lw=4, ls=:dash, label=&quot;Numerics&quot;)</code></pre><h2 id="Bibliography"><a class="docs-heading-anchor" href="#Bibliography">Bibliography</a><a id="Bibliography-1"></a><a class="docs-heading-anchor-permalink" href="#Bibliography" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-breuer"><a class="tag is-link" href="#citeref-breuer">breuer</a><blockquote><p>Breuer, H.;Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press, 2002.</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../sbm/">« The Spin-Boson Model</a><a class="docs-footer-nextpage" href="../timedep/">Time-dependent Hamiltonian »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+plot!(dat[&quot;data/times&quot;], ρ12, lw=4, ls=:dash, label=&quot;Numerics&quot;)</code></pre><h2 id="Bibliography"><a class="docs-heading-anchor" href="#Bibliography">Bibliography</a><a id="Bibliography-1"></a><a class="docs-heading-anchor-permalink" href="#Bibliography" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-breuer"><a class="tag is-link" href="#citeref-breuer">breuer</a><blockquote><p>Breuer, H.;Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press, 2002.</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../sbm/">« The Spin-Boson Model</a><a class="docs-footer-nextpage" href="../timedep/">Time-dependent Hamiltonian »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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index 90f134a..2a622ad 100644
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+++ b/docs/examples/sbm/index.html
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href="../../">MPSDynamics.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">Introduction</a></li><li><a class="tocitem" href="../../user-guide/">User Guide</a></li><li><a class="tocitem" href="../../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li class="is-active"><a class="tocitem" href>The Spin-Boson Model</a><ul class="internal"><li><a class="tocitem" href="#Context"><span>Context</span></a></li><li><a class="tocitem" href="#The-code"><span>The code</span></a></li></ul></li><li><a class="tocitem" href="../puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" 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Spin-Boson Model</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The Spin-Boson Model</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-Spin-Boson-Model"><a class="docs-heading-anchor" href="#The-Spin-Boson-Model">The Spin-Boson Model</a><a id="The-Spin-Boson-Model-1"></a><a class="docs-heading-anchor-permalink" href="#The-Spin-Boson-Model" title="Permalink"></a></h1><h2 id="Context"><a class="docs-heading-anchor" href="#Context">Context</a><a id="Context-1"></a><a class="docs-heading-anchor-permalink" href="#Context" title="Permalink"></a></h2><p>The Spin-Boson Model (SBM) is a prototypical model in the theory of open quantum systems where a two level system interacts linearly with a bosonic bath (<span>$\hbar = 1$</span>)</p><p class="math-container">\[	\hat{H} = \frac{\omega_0}{2}\hat{\sigma}_z + \Delta\hat{\sigma}_x + \int_0^{+\infty}\omega \hat{a}^\dagger_\omega\hat{a}_\omega \mathrm{d}\omega + \hat{\sigma}_x\int_0^{+\infty}\sqrt{J(\omega)}(\hat{a}_\omega + \hat{a}^\dagger_\omega)\mathrm{d}\omega\]</p><p>Even though this model is fairly simple it is physically very rich and it is not analytically solvable. For these reason it has become a test-bed for numerical methods simulating open quantum systems dynamics in the non-perturbative non-Markovian regime.</p><p>For instance when the SD is Ohmic, this model presents a phase transition between a so called localised and a delocalised phase for <span>$\alpha \approx 1.2$</span>.</p><p>Here we break out and comment the script in <code>MPSDynamics/examples/sbm_zero_temperature.jl</code> to show how to simulate this model with an Ohmic SD (hard cut-off) using the T-TEDOPA method as implemented in <code>MPSDynamics.jl</code>.</p><p>The T-TEDOPA method relies on a truncated chain mapping that transform the initial Hamiltonian into</p><p class="math-container">\[	\hat{H} = \frac{\omega_0}{2} \hat{\sigma}_z + \Delta \hat{\sigma}_x + c_0 \hat{\sigma}_x(\hat{b}_0^\dagger + \hat{b}_0) + \sum_{i=0}^{N-1} t_i (\hat{b}_{i+1}^\dagger \hat{b}_i + \mathrm{h.c.}) + \sum_{i=0}^{N-1} \epsilon_i \hat{b}_i^\dagger \hat{b}_i \]</p><h2 id="The-code"><a class="docs-heading-anchor" href="#The-code">The code</a><a id="The-code-1"></a><a class="docs-heading-anchor-permalink" href="#The-code" title="Permalink"></a></h2><p>First a multi-line comment introduces the model and the aim of the script.</p><pre><code class="language-julia hljs">#=
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>The Spin-Boson Model · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Introduction</a></li><li><a class="tocitem" href="../../user-guide/">User Guide</a></li><li><a class="tocitem" href="../../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li class="is-active"><a class="tocitem" href>The Spin-Boson Model</a><ul class="internal"><li><a class="tocitem" href="#Context"><span>Context</span></a></li><li><a class="tocitem" href="#The-code"><span>The code</span></a></li></ul></li><li><a class="tocitem" href="../puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="../../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../../methods/">Methods</a></li><li><a class="tocitem" href="../../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>The Spin-Boson Model</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The Spin-Boson Model</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/examples/sbm.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="The-Spin-Boson-Model"><a class="docs-heading-anchor" href="#The-Spin-Boson-Model">The Spin-Boson Model</a><a id="The-Spin-Boson-Model-1"></a><a class="docs-heading-anchor-permalink" href="#The-Spin-Boson-Model" title="Permalink"></a></h1><h2 id="Context"><a class="docs-heading-anchor" href="#Context">Context</a><a id="Context-1"></a><a class="docs-heading-anchor-permalink" href="#Context" title="Permalink"></a></h2><p>The Spin-Boson Model (SBM) is a prototypical model in the theory of open quantum systems where a two level system interacts linearly with a bosonic bath</p><p class="math-container">\[	\hat{H} = \frac{\omega_0}{2}\hat{\sigma}_z + \Delta\hat{\sigma}_x + \int_0^{+\infty}\omega \hat{a}^\dagger_\omega\hat{a}_\omega \mathrm{d}\omega + \hat{\sigma}_x\int_0^{+\infty}\sqrt{J(\omega)}(\hat{a}_\omega + \hat{a}^\dagger_\omega)\mathrm{d}\omega\]</p><p>Even though this model is fairly simple it is physically very rich and it is not analytically solvable. For these reason it has become a test-bed for numerical methods simulating open quantum systems dynamics in the non-perturbative non-Markovian regime.</p><p>For instance when the SD is Ohmic, this model presents a phase transition between a so called localised and a delocalised phase for <span>$\alpha \approx 1.2$</span>.</p><p>Here we break out and comment the script in <code>MPSDynamics/examples/sbm_zero_temperature.jl</code> to show how to simulate this model with an Ohmic SD (hard cut-off) using the T-TEDOPA method as implemented in <code>MPSDynamics.jl</code>.</p><p>The T-TEDOPA method relies on a truncated chain mapping that transform the initial Hamiltonian into</p><p class="math-container">\[	\hat{H} = \frac{\omega_0}{2} \hat{\sigma}_z + \Delta \hat{\sigma}_x + c_0 \hat{\sigma}_x(\hat{b}_0^\dagger + \hat{b}_0) + \sum_{i=0}^{N-1} t_i (\hat{b}_{i+1}^\dagger \hat{b}_i + \mathrm{h.c.}) + \sum_{i=0}^{N-1} \epsilon_i \hat{b}_i^\dagger \hat{b}_i \]</p><h2 id="The-code"><a class="docs-heading-anchor" href="#The-code">The code</a><a id="The-code-1"></a><a class="docs-heading-anchor-permalink" href="#The-code" title="Permalink"></a></h2><p>First a multi-line comment introduces the model and the aim of the script.</p><pre><code class="language-julia">#=
     Example of a zero-temperature Spin-Boson Model with an hard cut-off Ohmic spectral density J(ω) = 2αω when ω &lt; ωc and 0 otherwise
 
     The dynamics is simulated using the T-TEDOPA method that maps the normal modes environment into a non-uniform tight-binding chain.
@@ -7,7 +7,7 @@
     H = \\frac{ω_0}{2} σ_z + Δ σ_x + c_0 σ_x(b_0^\\dagger + b_0) + \\sum_{i=0}^{N-1} t_i (b_{i+1}^\\dagger b_i +h.c.) + \\sum_{i=0}^{N-1} ϵ_i b_i^\\dagger b_i 
 
     Two variants of the one-site Time Dependent Variational Principal (TDVP) are presented for the time evolution of the quantum state.
-=#</code></pre><p>We load the <code>MPSdynamics.jl</code> package to be able to perform the simulation, the <code>Plots.jl</code> one to plot the results, and the <code>LaTeXStrings.jl</code> one to be able to use <span>$\LaTeX$</span> in the plots.</p><pre><code class="language-julia hljs">using MPSDynamics, Plots, LaTeXStrings</code></pre><p>We then define variables for the physical parameters of the simulation. Among these, two are convergence parameters:</p><ul><li><code>d</code> is the number of states we retain for the truncated harmonic oscillators representation of environmental modes </li><li><code>N</code> is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results</li></ul><pre><code class="language-julia hljs">#----------------------------
+=#</code></pre><p>We load the <code>MPSdynamics.jl</code> package to be able to perform the simulation, the <code>Plots.jl</code> one to plot the results, and the <code>LaTeXStrings.jl</code> one to be able to use <span>$\LaTeX$</span> in the plots.</p><pre><code class="language-julia">using MPSDynamics, Plots, LaTeXStrings</code></pre><p>We then define variables for the physical parameters of the simulation. Among these, two are convergence parameters:</p><ul><li><code>d</code> is the number of states we retain for the truncated harmonic oscillators representation of environmental modes </li><li><code>N</code> is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results</li></ul><pre><code class="language-julia">#----------------------------
 # Physical parameters
 #----------------------------
 
@@ -23,7 +23,7 @@
 
 s = 1 # ohmicity
 
-cpars = chaincoeffs_ohmic(N, α, s) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0</code></pre><p>We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is <span>$\mathcal{O}(dt^3)$</span>. In this example we present two one-site implementation of TDVP that both preserves the unitarity of the evolution:</p><ul><li>the regular one-site method with the keyword <code>:TDVP1</code> where all the virtual bonds of the MPS have the same bond dimension <span>$D$</span></li><li>the adaptive method with the keyword <code>:DTDVP</code> where the bond dimension is locally increased at each time step if the TDVP projection error crosses a threshold value</li></ul><p>Logically the constant bond dimension of the MPS for TDVP1 and the threshold of the projection error for DTDVP are their respective convergence parameter. </p><pre><code class="language-julia hljs">#-----------------------
+cpars = chaincoeffs_ohmic(N, α, s) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0</code></pre><p>We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is <span>$\mathcal{O}(dt^3)$</span>. In this example we present two one-site implementation of TDVP that both preserves the unitarity of the evolution:</p><ul><li>the regular one-site method with the keyword <code>:TDVP1</code> where all the virtual bonds of the MPS have the same bond dimension <span>$D$</span></li><li>the adaptive method with the keyword <code>:DTDVP</code> where the bond dimension is locally increased at each time step if the TDVP projection error crosses a threshold value</li></ul><p>Logically the constant bond dimension of the MPS for TDVP1 and the threshold of the projection error for DTDVP are their respective convergence parameter. </p><pre><code class="language-julia">#-----------------------
 # Simulation parameters
 #-----------------------
 
@@ -37,7 +37,7 @@
 
 convparams = [2,4,6] # MPS bond dimension (1TDVP)
 
-# convparams = [1e-2, 1e-3, 1e-4] # threshold value of the projection error (DTDVP)</code></pre><p>Using <code>MPSDynamics.jl</code> built-in methods we define the SBM MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the &#39;local state&#39; of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.</p><pre><code class="language-julia hljs">#---------------------------
+# convparams = [1e-2, 1e-3, 1e-4] # threshold value of the projection error (DTDVP)</code></pre><p>Using <code>MPSDynamics.jl</code> built-in methods we define the SBM MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the &#39;local state&#39; of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.</p><pre><code class="language-julia">#---------------------------
 # MPO and initial state MPS
 #---------------------------
 
@@ -45,7 +45,7 @@
 
 ψ = unitcol(1,2) # Initial up-z system state 
 
-A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre><p>We then chose the observables that will be measured during the time evolution after being passed as an argument of the <a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> method.</p><pre><code class="language-julia hljs">#---------------------------
+A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre><p>We then chose the observables that will be measured during the time evolution after being passed as an argument of the <a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> method.</p><pre><code class="language-julia">#---------------------------
 # Definition of observables
 #---------------------------
 
@@ -53,7 +53,7 @@
 
 ob2 = OneSiteObservable(&quot;chain mode occupation&quot;, numb(d), (2,N+1))
 
-ob3 = TwoSiteObservable(&quot;SXdisp&quot;, sx, disp(d), [1], collect(2:N+1))</code></pre><p>We run the simulation from <code>t = 0</code> to <code>tfinal</code> with time-steps of size <code>dt</code> using the <code>method</code> method. The <code>convobs</code> observables will be measured and stored for the convergence parameters <code>convparams</code>, whereas the <code>obs</code> observables will only be measured and stored for the most precise convergence parameter.</p><pre><code class="language-julia hljs">#-------------
+ob3 = TwoSiteObservable(&quot;SXdisp&quot;, sx, disp(d), [1], collect(2:N+1))</code></pre><p>We run the simulation from <code>t = 0</code> to <code>tfinal</code> with time-steps of size <code>dt</code> using the <code>method</code> method. The <code>convobs</code> observables will be measured and stored for the convergence parameters <code>convparams</code>, whereas the <code>obs</code> observables will only be measured and stored for the most precise convergence parameter.</p><pre><code class="language-julia">#-------------
 # Simulation
 #------------
 
@@ -68,7 +68,7 @@
                 savebonddims = true, # this keyword argument enables the bond dimension at each time step to be saved when using DTDVP
                 save = true,
                 plot = true,
-                );</code></pre><p>Finally, we plot the simulation results that are stored in the <code>dat</code> dictionnary.</p><pre><code class="language-julia hljs">#----------
+                );</code></pre><p>Finally, we plot the simulation results that are stored in the <code>dat</code> dictionnary.</p><pre><code class="language-julia">#----------
 # Plots
 #----------
 
@@ -78,4 +78,4 @@
 
 method == :DTDVP &amp;&amp; heatmap(dat[&quot;data/times&quot;], collect(0:N+1), dat[&quot;data/bonddims&quot;], xlabel=L&quot;t&quot;,ylabel=&quot;bond index&quot;)
 
-heatmap(dat[&quot;data/times&quot;], collect(1:N), abs.(dat[&quot;data/SXdisp&quot;][1,:,:]), xlabel=L&quot;t&quot;,ylabel=&quot;chain mode&quot;)</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../nutshell/">« In a nutshell: MPSDynamics.jl for open quantum systems</a><a class="docs-footer-nextpage" href="../puredephasing/">Pure-Dephasing »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+heatmap(dat[&quot;data/times&quot;], collect(1:N), abs.(dat[&quot;data/SXdisp&quot;][1,:,:]), xlabel=L&quot;t&quot;,ylabel=&quot;chain mode&quot;)</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../nutshell/">« In a nutshell: MPSDynamics.jl for open quantum systems</a><a class="docs-footer-nextpage" href="../puredephasing/">Pure-Dephasing »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/docs/examples/timedep/index.html b/docs/examples/timedep/index.html
index 666f8a6..8bb4610 100644
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+++ b/docs/examples/timedep/index.html
@@ -1,7 +1,7 @@
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href>Time-dependent Hamiltonian</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Time-dependent Hamiltonian</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Time-dependent-Hamiltonian"><a class="docs-heading-anchor" href="#Time-dependent-Hamiltonian">Time-dependent Hamiltonian</a><a id="Time-dependent-Hamiltonian-1"></a><a class="docs-heading-anchor-permalink" href="#Time-dependent-Hamiltonian" title="Permalink"></a></h1><h2 id="Context"><a class="docs-heading-anchor" href="#Context">Context</a><a id="Context-1"></a><a class="docs-heading-anchor-permalink" href="#Context" title="Permalink"></a></h2><p>To simulate a drive or a laser pulse, a time-dependent Hamiltonian is often needed. Here we explain thanks to the script in <code>MPSDynamics/examples/sbm_Htimedependent.jl</code> how this type of Hamiltonian can be integrated within the method and applied on a wavefunction. We will take a Spin-Boson Model (SBM) Hamiltonian with an Ohmic spectral density. However, the time-dependency is not model specific and can be adapated for other models. For more information about the SBM, see the dedicated example. For this example, we simulate a drive of the form</p><p class="math-container">\[        \hat{H}_\text{drive}(t) = \epsilon \hat{\sigma}_x \sin(\omega_\text{drive} t)\]</p><p>with <span>$\epsilon = \frac{2 \pi}{T_\text{Rabi}}$</span> the amplitude of the drive, <span>$T_\text{Rabi}$</span> the Rabi period, and <span>$\omega_\text{drive}$</span> the frequency of the drive. The drive is set up to be on resonance with the two-level system.</p><h2 id="The-code"><a class="docs-heading-anchor" href="#The-code">The code</a><a id="The-code-1"></a><a class="docs-heading-anchor-permalink" href="#The-code" title="Permalink"></a></h2><p>First we load the <code>MPSdynamics.jl</code> package to be able to perform the simulation, the <code>Plots.jl</code> one to plot the results, and the <code>LaTeXStrings.jl</code> one to be able to use <span>$\LaTeX$</span> in the plots. The function <a href="../../methods/#MPSDynamics.disp-Tuple{Any, Any, Any}"><code>MPSDynamics.disp</code></a> is also imported.</p><pre><code class="language-julia hljs">using MPSDynamics, Plots, LaTeXStrings
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Time-dependent Hamiltonian · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Introduction</a></li><li><a class="tocitem" href="../../user-guide/">User Guide</a></li><li><a class="tocitem" href="../../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../puredephasing/">Pure-Dephasing</a></li><li class="is-active"><a class="tocitem" href>Time-dependent Hamiltonian</a><ul class="internal"><li><a class="tocitem" href="#Context"><span>Context</span></a></li><li><a class="tocitem" href="#The-code"><span>The code</span></a></li></ul></li><li><a class="tocitem" href="../anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="../../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../../methods/">Methods</a></li><li><a class="tocitem" href="../../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Time-dependent Hamiltonian</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Time-dependent Hamiltonian</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/examples/timedep.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Time-dependent-Hamiltonian"><a class="docs-heading-anchor" href="#Time-dependent-Hamiltonian">Time-dependent Hamiltonian</a><a id="Time-dependent-Hamiltonian-1"></a><a class="docs-heading-anchor-permalink" href="#Time-dependent-Hamiltonian" title="Permalink"></a></h1><h2 id="Context"><a class="docs-heading-anchor" href="#Context">Context</a><a id="Context-1"></a><a class="docs-heading-anchor-permalink" href="#Context" title="Permalink"></a></h2><p>To simulate a drive or a laser pulse, a time-dependent Hamiltonian is often needed. Here we explain thanks to the script in <code>MPSDynamics/examples/sbm_Htimedependent.jl</code> how this type of Hamiltonian can be integrated within the method and applied on a wavefunction. We will take a Spin-Boson Model (SBM) Hamiltonian with an Ohmic spectral density. However, the time-dependency is not model specific and can be adapated for other models. For more information about the SBM, see the dedicated example. For this example, we simulate a drive of the form</p><p class="math-container">\[        \hat{H}_\text{drive}(t) = \epsilon \hat{\sigma}_x \sin(\omega_\text{drive} t)\]</p><p>with <span>$\epsilon = \frac{2 \pi}{T_\text{Rabi}}$</span> the amplitude of the drive, <span>$T_\text{Rabi}$</span> the Rabi period, and <span>$\omega_\text{drive}$</span> the frequency of the drive. The drive is set up to be on resonance with the two-level system.</p><h2 id="The-code"><a class="docs-heading-anchor" href="#The-code">The code</a><a id="The-code-1"></a><a class="docs-heading-anchor-permalink" href="#The-code" title="Permalink"></a></h2><p>First we load the <code>MPSdynamics.jl</code> package to be able to perform the simulation, the <code>Plots.jl</code> one to plot the results, and the <code>LaTeXStrings.jl</code> one to be able to use <span>$\LaTeX$</span> in the plots. The function <a href="examples/@ref"><code>MPSDynamics.disp</code></a> is also imported.</p><pre><code class="language-julia">using MPSDynamics, Plots, LaTeXStrings
 
-import MPSDynamics: disp</code></pre><p>We then define variables for the physical parameters of the simulation. Among these, three are convergence parameters:</p><ul><li><code>d</code> is the number of states we retain for the truncated harmonic oscillators representation of environmental modes</li><li><code>N</code> is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results</li></ul><p>The variable <code>Ndrive</code> represents the site of the MPO where the operator of the time-dependent part acts. For this example, the two-level system is at the first site of the MPS. </p><pre><code class="language-julia hljs">#----------------------------
+import MPSDynamics: disp</code></pre><p>We then define variables for the physical parameters of the simulation. Among these, three are convergence parameters:</p><ul><li><code>d</code> is the number of states we retain for the truncated harmonic oscillators representation of environmental modes</li><li><code>N</code> is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results</li></ul><p>The variable <code>Ndrive</code> represents the site of the MPO where the operator of the time-dependent part acts. For this example, the two-level system is at the first site of the MPS. </p><pre><code class="language-julia">#----------------------------
 # Physical parameters
 #----------------------------
 
@@ -25,7 +25,7 @@
 
 ωdrive = ω0 # Frequency of the drive
 
-Ndrive = 1 #Number of the site on which the drive is applied</code></pre><p>We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is <span>$\mathcal{O}(dt^3)$</span>. In this example we present only one-site implementation of TDVP that preserves the unitarity of the evolution:</p><ul><li>the regular one-site method with the keyword <code>:TDVP1</code> where all the virtual bonds of the MPS have the same bond dimension <span>$D$</span></li></ul><p>Logically the constant bond dimension of the MPS for TDVP1 is the respective convergence parameter.</p><pre><code class="language-julia hljs">#-----------------------
+Ndrive = 1 #Number of the site on which the drive is applied</code></pre><p>We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is <span>$\mathcal{O}(dt^3)$</span>. In this example we present only one-site implementation of TDVP that preserves the unitarity of the evolution:</p><ul><li>the regular one-site method with the keyword <code>:TDVP1</code> where all the virtual bonds of the MPS have the same bond dimension <span>$D$</span></li></ul><p>Logically the constant bond dimension of the MPS for TDVP1 is the respective convergence parameter.</p><pre><code class="language-julia">#-----------------------
 # Simulation parameters
 #-----------------------
 
@@ -35,7 +35,7 @@
 
 method = :TDVP1 # time-evolution method
 
-D = [6] # MPS bond dimension</code></pre><p>Using <code>MPSDynamics.jl</code> built-in methods we define the SBM MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the &#39;local state&#39; of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.</p><p>In this part, the time-dependent terms of the MPO are stored in a list. This enables to add these terms to the MPO elements at each timestep and avoid the calculation of an entire new MPO. This way is not cumbersome since it adds a matrix sum at each time step.</p><pre><code class="language-julia hljs">#---------------------------
+D = [6] # MPS bond dimension</code></pre><p>Using <code>MPSDynamics.jl</code> built-in methods we define the SBM MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the &#39;local state&#39; of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.</p><p>In this part, the time-dependent terms of the MPO are stored in a list. This enables to add these terms to the MPO elements at each timestep and avoid the calculation of an entire new MPO. This way is not cumbersome since it adds a matrix sum at each time step.</p><pre><code class="language-julia">#---------------------------
 # MPO and initial state MPS
 #---------------------------
 
@@ -48,7 +48,7 @@
 
 ψ = unitcol(2,2) # Initial down-z system state 
 
-A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre><p>We then choose the observables that will be stored in the data and the <a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> arguments.</p><pre><code class="language-julia hljs">#---------------------------
+A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre><p>We then choose the observables that will be stored in the data and the <a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> arguments.</p><pre><code class="language-julia">#---------------------------
 # Definition of observables
 #---------------------------
 
@@ -57,7 +57,7 @@
 ob2 = OneSiteObservable(&quot;chain mode occupation&quot;, numb(d), (2,N+1))
 
 ob3 = TwoSiteObservable(&quot;SXdisp&quot;, sx, disp(d), [1], collect(2:N+1))
-</code></pre><p><a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> is called to perform the dynamics. The argument <code>timedep</code> is set up to true and <code>Ndrive</code> and <code>Htime</code> are provided in the kwargs. </p><pre><code class="language-julia hljs">#-------------
+</code></pre><p><a href="../../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> is called to perform the dynamics. The argument <code>timedep</code> is set up to true and <code>Ndrive</code> and <code>Htime</code> are provided in the kwargs. </p><pre><code class="language-julia">#-------------
 # Simulation
 #------------
 
@@ -74,9 +74,9 @@
                 verbose = false,
                 save = true,
                 plot = true,
-                );</code></pre><p>Eventually, the stored observables can be represented</p><pre><code class="language-julia hljs">
+                );</code></pre><p>Eventually, the stored observables can be represented</p><pre><code class="language-julia">
 #----------
 # Plots
 #----------
 
-plot(dat[&quot;data/times&quot;], dat[&quot;data/sz&quot;], label=&quot;Dmax = $(D...)&quot;, xlabel=L&quot;t&quot;,ylabel=L&quot;\sigma_z&quot;, title=&quot;&quot;)</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../puredephasing/">« Pure-Dephasing</a><a class="docs-footer-nextpage" href="../anderson-model/">The Anderson Impurity Model »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+plot(dat[&quot;data/times&quot;], dat[&quot;data/sz&quot;], label=&quot;Dmax = $(D...)&quot;, xlabel=L&quot;t&quot;,ylabel=L&quot;\sigma_z&quot;, title=&quot;&quot;)</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../puredephasing/">« Pure-Dephasing</a><a class="docs-footer-nextpage" href="../anderson-model/">The Anderson Impurity Model »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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(Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Introduction</a><ul class="internal"><li><a class="tocitem" href="#Installation"><span>Installation</span></a></li><li><a class="tocitem" href="#Table-of-Contents"><span>Table of Contents</span></a></li><li><a class="tocitem" href="#Citation"><span>Citation</span></a></li></ul></li><li><a class="tocitem" href="user-guide/">User Guide</a></li><li><a class="tocitem" href="nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li><li><a class="tocitem" href="examples/protontransfer/">Proton Transfer Model</a></li></ul></li><li><a class="tocitem" href="theory/">Theoretical Background</a></li><li><a class="tocitem" href="methods/">Methods</a></li><li><a class="tocitem" href="dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p>The <code>MPSDynamics.jl</code> package provides an easy to use interface for performing tensor network simulations on matrix product states (MPS) and tree tensor network (TTN) states. Written in the Julia programming language, <code>MPSDynamics.jl</code> is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle (TDVP) method for time evolution.  The package also provides strong support for the measurement of observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics.  The package has been developed with the aim of studying non-Markovian open system dynamics at finite temperature using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm (T-TEDOPA) based on environment chain mapping. However the methods implemented can equally be applied to other areas of physics.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>The package may be installed by typing the following into a Julia REPL</p><pre><code class="language-julia hljs">    ] add https://github.com/shareloqs/MPSDynamics.git</code></pre><h2 id="Table-of-Contents"><a class="docs-heading-anchor" href="#Table-of-Contents">Table of Contents</a><a id="Table-of-Contents-1"></a><a class="docs-heading-anchor-permalink" href="#Table-of-Contents" title="Permalink"></a></h2><ul><li><a href="#Introduction">Introduction</a></li><li class="no-marker"><ul><li><a href="#Installation">Installation</a></li><li><a href="#Table-of-Contents">Table of Contents</a></li><li><a href="#Citation">Citation</a></li></ul></li><li><a href="user-guide/#User-Guide">User Guide</a></li><li class="no-marker"><ul><li><a href="user-guide/#Initial-State">Initial State</a></li><li class="no-marker"><ul><li><a href="user-guide/#Matrix-Product-States">Matrix Product States</a></li><li><a href="user-guide/#Tree-Tensor-Networks">Tree Tensor Networks</a></li></ul></li><li><a href="user-guide/#Hamiltonian">Hamiltonian</a></li><li class="no-marker"><ul><li><a href="user-guide/#Build-in-Hamiltonian">Build-in Hamiltonian</a></li><li><a href="user-guide/#Convert-a-MPO-from-ITensor">Convert a MPO from ITensor</a></li><li><a href="user-guide/#Tailored-MPO">Tailored MPO</a></li><li><a href="user-guide/#Finite-Temperature-and-Custom-SD">Finite Temperature and Custom SD</a></li></ul></li><li><a href="user-guide/#Observables">Observables</a></li><li><a href="user-guide/#Time-Evolution">Time-Evolution</a></li><li class="no-marker"><ul><li><a href="user-guide/#One-site-TDVP">One-site TDVP</a></li><li><a href="user-guide/#Two-Site-TDVP">Two-Site TDVP</a></li><li><a href="user-guide/#Adaptive-One-Site-TDVP">Adaptive One-Site TDVP</a></li></ul></li><li><a href="user-guide/#Data-Storage">Data Storage</a></li></ul></li><li><a href="examples/sbm/#The-Spin-Boson-Model">The Spin-Boson Model</a></li><li class="no-marker"><ul><li><a href="examples/sbm/#Context">Context</a></li><li><a href="examples/sbm/#The-code">The code</a></li></ul></li><li><a href="examples/puredephasing/#Pure-Dephasing">Pure-Dephasing</a></li><li class="no-marker"><ul><li><a href="examples/puredephasing/#Context">Context</a></li><li><a href="examples/puredephasing/#The-code">The code</a></li><li><a href="examples/puredephasing/#Bibliography">Bibliography</a></li></ul></li><li><a href="examples/timedep/#Time-dependent-Hamiltonian">Time-dependent Hamiltonian</a></li><li class="no-marker"><ul><li><a href="examples/timedep/#Context">Context</a></li><li><a href="examples/timedep/#The-code">The code</a></li></ul></li><li><a href="examples/anderson-model/#The-Anderson-Impurity-Model">The Anderson Impurity Model</a></li><li class="no-marker"><ul><li><a href="examples/anderson-model/#Basics-of-the-fermionic-chain-mapping">Basics of the fermionic chain mapping</a></li><li class="no-marker"><ul><li><a href="examples/anderson-model/#Double-chain-MPO">Double chain MPO</a></li><li><a href="examples/anderson-model/#Interleaved-chain-MPO">Interleaved chain MPO</a></li></ul></li><li><a href="examples/anderson-model/#Code-implementation">Code implementation</a></li><li><a href="examples/anderson-model/#Bibliography">Bibliography</a></li></ul></li><li><a href="examples/bath-observables/#Inspecting-the-bath-by-undoing-the-chain-mapping">Inspecting the bath by undoing the chain mapping</a></li><li class="no-marker"><ul><li><a href="examples/bath-observables/#The-code">The code</a></li><li><a href="examples/bath-observables/#Bibliography">Bibliography</a></li></ul></li><li><a href="examples/protontransfer/#Proton-Transfer-Model">Proton Transfer Model</a></li><li class="no-marker"><ul><li><a href="examples/protontransfer/#Context">Context</a></li><li><a href="examples/protontransfer/#The-code">The code</a></li><li><a href="examples/protontransfer/#Bibliography">Bibliography</a></li></ul></li><li><a href="theory/#Theoretical-Background">Theoretical Background</a></li><li class="no-marker"><ul><li><a href="theory/#Chain-Mapping-of-bosonic-environments">Chain-Mapping of bosonic environments</a></li><li class="no-marker"><ul><li><a href="theory/#Zero-Temperature">Zero Temperature</a></li><li><a href="theory/#Finite-Temperature-with-T-TEDOPA">Finite Temperature with T-TEDOPA</a></li><li><a href="theory/#Finite-temperature-with-the-thermofield-transformation">Finite temperature with the thermofield transformation</a></li></ul></li><li><a href="theory/#Computation-of-the-chain-coefficients">Computation of the chain coefficients</a></li><li><a href="theory/#Tensor-Networks">Tensor Networks</a></li><li class="no-marker"><ul><li><a href="theory/#MPS">MPS</a></li><li><a href="theory/#MPO">MPO</a></li><li><a href="theory/#TTN">TTN</a></li></ul></li><li><a href="theory/#Time-Dependent-Variational-Principle">Time-Dependent Variational Principle</a></li><li><a href="theory/#Bibliography">Bibliography</a></li></ul></li><li><a href="methods/#List-of-all-methods">List of all methods</a></li><li class="no-marker"><ul><li><a href="methods/#Matrix-Product-State-(MPS)">Matrix Product State (MPS)</a></li><li><a href="methods/#Tree-Tensor-Network-(TTN)">Tree Tensor Network (TTN)</a></li><li><a href="methods/#Measure-and-Obervables">Measure and Obervables</a></li><li><a href="methods/#Models-and-Hamiltonians-(MPO)">Models and Hamiltonians (MPO)</a></li><li><a href="methods/#Chain-Mapping">Chain-Mapping</a></li><li><a href="methods/#Dynamics-propagation-function">Dynamics propagation function</a></li><li><a href="methods/#Advanced">Advanced</a></li></ul></li><li><a href="dev/#Developers">Developers</a></li><li class="no-marker"><ul><li><a href="dev/#Simulation-Workflow">Simulation Workflow</a></li><li><a href="dev/#How-to-Contribute">How to Contribute</a></li></ul></li></ul><h2 id="Citation"><a class="docs-heading-anchor" href="#Citation">Citation</a><a id="Citation-1"></a><a class="docs-heading-anchor-permalink" href="#Citation" title="Permalink"></a></h2><p>If you use the package in your research, please consider citing it. You can add the Zenodo record and the software paper to your BibTex file:</p><pre><code class="language-tex hljs">@misc{mpsdynamics_zenodo2021,
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Introduction · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href><img src="assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Introduction</a><ul class="internal"><li><a class="tocitem" href="#Installation"><span>Installation</span></a></li><li><a class="tocitem" href="#Table-of-Contents"><span>Table of Contents</span></a></li><li><a class="tocitem" href="#Citation"><span>Citation</span></a></li></ul></li><li><a class="tocitem" href="user-guide/">User Guide</a></li><li><a class="tocitem" href="nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="theory/">Theoretical Background</a></li><li><a class="tocitem" href="methods/">Methods</a></li><li><a class="tocitem" href="dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/index.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p>The <code>MPSDynamics.jl</code> package provides an easy to use interface for performing tensor network simulations on matrix product states (MPS) and tree tensor network (TTN) states. Written in the Julia programming language, <code>MPSDynamics.jl</code> is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle (TDVP) method for time evolution.  The package also provides strong support for the measurement of observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics.  The package has been developed with the aim of studying non-Markovian open system dynamics at finite temperature using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm (T-TEDOPA) based on environment chain mapping. However the methods implemented can equally be applied to other areas of physics.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The documentation is currently undergoing massive restructurations/improvement. It&#39;s a work in progress until the next release scheduled for May, 2024.</p></div></div><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>The package may be installed by typing the following into a Julia REPL</p><pre><code class="language-julia">    ] add https://github.com/shareloqs/MPSDynamics.git</code></pre><h2 id="Table-of-Contents"><a class="docs-heading-anchor" href="#Table-of-Contents">Table of Contents</a><a id="Table-of-Contents-1"></a><a class="docs-heading-anchor-permalink" href="#Table-of-Contents" title="Permalink"></a></h2><ul><li><a href="#Introduction">Introduction</a></li><ul><li><a href="#Installation">Installation</a></li><li><a href="#Table-of-Contents">Table of Contents</a></li><li><a href="#Citation">Citation</a></li></ul><li><a href="user-guide/#User-Guide">User Guide</a></li><ul><li><a href="user-guide/#Initial-State">Initial State</a></li><ul><li><a href="user-guide/#Matrix-Product-States">Matrix Product States</a></li><li><a href="user-guide/#Tree-Tensor-Networks">Tree Tensor Networks</a></li></ul><li><a href="user-guide/#Hamiltonian">Hamiltonian</a></li><ul><li><a href="user-guide/#Build-in-Hamiltonian">Build-in Hamiltonian</a></li><li><a href="user-guide/#Convert-a-MPO-from-ITensor">Convert a MPO from ITensor</a></li><li><a href="user-guide/#Tailored-MPO">Tailored MPO</a></li><li><a href="user-guide/#Finite-Temperature-and-Custom-SD">Finite Temperature and Custom SD</a></li></ul><li><a href="user-guide/#Observables">Observables</a></li><li><a href="user-guide/#Time-Evolution">Time-Evolution</a></li><ul><li><a href="user-guide/#One-site-TDVP">One-site TDVP</a></li><li><a href="user-guide/#Two-Site-TDVP">Two-Site TDVP</a></li><li><a href="user-guide/#Adaptive-One-Site-TDVP">Adaptive One-Site TDVP</a></li></ul><li><a href="user-guide/#Data-Storage">Data Storage</a></li></ul><li><a href="examples/sbm/#The-Spin-Boson-Model">The Spin-Boson Model</a></li><ul><li><a href="examples/sbm/#Context">Context</a></li><li><a href="examples/sbm/#The-code">The code</a></li></ul><li><a href="examples/puredephasing/#Pure-Dephasing">Pure-Dephasing</a></li><ul><li><a href="examples/puredephasing/#Context">Context</a></li><li><a href="examples/puredephasing/#The-code">The code</a></li><li><a href="examples/puredephasing/#Bibliography">Bibliography</a></li></ul><li><a href="examples/timedep/#Time-dependent-Hamiltonian">Time-dependent Hamiltonian</a></li><ul><li><a href="examples/timedep/#Context">Context</a></li><li><a href="examples/timedep/#The-code">The code</a></li></ul><li><a href="examples/anderson-model/#The-Anderson-Impurity-Model">The Anderson Impurity Model</a></li><ul><li><a href="examples/anderson-model/#Basics-of-the-fermionic-chain-mapping">Basics of the fermionic chain mapping</a></li><ul><li><a href="examples/anderson-model/#Double-chain-MPO">Double chain MPO</a></li><li><a href="examples/anderson-model/#Interleaved-chain-MPO">Interleaved chain MPO</a></li></ul><li><a href="examples/anderson-model/#Code-implementation">Code implementation</a></li><ul><li><a href="examples/anderson-model/#References">References</a></li></ul></ul><li><a href="examples/bath-observables/#Inspecting-the-bath-by-undoing-the-chain-mapping">Inspecting the bath by undoing the chain mapping</a></li><ul><li><a href="examples/bath-observables/#The-code">The code</a></li><li><a href="examples/bath-observables/#References">References</a></li></ul><li><a href="theory/#Theoretical-Background">Theoretical Background</a></li><ul><li><a href="theory/#Chain-Mapping-of-bosonic-environments">Chain-Mapping of bosonic environments</a></li><ul><li><a href="theory/#Zero-Temperature">Zero Temperature</a></li><li><a href="theory/#Finite-Temperature-with-T-TEDOPA">Finite Temperature with T-TEDOPA</a></li><li><a href="theory/#Finite-temperature-with-the-thermofield-transformation">Finite temperature with the thermofield transformation</a></li></ul><li><a href="theory/#Computation-of-the-chain-coefficients">Computation of the chain coefficients</a></li><li><a href="theory/#Tensor-Networks">Tensor Networks</a></li><ul><li><a href="theory/#MPS">MPS</a></li><li><a href="theory/#MPO">MPO</a></li><li><a href="theory/#TTN">TTN</a></li></ul><li><a href="theory/#Time-Dependent-Variational-Principle">Time-Dependent Variational Principle</a></li><li><a href="theory/#Bibliography">Bibliography</a></li></ul><li><a href="methods/#List-of-all-methods">List of all methods</a></li><li><a href="dev/#Developers">Developers</a></li><ul><li><a href="dev/#Simulation-Workflow">Simulation Workflow</a></li><li><a href="dev/#How-to-Contribute">How to Contribute</a></li></ul></ul><h2 id="Citation"><a class="docs-heading-anchor" href="#Citation">Citation</a><a id="Citation-1"></a><a class="docs-heading-anchor-permalink" href="#Citation" title="Permalink"></a></h2><p>If you use the package in your research, please consider citing it. You can add the Zenodo record to your BibTex file:</p><pre><code class="language-tex">@misc{mpsdynamics_zenodo2021,
 	title = {shareloqs/{MPSDynamics}},
 	shorttitle = {{MPSDynamics.jl}},
 	url = {https://zenodo.org/record/5106435},
@@ -8,20 +8,4 @@
 	author = {Dunnett, Angus and Lacroix, Thibaut and Le Dé, Brieuc and Riva, Angela},
 	year = {2021},
 	doi = {10.5281/zenodo.5106435},
-}
-
-@article{mpsdynamicsjl_2024,
-	title = {{MPSDynamics}.jl: {Tensor} network simulations for finite-temperature (non-{Markovian}) open quantum system dynamics},
-	volume = {161},
-	issn = {0021-9606},
-	shorttitle = {{MPSDynamics}.jl},
-	url = {https://doi.org/10.1063/5.0223107},
-	doi = {10.1063/5.0223107},
-	number = {8},
-	journal = {The Journal of Chemical Physics},
-	author = {Lacroix, Thibaut and Le Dé, Brieuc and Riva, Angela and Dunnett, Angus J. and Chin, Alex W.},
-	month = aug,
-	year = {2024},
-	pages = {084116},
-}
-</code></pre></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="user-guide/">User Guide »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+}</code></pre></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="user-guide/">User Guide »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/docs/methods/index.html b/docs/methods/index.html
index faec2df..5cbdbd4 100644
--- a/docs/methods/index.html
+++ b/docs/methods/index.html
@@ -1,5 +1,7 @@
 <!DOCTYPE html>
-<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Methods · MPSDynamics.jl</title><meta name="title" content="Methods · MPSDynamics.jl"/><meta property="og:title" content="Methods · MPSDynamics.jl"/><meta property="twitter:title" content="Methods · MPSDynamics.jl"/><meta name="description" content="Documentation for MPSDynamics.jl."/><meta property="og:description" content="Documentation for MPSDynamics.jl."/><meta property="twitter:description" content="Documentation for MPSDynamics.jl."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" 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id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Introduction</a></li><li><a class="tocitem" href="../user-guide/">User Guide</a></li><li><a class="tocitem" href="../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li><li><a class="tocitem" href="../examples/protontransfer/">Proton Transfer Model</a></li></ul></li><li><a class="tocitem" href="../theory/">Theoretical Background</a></li><li class="is-active"><a class="tocitem" href>Methods</a><ul class="internal"><li><a class="tocitem" href="#Matrix-Product-State-(MPS)"><span>Matrix Product State (MPS)</span></a></li><li><a class="tocitem" href="#Tree-Tensor-Network-(TTN)"><span>Tree Tensor Network (TTN)</span></a></li><li><a class="tocitem" href="#Measure-and-Obervables"><span>Measure and Obervables</span></a></li><li><a class="tocitem" href="#Models-and-Hamiltonians-(MPO)"><span>Models and Hamiltonians (MPO)</span></a></li><li><a class="tocitem" href="#Chain-Mapping"><span>Chain-Mapping</span></a></li><li><a class="tocitem" href="#Dynamics-propagation-function"><span>Dynamics propagation function</span></a></li><li><a class="tocitem" href="#Advanced"><span>Advanced</span></a></li></ul></li><li><a class="tocitem" href="../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Methods</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Methods</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="List-of-all-methods"><a class="docs-heading-anchor" href="#List-of-all-methods">List of all methods</a><a id="List-of-all-methods-1"></a><a class="docs-heading-anchor-permalink" href="#List-of-all-methods" title="Permalink"></a></h1><h2 id="Matrix-Product-State-(MPS)"><a class="docs-heading-anchor" href="#Matrix-Product-State-(MPS)">Matrix Product State (MPS)</a><a id="Matrix-Product-State-(MPS)-1"></a><a class="docs-heading-anchor-permalink" href="#Matrix-Product-State-(MPS)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Int64}" href="#MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Int64}"><code>MPSDynamics.apply1siteoperator!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">apply1siteoperator!(A, O, sites::Int)</code></pre><p>Apply an operator O on the MPS A. O is acting on only one site ::Int. The resulting MPS A is the MPS modified by the operator O.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Vector{Int64}}" href="#MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Vector{Int64}}"><code>MPSDynamics.apply1siteoperator!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">apply1siteoperator!(A, O, sites::Vector{Int})</code></pre><p>Apply an operator O on the MPS A. O is acting on several sites ::Vector{Int}. The resulting MPS A is the MPS modified by the operator O.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.applympo!-Tuple{Any, Any}" href="#MPSDynamics.applympo!-Tuple{Any, Any}"><code>MPSDynamics.applympo!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">applympo!(A, H)</code></pre><p>Apply an MPO H on the MPS A. H must have the same number of site than A. The resulting MPS A is the MPS modified by the MPO H.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.chainmps-Tuple{Int64, Int64, Int64}" href="#MPSDynamics.chainmps-Tuple{Int64, Int64, Int64}"><code>MPSDynamics.chainmps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">chainmps(N::Int, site::Int, numex::Int)</code></pre><p>Generate an MPS with <code>numex</code> excitations on <code>site</code></p><p>The returned MPS will have bond-dimensions and physical dimensions <code>numex+1</code></p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.chainmps-Tuple{Int64, Vector{Int64}, Int64}" href="#MPSDynamics.chainmps-Tuple{Int64, Vector{Int64}, Int64}"><code>MPSDynamics.chainmps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">chainmps(N::Int, sites::Vector{Int}, numex::Int)</code></pre><p>Generate an MPS with <code>numex</code> excitations of an equal super-position over <code>sites</code></p><p>The returned MPS will have bond-dimensions and physical dimensions <code>numex+1</code></p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.displacedchainmps-Tuple{Vector{Any}, Int64, Int64, Any}" href="#MPSDynamics.displacedchainmps-Tuple{Vector{Any}, Int64, Int64, Any}"><code>MPSDynamics.displacedchainmps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">displacedchainmps(A::Vector{Any}, N::Int, Nm::Int, γ::Any)</code></pre><p>Given a MPS A, return a MPS B where the <code>Nm</code>-long chain is displaced by <code>γ</code> without displacing the <code>N</code>-long system.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.electron2kmps" href="#MPSDynamics.electron2kmps"><code>MPSDynamics.electron2kmps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">electronkmps(N::Int, k::Vector{Int}, spin=:Up, chainparams=[fill(1.0,N), fill(1.0,N-1)])</code></pre><p>Generate an MPS with 2 electrons in k-states <code>k1</code> and <code>k2</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.electronkmps" href="#MPSDynamics.electronkmps"><code>MPSDynamics.electronkmps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">electronkmps(N::Int, k::Int, spin=:Up, chainparams=[fill(1.0,N), fill(1.0,N-1)])</code></pre><p>Generate an MPS for an electron with momentum <code>k</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.elementmpo-Tuple{Any, Vararg{Any}}" href="#MPSDynamics.elementmpo-Tuple{Any, Vararg{Any}}"><code>MPSDynamics.elementmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">elementmpo(M, el...)</code></pre><p>Return the element of the MPO <code>M</code> for the set of physical states <code>el...</code></p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.elementmps-Tuple{Any, Vararg{Any}}" href="#MPSDynamics.elementmps-Tuple{Any, Vararg{Any}}"><code>MPSDynamics.elementmps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">elementmps(A, el...)</code></pre><p>Return the element of the MPS <code>A</code> for the set of physical states <code>el...</code></p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; A = chainmps(6, [2,4], 1);
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Methods · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img src="../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Introduction</a></li><li><a class="tocitem" href="../user-guide/">User Guide</a></li><li><a class="tocitem" href="../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="../theory/">Theoretical Background</a></li><li class="is-active"><a class="tocitem" href>Methods</a></li><li><a class="tocitem" href="../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Methods</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Methods</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/methods.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="List-of-all-methods"><a class="docs-heading-anchor" href="#List-of-all-methods">List of all methods</a><a id="List-of-all-methods-1"></a><a class="docs-heading-anchor-permalink" href="#List-of-all-methods" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.OneSiteObservable-Tuple{Any, Any, Any}" href="#MPSDynamics.OneSiteObservable-Tuple{Any, Any, Any}"><code>MPSDynamics.OneSiteObservable</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">OneSiteObservable(name,op,sites)</code></pre><p>Computes the local expectation value of the one-site operator <code>op</code> on the specified sites. Used to define one-site observables that are obs and convobs parameters for the <code>runsim</code> function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/measure.jl#L11-L17">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.MPOtoVector-Tuple{ITensors.MPO}" href="#MPSDynamics.MPOtoVector-Tuple{ITensors.MPO}"><code>MPSDynamics.MPOtoVector</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">MPOtoVector(mpo::MPO)</code></pre><p>Convert an ITensors chain MPO into a form compatible with MPSDynamics</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/fundamentals.jl#L492-L497">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.addchild!-Tuple{MPSDynamics.Tree, Int64}" href="#MPSDynamics.addchild!-Tuple{MPSDynamics.Tree, Int64}"><code>MPSDynamics.addchild!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">addchild!(tree::Tree, id::Int)</code></pre><p>Add child to node <code>id</code> of <code>tree</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeBasics.jl#L40-L45">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.addchildren!-Tuple{MPSDynamics.Tree, Int64, Int64}" href="#MPSDynamics.addchildren!-Tuple{MPSDynamics.Tree, Int64, Int64}"><code>MPSDynamics.addchildren!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">addchildren!(tree::Tree, id::Int, n::Int)</code></pre><p>Add <code>n</code> children to node <code>id</code> of <code>tree</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeBasics.jl#L54-L59">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Int64}" href="#MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Int64}"><code>MPSDynamics.apply1siteoperator!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">apply1siteoperator!(A, O, sites::Int)</code></pre><p>Apply an operator O on the MPS A. O is acting on only one site ::Int. The resulting MPS A is the MPS modified by the operator O.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L534-L539">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Vector{Int64}}" href="#MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Vector{Int64}}"><code>MPSDynamics.apply1siteoperator!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">apply1siteoperator!(A, O, sites::Vector{Int})</code></pre><p>Apply an operator O on the MPS A. O is acting on several sites ::Vector{Int}. The resulting MPS A is the MPS modified by the operator O.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L521-L526">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.applympo!-Tuple{Any, Any}" href="#MPSDynamics.applympo!-Tuple{Any, Any}"><code>MPSDynamics.applympo!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">applympo!(A, H)</code></pre><p>Apply an MPO H on the MPS A. H must have the same number of site than A. The resulting MPS A is the MPS modified by the mpo H.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L542-L547">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.chaincoeffs_finiteT" href="#MPSDynamics.chaincoeffs_finiteT"><code>MPSDynamics.chaincoeffs_finiteT</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">chaincoeffs_finiteT(nummodes, β, ohmic=true; α, s, J, ωc=1, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=true)</code></pre><p>Generate chain coefficients <span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span> for a harmonic bath at the inverse temperature β.</p><p>By default a Ohmic spectral density <span>$J(ω) = 2αω_c (\frac{ω}{ω_c})^s θ(ω-ω_c)$</span> is considered. Users can provide their own spectral density.</p><p><strong>Arguments</strong></p><ul><li>nummodes: Number of bath modes</li><li>β: inverse temperature</li><li>ohmic: true if the spectral density is Ohmic, false if the user provides its own spectral density</li><li>α: Kondo parameter of the Ohmic spectral density </li><li>s: ohmicity</li><li>J: user-provided spectral density. Should be a function f(x,i) where x is the frequency and i ∈ {1,...,mc} labels the intervals on which the SD is defined</li><li>ωc: the maximum frequency of the Ohmic spectral density</li><li>mc: the number of component intervals</li><li>mp: the number of points in the discrete part of the measure (mp=0 if there is none)</li><li>iq: a parameter to be set equal to 1, if the user provides his or her own quadrature routine, and different from 1 otherwise</li><li>idelta: a parameter whose default value is 1, but is preferably set equal to 2, if iq=1 and the user provides Gauss-type quadrature routines</li><li>procedure: choice between the Stieltjes and the Lanczos procedure</li><li>AB: component intervals</li><li>Mmax: maximum number of integration points</li><li>save: if true the coefficients are saved</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/finitetemperature.jl#L6-L30">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.chaincoeffs_ohmic-Tuple{Any, Any, Any}" href="#MPSDynamics.chaincoeffs_ohmic-Tuple{Any, Any, Any}"><code>MPSDynamics.chaincoeffs_ohmic</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">chaincoeffs_ohmic(N, α, s; ωc=1, soft=false)</code></pre><p>Generate chain coefficients <span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span> for an Harmonic bath at zero temperature with a power law spectral density given by:</p><p>soft cutoff: <span>$J(ω) = 2αω_c (\frac{ω}{ω_c})^s \exp(-ω/ω_c)$</span> </p><p>hard cutoff: <span>$J(ω) = 2αω_c (\frac{ω}{ω_c})^s θ(ω-ω_c)$</span></p><p>The coefficients parameterise the chain Hamiltonian</p><p><span>$H = H_S + c_0 A_S⊗B_0+\sum_{i=0}^{N-1}t_i (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N} ϵ_ib_i^\dagger b_i$</span></p><p>which is unitarily equivalent (before the truncation to <code>N</code> sites) to</p><p><span>$H = H_S + A_S⊗\int_0^∞dω\sqrt{J(ω)}B_ω + \int_0^∞dωωb_ω^\dagger b_ω$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L483-L505">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.chainmps-Tuple{Int64, Int64, Int64}" href="#MPSDynamics.chainmps-Tuple{Int64, Int64, Int64}"><code>MPSDynamics.chainmps</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">chainmps(N::Int, site::Int, numex::Int)</code></pre><p>Generate an MPS with <code>numex</code> excitations on <code>site</code></p><p>The returned MPS will have bond-dimensions and physical dimensions <code>numex+1</code></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L221-L228">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.chainmps-Tuple{Int64, Vector{Int64}, Int64}" href="#MPSDynamics.chainmps-Tuple{Int64, Vector{Int64}, Int64}"><code>MPSDynamics.chainmps</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">chainmps(N::Int, sites::Vector{Int}, numex::Int)</code></pre><p>Generate an MPS with <code>numex</code> excitations of an equal super-position over <code>sites</code></p><p>The returned MPS will have bond-dimensions and physical dimensions <code>numex+1</code></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L248-L255">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.chainprop-Tuple{Any, Any}" href="#MPSDynamics.chainprop-Tuple{Any, Any}"><code>MPSDynamics.chainprop</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">chainprop(t, cparams...)</code></pre><p>Propagate an excitation placed initially on the first site of a tight-binding chain with parameters given by cparams for a time t and return occupation expectation for each site.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/fundamentals.jl#L328-L333">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.correlatedenvironmentmpo-Tuple{Vector, Int64, Int64}" href="#MPSDynamics.correlatedenvironmentmpo-Tuple{Vector, Int64, Int64}"><code>MPSDynamics.correlatedenvironmentmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">correlatedenvironmentmpo(R::Vector, Nm::Int, d::Int; chainparams, fnamecc::String, s=1, α=1, ωc=1, c_phonon=1, β=&quot;inf&quot;, issoft=false)</code></pre><p>Generate a MPO for a one-dimensional bosonic bath spatially correlated to a multi-component system </p><p><span>$H_B + H_int = \int_{-∞}^{+∞} dk ω_k b_k^\dagger b_k + ∑_j \int_{-∞}^{+∞}dk \sqrt{J(k)}(A_j b_k e^{i k R_j} + h.c.)$</span>.</p><p>The interactions between the system and the chain-mapped bath are long range, i.e. each site interacts with all the chain modes. The spectral density is assumed to be Ohmic <span>$J(ω) = 2αωc(ω/ωc)^s$</span>.</p><p><strong>Arguments</strong></p><ul><li><code>R</code>: List of system&#39;s components positions</li><li><code>Nm</code>: Number of chain modes. The actual number of mode will be doubled to account for the left and right moving excitations.</li><li><code>d</code>: Local Hilbert space dimension of the bath modes</li><li><code>chainparams</code>: chain parameters, of the form <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature.</li><li><code>fnamecc</code>: Path to a file containing pre-computed long-range coupling coefficient. If not provided, the coupling coefficients will be computed and stored.</li><li><code>s</code>: Ohmicity</li><li><code>α</code>: Kondo parameter</li><li><code>ωc</code>: Bath cut-off frequency</li><li><code>c_phonon</code>: Speed of sound in the bath</li><li><code>β</code>: Inverse temperature </li><li><code>issoft</code>: Is the cut-off of the Ohmic SD soft or hard?</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L993-L1017">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.displacedchainmps-Tuple{Vector{Any}, Int64, Int64, Any}" href="#MPSDynamics.displacedchainmps-Tuple{Vector{Any}, Int64, Int64, Any}"><code>MPSDynamics.displacedchainmps</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">displacedchainmps(A::Vector{Any}, N::Int, Nm::Int, γ::Any)
+
+Given a mps A, return a mps B where the `Nm`-long chain is displaced by `γ` without displacing the `N`-long system.</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L729-L733">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.dynamap-NTuple{4, Any}" href="#MPSDynamics.dynamap-NTuple{4, Any}"><code>MPSDynamics.dynamap</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">dynamap(ps1,ps2,ps3,ps4)</code></pre><p>Calulate complete dynamical map to time step at which ps1, ps2, ps3 and ps4 are specified.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/fundamentals.jl#L392-L397">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.electron2kmps" href="#MPSDynamics.electron2kmps"><code>MPSDynamics.electron2kmps</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">electronkmps(N::Int, k::Vector{Int}, spin=:Up, chainparams=[fill(1.0,N), fill(1.0,N-1)])</code></pre><p>Generate an MPS with 2 electrons in k-states <code>k1</code> and <code>k2</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L382-L387">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.electronkmps" href="#MPSDynamics.electronkmps"><code>MPSDynamics.electronkmps</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">electronkmps(N::Int, k::Int, spin=:Up, chainparams=[fill(1.0,N), fill(1.0,N-1)])</code></pre><p>Generate an MPS for an electron with momentum <code>k</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L346-L351">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.elementmpo-Tuple{Any, Vararg{Any}}" href="#MPSDynamics.elementmpo-Tuple{Any, Vararg{Any}}"><code>MPSDynamics.elementmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">elementmpo(M, el...)</code></pre><p>Return the element of the MPO <code>M</code> for the set of physical states <code>el...</code></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L506-L510">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.elementmps-Tuple{Any, Vararg{Any}}" href="#MPSDynamics.elementmps-Tuple{Any, Vararg{Any}}"><code>MPSDynamics.elementmps</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">elementmps(A, el...)</code></pre><p>Return the element of the MPS <code>A</code> for the set of physical states <code>el...</code></p><p><strong>Examples</strong></p><pre><code class="language-julia-repl">julia&gt; A = chainmps(6, [2,4], 1);
 
 julia&gt; elementmps(A, 1, 2, 1, 1, 1, 1)
 0.7071067811865475
@@ -11,28 +13,48 @@
 0.0
 
 julia&gt; elementmps(A, 1, 1, 1, 1, 1, 1)
-0.0</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.entanglemententropy-Tuple{Any}" href="#MPSDynamics.entanglemententropy-Tuple{Any}"><code>MPSDynamics.entanglemententropy</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">entanglemententropy(A)</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the entanglement entropy for a bipartite cut for every bond.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.modemps" href="#MPSDynamics.modemps"><code>MPSDynamics.modemps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">modemps(N::Int, k::Vector{Int}, numex::Int, chainparams=[fill(1.0,N), fill(1.0,N-1)])</code></pre><p>Generate an MPS with <code>numex</code> excitations of an equal superposition of modes <code>k</code> of a bosonic tight-binding chain.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.modemps" href="#MPSDynamics.modemps"><code>MPSDynamics.modemps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">modemps(N::Int, k::Int, numex::Int, chainparams=[fill(1.0,N), fill(1.0,N-1)])</code></pre><p>Generate an MPS with <code>numex</code> excitations of mode <code>k</code> of a bosonic tight-binding chain. </p><p><code>chainparams</code> takes the form <code>[e::Vector, t::Vector]</code> where <code>e</code> are the on-site energies and <code>t</code> are the hoppping parameters.</p><p>The returned MPS will have bond-dimensions and physical dimensions <code>numex+1</code></p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.mpsembed!-Tuple{Vector, Int64}" href="#MPSDynamics.mpsembed!-Tuple{Vector, Int64}"><code>MPSDynamics.mpsembed!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">mpsembed!(A::Vector, Dmax::Int)</code></pre><p>Embed MPS <code>A</code> in manifold of max bond-dimension <code>Dmax</code></p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.mpsleftnorm!" href="#MPSDynamics.mpsleftnorm!"><code>MPSDynamics.mpsleftnorm!</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">mpsleftnorm!(A::Vector, jq::Int=length(A))</code></pre><p>Left orthoganalise MPS <code>A</code> up to site <code>jq</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.mpsmixednorm!-Tuple{Vector, Int64}" href="#MPSDynamics.mpsmixednorm!-Tuple{Vector, Int64}"><code>MPSDynamics.mpsmixednorm!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">mpsmixednorm!(A::Vector, OC::Int)</code></pre><p>Put MPS <code>A</code> into mixed canonical form with orthogonality centre on site <code>OC</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.mpsrightnorm!" href="#MPSDynamics.mpsrightnorm!"><code>MPSDynamics.mpsrightnorm!</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">mpsrightnorm!(A::Vector, jq::Int=1)</code></pre><p>Right orthoganalise MPS <code>A</code> up to site <code>jq</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.multiply-Tuple{Vector, Vector}" href="#MPSDynamics.multiply-Tuple{Vector, Vector}"><code>MPSDynamics.multiply</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">multiply(M1::Vector, M2::Vector)</code></pre><p>Calculates M1*M2 where M1 and M2 are MPOs</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.normmps-Tuple{Vector}" href="#MPSDynamics.normmps-Tuple{Vector}"><code>MPSDynamics.normmps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">normmps(A::Vector; mpsorthog=:None)</code></pre><p>Calculate norm of MPS <code>A</code>.</p><p>Setting <code>mpsorthog</code>=<code>:Right</code>/<code>:Left</code> will calculate the norm assuming right/left canonical form. Setting <code>mpsorthog=OC::Int</code> will cause the norm to be calculated assuming the orthoganility center is on site <code>OC</code>. If mpsorthog is <code>:None</code> the norm will be calculated as an MPS-MPS product.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.orthcentersmps-Tuple{Vector}" href="#MPSDynamics.orthcentersmps-Tuple{Vector}"><code>MPSDynamics.orthcentersmps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">orthcentersmps(A::Vector)</code></pre><p>Compute the orthoganality centres of MPS <code>A</code>.</p><p>Return value is a list in which each element is the corresponding site tensor of <code>A</code> with the orthogonality centre on that site. Assumes <code>A</code> is right normalised.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.physdims-Tuple{Vector}" href="#MPSDynamics.physdims-Tuple{Vector}"><code>MPSDynamics.physdims</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">physdims(M)</code></pre><p>Return the physical dimensions of an MPS or MPO <code>M</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.productstatemps" href="#MPSDynamics.productstatemps"><code>MPSDynamics.productstatemps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">productstatemps(physdims::Dims, Dmax=1; state=:Vacuum, mpsorthog=:Right)</code></pre><p>Return an MPS representing a product state with local Hilbert space dimensions given by <code>physdims</code>.</p><p>By default all bond-dimensions will be 1 since the state is a product state. However, to embed the product state in a manifold of greater bond-dimension, <code>Dmax</code> can be set accordingly.</p><p>The indvidual states of the MPS sites can be provided by setting <code>state</code> to a list of column vectors. Setting <code>state=:Vacuum</code> will produce an MPS in the vacuum state (where the state of each site is represented by a column vector with a 1 in the first row and zeros elsewhere). Setting <code>state=:FullOccupy</code> will produce an MPS in which each site is fully occupied (ie. a column vector with a 1 in the last row and zeros elsewhere).</p><p>The argument <code>mpsorthog</code> can be used to set the gauge of the resulting MPS.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; ψ = unitcol(1,2); d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1
+0.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L473-L495">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.entanglemententropy-Tuple{Any}" href="#MPSDynamics.entanglemententropy-Tuple{Any}"><code>MPSDynamics.entanglemententropy</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">entanglemententropy(A)</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the entanglement entropy for a bipartite cut for every bond.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L422-L428">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.findchainlength-Tuple{Any, Any}" href="#MPSDynamics.findchainlength-Tuple{Any, Any}"><code>MPSDynamics.findchainlength</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">findchainlength(T, cparams...; eps=10^-6)</code></pre><p>Estimate length of chain required for a particular set of chain parameters by calculating how long an excitation on the first site takes to reach the end. The chain length is given as the length required for the excitation to have just reached the last site after time T. The initial number of sites in cparams has to be larger than the findchainlength result.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/fundamentals.jl#L298-L305">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.findchild-Tuple{MPSDynamics.TreeNode, Int64}" href="#MPSDynamics.findchild-Tuple{MPSDynamics.TreeNode, Int64}"><code>MPSDynamics.findchild</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">findchild(node::TreeNode, id::Int)</code></pre><p>Return integer corresponding to the which number child site <code>id</code> is of <code>node</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeBasics.jl#L222-L227">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.gauss-Tuple{Any, Any}" href="#MPSDynamics.gauss-Tuple{Any, Any}"><code>MPSDynamics.gauss</code></a> — <span class="docstring-category">Method</span></header><section><div><p>GAUSS Gauss quadrature rule.</p><pre><code class="language-none">Given a weight function w encoded by the nx2 array ab of the
+first n recurrence coefficients for the associated orthogonal
+polynomials, the first column of ab containing the n alpha-
+coefficients and the second column the n beta-coefficients,
+the call xw = GAUSS(n,ab) generates the nodes and weights xw of
+the n-point Gauss quadrature rule for the weight function w.
+The nodes, in increasing order, are stored in the first
+column, the n corresponding weights in the second column, of
+the nx2 array xw.</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/ChainOhmT/gauss.jl#L3-L15">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.hbathchain-Tuple{Int64, Int64, Any, Vararg{Any}}" href="#MPSDynamics.hbathchain-Tuple{Int64, Int64, Any, Vararg{Any}}"><code>MPSDynamics.hbathchain</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">hbathchain(N::Int, d::Int, chainparams, longrangecc...; tree=false, reverse=false, coupletox=false)</code></pre><p>Generate MPO representing a tight-binding chain of <code>N</code> oscillators with <code>d</code> Fock states each. Chain parameters are supplied in the standard form: <code>chainparams</code> <span>$=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>. The output does not itself represent a complete MPO but will possess an end which is <em>open</em> and should be attached to another tensor site, usually representing the <em>system</em>.</p><p><strong>Arguments</strong></p><ul><li><code>reverse</code>: If <code>reverse=true</code> create a chain were the last (i.e. Nth) site is the site which couples to the system</li><li><code>coupletox</code>: Used to choose the form of the system coupling. <code>coupletox=true</code> gives a non-number conserving coupling of the form <span>$H_{\text{I}}= A_{\text{S}}(b_{0}^\dagger + b_0)$</span> where <span>$A_{\text{S}}$</span> is a system operator, while <code>coupletox=false</code> gives the number-converving coupling <span>$H_{\text{I}}=(A_{\text{S}} b_{0}^\dagger + A_{\text{S}}^\dagger b_0)$</span></li><li><code>tree</code>: If <code>true</code> the resulting chain will be of type <code>TreeNetwork</code>; useful for construcing tree-MPOs </li></ul><p><strong>Example</strong></p><p>One can constuct a system site tensor to couple to a chain by using the function <code>up</code> to populate the tensor. For example, to construct a system site with Hamiltonian <code>Hs</code> and coupling operator <code>As</code>, the system tensor <code>M</code> is constructed as follows for a non-number conserving interaction:</p><pre><code class="language-julia">u = one(Hs) # system identity
+M = zeros(1,3,2,2)
+M[1, :, :, :] = up(Hs, As, u)</code></pre><p>The full MPO can then be constructed with:</p><pre><code class="language-julia">Hmpo = [M, hbathchain(N, d, chainparams, coupletox=true)...]</code></pre><p>Similarly for a number conserving interaction the site tensor would look like:</p><pre><code class="language-julia">u = one(Hs) # system identity
+M = zeros(1,4,2,2)
+M[1, :, :, :] = up(Hs, As, As&#39;, u)</code></pre><p>And the full MPO would be</p><pre><code class="language-julia">Hmpo = [M, hbathchain(N, d, chainparams; coupletox=false)...]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L180-L216">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.heisenbergmpo" href="#MPSDynamics.heisenbergmpo"><code>MPSDynamics.heisenbergmpo</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">heisenbergmpo(N::Int, J=1.0) = xyzmpo(N; Jx=J)</code></pre><p>Generate MPO for the <code>N</code>-spin Heisenberg XXX model, defined by the Hamiltonian</p><p><span>$H = \sum_{n=1}^{N-1} -J σ_x^{n} σ_x^{n+1} - J σ_y^{n} σ_y^{n+1} - J σ_z^{n} σ_z^{n+1}$</span></p><p>with <span>$σ_x^{n}, σ_y^{n}, σ_z^{n}$</span> the Pauli spin-1/2 matrices of the <span>$n^\text{th}$</span> site.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L55-L67">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.ibmmpo-NTuple{4, Any}" href="#MPSDynamics.ibmmpo-NTuple{4, Any}"><code>MPSDynamics.ibmmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">ibmmpo(ω0, d, N, chainparams; tree=false)</code></pre><p>Generate MPO for a spin-1/2 coupled to a chain of harmonic oscillators with the interacting boson model (IBM), defined by the Hamiltonian</p><p><span>$H = \frac{ω_0}{2}σ_z +  c_0σ_z(b_0^\dagger+b_0) + \sum_{i=0}^{N-1} t_i (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N} ϵ_ib_i^\dagger b_i$</span>.</p><p>The spin is on site 1 of the MPS and the bath modes are to the right.</p><p>This Hamiltonain is unitarily equivalent (before the truncation to <code>N</code> sites) to the spin-boson Hamiltonian defined by</p><p><span>$H =  \frac{ω_0}{2}σ_z + σ_z\int_0^∞ dω\sqrt{\frac{J(ω)}{π}}(b_ω^\dagger+b_ω) + \int_0^∞ dω ωb_ω^\dagger b_ω$</span>.</p><p>The chain parameters, supplied by <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L571-L590">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.isingmpo-Tuple{Int64}" href="#MPSDynamics.isingmpo-Tuple{Int64}"><code>MPSDynamics.isingmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">isingmpo(N; J=1.0, h=1.0)</code></pre><p>Generate MPO for the <code>N</code>-spin 1D Ising model with external field <span>$\vec{h} = (0,0,h)$</span>, defined by the Hamiltonian</p><p><span>$H = \sum_{n=1}^{N-1} -J_x σ_x^{n} σ_x^{n+1} + \sum_{n=1}^{N}(- h_z σ_z^{n})$</span></p><p>with <span>$σ_x^{n}, σ_y^{n}, σ_z^{n}$</span> the Pauli spin-1/2 matrices of the <span>$n^\text{th}$</span> site.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L42-L53">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.lanczos-Tuple{Any, Any}" href="#MPSDynamics.lanczos-Tuple{Any, Any}"><code>MPSDynamics.lanczos</code></a> — <span class="docstring-category">Method</span></header><section><div><p>LANCZOS Lanczos algorithm.</p><pre><code class="language-none">Given the discrete inner product whose nodes are contained
+in the first column, and whose weights are contained in the
+second column, of the nx2 array xw, the call ab=LANCZOS(n,xw)
+generates the first n recurrence coefficients ab of the
+corresponding discrete orthogonal polynomials. The n alpha-
+coefficients are stored in the first column, the n beta-
+coefficients in the second column, of the nx2 array ab.
+
+The script is adapted from the routine RKPW in
+W.B. Gragg and W.J. Harrod, ``The numerically stable
+reconstruction of Jacobi matrices from spectral data&#39;&#39;,
+Numer. Math. 44 (1984), 317-335.</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/ChainOhmT/lanczos.jl#L1-L16">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.longrange_isingmpo" href="#MPSDynamics.longrange_isingmpo"><code>MPSDynamics.longrange_isingmpo</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">longrange_isingmpo(N::Int, α::Float64=0.; J=1.0, h=1.0) = longrange_xyzmpo(N, α; Jx=J, Jy=0., Jz=0., hz=h, hx=0.)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L121-L124">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.longrange_xyzmpo" href="#MPSDynamics.longrange_xyzmpo"><code>MPSDynamics.longrange_xyzmpo</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">longrange_xyzmpo(N::Int, α::Float64=0.; Jx=1.0, Jy=Jx, Jz=Jx, hx=0., hz=0.)</code></pre><p>Gennerate MPO for the <code>N</code>-spin long-range XYZ model with external field <span>$\vec{h}=(h_x, 0, h_z)$</span>, , defined by the Hamiltonian</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L83-L89">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.measure-Tuple{Any, OneSiteObservable}" href="#MPSDynamics.measure-Tuple{Any, OneSiteObservable}"><code>MPSDynamics.measure</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">measure(A, O; kwargs...)</code></pre><p>Measure observable <code>O</code> on mps state <code>A</code></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/measure.jl#L136-L141">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Vector{Int64}}" href="#MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Vector{Int64}}"><code>MPSDynamics.measure1siteoperator</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">measure1siteoperator(A::Vector, O, sites::Vector{Int})</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for every site or just one if it is specified.</p><p>For calculating operators on single sites this will be more efficient if the site is on the left of the mps.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/measure.jl#L191-L199">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.measure1siteoperator-Tuple{Vector, Any}" href="#MPSDynamics.measure1siteoperator-Tuple{Vector, Any}"><code>MPSDynamics.measure1siteoperator</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">measure1siteoperator(A::Vector, O)</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for every site.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/measure.jl#L252-L257">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.measure2siteoperator-Tuple{Vector, Any, Any, Int64, Int64}" href="#MPSDynamics.measure2siteoperator-Tuple{Vector, Any, Any, Int64, Int64}"><code>MPSDynamics.measure2siteoperator</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia"> measure2siteoperator(A::Vector, M1, M2, j1, j2)</code></pre><p>Caculate expectation of M1*M2 where M1 acts on site j1 and M2 acts on site j2, assumes A is right normalised.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/measure.jl#L291-L296">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.modemps" href="#MPSDynamics.modemps"><code>MPSDynamics.modemps</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">modemps(N::Int, k::Vector{Int}, numex::Int, chainparams=[fill(1.0,N), fill(1.0,N-1)])</code></pre><p>Generate an MPS with <code>numex</code> excitations of an equal superposition of modes <code>k</code> of a bosonic tight-binding chain.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L313-L318">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.modemps" href="#MPSDynamics.modemps"><code>MPSDynamics.modemps</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">modemps(N::Int, k::Int, numex::Int, chainparams=[fill(1.0,N), fill(1.0,N-1)])</code></pre><p>Generate an MPS with <code>numex</code> excitations of mode <code>k</code> of a bosonic tight-binding chain. </p><p><code>chainparams</code> takes the form <code>[e::Vector, t::Vector]</code> where <code>e</code> are the on-site energies and <code>t</code> are the hoppping parameters.</p><p>The returned MPS will have bond-dimensions and physical dimensions <code>numex+1</code></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L276-L286">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.mpsembed!-Tuple{Vector, Int64}" href="#MPSDynamics.mpsembed!-Tuple{Vector, Int64}"><code>MPSDynamics.mpsembed!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">mpsembed(A::Vector, Dmax::Int)</code></pre><p>Embed MPS <code>A</code> in manifold of max bond-dimension <code>Dmax</code></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L699-L704">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.mpsleftnorm!" href="#MPSDynamics.mpsleftnorm!"><code>MPSDynamics.mpsleftnorm!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">mpsleftnorm!(A::Vector, jq::Int=length(A))</code></pre><p>Left orthoganalise MPS <code>A</code> up to site <code>jq</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L69-L74">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.mpsmixednorm!-Tuple{MPSDynamics.TreeNetwork, Int64}" href="#MPSDynamics.mpsmixednorm!-Tuple{MPSDynamics.TreeNetwork, Int64}"><code>MPSDynamics.mpsmixednorm!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">mpsmixednorm!(A::TreeNetwork, id::Int)</code></pre><p>Normalise tree-MPS <code>A</code> such that orthogonality centre is on site <code>id</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeTDVP.jl#L67-L72">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.mpsmixednorm!-Tuple{Vector, Int64}" href="#MPSDynamics.mpsmixednorm!-Tuple{Vector, Int64}"><code>MPSDynamics.mpsmixednorm!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">mpsmixednorm!(A::Vector, OC::Int)</code></pre><p>Put MPS <code>A</code> into mixed canonical form with orthogonality centre on site <code>OC</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L84-L89">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.mpsmoveoc!-Tuple{MPSDynamics.TreeNetwork, Int64}" href="#MPSDynamics.mpsmoveoc!-Tuple{MPSDynamics.TreeNetwork, Int64}"><code>MPSDynamics.mpsmoveoc!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">mpsmoveoc!(A::TreeNetwork, id::Int)</code></pre><p>Move the orthogonality centre of right normalised tree-MPS <code>A</code> to site <code>id</code>.</p><p>This function will be more efficient than using <code>mpsmixednorm!</code> if the tree-MPS is already right-normalised.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeTDVP.jl#L79-L86">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.mpsrightnorm!" href="#MPSDynamics.mpsrightnorm!"><code>MPSDynamics.mpsrightnorm!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">mpsrightnorm!(A::Vector, jq::Int=1)</code></pre><p>Right orthoganalise MPS <code>A</code> up to site <code>jq</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L52-L57">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.mpsrightnorm!-Tuple{MPSDynamics.TreeNetwork}" href="#MPSDynamics.mpsrightnorm!-Tuple{MPSDynamics.TreeNetwork}"><code>MPSDynamics.mpsrightnorm!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">mpsrightnorm!(A::TreeNetwork)</code></pre><p>When applied to a tree-MPS, right normalise towards head-node.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeTDVP.jl#L59-L64">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.mpsshiftoc!-Tuple{MPSDynamics.TreeNetwork, Int64}" href="#MPSDynamics.mpsshiftoc!-Tuple{MPSDynamics.TreeNetwork, Int64}"><code>MPSDynamics.mpsshiftoc!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">mpsshiftoc!(A::TreeNetwork, newhd::Int)</code></pre><p>Shift the orthogonality centre by one site, setting new head-node <code>newhd</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeTDVP.jl#L93-L98">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.multiply-Tuple{Vector, Vector}" href="#MPSDynamics.multiply-Tuple{Vector, Vector}"><code>MPSDynamics.multiply</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">multiply(M1::Vector, M2::Vector)</code></pre><p>Calculates M1*M2 where M1 and M2 are MPOs</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L680-L685">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.nearestneighbourmpo" href="#MPSDynamics.nearestneighbourmpo"><code>MPSDynamics.nearestneighbourmpo</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">nearestneighbourmpo(N::Int, h0, A, Ad = A&#39;)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L645-L652">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.nearestneighbourmpo" href="#MPSDynamics.nearestneighbourmpo"><code>MPSDynamics.nearestneighbourmpo</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">nearestneighbourmpo(tree_::Tree, h0, A, Ad = A&#39;)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L669-L676">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.normmps-Tuple{MPSDynamics.TreeNetwork}" href="#MPSDynamics.normmps-Tuple{MPSDynamics.TreeNetwork}"><code>MPSDynamics.normmps</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">normmps(net::TreeNetwork; mpsorthog=:None)</code></pre><p>When applied to a tree-MPS <code>mpsorthog=:Left</code> is not defined.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeTDVP.jl#L184-L189">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.normmps-Tuple{Vector}" href="#MPSDynamics.normmps-Tuple{Vector}"><code>MPSDynamics.normmps</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">normmps(A::Vector; mpsorthog=:None)</code></pre><p>Calculate norm of MPS <code>A</code>.</p><p>Setting <code>mpsorthog</code>=<code>:Right</code>/<code>:Left</code> will calculate the norm assuming right/left canonical form. Setting <code>mpsorthog=OC::Int</code> will cause the norm to be calculated assuming the orthoganility center is on site <code>OC</code>. If mpsorthog is <code>:None</code> the norm will be calculated as an MPS-MPS product.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L21-L30">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.orthcentersmps-Tuple{Vector}" href="#MPSDynamics.orthcentersmps-Tuple{Vector}"><code>MPSDynamics.orthcentersmps</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">orthcentersmps(A)</code></pre><p>Compute the orthoganality centres of MPS <code>A</code>.</p><p>Return value is a list in which each element is the corresponding site tensor of <code>A</code> with the orthoganility centre on that site. Assumes <code>A</code> is right normalised.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L1-L9">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.physdims-Tuple{Vector}" href="#MPSDynamics.physdims-Tuple{Vector}"><code>MPSDynamics.physdims</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">physdims(M)</code></pre><p>Return the physical dimensions of an MPS or MPO <code>M</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L621-L625">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.productstatemps" href="#MPSDynamics.productstatemps"><code>MPSDynamics.productstatemps</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">productstatemps(physdims::Dims, Dmax=1; state=:Vacuum, mpsorthog=:Right)</code></pre><p>Return an MPS representing a product state with local Hilbert space dimensions given by <code>physdims</code>.</p><p>By default all bond-dimensions will be 1 since the state is a product state. However, to embed the product state in a manifold of greater bond-dimension, <code>Dmax</code> can be set accordingly.</p><p>The indvidual states of the MPS sites can be provided by setting <code>state</code> to a list of column vectors. Setting <code>state=:Vacuum</code> will produce an MPS in the vacuum state (where the state of each site is represented by a column vector with a 1 in the first row and zeros elsewhere). Setting <code>state=:FullOccupy</code> will produce an MPS in which each site is fully occupied (ie. a column vector with a 1 in the last row and zeros elsewhere).</p><p>The argument <code>mpsorthog</code> can be used to set the gauge of the resulting MPS.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl">julia&gt; ψ = unitcol(1,2); d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1
 
 julia&gt; cpars = chaincoeffs_ohmic(N, α, s)
 
 julia&gt; H = spinbosonmpo(ω0, Δ, d, N, cpars)
 
-julia&gt; A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.productstatemps" href="#MPSDynamics.productstatemps"><code>MPSDynamics.productstatemps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">productstatemps(N::Int, d::Int, Dmax=1; state=:Vacuum, mpsorthog=:Right)</code></pre><p>Return an <code>N</code>-site MPS with all local Hilbert space dimensions given by <code>d</code>. </p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.randmps" href="#MPSDynamics.randmps"><code>MPSDynamics.randmps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">randmps(N::Int, d::Int, Dmax::Int, T=Float64)</code></pre><p>Construct a random, <code>N</code>-site, right-normalised MPS with all local Hilbert space dimensions given by <code>d</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.randmps-Union{Tuple{N}, Tuple{Tuple{Vararg{Int64, N}}, Int64}, Tuple{Tuple{Vararg{Int64, N}}, Int64, Type{&lt;:Number}}} where N" href="#MPSDynamics.randmps-Union{Tuple{N}, Tuple{Tuple{Vararg{Int64, N}}, Int64}, Tuple{Tuple{Vararg{Int64, N}}, Int64, Type{&lt;:Number}}} where N"><code>MPSDynamics.randmps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">randmps(physdims::Dims{N}, Dmax::Int, T::Type{&lt;:Number} = Float64) where {N}</code></pre><p>Construct a random, right-normalised MPS with local Hilbert space dimensions given by <code>physdims</code> and max bond-dimension given by <code>Dmax</code>. </p><p><code>T</code> specifies the element type, eg. use <code>T=ComplexF64</code> for a complex valued MPS.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.reversempo!-Tuple{Any}" href="#MPSDynamics.reversempo!-Tuple{Any}"><code>MPSDynamics.reversempo!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">reversempo!(M)</code></pre><p>Reverse the left and right dimensions of the MPO M. The resulting MPO M is the reversed MPO.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.reversempo-Tuple{Any}" href="#MPSDynamics.reversempo-Tuple{Any}"><code>MPSDynamics.reversempo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">reversempo(M)</code></pre><p>Reverse the left and right dimensions of the MPO M.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.reversemps!-Tuple{Any}" href="#MPSDynamics.reversemps!-Tuple{Any}"><code>MPSDynamics.reversemps!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">reversemps!(A)</code></pre><p>Reverse the left and right dimensions of the MPS A. The resulting MPS A is the reversed MPS.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.reversemps-Tuple{Any}" href="#MPSDynamics.reversemps-Tuple{Any}"><code>MPSDynamics.reversemps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">reversemps(A)</code></pre><p>Reverse the left and right bond-dimensions of the MPS A.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.svdmps-Tuple{Any}" href="#MPSDynamics.svdmps-Tuple{Any}"><code>MPSDynamics.svdmps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">svdmps(A)</code></pre><p>For a right normalised mps <code>A</code> compute the full svd spectrum for a bipartition at every bond.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.reshape-Tuple{Number, Vararg{Any}}" href="#Base.reshape-Tuple{Number, Vararg{Any}}"><code>Base.reshape</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">Base.reshape(x::Number, dims...)</code></pre><p>Reshape any matrix with dimensions &quot;dims&quot;</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; A = rand(2,3,2)
+julia&gt; A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L134-L161">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.productstatemps" href="#MPSDynamics.productstatemps"><code>MPSDynamics.productstatemps</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">productstatemps(N::Int, d::Int, Dmax=1; state=:Vacuum, mpsorthog=:Right)</code></pre><p>Return an <code>N</code>-site MPS with all local Hilbert space dimensions given by <code>d</code>. </p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L212-L217">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.protontransfermpo-NTuple{11, Any}" href="#MPSDynamics.protontransfermpo-NTuple{11, Any}"><code>MPSDynamics.protontransfermpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">protontransfermpo(ω0e,ω0k,x0e,x0k, Δ, dRC, d, N, chainparams, RCparams, λreorg)</code></pre><p>Generate a MPO for a system described in space with a reaction coordinate (RC) tensor. The RC tensor is coupled to a bosonic bath, taking into account the induced reorganization energy. </p><p><span>$H_S + H_RC + H_int^{S-RC} = \omega^0_{e} |e\rangle \langle e| + \omega^0_{k} |k\rangle \langle k| + \Delta (|e\rangle \langle k| + |k\rangle \langle e|) + \omega_{RC} (d^{\dagger}d + \frac{1}{2}) + g_{e} |e\rangle \langle e|( d + d^{\dagger})+ g_{k} |k \rangle \langle k|( d + d^{\dagger})$</span> <span>$H_B + H_int^{RC-B} = \int_{-∞}^{+∞} dk ω_k b_k^\dagger b_k - (d + d^{\dagger})\int_0^∞ dω\sqrt{J(ω)}(b_ω^\dagger+b_ω) + \lambda_{reorg}(d + d^{\dagger})^2$</span>. <span>$\lambda_{reorg} = \int \frac{J(\omega)}{\omega}d\omega$</span>.</p><p><strong>Arguments</strong></p><ul><li><code>ω0e</code>: enol energy at x=0 </li><li><code>ω0k</code>: keto energy at x=0</li><li><code>x0e</code>: enol equilibrium displacement</li><li><code>x0k</code>: keto equilibrium displacement </li><li><code>Δ</code>: direct coupling between enol and keto</li><li><code>dRC</code>: fock space of the RC tensor </li><li><code>d</code>: number of Fock states of the chain modes</li><li><code>N</code>: length of the chain</li><li><code>chainparams</code>: chain parameters, of the form <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature. </li><li><code>RCparams</code>: RC tensor parameter, of the form <code>RCparams</code>=<span>$[ωRC,-g/x]$</span> </li><li><code>λreorg</code>: reorganization energy</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L1338-L1367">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.puredephasingmpo-NTuple{4, Any}" href="#MPSDynamics.puredephasingmpo-NTuple{4, Any}"><code>MPSDynamics.puredephasingmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">puredephasingmpo(ΔE, dchain, Nchain, chainparams; tree=false)
 
-julia&gt; B = reshape(A,4,3)
-
-julia&gt; size(B) == (4,3) 
-</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.svdtrunc-Tuple{Any}" href="#MPSDynamics.svdtrunc-Tuple{Any}"><code>MPSDynamics.svdtrunc</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">U, S, Vd = svdtrunc(A; truncdim = max(size(A)...), truncerr = 0.)</code></pre><p>Perform a truncated SVD, with maximum number of singular values to keep equal to <code>truncdim</code> or truncating any singular values smaller than <code>truncerr</code>. If both options are provided, the smallest number of singular values will be kept. Unlike the SVD in Julia, this returns matrix U, a diagonal matrix (not a vector) S, and Vt such that A ≈ U * S * Vt</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.MPOtoVector-Tuple{ITensors.ITensorMPS.MPO}" href="#MPSDynamics.MPOtoVector-Tuple{ITensors.ITensorMPS.MPO}"><code>MPSDynamics.MPOtoVector</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">MPOtoVector(mpo::MPO)</code></pre><p>Convert an ITensors chain MPO into a form compatible with MPSDynamics</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.chainprop-Tuple{Any, Any}" href="#MPSDynamics.chainprop-Tuple{Any, Any}"><code>MPSDynamics.chainprop</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">chainprop(t, cparams)</code></pre><p>Propagate an excitation placed initially on the first site of a tight-binding chain with parameters given by cparams for a time t and return occupation expectation for each site.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.cosineh-Tuple{Any, Any}" href="#MPSDynamics.cosineh-Tuple{Any, Any}"><code>MPSDynamics.cosineh</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">cosineh(omega, bet)</code></pre><p>Calculates the hyperbolic cosine function function based on the input parameters,  for the Bogoliubov transformation necessary for the thermofield transformation.</p><p><strong>Arguments</strong></p><ul><li><code>omega::Float64</code>: The frequency parameter.</li><li><code>bet::Float64</code>: The beta parameter.</li></ul><p><strong>Returns</strong></p><ul><li><code>Float64</code>: The result of the modified cosine function.</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.disp-Tuple{Any, Any, Any}" href="#MPSDynamics.disp-Tuple{Any, Any, Any}"><code>MPSDynamics.disp</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">disp(d,ωvib,m)</code></pre><p>Displacement operator  <span>$X = \frac{\sqrt{2}}{2\sqrt{m \omega_{vib}}}(a + a^{\dagger})$</span></p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.disp-Tuple{Any}" href="#MPSDynamics.disp-Tuple{Any}"><code>MPSDynamics.disp</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">disp(d)</code></pre><p>Mass and frequency-weighted displacement operator  <span>$X = \frac{1}{2}(a + a^{\dagger})$</span></p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.dynamap-NTuple{4, Any}" href="#MPSDynamics.dynamap-NTuple{4, Any}"><code>MPSDynamics.dynamap</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">dynamap(ps1,ps2,ps3,ps4)</code></pre><p>Calculate complete dynamical map to time step at which ps1, ps2, ps3 and ps4 are specified.</p><p><strong>Arguments</strong></p><ul><li><code>ps1</code> : time evolved system density matrix starting from initial state up</li><li><code>ps2</code> : time evolved system density matrix starting from initial state down</li><li><code>ps3</code> : time evolved system density matrix starting from initial state (up + down)/sqrt(2)</li><li><code>ps4</code> : time evolved system density matrix starting from initial state (up - i*down)/sqrt(2)</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.eigenchain-Tuple{Any}" href="#MPSDynamics.eigenchain-Tuple{Any}"><code>MPSDynamics.eigenchain</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">eigenchain(cparams; nummodes=nothing)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.endsiteocc-Tuple{Any, Any}" href="#MPSDynamics.endsiteocc-Tuple{Any, Any}"><code>MPSDynamics.endsiteocc</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">endsiteocc(t, cparams)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.entropy-Tuple{Any}" href="#MPSDynamics.entropy-Tuple{Any}"><code>MPSDynamics.entropy</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">entropy(rho)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.findchainlength" href="#MPSDynamics.findchainlength"><code>MPSDynamics.findchainlength</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">findchainlength(T, ωc::Float64, β=Inf)</code></pre><p>Estimate length of chain using universal asymptotic properties of chain mapped environments given the simulation time T, the bath cut-off frequency ωc, and the inverse temperature β.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.findchainlength-Tuple{Any, Vector}" href="#MPSDynamics.findchainlength-Tuple{Any, Vector}"><code>MPSDynamics.findchainlength</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">findchainlength(T, cparams::Vector; eps=10^-6, verbose=false)</code></pre><p>Estimate length of chain required for a particular set of chain parameters by calculating how long an excitation on the first site takes to reach the end. The chain length is given as the length required for the excitation to have just reached the last site after time T. The initial number of sites in cparams has to be larger than the findchainlength result.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measurecorrs-Tuple{Any, Vector, Vector}" href="#MPSDynamics.measurecorrs-Tuple{Any, Vector, Vector}"><code>MPSDynamics.measurecorrs</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measurecorrs(oper, , e::Vector, t::Vector)</code></pre><p><strong>Parameters</strong></p><p><code>oper</code>: Square matrix (Matrix{Float64}) representing the operator to be transformed. <code>e</code>: Vector (Vector{Float64}) of diagonal (on-site energy) chain coefficients. <code>t</code>: Vector (Vector{Float64}) of off-diagonal (hopping terms) chain coefficients.</p><p><strong>Returns</strong></p><p>Matrix{Float64}: This matrix is the operator <code>oper</code> transformed back from the chain  representation to the representation corresponding to the extended bath. The resulting  operator represents quantities like mode occupations or other properties in the basis  of environmental modes associated with specific frequencies <span>$\omega_i$</span>.</p><p><strong>Description</strong></p><p>This function performs a basis transformation of the operator <code>oper</code>. Specifically,  this transformation reverses the unitary transformation that maps the extended bath Hamiltonian into the chain representation. </p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measuremodes" href="#MPSDynamics.measuremodes"><code>MPSDynamics.measuremodes</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measuremodes(adaga, e=1.0, t=1.0)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measuremodes-Tuple{Any, Tuple{Int64, Int64}, AbstractArray}" href="#MPSDynamics.measuremodes-Tuple{Any, Tuple{Int64, Int64}, AbstractArray}"><code>MPSDynamics.measuremodes</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measuremodes(A, chainsection::Tuple{Int64,Int64}, U::AbstractArray)</code></pre><p>for longer chains it can be worth calculating U in advance</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measuremodes-Tuple{Any, Tuple{Int64, Int64}, Vector{Float64}, Vector{Float64}}" href="#MPSDynamics.measuremodes-Tuple{Any, Tuple{Int64, Int64}, Vector{Float64}, Vector{Float64}}"><code>MPSDynamics.measuremodes</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measuremodes(A, chainsection::Tuple{Int64,Int64}, e::Array{Float64,1}, t::Array{Float64,1})</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measuremodes-Tuple{Any, Vector, Vector}" href="#MPSDynamics.measuremodes-Tuple{Any, Vector, Vector}"><code>MPSDynamics.measuremodes</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measuremodes(adaga, e::Vector, t::Vector)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.physical_occup-NTuple{6, Any}" href="#MPSDynamics.physical_occup-NTuple{6, Any}"><code>MPSDynamics.physical_occup</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">physical_occup(corr_constr, corr_destr, omega, occup, b, M)</code></pre><p>Calculates the physical occupation based on correlation matrices, omega values,  and other parameters. The physical occupation in the original frequency environment is computed by reverting the thermofield transformation.</p><p><strong>Arguments</strong></p><ul><li><code>corr_constr::Matrix{ComplexF64}</code>: The correlation construction matrix.</li><li><code>corr_destr::Matrix{ComplexF64}</code>: The correlation destruction matrix.</li><li><code>omega::Vector{Float64}</code>: The omega values.</li><li><code>occup::Matrix{Float64}</code>: The occupation matrix.</li><li><code>b::Float64</code>: The beta parameter.</li><li><code>M::Int</code>: The number of points for interpolation.</li></ul><p><strong>Returns</strong></p><ul><li><code>Vector{Float64}</code>: The physical occupation values.</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.randisometry-Tuple{Type, Int64, Int64}" href="#MPSDynamics.randisometry-Tuple{Type, Int64, Int64}"><code>MPSDynamics.randisometry</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">randisometry([T=Float64], dims...)</code></pre><p>Construct a random isometry</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.rmsd-Tuple{Any, Any}" href="#MPSDynamics.rmsd-Tuple{Any, Any}"><code>MPSDynamics.rmsd</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">rmsd(ob1, ob2)</code></pre><p>Calculate the root mean squared difference between two measurements of an observable over the same time period.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.sineh-Tuple{Any, Any}" href="#MPSDynamics.sineh-Tuple{Any, Any}"><code>MPSDynamics.sineh</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">sineh(omega, bet)</code></pre><p>Calculates the hyperbolic sine function function based on the input parameters,  for the Bogoliubov transformation necessary for the thermofield transformation.</p><p><strong>Arguments</strong></p><ul><li><code>omega::Float64</code>: The frequency parameter.</li><li><code>bet::Float64</code>: The beta parameter.</li></ul><p><strong>Returns</strong></p><ul><li><code>Float64</code>: The result of the modified cosine function.</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.therHam-Tuple{Any, Any, Any}" href="#MPSDynamics.therHam-Tuple{Any, Any, Any}"><code>MPSDynamics.therHam</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs"> therHam(psi, site1, site2)</code></pre><p>Calculates Hβ such that ρ = e^(-βH) for some density matrix ρ obatined from tracing out everything outside the range [site1,site2] in the MPS psi</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.thermaloccupations-Tuple{Any, Vararg{Any}}" href="#MPSDynamics.thermaloccupations-Tuple{Any, Vararg{Any}}"><code>MPSDynamics.thermaloccupations</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">thermaloccupations(β, cparams...)</code></pre></div></section></article><h2 id="Tree-Tensor-Network-(TTN)"><a class="docs-heading-anchor" href="#Tree-Tensor-Network-(TTN)">Tree Tensor Network (TTN)</a><a id="Tree-Tensor-Network-(TTN)-1"></a><a class="docs-heading-anchor-permalink" href="#Tree-Tensor-Network-(TTN)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.addchild!-Tuple{MPSDynamics.Tree, Int64}" href="#MPSDynamics.addchild!-Tuple{MPSDynamics.Tree, Int64}"><code>MPSDynamics.addchild!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">addchild!(tree::Tree, id::Int)</code></pre><p>Add child to node <code>id</code> of <code>tree</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.addchildren!-Tuple{MPSDynamics.Tree, Int64, Int64}" href="#MPSDynamics.addchildren!-Tuple{MPSDynamics.Tree, Int64, Int64}"><code>MPSDynamics.addchildren!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">addchildren!(tree::Tree, id::Int, n::Int)</code></pre><p>Add <code>n</code> children to node <code>id</code> of <code>tree</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.findchild-Tuple{MPSDynamics.TreeNode, Int64}" href="#MPSDynamics.findchild-Tuple{MPSDynamics.TreeNode, Int64}"><code>MPSDynamics.findchild</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">findchild(node::TreeNode, id::Int)</code></pre><p>Return integer corresponding to the which number child site <code>id</code> is of <code>node</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.randtree-Tuple{Int64, Int64}" href="#MPSDynamics.randtree-Tuple{Int64, Int64}"><code>MPSDynamics.randtree</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">randtree(numnodes::Int, maxdegree::Int)</code></pre><p>Construct a random tree with <code>nummodes</code> modes and max degree <code>maxdegree</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure1siteoperator-Tuple{MPSDynamics.TreeNetwork, Any, Int64}" href="#MPSDynamics.measure1siteoperator-Tuple{MPSDynamics.TreeNetwork, Any, Int64}"><code>MPSDynamics.measure1siteoperator</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measure1siteoperator(net::TreeNetwork, O, id::Int)</code></pre><p>For a Tree, compute the local expectation value of a one-site operator O for site id.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure1siteoperator-Tuple{MPSDynamics.TreeNetwork, Any, Tuple{Int64, Int64}}" href="#MPSDynamics.measure1siteoperator-Tuple{MPSDynamics.TreeNetwork, Any, Tuple{Int64, Int64}}"><code>MPSDynamics.measure1siteoperator</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measure1siteoperator(net::TreeNetwork, O, sites::Tuple{Int,Int})</code></pre><p>For a Tree, compute the local expectation value of a one-site operator O for the specified site range.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure2siteoperator-Tuple{MPSDynamics.TreeNetwork, Any, Any, Tuple{Int64, Int64}}" href="#MPSDynamics.measure2siteoperator-Tuple{MPSDynamics.TreeNetwork, Any, Any, Tuple{Int64, Int64}}"><code>MPSDynamics.measure2siteoperator</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measure2siteoperator(net::TreeNetwork, O1, O2, sites::Tuple{Int,Int})</code></pre><p>For a Tree, compute the local expectation value of two one-site operators O1 and O2 for the specified site range.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.bonddims-Tuple{MPSDynamics.TreeNetwork}" href="#MPSDynamics.bonddims-Tuple{MPSDynamics.TreeNetwork}"><code>MPSDynamics.bonddims</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">bonddims(A::TreeNetwork)</code></pre><p>Return the bon-dimension of a tree-MPS <code>A</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.mpsembed!-Tuple{MPSDynamics.TreeNetwork, Int64}" href="#MPSDynamics.mpsembed!-Tuple{MPSDynamics.TreeNetwork, Int64}"><code>MPSDynamics.mpsembed!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">mpsembed(A::TreeNetwork, Dmax::Int)</code></pre><p>Embed tree-MPS <code>A</code> in manifold of max bond-dimension <code>Dmax</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.mpsmixednorm!-Tuple{MPSDynamics.TreeNetwork, Int64}" href="#MPSDynamics.mpsmixednorm!-Tuple{MPSDynamics.TreeNetwork, Int64}"><code>MPSDynamics.mpsmixednorm!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">mpsmixednorm!(A::TreeNetwork, id::Int)</code></pre><p>Normalise tree-MPS <code>A</code> such that orthogonality centre is on site <code>id</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.mpsmoveoc!-Tuple{MPSDynamics.TreeNetwork, Int64}" href="#MPSDynamics.mpsmoveoc!-Tuple{MPSDynamics.TreeNetwork, Int64}"><code>MPSDynamics.mpsmoveoc!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">mpsmoveoc!(A::TreeNetwork, id::Int)</code></pre><p>Move the orthogonality centre of right normalised tree-MPS <code>A</code> to site <code>id</code>.</p><p>This function will be more efficient than using <code>mpsmixednorm!</code> if the tree-MPS is already right-normalised.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.mpsrightnorm!-Tuple{MPSDynamics.TreeNetwork}" href="#MPSDynamics.mpsrightnorm!-Tuple{MPSDynamics.TreeNetwork}"><code>MPSDynamics.mpsrightnorm!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">mpsrightnorm!(A::TreeNetwork)</code></pre><p>When applied to a tree-MPS, right normalise towards head-node.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.mpsshiftoc!-Tuple{MPSDynamics.TreeNetwork, Int64}" href="#MPSDynamics.mpsshiftoc!-Tuple{MPSDynamics.TreeNetwork, Int64}"><code>MPSDynamics.mpsshiftoc!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">mpsshiftoc!(A::TreeNetwork, newhd::Int)</code></pre><p>Shift the orthogonality centre by one site, setting new head-node <code>newhd</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.normmps-Tuple{MPSDynamics.TreeNetwork}" href="#MPSDynamics.normmps-Tuple{MPSDynamics.TreeNetwork}"><code>MPSDynamics.normmps</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">normmps(net::TreeNetwork; mpsorthog=:None)</code></pre><p>When applied to a tree-MPS <code>mpsorthog=:Left</code> is not defined.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.physdims-Tuple{MPSDynamics.TreeNetwork}" href="#MPSDynamics.physdims-Tuple{MPSDynamics.TreeNetwork}"><code>MPSDynamics.physdims</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">physdims(M::TreeNetwork)</code></pre><p>Return the physical dimensions of a tree-MPS or tree-MPO <code>M</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.productstatemps" href="#MPSDynamics.productstatemps"><code>MPSDynamics.productstatemps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">productstatemps(tree_::Tree, physdims::Dims, Dmax::Int=1; state=:Vacuum)</code></pre><p>Return a tree-MPS representing a product state with local Hilbert space dimensions given by <code>physdims</code>.</p><p>By default all bond-dimensions will be 1 since the state is a product state. However, to embed the product state in a manifold of greater bond-dimension, <code>Dmax</code> can be set accordingly.</p><p>The indvidual states of the MPS sites can be provided by setting <code>state</code> to a list of column vectors. Setting <code>state=:Vacuum</code> will produce an MPS in the vacuum state (where the state of each site is represented by a column vector with a 1 in the first row and zeros elsewhere). Setting <code>state=:FullOccupy</code> will produce an MPS in which each site is fully occupied (ie. a column vector with a 1 in the last row and zeros elsewhere).</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; ψ = unitcol(1,2); d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1
-
-julia&gt; cpars = chaincoeffs_ohmic(N, α, s)
+Generate MPO for a pure dephasing model, defined by the Hamiltonian
+``H = \frac{ΔE}{2} σ_z +  \frac{σ_z}{2} c_0 (b_0^\dagger + b_0) + \sum_{i=0}^{N-1} t_i (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N-1} ϵ_i b_i^\dagger b_i  ``
 
-julia&gt; H = spinbosonmpo(ω0, Δ, d, N, cpars, tree=true)
+The spin is on site 1 of the MPS and the bath modes are to the right.
 
-julia&gt; A = productstatemps(H.tree, physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # tree-MPS representation of |ψ&gt;|Vacuum&gt;</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.randmps" href="#MPSDynamics.randmps"><code>MPSDynamics.randmps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">randmps(tree::Tree, physdims, Dmax::Int, T::Type{&lt;:Number} = Float64)</code></pre><p>Construct a random, right-normalised, tree-MPS, with structure given by tree and max bond-dimension given by <code>Dmax</code>.</p><p>The local Hilbert space dimensions are specified by physdims which can either be of type <code>Dims{length(tree)}</code>, specifying the dimension of each site, or of type <code>Int</code>, in which case the same local dimension is used for every site.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.tdvp1sweep!-Tuple{Any, MPSDynamics.TreeNetwork, MPSDynamics.TreeNetwork, Vector, Int64}" href="#MPSDynamics.tdvp1sweep!-Tuple{Any, MPSDynamics.TreeNetwork, MPSDynamics.TreeNetwork, Vector, Int64}"><code>MPSDynamics.tdvp1sweep!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">tdvp1sweep!(dt, A::TreeNetwork, M::TreeNetwork, F::Vector, id::Int; verbose=false, kwargs...)</code></pre><p>Propagates the tree-MPS A with the tree-MPO M following the 1-site TDVP method. The sweep is done back and forth with a time step dt/2. F represents the merged left and right parts of the site being propagated.  </p></div></section></article><h2 id="Measure-and-Obervables"><a class="docs-heading-anchor" href="#Measure-and-Obervables">Measure and Obervables</a><a id="Measure-and-Obervables-1"></a><a class="docs-heading-anchor-permalink" href="#Measure-and-Obervables" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.OneSiteObservable-Tuple{Any, Any, Any}" href="#MPSDynamics.OneSiteObservable-Tuple{Any, Any, Any}"><code>MPSDynamics.OneSiteObservable</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">OneSiteObservable(name,op,sites)</code></pre><p>Computes the local expectation value of the one-site operator <code>op</code> on the specified sites. Used to define one-site observables that are obs and convobs parameters for the <code>runsim</code> function.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.OneSiteObservable-Tuple{Any, Any}" href="#MPSDynamics.OneSiteObservable-Tuple{Any, Any}"><code>MPSDynamics.OneSiteObservable</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">OneSiteObservable(name,op)</code></pre><p>Computes the local expectation value of the one-site operator <code>op</code> on the every site. Used to define one-site observables that are obs and convobs parameters for the <code>runsim</code> function.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.TwoSiteObservable" href="#MPSDynamics.TwoSiteObservable"><code>MPSDynamics.TwoSiteObservable</code></a> — <span class="docstring-category">Type</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">TwoSiteObservable(name,op1,op2,sites1=nothing,sites2=nothing)</code></pre><p>Computes the local expectation value of operators <code>op1</code> and <code>op2</code> where <code>op1</code> acts on sites1 and <code>op2</code> acts on sites2. Used to define several-site observables that are obs and convobs parameters for the <code>runsim</code> function.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure-Tuple{Any, OneSiteObservable}" href="#MPSDynamics.measure-Tuple{Any, OneSiteObservable}"><code>MPSDynamics.measure</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measure(A, O; kwargs...)</code></pre><p>Measure observable <code>O</code> on mps state <code>A</code></p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure-Tuple{Vector, Vector}" href="#MPSDynamics.measure-Tuple{Vector, Vector}"><code>MPSDynamics.measure</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs"> measure(A::Vector, Os::Vector; kwargs...)</code></pre><p>Caculate expectation of Os on MPS A.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Int64}" href="#MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Int64}"><code>MPSDynamics.measure1siteoperator</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measure1siteoperator(A::Vector, O, site::Int)</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for a single site.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Tuple{Int64, Int64}}" href="#MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Tuple{Int64, Int64}}"><code>MPSDynamics.measure1siteoperator</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measure1siteoperator(A::Vector, O, chainsection::Tuple{Int64,Int64})</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for a chainsection.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Vector{Int64}}" href="#MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Vector{Int64}}"><code>MPSDynamics.measure1siteoperator</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measure1siteoperator(A::Vector, O, sites::Vector{Int})</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for every site or just one if it is specified.</p><p>For calculating operators on single sites this will be more efficient if the site is on the left of the mps.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure1siteoperator-Tuple{Vector, Any}" href="#MPSDynamics.measure1siteoperator-Tuple{Vector, Any}"><code>MPSDynamics.measure1siteoperator</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measure1siteoperator(A::Vector, O)</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for every site.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure2siteoperator-Tuple{Vector, Any, Any, Int64, Int64}" href="#MPSDynamics.measure2siteoperator-Tuple{Vector, Any, Any, Int64, Int64}"><code>MPSDynamics.measure2siteoperator</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs"> measure2siteoperator(A::Vector, M1, M2, j1, j2)</code></pre><p>Caculate expectation of M1*M2 where M1 acts on site j1 and M2 acts on site j2, assumes A is right normalised.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure2siteoperator-Tuple{Vector, Any, Any, Vector{Int64}, Vector{Int64}}" href="#MPSDynamics.measure2siteoperator-Tuple{Vector, Any, Any, Vector{Int64}, Vector{Int64}}"><code>MPSDynamics.measure2siteoperator</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs"> measure2siteoperator(A::Vector, M1, M2, sites1::Vector{Int}, sites2::Vector{Int})</code></pre><p>Caculate expectation of M1*M2 where M1 acts on sites1 and M2 acts on sites2, assumes A is right normalised.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measurempo-Tuple{Vector, Vector, Tuple{Int64, Int64}}" href="#MPSDynamics.measurempo-Tuple{Vector, Vector, Tuple{Int64, Int64}}"><code>MPSDynamics.measurempo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measurempo(A::Vector, M::Vector, sites::Tuples{Int,Int})</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the local expectation value of the MPO M on specified sites.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measurempo-Tuple{Vector, Vector}" href="#MPSDynamics.measurempo-Tuple{Vector, Vector}"><code>MPSDynamics.measurempo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measurempo(A::Vector, M::Vector)</code></pre><p>For a list of tensors <code>A</code> representing a right orthonormalized MPS, compute the local expectation value of the MPO M on every site.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.rhoreduced_1site" href="#MPSDynamics.rhoreduced_1site"><code>MPSDynamics.rhoreduced_1site</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs"> rhoreduced_1site(A::Vector, site::Int=1)</code></pre><p>Caculate the reduced density matrix of the MPS A at the specified site.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.rhoreduced_2sites-Tuple{Vector, Tuple{Int64, Int64}}" href="#MPSDynamics.rhoreduced_2sites-Tuple{Vector, Tuple{Int64, Int64}}"><code>MPSDynamics.rhoreduced_2sites</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs"> rhoreduced_2sites(A::Vector, site::Tuple{Int, Int})</code></pre><p>Caculate the reduced density matrix of the MPS A of two neigbour sites. The resulting dimensions will be the four physical dimensions in total,  corresponding to the dimensions  of the two sites</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.measure-Tuple{Any, FockError}" href="#MPSDynamics.measure-Tuple{Any, FockError}"><code>MPSDynamics.measure</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">measure(A, obs::FockError; t=0, kwargs...)</code></pre><p>Return the measure of the observable obs on the MPS A. </p></div></section></article><h2 id="Models-and-Hamiltonians-(MPO)"><a class="docs-heading-anchor" href="#Models-and-Hamiltonians-(MPO)">Models and Hamiltonians (MPO)</a><a id="Models-and-Hamiltonians-(MPO)-1"></a><a class="docs-heading-anchor-permalink" href="#Models-and-Hamiltonians-(MPO)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.chaincoeffs_ohmic-Tuple{Any, Any, Any}" href="#MPSDynamics.chaincoeffs_ohmic-Tuple{Any, Any, Any}"><code>MPSDynamics.chaincoeffs_ohmic</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">chaincoeffs_ohmic(N, α, s; ωc=1, soft=false)</code></pre><p>Generate chain coefficients <span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span> for an Harmonic bath at zero temperature with a power law spectral density given by:</p><p>soft cutoff: <span>$J(ω) = 2αω_c (\frac{ω}{ω_c})^s \exp(-ω/ω_c)$</span> </p><p>hard cutoff: <span>$J(ω) = 2αω_c (\frac{ω}{ω_c})^s θ(ω-ω_c)$</span></p><p>The coefficients parameterise the chain Hamiltonian</p><p><span>$H = H_S + c_0 A_S⊗B_0+\sum_{i=0}^{N-1}t_i (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N} ϵ_ib_i^\dagger b_i$</span></p><p>which is unitarily equivalent (before the truncation to <code>N</code> sites) to</p><p><span>$H = H_S + A_S⊗\int_0^∞dω\sqrt{J(ω)}B_ω + \int_0^∞dωωb_ω^\dagger b_ω$</span>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.correlatedenvironmentmpo-Tuple{Vector, Int64, Int64}" href="#MPSDynamics.correlatedenvironmentmpo-Tuple{Vector, Int64, Int64}"><code>MPSDynamics.correlatedenvironmentmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">correlatedenvironmentmpo(R::Vector, Nm::Int, d::Int; chainparams, fnamecc::String, s=1, α=1, ωc=1, c_phonon=1, β=&quot;inf&quot;, issoft=false)</code></pre><p>Generate a MPO for a one-dimensional bosonic bath spatially correlated to a multi-component system </p><p><span>$H_B + H_{int} = \int_{-∞}^{+∞} dk ω_k b_k^\dagger b_k + ∑_j \int_{-∞}^{+∞}dk \sqrt{J(k)}(A_j b_k e^{i k R_j} + h.c.)$</span>.</p><p>The interactions between the system and the chain-mapped bath are long range, i.e. each site interacts with all the chain modes. The spectral density is assumed to be Ohmic <span>$J(ω) = 2αωc(ω/ωc)^s$</span>.</p><p><strong>Arguments</strong></p><ul><li><code>R</code>: List of system&#39;s components positions</li><li><code>Nm</code>: Number of chain modes. The actual number of mode will be doubled to account for the left and right moving excitations.</li><li><code>d</code>: Local Hilbert space dimension of the bath modes</li><li><code>chainparams</code>: chain parameters, of the form <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature.</li><li><code>fnamecc</code>: Path to a file containing pre-computed long-range coupling coefficient. If not provided, the coupling coefficients will be computed and stored.</li><li><code>s</code>: Ohmicity</li><li><code>α</code>: Kondo parameter</li><li><code>ωc</code>: Bath cut-off frequency</li><li><code>c_phonon</code>: Speed of sound in the bath</li><li><code>β</code>: Inverse temperature </li><li><code>issoft</code>: Is the cut-off of the Ohmic SD soft or hard?</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.hbathchain-Tuple{Int64, Int64, Any, Vararg{Any}}" href="#MPSDynamics.hbathchain-Tuple{Int64, Int64, Any, Vararg{Any}}"><code>MPSDynamics.hbathchain</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">hbathchain(N::Int, d::Int, chainparams, longrangecc...; tree=false, reverse=false, coupletox=false)</code></pre><p>Generate MPO representing a tight-binding chain of <code>N</code> oscillators with <code>d</code> Fock states each. Chain parameters are supplied in the standard form: <code>chainparams</code> <span>$=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>. The output does not itself represent a complete MPO but will possess an end which is <em>open</em> and should be attached to another tensor site, usually representing the <em>system</em>.</p><p><strong>Arguments</strong></p><ul><li><code>reverse</code>: If <code>reverse=true</code> create a chain were the last (i.e. Nth) site is the site which couples to the system</li><li><code>coupletox</code>: Used to choose the form of the system coupling. <code>coupletox=true</code> gives a non-number conserving coupling of the form <span>$H_{\text{I}}= A_{\text{S}}(b_{0}^\dagger + b_0)$</span> where <span>$A_{\text{S}}$</span> is a system operator, while <code>coupletox=false</code> gives the number-converving coupling <span>$H_{\text{I}}=(A_{\text{S}} b_{0}^\dagger + A_{\text{S}}^\dagger b_0)$</span></li><li><code>tree</code>: If <code>true</code> the resulting chain will be of type <code>TreeNetwork</code>; useful for construcing tree-MPOs </li></ul><p><strong>Example</strong></p><p>One can constuct a system site tensor to couple to a chain by using the function <code>up</code> to populate the tensor. For example, to construct a system site with Hamiltonian <code>Hs</code> and coupling operator <code>As</code>, the system tensor <code>M</code> is constructed as follows for a non-number conserving interaction:</p><pre><code class="language-julia hljs">u = one(Hs) # system identity
-M = zeros(1,3,2,2)
-M[1, :, :, :] = up(Hs, As, u)</code></pre><p>The full MPO can then be constructed with:</p><pre><code class="language-julia hljs">Hmpo = [M, hbathchain(N, d, chainparams, coupletox=true)...]</code></pre><p>Similarly for a number conserving interaction the site tensor would look like:</p><pre><code class="language-julia hljs">u = one(Hs) # system identity
-M = zeros(1,4,2,2)
-M[1, :, :, :] = up(Hs, As, As&#39;, u)</code></pre><p>And the full MPO would be</p><pre><code class="language-julia hljs">Hmpo = [M, hbathchain(N, d, chainparams; coupletox=false)...]</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.heisenbergmpo" href="#MPSDynamics.heisenbergmpo"><code>MPSDynamics.heisenbergmpo</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">heisenbergmpo(N::Int, J=1.0) = xyzmpo(N; Jx=J)</code></pre><p>Generate MPO for the <code>N</code>-spin Heisenberg XXX model, defined by the Hamiltonian</p><p><span>$H = \sum_{n=1}^{N-1} -J σ_x^{n} σ_x^{n+1} - J σ_y^{n} σ_y^{n+1} - J σ_z^{n} σ_z^{n+1}$</span></p><p>with <span>$σ_x^{n}, σ_y^{n}, σ_z^{n}$</span> the Pauli spin-1/2 matrices of the <span>$n^\text{th}$</span> site.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.ibmmpo-NTuple{4, Any}" href="#MPSDynamics.ibmmpo-NTuple{4, Any}"><code>MPSDynamics.ibmmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">ibmmpo(ω0, d, N, chainparams; tree=false)</code></pre><p>Generate MPO for a spin-1/2 coupled to a chain of harmonic oscillators with the interacting boson model (IBM), defined by the Hamiltonian</p><p><span>$H = \frac{ω_0}{2}σ_z +  c_0σ_z(b_0^\dagger+b_0) + \sum_{i=0}^{N-1} t_i (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N} ϵ_ib_i^\dagger b_i$</span>.</p><p>The spin is on site 1 of the MPS and the bath modes are to the right.</p><p>This Hamiltonain is unitarily equivalent (before the truncation to <code>N</code> sites) to the spin-boson Hamiltonian defined by</p><p><span>$H =  \frac{ω_0}{2}σ_z + σ_z\int_0^∞ dω\sqrt{J(ω)}(b_ω^\dagger+b_ω) + \int_0^∞ dω ωb_ω^\dagger b_ω$</span>.</p><p>The chain parameters, supplied by <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.interleaved_tightbinding_mpo-NTuple{4, Any}" href="#MPSDynamics.interleaved_tightbinding_mpo-NTuple{4, Any}"><code>MPSDynamics.interleaved_tightbinding_mpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">interleaved_tightbinding_mpo(N, ϵd, chainparams1, chainparams2)</code></pre><p>Generate MPO for a tight-binding chain of N fermionic sites with a single impurity site (fermionic as well) of energy ϵd. The impurity is coupled to two leads, each described by a set of chain parameters.  The interactions are next-nearest-neighbour, with the first site corresponding to the impurity, and the two chains organised in an interleaved fashion.</p><p><strong>Arguments</strong></p><ul><li><code>N::Int</code>: number of sites in the chain</li><li><code>ϵd::Real</code>: energy of the impurity site at the first site, as Ed - μ, where μ is the chemical potential</li><li>chainparams1::Array{Real,1}: chain parameters for the first lead</li><li>chainparams2::Array{Real,1}: chain parameters for the second lead</li></ul><p>The chain parameters are given in the standard form: <code>chainparams</code> <span>$=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.isingmpo-Tuple{Int64}" href="#MPSDynamics.isingmpo-Tuple{Int64}"><code>MPSDynamics.isingmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">isingmpo(N; J=1.0, h=1.0)</code></pre><p>Generate MPO for the <code>N</code>-spin 1D Ising model with external field <span>$\vec{h} = (0,0,h)$</span>, defined by the Hamiltonian</p><p><span>$H = \sum_{n=1}^{N-1} -J_x σ_x^{n} σ_x^{n+1} + \sum_{n=1}^{N}(- h_z σ_z^{n})$</span></p><p>with <span>$σ_x^{n}, σ_y^{n}, σ_z^{n}$</span> the Pauli spin-1/2 matrices of the <span>$n^\text{th}$</span> site.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.longrange_isingmpo" href="#MPSDynamics.longrange_isingmpo"><code>MPSDynamics.longrange_isingmpo</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">longrange_isingmpo(N::Int, α::Float64=0.; J=1.0, h=1.0) = longrange_xyzmpo(N, α; Jx=J, Jy=0., Jz=0., hz=h, hx=0.)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.longrange_xyzmpo" href="#MPSDynamics.longrange_xyzmpo"><code>MPSDynamics.longrange_xyzmpo</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">longrange_xyzmpo(N::Int, α::Float64=0.; Jx=1.0, Jy=Jx, Jz=Jx, hx=0., hz=0.)</code></pre><p>Gennerate MPO for the <code>N</code>-spin long-range XYZ model with external field <span>$\vec{h}=(h_x, 0, h_z)$</span>, , defined by the Hamiltonian</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.multisitempo" href="#MPSDynamics.multisitempo"><code>MPSDynamics.multisitempo</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">multisitempo(N::Int, E=[]::Vector, J=[]::Vector, As=[]::Vector)</code></pre><p>Generate a MPO for a N-site multi-component system to be coupled with a correlated environment.</p><p><strong>Arguments:</strong></p><ul><li><code>N</code>: number of sites</li><li><code>E</code>: list of on-site energies</li><li><code>J</code>: list of sites tunnelling energies</li><li><code>As</code>: list of system operators coupling to the environment</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.nearestneighbourmpo" href="#MPSDynamics.nearestneighbourmpo"><code>MPSDynamics.nearestneighbourmpo</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">nearestneighbourmpo(N::Int, h0, A, Ad = A&#39;)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.nearestneighbourmpo" href="#MPSDynamics.nearestneighbourmpo"><code>MPSDynamics.nearestneighbourmpo</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">nearestneighbourmpo(tree_::Tree, h0, A, Ad = A&#39;)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.protontransfermpo-NTuple{11, Any}" href="#MPSDynamics.protontransfermpo-NTuple{11, Any}"><code>MPSDynamics.protontransfermpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">protontransfermpo(ω0e,ω0k,x0e,x0k, Δ, dRC, d, N, chainparams, RCparams, λreorg)</code></pre><p>Generate a MPO for a system described in space with a reaction coordinate (RC) tensor. The Hamiltonian of the two-level system and of the reaction coordinate tensor reads </p><p><span>$H_S + H_{RC} + H_{int}^{S-RC} = \omega^0_{e} |e\rangle \langle e| + \omega^0_{k} |k\rangle \langle k| + \Delta (|e\rangle \langle k| + |k\rangle \langle e|) + \omega_{RC} (d^{\dagger}d + \frac{1}{2}) + g_{e} |e\rangle \langle e|( d + d^{\dagger})+ g_{k} |k \rangle \langle k|( d + d^{\dagger})$</span> The RC tensor is coupled to a bosonic bath, taking into account the induced reorganization energy <span>$H_B + H_{int}^{RC-B} = \int_{-∞}^{+∞} dk ω_k b_k^\dagger b_k - (d + d^{\dagger})\int_0^∞ dω\sqrt{J(ω)}(b_ω^\dagger+b_ω) + \lambda_{reorg}(d + d^{\dagger})^2$</span> with <span>$\lambda_{reorg} = \int \frac{J(\omega)}{\omega}d\omega.$</span></p><p><strong>Arguments</strong></p><ul><li><code>ω0e</code>: enol energy at reaction coordinate value x=0 </li><li><code>ω0k</code>: keto energy at reaction coordinate value x=0</li><li><code>x0e</code>: enol equilibrium displacement</li><li><code>x0k</code>: keto equilibrium displacement </li><li><code>Δ</code>: direct coupling between enol and keto</li><li><code>dRC</code>: fock space of the RC tensor </li><li><code>d</code>: number of Fock states of the chain modes</li><li><code>N</code>: length of the bosonic chain</li><li><code>chainparams</code>: chain parameters, of the form <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature. </li><li><code>RCparams</code>: RC tensor parameter, of the form <code>RCparams</code>=<span>$[ω_RC,-g/x]$</span> </li><li><code>λreorg</code>: reorganization energy</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.puredephasingmpo-NTuple{4, Any}" href="#MPSDynamics.puredephasingmpo-NTuple{4, Any}"><code>MPSDynamics.puredephasingmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">puredephasingmpo(ΔE, dchain, Nchain, chainparams; tree=false)</code></pre><p>Generate MPO for a pure dephasing model, defined by the Hamiltonian <span>$H = \frac{ΔE}{2} σ_z +  \frac{σ_z}{2} c_0 (b_0^\dagger + b_0) + \sum_{i=0}^{N-1} t_i (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N-1} ϵ_i b_i^\dagger b_i$</span></p><p>The spin is on site 1 of the MPS and the bath modes are to the right.</p><p><strong>Arguments</strong></p><ul><li><code>ΔE::Real</code>: energy splitting of the spin</li><li><code>dchain::Int</code>: physical dimension of the chain sites truncated Hilbert spaces</li><li><code>Nchain::Int</code>: number of sites in the chain</li><li><code>chainparams::Array{Real,1}</code>: chain parameters for the bath chain. The chain parameters are given in the standard form: <code>chainparams</code> <span>$=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>.</li><li><code>tree::Bool</code>: if true, return a <code>TreeNetwork</code> object, otherwise return a vector of MPO tensors</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.readchaincoeffs-Tuple{Any, Vararg{Any}}" href="#MPSDynamics.readchaincoeffs-Tuple{Any, Vararg{Any}}"><code>MPSDynamics.readchaincoeffs</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">readchaincoeffs(fdir, params...)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.spinbosonmpo-NTuple{5, Any}" href="#MPSDynamics.spinbosonmpo-NTuple{5, Any}"><code>MPSDynamics.spinbosonmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">spinbosonmpo(ω0, Δ, d, N, chainparams; rwa=false, tree=false)</code></pre><p>Generate MPO for a spin-1/2 coupled to a chain of harmonic oscillators, defined by the Hamiltonian</p><p><span>$H = \frac{ω_0}{2}σ_z + Δσ_x + c_0σ_x(b_0^\dagger+b_0) + \sum_{i=0}^{N-1} t_i (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N} ϵ_ib_i^\dagger b_i$</span>.</p><p>The spin is on site 1 of the MPS and the bath modes are to the right.</p><p>This Hamiltonain is unitarily equivalent (before the truncation to <code>N</code> sites) to the spin-boson Hamiltonian defined by</p><p><span>$H =  \frac{ω_0}{2}σ_z + Δσ_x + σ_x\int_0^∞ dω\sqrt{J(ω)}(b_ω^\dagger+b_ω) + \int_0^∞ dω ωb_ω^\dagger b_ω$</span>.</p><p>The chain parameters, supplied by <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature.</p><p>The rotating wave approximation can be made by setting <code>rwa=true</code>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.spinchainmpo-Tuple{Int64}" href="#MPSDynamics.spinchainmpo-Tuple{Int64}"><code>MPSDynamics.spinchainmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">spinchainmpo(N::Int; J=1.0, hz=1.0, hx=0.0, i=div(N,2))</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.tightbinding_mpo-NTuple{4, Any}" href="#MPSDynamics.tightbinding_mpo-NTuple{4, Any}"><code>MPSDynamics.tightbinding_mpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">tightbinding_mpo(N, ϵd, chainparams1, chainparams2)</code></pre><p>Generate MPO for a tight-binding chain of N fermionic sites with a single impurity site (fermionic as well)  of energy ϵd at the center. The impurity is coupled to two leads, each described by a set of chain parameters. The interactions are nearest-neighbour, with the first N/2-1 sites corresponding to the first lead, the Nth site corresponding to the impurity, and the rest of the sites corresponding to the second lead.</p><p><strong>Arguments</strong></p><ul><li><code>N::Int</code>: number of sites in the chain</li><li><code>ϵd::Real</code>: energy of the impurity site at the center, as Ed - μ, where μ is the chemical potential</li><li><code>chainparams1::Array{Real,1}</code>: chain parameters for the first lead</li><li><code>chainparams2::Array{Real,1}</code>: chain parameters for the second lead</li></ul><p>The chain parameters are given in the standard form: <code>chainparams</code> <span>$=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.tightbindingmpo-Tuple{Int64, Int64}" href="#MPSDynamics.tightbindingmpo-Tuple{Int64, Int64}"><code>MPSDynamics.tightbindingmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">tightbindingmpo(N::Int, d::Int; J=1.0, e=1.0)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.tunnelingmpo-Tuple{Any, Any, Any, Any, Any, Int64, Int64}" href="#MPSDynamics.tunnelingmpo-Tuple{Any, Any, Any, Any, Any, Int64, Int64}"><code>MPSDynamics.tunnelingmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">tunnelingmpo(ϵ, delta, α, s, β, d::Int, nummodes::Int; tree=false, ωc=1)</code></pre></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.twobathspinmpo" href="#MPSDynamics.twobathspinmpo"><code>MPSDynamics.twobathspinmpo</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">twobathspinmpo(ω0, Δ, Nl, Nr, dl, dr, chainparamsl=[fill(1.0,N),fill(1.0,N-1), 1.0], chainparamsr=chainparamsl; tree=false)</code></pre><p>Generate MPO for a spin-1/2 coupled to two chains of harmonic oscillators, defined by the Hamiltonian</p><p><span>$H = \frac{ω_0}{2}σ_z + Δσ_x + c_0^rσ_x(b_0^\dagger+b_0) + \sum_{i=0}^{N_r-1} t_i^r (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N_r} ϵ_i^rb_i^\dagger b_i + c_0^lσ_x(d_0^\dagger+d_0) + \sum_{i=0}^{N_l-1} t_i^l (d_{i+1}^\dagger d_i +h.c.) + \sum_{i=0}^{N_l} ϵ_i^l d_i^\dagger d_i$</span>.</p><p>The spin is on site <span>$N_l + 1$</span> of the MPS, surrounded by the left chain modes and the right chain modes.</p><p>This Hamiltonain is unitarily equivalent (before the truncation to <code>N</code> sites) to the spin-boson Hamiltonian defined by</p><p><span>$H =  \frac{ω_0}{2}σ_z + Δσ_x + σ_x\int_0^∞ dω\sqrt{J(ω)}(b_ω^\dagger+b_ω) + \int_0^∞ dω ωb_ω^\dagger b_ωi + σ_x\int_0^∞ dω\sqrt{J^l(ω)}(d_ω^\dagger+d_ω) + \int_0^∞ dω ωd_ω^\dagger d_ω$</span>.</p><p>The chain parameters, supplied by <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature. The two chains can have a different spectral density.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.xxzmpo" href="#MPSDynamics.xxzmpo"><code>MPSDynamics.xxzmpo</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">xxzmpo(N::Int, Δ = 1.0, J=1.0) = xyzmpo(N; Jx=J, Jy=J, Jz=J*Δ)</code></pre><p>Generate MPO for the <code>N</code>-spin XXZ model, defined by the Hamiltonian</p><p><span>$H = \sum_{n=1}^{N-1} -J σ_x^{n} σ_x^{n+1} - J σ_y^{n} σ_y^{n+1} - \Delta J σ_z^{n} σ_z^{n+1}$</span></p><p>with <span>$σ_x^{n}, σ_y^{n}, σ_z^{n}$</span> the Pauli spin-1/2 matrices of the <span>$n^\text{th}$</span> site.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.xyzmpo-Tuple{Int64}" href="#MPSDynamics.xyzmpo-Tuple{Int64}"><code>MPSDynamics.xyzmpo</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">xyzmpo(N::Int; Jx=1.0, Jy=Jx, Jz=Jx, hx=0., hz=0.)</code></pre><p>Generate MPO for the <code>N</code>-spin XYZ model with external field <span>$\vec{h}=(h_x, 0, h_z)$</span>, , defined by the Hamiltonian</p><p><span>$H = \sum_{n=1}^{N-1} -J_x σ_x^{n} σ_x^{n+1} - J_y σ_y^{n} σ_y^{n+1} - J_z σ_z^{n} σ_z^{n+1} + \sum_{n=1}^{N}(- h_x σ_x^{n} - h_z σ_z^{n})$</span></p><p>with <span>$σ_x^{n}, σ_y^{n}, σ_z^{n}$</span> the Pauli spin-1/2 matrices of the <span>$n^\text{th}$</span> site.</p></div></section></article><h2 id="Chain-Mapping"><a class="docs-heading-anchor" href="#Chain-Mapping">Chain-Mapping</a><a id="Chain-Mapping-1"></a><a class="docs-heading-anchor-permalink" href="#Chain-Mapping" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.chaincoeffs_fermionic-Tuple{Any, Any, Any}" href="#MPSDynamics.chaincoeffs_fermionic-Tuple{Any, Any, Any}"><code>MPSDynamics.chaincoeffs_fermionic</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">chaincoeffs_fermionic(nummodes, β, chain; ϵ=x, ωc=1, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=true)</code></pre><p>Generate chain coefficients <span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span> for a fermionic bath at the inverse temperature β.</p><p><strong>Arguments</strong></p><ul><li>nummodes: Number of bath modes</li><li>β: inverse temperature</li><li>chain: 1 if the chain modes are empty, 2 if the chain modes are filled</li><li>ϵ: user-provided dispersion relation. Should be a function f(x) where x is the wavenumber</li><li>J: user-provided spectral density. Should be a function f(x) where x is the wavenumber</li><li>ωc: the maximum frequency allowwed in the spectral density</li><li>mc: the number of component intervals</li><li>mp: the number of points in the discrete part of the measure (mp=0 if there is none)</li><li>iq: a parameter to be set equal to 1, if the user provides his or her own quadrature routine, and different from 1 otherwise</li><li>idelta: a parameter whose default value is 1, but is preferably set equal to 2, if iq=1 and the user provides Gauss-type quadrature routines</li><li>procedure: choice between the Stieltjes and the Lanczos procedure</li><li>AB: component intervals</li><li>Mmax: maximum number of integration points</li><li>save: if true the coefficients are saved</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.chaincoeffs_finiteT" href="#MPSDynamics.chaincoeffs_finiteT"><code>MPSDynamics.chaincoeffs_finiteT</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">chaincoeffs_finiteT(nummodes, β, ohmic=true; α, s, J, ωc=1, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=true)</code></pre><p>Generate chain coefficients <span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span> for a harmonic bath at the inverse temperature β.</p><p>By default a Ohmic spectral density <span>$J(ω) = 2αω_c (\frac{ω}{ω_c})^s θ(ω-ω_c)$</span> is considered. Users can provide their own spectral density.</p><p><strong>Arguments</strong></p><ul><li>nummodes: Number of bath modes</li><li>β: inverse temperature</li><li>ohmic: true if the spectral density is Ohmic, false if the user provides its own spectral density</li><li>α: Kondo parameter of the Ohmic spectral density </li><li>s: ohmicity</li><li>J: user-provided spectral density. Should be a function f(x,i) where x is the frequency and i ∈ {1,...,mc} labels the intervals on which the SD is defined</li><li>ωc: the maximum frequency of the Ohmic spectral density</li><li>mc: the number of component intervals</li><li>mp: the number of points in the discrete part of the measure (mp=0 if there is none)</li><li>iq: a parameter to be set equal to 1, if the user provides his or her own quadrature routine, and different from 1 otherwise</li><li>idelta: a parameter whose default value is 1, but is preferably set equal to 2, if iq=1 and the user provides Gauss-type quadrature routines</li><li>procedure: choice between the Stieltjes and the Lanczos procedure</li><li>AB: component intervals</li><li>Mmax: maximum number of integration points</li><li>save: if true the coefficients are saved</li></ul></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.gauss-Tuple{Any, Any}" href="#MPSDynamics.gauss-Tuple{Any, Any}"><code>MPSDynamics.gauss</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">gauss(N,ab)</code></pre><p>Gauss quadrature rule for <code>N</code> sites on an interval <code>ab</code>. Given a weight function w encoded by the <code>N</code>x2 array <code>ab</code> of the first <code>N</code> recurrence coefficients for the associated orthogonal polynomials, the first column of <code>ab</code> containing the <code>N</code> alpha-coefficients and the second column the <code>N</code> beta-coefficients, the call xw = gauss(N,ab) generates the nodes and weights xw of the <code>N</code>-point Gauss quadrature rule for the weight function w. The nodes, in increasing order, are stored in the first column, the n corresponding weights in the second column, of the <code>N</code>x2 array xw.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.lanczos-Tuple{Any, Any}" href="#MPSDynamics.lanczos-Tuple{Any, Any}"><code>MPSDynamics.lanczos</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">lanczos(N,xw)</code></pre><p>Given the discrete inner product whose nodes are contained in the first column, and whose weights are contained in the second column, of the <code>N</code>x2 array <code>xw</code>, the call ab=lanczos(N,xw) generates the first <code>N</code> recurrence coefficients ab of the corresponding discrete orthogonal polynomials. The <code>N</code> alpha-coefficients are stored in the first column, the <code>N</code> beta-coefficients in the second column, of the <code>N</code>x2 array ab.</p><p>The script is adapted from the routine RKPW in W.B. Gragg and W.J. Harrod, ``The numerically stable reconstruction of Jacobi matrices from spectral data&#39;&#39;, Numer. Math. 44 (1984), 317-335.</p></div></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.stieltjes-Tuple{Any, Any}" href="#MPSDynamics.stieltjes-Tuple{Any, Any}"><code>MPSDynamics.stieltjes</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">stieltjes(N,xw)</code></pre><p>Discretized Stieltjes procedure. Given the discrete inner product whose nodes are contained in the first column, and whose weights are contained in the second column, of the <code>N</code>x2 array <code>xw</code>, the call ab=stieltjes(N,xw) generates the first <code>N</code> recurrence coefficients ab of the corresponding discrete orthogonal polynomials. The <code>N</code> alpha- coefficients are stored in the first column, the <code>N</code> beta-coefficients in the second column, of the <code>N</code>x2 array ab.</p></div></section></article><h2 id="Dynamics-propagation-function"><a class="docs-heading-anchor" href="#Dynamics-propagation-function">Dynamics propagation function</a><a id="Dynamics-propagation-function-1"></a><a class="docs-heading-anchor-permalink" href="#Dynamics-propagation-function" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MPSDynamics.runsim-NTuple{4, Any}" href="#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">runsim(dt, tmax, A, H; 
+### Arguments
+* `ΔE::Real`: energy splitting of the spin
+* `dchain::Int`: physical dimension of the chain sites truncated Hilbert spaces
+* `Nchain::Int`: number of sites in the chain
+* `chainparams::Array{Real,1}`: chain parameters for the bath chain. The chain parameters are given in the standard form: `chainparams` ``=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]``.
+* `tree::Bool`: if true, return a `TreeNetwork` object, otherwise return a vector of MPO tensors</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L708-L722">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.randisometry-Tuple{Type, Int64, Int64}" href="#MPSDynamics.randisometry-Tuple{Type, Int64, Int64}"><code>MPSDynamics.randisometry</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">randisometry([T=Float64], dims...)</code></pre><p>Construct a random isometry</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/fundamentals.jl#L457-L460">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.randmps" href="#MPSDynamics.randmps"><code>MPSDynamics.randmps</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">randmps(tree::Tree, physdims, Dmax::Int, T::Type{&lt;:Number} = Float64)</code></pre><p>Construct a random, right-normalised, tree-MPS, with structure given by tree and max bond-dimension given by <code>Dmax</code>.</p><p>The local Hilbert space dimensions are specified by physdims which can either be of type <code>Dims{length(tree)}</code>, specifying the dimension of each site, or of type <code>Int</code>, in which case the same local dimension is used for every site.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeTDVP.jl#L128-L135">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.randmps" href="#MPSDynamics.randmps"><code>MPSDynamics.randmps</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">randmps(N::Int, d::Int, Dmax::Int, T=Float64)</code></pre><p>Construct a random, <code>N</code>-site, right-normalised MPS with all local Hilbert space dimensions given by <code>d</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L126-L131">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.randmps-Union{Tuple{N}, Tuple{Tuple{Vararg{Int64, N}}, Int64}, Tuple{Tuple{Vararg{Int64, N}}, Int64, Type{&lt;:Number}}} where N" href="#MPSDynamics.randmps-Union{Tuple{N}, Tuple{Tuple{Vararg{Int64, N}}, Int64}, Tuple{Tuple{Vararg{Int64, N}}, Int64, Type{&lt;:Number}}} where N"><code>MPSDynamics.randmps</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">randmps(physdims::Dims{N}, Dmax::Int, T::Type{&lt;:Number} = Float64) where {N}</code></pre><p>Construct a random, right-normalised MPS with local Hilbert space dimensions given by <code>physdims</code> and max bond-dimension given by <code>Dmax</code>. </p><p><code>T</code> specifies the element type, eg. use <code>T=ComplexF64</code> for a complex valued MPS.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L95-L103">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.randtree-Tuple{Int64, Int64}" href="#MPSDynamics.randtree-Tuple{Int64, Int64}"><code>MPSDynamics.randtree</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">randtree(numnodes::Int, maxdegree::Int)</code></pre><p>Construct a random tree with <code>nummodes</code> modes and max degree <code>maxdegree</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/treeBasics.jl#L353-L358">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.readchaincoeffs-Tuple{Any, Vararg{Any}}" href="#MPSDynamics.readchaincoeffs-Tuple{Any, Vararg{Any}}"><code>MPSDynamics.readchaincoeffs</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">readchaincoeffs(fdir, params...)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L543-L549">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.rmsd-Tuple{Any, Any}" href="#MPSDynamics.rmsd-Tuple{Any, Any}"><code>MPSDynamics.rmsd</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">rmsd(dat1::Vector{Float64}, dat2::Vector{Float64})</code></pre><p>Calculate the root mean squared difference between two measurements of an observable over the same time period.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/fundamentals.jl#L378-L383">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.runsim-NTuple{4, Any}" href="#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">runsim(dt, tmax, A, H; 
 	method=:TDVP1, 
 	machine=LocalMachine(), 
 	params=[], 
@@ -45,4 +67,25 @@
             unid=randstring(5),
             name=nothing,
 	kwargs...
-	)</code></pre><p>Propagate the MPS <code>A</code> with the MPO <code>H</code> up to time <code>tmax</code> in time steps of <code>dt</code>. The final MPS is returned to <code>A</code> and the measurement data is returned to <code>dat</code> </p><p><strong>Arguments</strong></p><ul><li><code>method</code>: Several methods are implemented in MPSDynamics. <code>:TDVP1</code> refers to 1-site TDVP on tree and chain MPS, <code>:TDVP2</code> refers to 2-site TDVP on chain MPS, <code>:DTDVP</code> refers to a variant of 1-site TDVP with dynamics bond-dimensions on chain MPS</li><li><code>machine</code>: <code>LocalMachine()</code> points local ressources, <code>RemoteMachine()</code> points distant ressources</li><li><code>params</code>: list of parameters written in the log.txt file to describe the dynamics. Can be listed with @LogParams(). </li><li><code>obs</code>: list of observables that will be measured at every time step for the most accurate convergence parameter supplied to <code>convparams</code> </li><li><code>convobs</code>: list of observables that will be measure at every time step for every convergence parameter supplied to <code>convparams</code> </li><li><code>convparams</code>: list of convergence parameter with which the propagation will be calculated for every parameter. At each parameter, <code>convobs</code> are measured while <code>obs</code> are measured only for the most accurate dynamics</li><li><code>save</code>: Used to choose whether the data will also be saved to a file  </li><li><code>plot</code>: Used to choose whether plots for 1D observables will be automatically generated and saved along with the data</li><li><code>savedir</code>: Used to specify the path where resulting files are stored</li><li><code>unid</code>: Used to specify the name of the directory containing the resulting files</li><li><code>name</code>: Used to describe the calculation. This name will appear in the log.txt file</li></ul></div></section></article><h2 id="Advanced"><a class="docs-heading-anchor" href="#Advanced">Advanced</a><a id="Advanced-1"></a><a class="docs-heading-anchor-permalink" href="#Advanced" title="Permalink"></a></h2></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../theory/">« Theoretical Background</a><a class="docs-footer-nextpage" href="../dev/">Developers »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+	)</code></pre><p>Propagate the MPS <code>A</code> with the MPO <code>H</code> up to time <code>tmax</code> in time steps of <code>dt</code>. The final MPS is returned to <code>A</code> and the measurement data is returned to <code>dat</code> </p><p><strong>Arguments</strong></p><ul><li><code>method</code>: Several methods are implemented in MPSDynamics. <code>:TDVP1</code> refers to 1-site TDVP on tree and chain MPS, <code>:TDVP2</code> refers to 2-site TDVP on chain MPS, <code>:DTDVP</code> refers to a variant of 1-site TDVP with dynamics bond-dimensions on chain MPS</li><li><code>machine</code>: <code>LocalMachine()</code> points local ressources, <code>RemoteMachine()</code> points distant ressources</li><li><code>params</code>: list of parameters written in the log.txt file to describe the dynamics. Can be listed with @LogParams(). </li><li><code>obs</code>: list of observables that will be measured at every time step for the most accurate convergence parameter supplied to <code>convparams</code> </li><li><code>convobs</code>: list of observables that will be measure at every time step for every convergence parameter supplied to <code>convparams</code> </li><li><code>convparams</code>: list of convergence parameter with which the propagation will be calculated for every parameter. At each parameter, <code>convobs</code> are measured while <code>obs</code> are measured only for the most accurate dynamics</li><li><code>save</code>: Used to choose whether the data will also be saved to a file  </li><li><code>plot</code>: Used to choose whether plots for 1D observables will be automatically generated and saved along with the data</li><li><code>savedir</code>: Used to specify the path where resulting files are stored</li><li><code>unid</code>: Used to specify the name of the directory containing the resulting files</li><li><code>name</code>: Used to describe the calculation. This name will appear in the log.txt file</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/MPSDynamics.jl#L33-L65">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.spinbosonmpo-NTuple{5, Any}" href="#MPSDynamics.spinbosonmpo-NTuple{5, Any}"><code>MPSDynamics.spinbosonmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">spinbosonmpo(ω0, Δ, d, N, chainparams; rwa=false, tree=false)</code></pre><p>Generate MPO for a spin-1/2 coupled to a chain of harmonic oscillators, defined by the Hamiltonian</p><p><span>$H = \frac{ω_0}{2}σ_z + Δσ_x + c_0σ_x(b_0^\dagger+b_0) + \sum_{i=0}^{N-1} t_i (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N} ϵ_ib_i^\dagger b_i$</span>.</p><p>The spin is on site 1 of the MPS and the bath modes are to the right.</p><p>This Hamiltonain is unitarily equivalent (before the truncation to <code>N</code> sites) to the spin-boson Hamiltonian defined by</p><p><span>$H =  \frac{ω_0}{2}σ_z + Δσ_x + σ_x\int_0^∞ dω\sqrt{J(ω)}(b_ω^\dagger+b_ω) + \int_0^∞ dω ωb_ω^\dagger b_ω$</span>.</p><p>The chain parameters, supplied by <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature.</p><p>The rotating wave approximation can be made by setting <code>rwa=true</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L398-L419">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.spinchainmpo-Tuple{Int64}" href="#MPSDynamics.spinchainmpo-Tuple{Int64}"><code>MPSDynamics.spinchainmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">spinchainmpo(N::Int; J=1.0, hz=1.0, hx=0.0, i=div(N,2))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L127-L131">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.stieltjes-Tuple{Any, Any}" href="#MPSDynamics.stieltjes-Tuple{Any, Any}"><code>MPSDynamics.stieltjes</code></a> — <span class="docstring-category">Method</span></header><section><div><p>STIELTJES Discretized Stieltjes procedure.</p><pre><code class="language-none">Given the discrete inner product whose nodes are contained
+in the first column, and whose weights are contained in the
+second column, of the nx2 array xw, the call ab=STIELTJES(n,xw)
+generates the first n recurrence coefficients ab of the
+corresponding discrete orthogonal polynomials. The n alpha-
+coefficients are stored in the first column, the n beta-
+coefficients in the second column, of the nx2 array ab.</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/ChainOhmT/stieltjes.jl#L1-L11">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.svdmps-Tuple{Any}" href="#MPSDynamics.svdmps-Tuple{Any}"><code>MPSDynamics.svdmps</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">svdmps(A)</code></pre><p>For a right normalised mps <code>A</code> compute the full svd spectrum for a bipartition at every bond.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/mpsBasics.jl#L451-L455">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.svdtrunc-Tuple{Any}" href="#MPSDynamics.svdtrunc-Tuple{Any}"><code>MPSDynamics.svdtrunc</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">U, S, Vd = svdtrunc(A; truncdim = max(size(A)...), truncerr = 0.)</code></pre><p>Perform a truncated SVD, with maximum number of singular values to keep equal to <code>truncdim</code> or truncating any singular values smaller than <code>truncerr</code>. If both options are provided, the smallest number of singular values will be kept. Unlike the SVD in Julia, this returns matrix U, a diagonal matrix (not a vector) S, and Vt such that A ≈ U * S * Vt</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/tensorOps.jl#L594-L601">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.tightbinding_mpo-NTuple{4, Any}" href="#MPSDynamics.tightbinding_mpo-NTuple{4, Any}"><code>MPSDynamics.tightbinding_mpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">tightbinding_mpo(N, ϵd, chainparams1, chainparams2)
+
+Generate MPO for a tight-binding chain of N fermionic sites with a single impurity site (fermionic as well) 
+of energy ϵd at the center. The impurity is coupled to two leads, each described by a set of chain parameters.
+The interactions are nearest-neighbour, with the first N/2-1 sites corresponding to the first lead,
+the Nth site corresponding to the impurity, and the rest of the sites corresponding to the second
+lead.
+
+# Arguments
+
+* `N::Int`: number of sites in the chain
+* `ϵd::Real`: energy of the impurity site at the center, as Ed - μ, where μ is the chemical potential
+* chainparams1::Array{Real,1}: chain parameters for the first lead
+* chainparams2::Array{Real,1}: chain parameters for the second lead
+
+The chain parameters are given in the standard form: `chainparams` ``=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]``.</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L737-L756">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.tightbindingmpo-Tuple{Int64, Int64}" href="#MPSDynamics.tightbindingmpo-Tuple{Int64, Int64}"><code>MPSDynamics.tightbindingmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">tightbindingmpo(N::Int, d::Int; J=1.0, e=1.0)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L156-L161">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.tunnelingmpo-Tuple{Any, Any, Any, Any, Any, Int64, Int64}" href="#MPSDynamics.tunnelingmpo-Tuple{Any, Any, Any, Any, Any, Int64, Int64}"><code>MPSDynamics.tunnelingmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">tunnelingmpo(ϵ, delta, α, s, β, d::Int, nummodes::Int; tree=false, ωc=1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L613-L620">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.twobathspinmpo" href="#MPSDynamics.twobathspinmpo"><code>MPSDynamics.twobathspinmpo</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">twobathspinmpo(ω0, Δ, Nl, Nr, dl, dr, chainparamsl=[fill(1.0,N),fill(1.0,N-1), 1.0], chainparamsr=chainparamsl; tree=false)</code></pre><p>Generate MPO for a spin-1/2 coupled to two chains of harmonic oscillators, defined by the Hamiltonian</p><p><span>$H = \frac{ω_0}{2}σ_z + Δσ_x + c_0^rσ_x(b_0^\dagger+b_0) + \sum_{i=0}^{N_r-1} t_i^r (b_{i+1}^\dagger b_i +h.c.) + \sum_{i=0}^{N_r} ϵ_i^rb_i^\dagger b_i + c_0^lσ_x(d_0^\dagger+d_0) + \sum_{i=0}^{N_l-1} t_i^l (d_{i+1}^\dagger d_i +h.c.) + \sum_{i=0}^{N_l} ϵ_i^l d_i^\dagger d_i$</span>.</p><p>The spin is on site <span>$N_l + 1$</span> of the MPS, surrounded by the left chain modes and the right chain modes.</p><p>This Hamiltonain is unitarily equivalent (before the truncation to <code>N</code> sites) to the spin-boson Hamiltonian defined by</p><p><span>$H =  \frac{ω_0}{2}σ_z + Δσ_x + σ_x\int_0^∞ dω\sqrt{\frac{J(ω)}{π}}(b_ω^\dagger+b_ω) + \int_0^∞ dω ωb_ω^\dagger b_ωi + σ_x\int_0^∞ dω\sqrt{\frac{J^l(ω)}{π}}(d_ω^\dagger+d_ω) + \int_0^∞ dω ωd_ω^\dagger d_ω$</span>.</p><p>The chain parameters, supplied by <code>chainparams</code>=<span>$[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]$</span>, can be chosen to represent any arbitrary spectral density <span>$J(ω)$</span> at any temperature. The two chains can have a different spectral density.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L446-L465">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.xxzmpo" href="#MPSDynamics.xxzmpo"><code>MPSDynamics.xxzmpo</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">xxzmpo(N::Int, Δ = 1.0, J=1.0) = xyzmpo(N; Jx=J, Jy=J, Jz=J*Δ)</code></pre><p>Generate MPO for the <code>N</code>-spin XXZ model, defined by the Hamiltonian</p><p><span>$H = \sum_{n=1}^{N-1} -J σ_x^{n} σ_x^{n+1} - J σ_y^{n} σ_y^{n+1} - \Delta J σ_z^{n} σ_z^{n+1}$</span></p><p>with <span>$σ_x^{n}, σ_y^{n}, σ_z^{n}$</span> the Pauli spin-1/2 matrices of the <span>$n^\text{th}$</span> site.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L69-L80">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="MPSDynamics.xyzmpo-Tuple{Int64}" href="#MPSDynamics.xyzmpo-Tuple{Int64}"><code>MPSDynamics.xyzmpo</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia">xyzmpo(N::Int; Jx=1.0, Jy=Jx, Jz=Jx, hx=0., hz=0.)</code></pre><p>Generate MPO for the <code>N</code>-spin XYZ model with external field <span>$\vec{h}=(h_x, 0, h_z)$</span>, , defined by the Hamiltonian</p><p><span>$H = \sum_{n=1}^{N-1} -J_x σ_x^{n} σ_x^{n+1} - J_y σ_y^{n} σ_y^{n+1} - J_z σ_z^{n} σ_z^{n+1} + \sum_{n=1}^{N}(- h_x σ_x^{n} - h_z σ_z^{n})$</span></p><p>with <span>$σ_x^{n}, σ_y^{n}, σ_z^{n}$</span> the Pauli spin-1/2 matrices of the <span>$n^\text{th}$</span> site.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/shareloqs/MPSDynamics/blob/f2c882367fc245f62fa753becf0192fafb7710fa/src/models.jl#L4-L15">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../theory/">« Theoretical Background</a><a class="docs-footer-nextpage" href="../dev/">Developers »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. 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chain mapping</a></li><li><a class="tocitem" href="../examples/protontransfer/">Proton Transfer Model</a></li></ul></li><li><a class="tocitem" href="../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../methods/">Methods</a></li><li><a class="tocitem" href="../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>In a nutshell: MPSDynamics.jl for open quantum systems</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>In a nutshell: MPSDynamics.jl for open quantum systems</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="In-a-nutshell:-MPSDynamics.jl-for-open-quantum-systems"><a class="docs-heading-anchor" href="#In-a-nutshell:-MPSDynamics.jl-for-open-quantum-systems">In a nutshell: MPSDynamics.jl for open quantum systems</a><a id="In-a-nutshell:-MPSDynamics.jl-for-open-quantum-systems-1"></a><a class="docs-heading-anchor-permalink" href="#In-a-nutshell:-MPSDynamics.jl-for-open-quantum-systems" title="Permalink"></a></h1><p><code>MPSDynamics.jl</code> was originally developed to perform tensor network simulations of open quantum systems by solving the Schrödinger equation for the closed {System + Environment}.  The key idea to construct the simulations is to reformulate the open quantum system Hamiltonian as a one-dimensional many-body problem with nearest-neighbor interactions.  The dynamics are then efficiently simulated with <a href="../theory/#Tensor-Networks">Tensor Networks</a> methods. </p><p><img src="../examples/mappings-1.png" alt="Sketch of the different ingredients behind MPSDynamics"/></p><p>The starting point is a system linearly coupled to a bosonic environment, where the coupling between system and environment is specified by the spectral density function <span>$J(\omega)$</span>: the prototypical example being <a href="../examples/sbm/#The-Spin-Boson-Model">The Spin-Boson Model</a>. At finite temperature, the initial state of the environment will be a thermal state (mixed), requiring density matrix formalism and thus making the problem more complex. To circumvent this issue, we consider instead an <em>extended environment</em>, with coupling to the system characterized by a thermalized spectral density function <span>$J(\omega,\beta)$</span>.  The two environments induce the same reduced dynamics on the system, enabling us to deal with pure states only<sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>. The vacuum state (pure) of the extended environment corresponds to the thermal state of the original environment. </p><p>To exploit the computational efficiency of the matrix product states (MPS) description of pure states, we apply the so called <a href="../theory/#Chain-Mapping-of-bosonic-environments">Chain-Mapping of bosonic environments</a> —a unitary transformation dependent on <span>$J(\omega,\beta)$</span>— to the Spin-Boson Hamiltonian, mapping it on a chain Hamiltonian with nearest-neighbor interactions. This enables efficient representation of the full {System + Environment} state and its time evolution using the <a href="../theory/#Time-Dependent-Variational-Principle">Time-Dependent Variational Principle</a>.</p><p>To go further, you will find in this documentation:</p><ul><li>a <a href="../user-guide/#User-Guide">User Guide</a> explaining how to use the package</li><li>several examples illustrating different features of the package: adaptive time-evolution methods in <a href="../examples/sbm/#The-Spin-Boson-Model">The Spin-Boson Model</a>, <a href="../examples/timedep/#Time-dependent-Hamiltonian">Time-dependent Hamiltonian</a>, or fermionic environments with <a href="../examples/anderson-model/#The-Anderson-Impurity-Model">The Anderson Impurity Model</a></li><li>a <a href="../theory/#Theoretical-Background">Theoretical Background</a> covering the chain mapping procedure, flying over the basics of tensor networks, and summarizing the TDVP methods</li></ul><p>If this package was useful in your work, do not forget <a href="../#Citation">Citation</a>. And if you would like to get involved in its development, you can find out <a href="../dev/#How-to-Contribute">How to Contribute</a>.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>Moreover, the dynamics of the original environment can still be recovered, see <a href="../examples/bath-observables/#Inspecting-the-bath-by-undoing-the-chain-mapping">Inspecting the bath by undoing the chain mapping</a></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../user-guide/">« User Guide</a><a class="docs-footer-nextpage" href="../examples/sbm/">The Spin-Boson Model »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. 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+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>In a nutshell: MPSDynamics.jl for open quantum systems · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img src="../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Introduction</a></li><li><a class="tocitem" href="../user-guide/">User Guide</a></li><li class="is-active"><a class="tocitem" href>In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../methods/">Methods</a></li><li><a class="tocitem" href="../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>In a nutshell: MPSDynamics.jl for open quantum systems</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>In a nutshell: MPSDynamics.jl for open quantum systems</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/nutshell.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="In-a-nutshell:-MPSDynamics.jl-for-open-quantum-systems"><a class="docs-heading-anchor" href="#In-a-nutshell:-MPSDynamics.jl-for-open-quantum-systems">In a nutshell: MPSDynamics.jl for open quantum systems</a><a id="In-a-nutshell:-MPSDynamics.jl-for-open-quantum-systems-1"></a><a class="docs-heading-anchor-permalink" href="#In-a-nutshell:-MPSDynamics.jl-for-open-quantum-systems" title="Permalink"></a></h1><p><code>MPSDynamics.jl</code> was originally developed to perform tensor network simulations of open quantum systems by solving the Schrödinger equation for the closed {System + Environment}.  The key idea to construct the simulations is to reformulate the open quantum system Hamiltonian as a one-dimensional many-body problem with nearest-neighbor interactions.  The dynamics are then efficiently simulated with <a href="../theory/#Tensor-Networks">Tensor Networks</a> methods. </p><p><img src="../examples/mappings-1.png" alt="Sketch of the different ingredients behind MPSDynamics"/></p><p>The starting point is a system linearly coupled to a bosonic environment, where the coupling between system and environment is specified by the spectral density function <span>$J(\omega)$</span>: the prototypical example being <a href="../examples/sbm/#The-Spin-Boson-Model">The Spin-Boson Model</a>. At finite temperature, the initial state of the environment will be a thermal state (mixed), requiring density matrix formalism and thus making the problem more complex. To circumvent this issue, we consider instead an <em>extended environment</em>, with coupling to the system characterized by a thermalized spectral density function <span>$J(\omega,\beta)$</span>.  The two environments induce the same reduced dynamics on the system, enabling us to deal with pure states only<sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>. The vacuum state (pure) of the extended environment corresponds to the thermal state of the original environment. </p><p>To exploit the computational efficiency of the matrix product states (MPS) description of pure states, we apply the so called <a href="../theory/#Chain-Mapping-of-bosonic-environments">Chain-Mapping of bosonic environments</a> —a unitary transformation dependent on <span>$J(\omega,\beta)$</span>— to the Spin-Boson Hamiltonian, mapping it on a chain Hamiltonian with nearest-neighbor interactions. This enables efficient representation of the full {System + Environment} state and its time evolution using the <a href="../theory/#Time-Dependent-Variational-Principle">Time-Dependent Variational Principle</a>.</p><p>To go further, you will find in this documentation:</p><ul><li>a <a href="../user-guide/#User-Guide">User Guide</a> explaining how to use the package</li><li>several examples illustrating different features of the package: adaptive time-evolution methods, <a href="../examples/timedep/#Time-dependent-Hamiltonian">Time-dependent Hamiltonian</a>, or fermionic environments with <a href="../examples/anderson-model/#The-Anderson-Impurity-Model">The Anderson Impurity Model</a></li><li>a <a href="../theory/#Theoretical-Background">Theoretical Background</a> covering the chain mapping procedure, flying over the basics of tensor networks, and summarizing the TDVP methods</li></ul><p>If this package was useful in your work, do not forget <a href="../#Citation">Citation</a>. And if you would like to get involved in its development, you can find out <a href="../dev/#How-to-Contribute">How to Contribute</a>.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>Moreover, the dynamics of the original environment can still be recovered, see <a href="../examples/bath-observables/#Inspecting-the-bath-by-undoing-the-chain-mapping">Inspecting the bath by undoing the chain mapping</a></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../user-guide/">« User Guide</a><a class="docs-footer-nextpage" href="../examples/sbm/">The Spin-Boson Model »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Search · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img src="../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Introduction</a></li><li><a class="tocitem" href="../user-guide/">User Guide</a></li><li><a class="tocitem" href="../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../methods/">Methods</a></li><li><a class="tocitem" href="../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Search</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Search</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article><p id="documenter-search-info">Loading search...</p><ul id="documenter-search-results"></ul></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body><script src="../search_index.js"></script><script src="../assets/search.js"></script></html>
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","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"(Image: Sketch of the different ingredients behind MPSDynamics)","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"The starting point is a system linearly coupled to a bosonic environment, where the coupling between system and environment is specified by the spectral density function J(omega): the prototypical example being The Spin-Boson Model. At finite temperature, the initial state of the environment will be a thermal state (mixed), requiring density matrix formalism and thus making the problem more complex. To circumvent this issue, we consider instead an extended environment, with coupling to the system characterized by a thermalized spectral density function J(omegabeta).  The two environments induce the same reduced dynamics on the system, enabling us to deal with pure states only[1]. The vacuum state (pure) of the extended environment corresponds to the thermal state of the original environment. ","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"To exploit the computational efficiency of the matrix product states (MPS) description of pure states, we apply the so called Chain-Mapping of bosonic environments —a unitary transformation dependent on J(omegabeta)— to the Spin-Boson Hamiltonian, mapping it on a chain Hamiltonian with nearest-neighbor interactions. This enables efficient representation of the full {System + Environment} state and its time evolution using the Time-Dependent Variational Principle.","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"To go further, you will find in this documentation:","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"a User Guide explaining how to use the package\nseveral examples illustrating different features of the package: adaptive time-evolution methods in The Spin-Boson Model, Time-dependent Hamiltonian, or fermionic environments with The Anderson Impurity Model\na Theoretical Background covering the chain mapping procedure, flying over the basics of tensor networks, and summarizing the TDVP methods","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"If this package was useful in your work, do not forget Citation. And if you would like to get involved in its development, you can find out How to Contribute.","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"[1]: Moreover, the dynamics of the original environment can still be recovered, see Inspecting the bath by undoing the chain mapping","category":"page"},{"location":"user-guide/#User-Guide","page":"User Guide","title":"User Guide","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Here we explain the different steps to perform a simulation.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Examples with detailed explanations can be found in the Examples section.","category":"page"},{"location":"user-guide/#Initial-State","page":"User Guide","title":"Initial State","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The initial many-body state of the {System + Environment} can be described easily as Matrix Product States (MPS) or Tree Tensor Networks (TTN) state (e.g. when a system is coupled to several environments).","category":"page"},{"location":"user-guide/#Matrix-Product-States","page":"User Guide","title":"Matrix Product States","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"A MPS can be initialized with several methods.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The productstatemps method enables  to instantiate arbitrary MPS of fixed uniform bond dimension with non-uniform physical dimensions. The individual states of the MPS sites can be provided by setting state to a list of column vectors.  Setting state=:Vacuum will produce an MPS in the vacuum state.  Setting state=:FullOccupy will produce an MPS in which each site is fully occupied. The gauge of the MPS can also be set using a keyword argument.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"julia> ψ = unitcol(1,2) # system initial state\n\njulia> d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1 \n\njulia> cpars = chaincoeffs_ohmic(N, α, s) # chain coefficient for an Ohmic spectral density\n\njulia> H = spinbosonmpo(ω0, Δ, d, N, cpars)\n\njulia> A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Alternatively, a chain with a specified number of excitation localised on one site, or delocalized accross several sites can be generated with MPSDynamics.chainmps.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Random MPS can also be generated with the randmps method.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"For the case of fermionic states (which need to be anti-symmetrized), the MPSDynamics.electronkmps method generate an MPS for an electron with momentum k, and the MPSDynamics.electron2kmps generate an MPS with 2 electrons in k-states k1 and k2.","category":"page"},{"location":"user-guide/#Tree-Tensor-Networks","page":"User Guide","title":"Tree Tensor Networks","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"TTN are a generalization of MPS that allows more ramifications to the one dimensional shape of the MSP. This is useful when, for instance, a system is interacting with more than two environments. Tensors of TreeNetwork are usually referred as nodes, while the first node, represented as the leftmost node is called the head node. The Tree structure is built such that a node has always one parent leg pointing to the left and its child legs pointing to the right. Most of the methods for MPS states are overloaded for TreeNetwork. For example, the productstatemps method tolerates also TTN state and a simple TreeNetwork with only one branch can be initialized in the same way than with MPS.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"julia> ψ = unitcol(1,2)\n\njulia> d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1\n\njulia> cpars = chaincoeffs_ohmic(N, α, s)\n\njulia> H = spinbosonmpo(ω0, Δ, d, N, cpars, tree=true)\n\njulia> A = productstatemps(H.tree, physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # tree-MPS representation of |ψ>|Vacuum>","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Building a specific tree-MPO such as MPSDynamics.nearestneighbourmpo adds no conceptual complexity and MPS methods have the option to built it as a TreeNetwork.","category":"page"},{"location":"user-guide/#Hamiltonian","page":"User Guide","title":"Hamiltonian","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"In order to perform time evolution and have access to the dynamics of the many-body state a Hamiltonian needs to be specified in the form of a Matrix Product Operator (MPO) of as a tree tensor network. Either way, this can be done by using a Build-in Hamiltonian, Convert a MPO from ITensor, or creating a Tailored MPO.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"In the context of Open Quantum Systems, custom chain coefficients for the environment can be generated for finite temperature simulations, and/or user provided spectral densities (SDs).","category":"page"},{"location":"user-guide/#Build-in-Hamiltonian","page":"User Guide","title":"Build-in Hamiltonian","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"MPSDynamics provides several topical Hamiltonians directly in the form of MPO or Tree Tensor Networks such as the Ising model MPSDynamics.isingmpo, the XYZ Hamiltonian MPSDynamics.xyzmpo, the Spin Boson Model MPSDynamics.spinbosonmpo, a spin coupled to two bosonic baths MPSDynamics.twobathspinmpo, nearest neighbour interactions Hamiltonian MPSDynamics.nearestneighbourmpo, the independent boson model MPSDynamics.ibmmpo, (non-)uniform tight-binding chain Hamiltonian MPSDynamics.tightbindingmpo.","category":"page"},{"location":"user-guide/#Convert-a-MPO-from-ITensor","page":"User Guide","title":"Convert a MPO from ITensor","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The method MPSDynamics.MPOtoVector converts an ITensors chain MPO into a form compatible with MPSDynamics.","category":"page"},{"location":"user-guide/#Tailored-MPO","page":"User Guide","title":"Tailored MPO","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"One can also construct MPO tailored to the problem one is interested in. MPOs are fundamentally a lists of rank-4 tensors such that the right bond dimension of the nth tensor must be equal to the left bond dimension of the n+1th tensor; and the dimension of the physical bonds of the nth tensor must be equal to the corresponding physical bond on the MPS.","category":"page"},{"location":"user-guide/#Finite-Temperature-and-Custom-SD","page":"User Guide","title":"Finite Temperature and Custom SD","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"In the T-TEDOPA framework (see Theoretical Background for more details) finite temperature simulations are done with an effective pure state description of the system and the environment where the coupling coefficients (or the SD) is temperature-dependent.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The corresponding chain coefficients for an Ohmic or a user provided spectral density (that can thus in pratice be either at zero or finite temperature) are computed with the chaincoeffs_finiteT. This method is based on the ORTHOPOL routines[Gautschi]","category":"page"},{"location":"user-guide/#Observables","page":"User Guide","title":"Observables","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"System and environment observables can be computed, as well as system-and-environment 'non-local' observables.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Observables that will be passed to MPSDynamics.runsim to have their expectation value computated at each time step are defined with the OneSiteObservable and TwoSiteObservable.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"One-site and two-site obsevables work similarly, they need to be given a name, an (pair of) operator(s) and the (list of) site(s) on which they are evaluated.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"For instance one can calculated the average number of excitation in each of the N environmental modes","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"    ob = OneSiteObservable(\"chain mode occupation\", numb(d), (2,N+1))","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"It is also possible to measure composite system/environment observables, for example","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"    ob = TwoSiteObservable(\"SXdisp\", sx, disp(d), [1], collect(2:N+1))","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"which measure the correlation between the spin in the x-direction and the displacement of the bath modes.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Purely environmental 'non-local' observables such as ","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"    ob = TwoSiteObservable(\"bath coherence\", crea(d), anih(d), collect(2:N+1), collect(2:N+1))","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"that computes all the chain mode coherences langlehata_n^daggerhata_mrangle (and the chain mode occupation when n = m).","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"We note that if one knows the coherences and populations of all the chain modes, it is then possible to reconstruct the populations of the normal mode environment.","category":"page"},{"location":"user-guide/#Time-Evolution","page":"User Guide","title":"Time-Evolution","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Simulation are performed when calling the MPSDynamics.runsim function where the time-evolution method should be specified.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The time-evolution methods currently implemented belong to the familly of Time-Dependent Variational Principle (TDVP). The central point of this method, in the modern tensor networks formulation, is that instead of solving the Schrödinger equation and then truncating the MPS representation of the quantum state, one can solve the equations of motion projected into a space of restricted bond dimension.  The major advantage of this method is that it naturally preserves the unitarity of the time evolution and conserves the energy (except in its two-site implementation). Three variants of TDVP are implemented in the MPSDynamics.jl package:","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"one-site TDVP (TDVP1): fixed bond dimension, preserves unitarity, conserves energy, scales as mathcalO(D^2 d^2 w^2 + D^3 dw + D^3 d^2 ) where D is the MPS (or TTN) bond dimension, d the local dimension, w the MPO bond dimension.\ntwo-site TDVP (TDVP2): adaptive bond dimension, breaks unitarity, scales as mathcalO(D^2d^3 w^2 + D^3 d^2 w + D^3 d^3 ).\nadaptive one-site TDVP (DTDVP): adaptive bond dimension, preserves unitarity, conserves energy.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Local one-site and two-site observables, as well as non-local two-site observables, can be efficiently computed for each time-step of the time-evolution with the three methods.","category":"page"},{"location":"user-guide/#One-site-TDVP","page":"User Guide","title":"One-site TDVP","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This implementation of the TDVP can be used for MPS and TTN. It performs one-site updates of the tensor network tensors while keeping the bond dimension fixed. Because no reduction of the dimensionality is performed (for instnce by throwing away some simgular values), it naturally preserves unitarity and hence conserves energy. The otherside of the coin is that simulation might be slower because the bond dimension is large at all time.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The convergence parameter of the simulation is thus the bond dimension D.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This method can be used with in MPSDynamics.runsim with the key word argument method=:TDVP1.","category":"page"},{"location":"user-guide/#Two-Site-TDVP","page":"User Guide","title":"Two-Site TDVP","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This version of the TDVP evolves two-site tensors than can be split back into two individual site tensors by applying an SVD. The bond dimension between two neighbouring sites can then be truncated to any value  r by throwing away any singular values that fall below some threshold.  In this way, the MPS bond dimension will grow dynamically throughout the course of the evolution to capture entanglement, as and when it emerges. The truncation entails a loss of unitarity.  Indeed, for a nearest-neighbour Hamiltonian, applying the two-site projector will not entail a projection error, leading to a scheme that is almost identical to TEBD,wherein the error arises solely from the truncation. Furthermore, 2TDVP scales poorly with the local physical dimension, and is known to have issues with long-range interactions.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The convergence parameter is the threshold of the SVD.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This method can be used with in MPSDynamics.runsim with the key word argument method=:TDVP2.","category":"page"},{"location":"user-guide/#Adaptive-One-Site-TDVP","page":"User Guide","title":"Adaptive One-Site TDVP","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Ahead of any time evolution, one computes new bond dimensions of the MPS if the relative rate of change of the TDVP projection error with respect to the bond dimension is larger than a chosen precision p. Then, TDVP1 is performed, using projectors with sub-spaces expanded accordingly, to produce an MPS evolved by one time step with the new,increased, bond dimensions. This version is faster than TDVP1 due to the acceleration gained from having more optimised bond dimensions; the bond update step is cheap and so its cost should not normally outweigh this advantage. However, two things need to be noticed:","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"DTDVP can get stuck into local minima from the initial time. If the bond dimension does not change during the time-evolution, consider embeding your MPS in a larger manifold with MPSDynamics.mpsembed! before time-evolving it.\nFor models with analytical solutions that are MPS of a given bond dimension (such as the independent boson model), DTDVP can overshoot the analytical bond dimension because the relative rate of change of the projection error becomes dominated by random numerical fluctuations.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The convergence parameter is a threshold value p for the rate of change of the projection error with respect to the bond dimension.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This method can be used with in MPSDynamics.runsim with the key word argument method=:DTDVP.","category":"page"},{"location":"user-guide/#Data-Storage","page":"User Guide","title":"Data Storage","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The data (i.e. observables time-series) is stored in the JLD format which is based on HDF5.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The HDF5 format is natively supported across many platforms and languages (e.g. Python, or Mathematica).","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"For the data to be saved to a file after a run, the keyword argument save=true needs to be used in runsim.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The directory where the data should be saved can be chosen by setting a path with the savedir keyword argument.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"A plot keyword argument can also be used to choose whether plots for 1D observables will be automatically generated and saved along with the data. ","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Loading the data in Julia using the JLD.jl package will recover the full type information of the Julia variables that were stored.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"julia> using JLD\n\njulia> dat = load(\"filename.jld\")\nDict{String, Any} with 2 entries:\n  \"parameters\" => Dict{String, Any}(\"tmax\"=>0.2, \"method\"=>:DTDVP, \"dt\"=>0.0005…\n  \"data\"       => Dict{String, Any}(\"bonddims\"=>[1 1 … 1 1; 1 2 … 2 2; … ; 1 1 …","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"If the data is loaded in an variable dat the structure is the folowing:","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"the parameters entry contains a dictionary where all the simulation parameters are stored\nthe data entry contains a dictionary where are stored simulation time, the observables and (whenever relevent) the bond dimension of the state at each time steps. ","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"julia> dat[\"parameters\"]\nDict{String, Any} with 11 entries:\n  \"tmax\"       => 0.2\n  \"method\"     => :DTDVP\n  \"dt\"         => 0.0005\n  \"name\"       => \"my model\"\n  \"Δ\"          => 0.0\n  \"β\"          => 0.0186854\n  \"N\"          => 300.0\n  \"d\"          => 15.0\n  \"unid\"       => \"Ovzm6\"\n  \"ω0\"         => 0.0\n  \"convparams\" => 0.0005\n\njulia> dat[\"data\"]\nDict{String, Any} with 6 entries:\n  \"bonddims\" => [1 1 … 1 1; 1 2 … 2 2; … ; 1 1 … 7 7; 1 1 … 1 1]\n  \"sx\"       => [1.0, 0.912818, 0.741759, 0.605797, 0.528792, 0.492497, 0.47976…\n  \"sz\"       => [0.0, 0.0871825, 0.25824, 0.394201, 0.471207, 0.507503, 0.52023…\n  \"nchain\"   => [0.0 0.23466 … 1.84319 1.76098; 0.0 0.00231507 … 0.83105 0.9033…\n  \"sy\"       => [0.0, -0.0133489, -0.0588887, -0.0858181, -0.0759996, -0.048539…\n  \"times\"    => [0.0, 0.0005, 0.001, 0.0015, 0.002, 0.0025, 0.003, 0.0035, 0.00…\n","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"[Gautschi]: Gautschi, W. Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 20, 21–62 (1994).","category":"page"},{"location":"examples/timedep/#Time-dependent-Hamiltonian","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"","category":"section"},{"location":"examples/timedep/#Context","page":"Time-dependent Hamiltonian","title":"Context","text":"","category":"section"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"To simulate a drive or a laser pulse, a time-dependent Hamiltonian is often needed. Here we explain thanks to the script in MPSDynamics/examples/sbm_Htimedependent.jl how this type of Hamiltonian can be integrated within the method and applied on a wavefunction. We will take a Spin-Boson Model (SBM) Hamiltonian with an Ohmic spectral density. However, the time-dependency is not model specific and can be adapated for other models. For more information about the SBM, see the dedicated example. For this example, we simulate a drive of the form","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"        hatH_textdrive(t) = epsilon hatsigma_x sin(omega_textdrive t)","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"with epsilon = frac2 piT_textRabi the amplitude of the drive, T_textRabi the Rabi period, and omega_textdrive the frequency of the drive. The drive is set up to be on resonance with the two-level system.","category":"page"},{"location":"examples/timedep/#The-code","page":"Time-dependent Hamiltonian","title":"The code","text":"","category":"section"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"First we load the MPSdynamics.jl package to be able to perform the simulation, the Plots.jl one to plot the results, and the LaTeXStrings.jl one to be able to use LaTeX in the plots. The function MPSDynamics.disp is also imported.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"using MPSDynamics, Plots, LaTeXStrings\n\nimport MPSDynamics: disp","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"We then define variables for the physical parameters of the simulation. Among these, three are convergence parameters:","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"d is the number of states we retain for the truncated harmonic oscillators representation of environmental modes\nN is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"The variable Ndrive represents the site of the MPO where the operator of the time-dependent part acts. For this example, the two-level system is at the first site of the MPS. ","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#----------------------------\n# Physical parameters\n#----------------------------\n\nd = 4 # number of Fock states of the chain modes\n\nN = 60 # length of the chain\n\nα = 0.005 # coupling strength\n\nΔ = 0.0 # tunneling \n\nω0 = 0.8 # TLS gap\n\ns = 1 # ohmicity\n\ncpars = chaincoeffs_ohmic(N, α, s) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0\n\nTrabi = 30.0 # Rabi period of the drive\n\nϵ = 2*pi / Trabi # Intensity of the drive\n\nωdrive = ω0 # Frequency of the drive\n\nNdrive = 1 #Number of the site on which the drive is applied","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is mathcalO(dt^3). In this example we present only one-site implementation of TDVP that preserves the unitarity of the evolution:","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"the regular one-site method with the keyword :TDVP1 where all the virtual bonds of the MPS have the same bond dimension D","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"Logically the constant bond dimension of the MPS for TDVP1 is the respective convergence parameter.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#-----------------------\n# Simulation parameters\n#-----------------------\n\ndt = 0.5 # time step\n\ntfinal = 100.0 # simulation time\n\nmethod = :TDVP1 # time-evolution method\n\nD = [6] # MPS bond dimension","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"Using MPSDynamics.jl built-in methods we define the SBM MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the 'local state' of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"In this part, the time-dependent terms of the MPO are stored in a list. This enables to add these terms to the MPO elements at each timestep and avoid the calculation of an entire new MPO. This way is not cumbersome since it adds a matrix sum at each time step.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#---------------------------\n# MPO and initial state MPS\n#---------------------------\n\ntimelist = collect(0:dt:tfinal)\nnumsteps = length(timelist)-1\n\nHt = [ϵ*sx*sin(ωdrive*tstep) for tstep in timelist] # Time-dependent Hamiltonian term\n\nH = spinbosonmpo(ω0, Δ, d, N, cpars) # MPO representation of the Hamiltonian\n\nψ = unitcol(2,2) # Initial down-z system state \n\nA = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"We then choose the observables that will be stored in the data and the MPSDynamics.runsim arguments.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#---------------------------\n# Definition of observables\n#---------------------------\n\nob1 = OneSiteObservable(\"sz\", sz, 1)\n\nob2 = OneSiteObservable(\"chain mode occupation\", numb(d), (2,N+1))\n\nob3 = TwoSiteObservable(\"SXdisp\", sx, disp(d), [1], collect(2:N+1))\n","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"MPSDynamics.runsim is called to perform the dynamics. The argument timedep is set up to true and Ndrive and Htime are provided in the kwargs. ","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#-------------\n# Simulation\n#------------\n\nA, dat = runsim(dt, tfinal, A, H;\n                name = \"Driving field on ohmic spin boson model\",\n                method = method,\n                obs = [ob1],\n                convobs = [ob1],\n                params = @LogParams(N, d, α, Δ, ω0, s),\n                convparams = D,\n                timedep = true, # the Hamiltonian is time dependent\n                Ndrive = Ndrive, # the first site of the MPS/MPO (i.e. the system) is concerned\n                Htime = Ht, # list of time-dependent terms\n                verbose = false,\n                save = true,\n                plot = true,\n                );","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"Eventually, the stored observables can be represented","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"\n#----------\n# Plots\n#----------\n\nplot(dat[\"data/times\"], dat[\"data/sz\"], label=\"Dmax = $(D...)\", xlabel=L\"t\",ylabel=L\"\\sigma_z\", title=\"\")","category":"page"},{"location":"dev/#Developers","page":"Developers","title":"Developers","text":"","category":"section"},{"location":"dev/#Simulation-Workflow","page":"Developers","title":"Simulation Workflow","text":"","category":"section"},{"location":"dev/","page":"Developers","title":"Developers","text":"(Image: Flowchart of the simulation workflow)","category":"page"},{"location":"dev/#How-to-Contribute","page":"Developers","title":"How to Contribute","text":"","category":"section"},{"location":"dev/","page":"Developers","title":"Developers","text":"Contributions are welcome! Don't hesitate to contact us if you","category":"page"},{"location":"dev/","page":"Developers","title":"Developers","text":"found a bug;\nhave a suggestion on how to improve the code and/or documentation;\nwould like to get involved in writing code and/or documentation.","category":"page"},{"location":"dev/","page":"Developers","title":"Developers","text":"You can contact us by raising an issue on Github, or by writing to one of the developers.","category":"page"},{"location":"examples/protontransfer/#Proton-Transfer-Model","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"","category":"section"},{"location":"examples/protontransfer/#Context","page":"Proton Transfer Model","title":"Context","text":"","category":"section"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"The MPS formalism can also be used for physical chemistry problems. One development done with the MPSDynamics.jl package is the introduction of a reaction coordinate tensor, allowing the system to be described in space [lede_ESIPT_2024]. It can model an electronic system with discretized states being described along a reaction coordinate. The introduction of a reaction coordinate allows to recover the well-known doublewell viewpoint and wavepacket dynamics can be analyzed with the reduced density matrix. ","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"Here is an illustrative example of two electronic configurations undergoing a tautomerization : the enol named erangle and the keto named krangle (hbar = 1) :","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"H_S = omega^0_e erangle langle e + omega^0_k krangle langle k + Delta (erangle langle k + krangle langle e) ","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"This description requires the tensor of the system to be linked to another tensor representing an harmonic oscillator. This reaction coordinate oscillator is expressed as RC and represents the reaction path of the proton.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"H_RC + H_int^S-RC = omega_RC (d^daggerd + frac12) + g_e erangle langle e( d + d^dagger)+ g_k k rangle langle k( d + d^dagger)","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"The coefficients g are set up in order to recover the doublewell formalism. A bath is introduced in the description and can represent intramolecular vibrations, damping the dynamics.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"H_B + H_int^RC-B = int_-^+ mathrmdk omega_k b_k^dagger b_k - (d + d^dagger)int_0^ mathrmdomega sqrtJ(omega)(b_omega^dagger+b_ω) + lambda_textreorg(d + d^dagger)^2","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"with the reorganization energy taken into account that reads","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"lambda_textreorg = int fracJ(omega)omegamathrmdomega","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":".","category":"page"},{"location":"examples/protontransfer/#The-code","page":"Proton Transfer Model","title":"The code","text":"","category":"section"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"First we load the MPSdynamics.jl package to be able to perform the simulation, the Plots.jl one to plot the results, and the LaTeXStrings.jl one to be able to use LaTeX in the plots. ColorSchemes.jl is uploaded for colors in plots, PolyChaos.jl is needed to calculate the Hermite Polynomials that are necessary for the translation of the reduced density matrix from the Fock space dimension to the x space dimension. Eventually, LinearAlgebra is loaded for the LinearAlgebra.diag function for post-processing. ","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"using MPSDynamics, Plots, LaTeXStrings, ColorSchemes, PolyChaos, LinearAlgebra","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"We then define variables for the physical parameters of the simulation. The argument isolated can be set up to true to simulate only the electronic system with the RC oscillator. isolated=false adds an environment with the usual parameters d N and α that can be changed","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#----------------------------\n# Physical parameters\n#----------------------------\n# Enol / Keto\n\nω0e= 0.8  # Enol energy at x=0\nω0k= 0.8  # Keto energy at x=0\n\nx0e = -0.25 # Enol Equilibrium displacement \nx0k = 0.25 # Keto Equilibrium displacement \n\nΔ = 0.05 # Direct coupling between enol and keto\n\nm=1.83E3 # Mass of the reaction coordinate particle. 1.83E3 is the proton mass in atomic units. \nħ=1 # Atomic Units convention\n\ndFockRC = 25 # Fock space of the RC tensor\n\nωRC = 0.0347 # Frequency of the RC tensor\n\nγ = sqrt(m*ωRC/2)*x0e # γ = displacement RC ; γ = \\sqrt{m\\omega/2} x_disp\n\ncparsRC = [ωRC,ωRC*sqrt(m*ωRC/2)] # Energy and g RC coupling parameter\n\nisolated = true # isolated = true : no environment ; isolated = false : system coupled to a bosonic environment\n# Creates the chain depending on the isolated condition. The parameters for isolated = false can be modified as desired.\nif isolated\n    d=1; N=2; α = 0.0  # number of Fock states of the chain modes ; length of the chain ; coupling strength\nelse\n    d=4; N=40; α = 0.008 # number of Fock states of the chain modes ; length of the chain ; coupling strength\nend\n\ns = 1 # ohmicity\n\nωc = 2*ωRC # Cut-off of the spectral density J(ω)\n\nλreorg = (2*α*ωc)/s # Reorganisation Energy taken into account in the Hamiltonian\n\ncpars = chaincoeffs_ohmic(N, α, s; ωc=ωc) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is mathcalO(dt^3). In this example we present only the regular one-site method with the keyword :TDVP1 where all the virtual bonds of the MPS have the same bond dimension D.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"Logically the constant bond dimension of the MPS for TDVP1 is the respective convergence parameter.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#-----------------------\n# Simulation parameters\n#-----------------------\n\ndt = 10.0 # time step\n\ntfinal = 2000.0 # simulation time\n\nnumsteps = length(collect(0:dt:tfinal))-1\n\nmethod = :TDVP1 # time-evolution method\n\nD = 5 # MPS bond dimension","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"This initial state is a product state between the system, the RC oscillator and the chain. While the chain is initially in the vacuum, the RC oscillator is initialised coherently representing a displaced state. The electronic system is diagonalized at this chosen displacement value in order to mainly initialize it in the adiabatic LOW state","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"The energy of the starting system is printed thanks to the function MPSDynamics.measurempo.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#---------------------------\n# MPO and initial state MPS\n#---------------------------\n\n# Initial electronic system in the left well, at mean displacement . It results to a superposition of 1 and 2\n# This state is obtained with the diagonalization of the diabatic Hamiltonian at x=γ\nψ = zeros(2)\nX2 = zeros(2)\n\nAe = ω0e  - 0.5*m*ωRC^2*(x0e)^2\nAk = ω0k  - 0.5*m*ωRC^2*(x0k)^2\nxini = γ/(sqrt(m*ωRC/2))\n\nAdumb = (ω0e-ω0k+0.5*m*ωRC^2*((xini-x0e)^2-(xini-x0k)^2))\nBdumb = sqrt((Adumb)^2 + 4*Δ^2)\nX2[1] = (Adumb-Bdumb)/(2*Δ)\nX2[2] = 1\n\nmod_X2 = sqrt(sum(X2[i]^2 for i=1:2))\n\nψ[1] = X2[1] / mod_X2\nψ[2] = X2[2] / mod_X2\n\n#### Coherent state . Initialise the RC oscillator at displacement γ thanks to the Taylor expression of the displacement operator\n\ncoherent_RC = [fill(unitcol(1,dFockRC), 1)...]\nchainpop_RCini =zeros(dFockRC,2)\n\nchainpop_RCini[1,2]=1\nchainpop_RCini[1,1]=exp(-(γ^2)/2)\n\nfor j=2:dFockRC            \n    chainpop_RCini[j,1]=chainpop_RCini[j-1,1]*(γ/sqrt(j-1))\nend\n\ncoherent_RC[1] = hcat(chainpop_RCini[:,1]...) #Manipulation to get good format for MPS, does not change value\n\nH = protontransfermpo(ω0e, ω0k, x0e, x0k, Δ, dFockRC, d, N, cpars, cparsRC, λreorg)\n\nA = productstatemps(physdims(H), state=[ψ, coherent_RC..., fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Displaced RC>|Vacuum>\n\nEini = measurempo(A,H)\nprint(\"\\n Energy =\",Eini)","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"We then choose the observables that will be stored in the data and the MPSDynamics.runsim arguments. The main analyzed data will be here the reduced density matrix.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#---------------------------\n# Definition of observables\n#---------------------------\n\nob1 = OneSiteObservable(\"sz\", sz, 1)\n#-------------\nob2 = OneSiteObservable(\"RC displacement\", MPSDynamics.disp(dFockRC,ωRC,m), 2)","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"MPSDynamics.runsim is called to perform the dynamics. The argument Nrho is set up to 2 to specifiy the MPS site of the reduced density matrix. ","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#-------------\n# Simulation\n#------------\n\nA, dat = runsim(dt, tfinal, A, H;\n                name = \"proton transfer and tunneling at zero temperature\",\n                method = method,\n                obs = [ob1, ob2],\n                convobs = [ob1, ob2],\n                params = @LogParams(N, d, α, s),\n                convparams = D,\n                Nrho = 2, # Need to precise that the reduced density matrix is\n                # calculated for the 2nd tensor. The value by default is 1. \n                reduceddensity= true,\n                verbose = false,\n                save = true,\n                plot = false,\n                );","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"Plot parameters are chosen. Among them, xlist_rho determines the resolution of the reduced density matrix in space. ","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#-----------------------\n# Plot parameters\n#-----------------------\n\nxlist_rho =collect(-1.2:0.05:1.2) # x definition for the reduced density matrix expressed in space\n# other values of xlist_rho can be chosen to gain numerical time\n\npalette = ColorSchemes.okabe_ito # color palette for plots","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"We begin by representing the doublewell energy landscape described by the parameters.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#-------------\n# Plots\n#------------\n\n#### Plot the doublewell in the adiabatic basis. It is obtained with the diagonalization of the diabatic space Hamiltonian #####\n\nfunction fp(ɸ)\n    H11 = Ae + 0.5*m*ωRC^2*(ɸ-x0e)^2\n    H22 = Ak + 0.5*m*ωRC^2*(ɸ-x0k)^2 \n    H12 = Δ\n    λp = ((H11+H22)/2)+0.5*sqrt((H11-H22)^2+4*H12^2)\n    return λp\nend\n\nfunction fm(ɸ)\n    H11 = Ae + 0.5*m*ωRC^2*(ɸ-x0e)^2\n    H22 = Ak + 0.5*m*ωRC^2*(ɸ-x0k)^2\n    H12 = Δ\n    λm = ((H11+H22)/2)-0.5*sqrt((H11-H22)^2+4*H12^2)\n    return λm\nend\n\nfunction gaussian(x)\n    return Eini .+((1/0.06*sqrt(2*pi)).*exp.(-(x.+(0.25)).^2 ./(2*0.06^2))) .*0.001\nend\n\nɸ = [i for i in -0.75:0.001:0.75]\nɸwp = [i for i in -0.45:0.001:-0.05]\nxmin = -0.75 ; xmax = 0.75 ; ymin = 0.7 ; ymax = 0.9 \n\nplt=plot(ɸ,fm, label=\"LOW\",xlabel=\"Reaction Coordinate (arb. units)\",ylabel=L\"$\\omega$ (arb. units)\",linewidth=4,thickness_scaling = 1, bg_legend=:transparent, legendfont=16, legendfontsize=16, xguidefontsize=16, yguidefontsize=16, tickfontsize=(12), xlims=(xmin,xmax),ylims=(ymin,ymax),xticks=(xmin:0.25:xmax),yticks=(ymin:0.05:ymax),legend=(0.6,0.6), grid=false,linecolor =palette[8])\n\nplt=plot!(ɸ,fp,label=\"UP\",lc =palette[3], linewidth=3,thickness_scaling = 1)\n\nplt = hline!([Eini], lw=2,  lc=:black, ls=:dash, label=L\"\\omega_{system}\")\n\nplt=plot!(ɸwp,gaussian,linecolor=:black,linewidth=3,  label=\"\", fillrange = (Eini , gaussian(ɸwp)),c = :black)\n\ndisplay(plt)","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"The stored observables can be showed as usual.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"##### Results #####\n\ndisplay(plot(dat[\"data/times\"], dat[\"data/RC displacement\"],label=\"<X> (arb. units)\", linecolor =:black, xlabel=\"Time (arb. units)\",ylabel=\"<X>\", title=\"\", linewidth=4, legend=:none, tickfontsize=10,xlims=(0,2000)))","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"Eventually, the reduced density matrix is analyzed. Initially expressed in the Fock dimension, the eigenvectors of the RC oscillator that are constructed here allows us to translate it into the reaction coordinate space dimension.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#### Reduced Density Matrix in space#####\n\n# Loads the Hermite coefficients to calculate the eigenstates of the harmonic oscillator. The coefficient 2 comes from the definition of the physics field. \nαherm, βherm = 2.0 .*rm_hermite(dFockRC)\n\n#Fct building the nth harmonic eigenstate at coordinate x\n# sqrt(m*ωRC/ħ)*sqrt(2)*x in the Hermite function to recover the form of the physical Hermite polynomial in the atomic units\nfunction Ψ(n,x)\nreturn Acoef(n)*(m*ωRC/(ħ*π))^(1/4)*exp(-m*ωRC*(x)^2/(2*ħ))*PolyChaos.evaluate(n,(sqrt(m*ωRC/ħ)*sqrt(2)*x),αherm,βherm)\nend\n\n\n#Function to compute the pre factor of Ψ(n,x)\n  function Acoef(n)\n    if n == 0\n        return 1.0\n    else\n        return (1/sqrt(n*2))*2^(1/2)*Acoef(n-1)\n    end\nend\n\n#### Calculation of functions to express the reduced densty matrix in space. These functions build the matrix of oscillator eigenstates\n# Ψ(i,x) Ψ(j,x) ∀i,j ∈ dFockRC^2 and ∀ x ∈ xlist_rho\nPmatrix = zeros(length(xlist_rho),dFockRC)\nPmatrixnormed = zeros(length(xlist_rho),dFockRC)\nnormalization = zeros(dFockRC)\n\nfor n=1:dFockRC\n    for x=1:length(xlist_rho)\n        Pmatrix[x,n] = Ψ(n-1,xlist_rho[x])\n    end\nend\n\nfor n=1:dFockRC\n    normalization[n] = sqrt(sum(Pmatrix[k,n]^2 for k=1:length(xlist_rho)))\nend\n\n\nfor n=1:dFockRC\n    Pmatrixnormed[:,n] = Pmatrix[:,n]/normalization[n]\nend\n\n### Converts the reduced density matrix expressed in Fock states in a reduced density matrix expressed in space.\nρx=zeros(ComplexF64,length(xlist_rho),length(xlist_rho),length(dat[\"data/times\"]))\nfor t=1:length(dat[\"data/times\"])\n    print(\" \\n t = \", dat[\"data/times\"][t])\n    for x=1:length(xlist_rho)\n        for y=1:length(xlist_rho)\n            for n=1:dFockRC\n                for m=1:dFockRC\n                    ρx[x,y,t] +=  Pmatrixnormed[x,n]*Pmatrixnormed[y,m]*dat[\"data/Reduced ρ\"][n,m,t]\n                end\n            end\n        end\n    end\nend","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"The reduced density matrixi in space can then be illustrated over time. The diagonal elements represent the population while off-diagonal elements are coherences.","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#Creates a GIF of the diagonal elements of the reduced density over time, ie the population dynamics. Here, the reduced density matrix is expressed in space.\n\nanim = @animate for t=1:length(dat[\"data/times\"])\n    k=(dat[\"data/times\"][t])\n    print(\"\\n k = \", k)\n    plot(xlist_rho, abs.(diag(ρx[:,:,t])), title=\"Time = $k (arb. units)\", xlabel=L\"Diag( $||\\rho_{RC}(x,x)||$ )\", ylabel=\"Amplitude\", xlims=(-0.5, 0.5), ylims=(0,0.25),legend=:none)\nend\ndisplay(gif(anim, \"gif_population.gif\", fps = 2.5))\n","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"#Creates a GIF of the reduced density expressed over time. Here, the reduced density matrix is expressed in space. Diagonal elements are populations \n#whereas anti-diagonal elements  represent coherences\n\nanim = @animate for t=1:length(dat[\"data/times\"])\n    k=(dat[\"data/times\"][t])\n    print(\"\\n k = \", k)\n    (plot(heatmap(xlist_rho, xlist_rho, abs.(ρx[:,:,t]), c=:Blues ), title=\"Time = $k (arb. units)\", xlabel=L\"$x$ (arb. units)\",ylabel=L\"$x$ (arb. units)\", tickfontsize=(12),colorbar_title = L\"||\\rho_{RC}(x,x)||\", legend=:none, xlims=(-0.5, 0.5), ylims=(-0.5,0.5), clims=(0,0.18),aspect_ratio=:equal))\nend\ndisplay(gif(anim, \"gif_reducedrho.gif\", fps = 2.5))","category":"page"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"________________","category":"page"},{"location":"examples/protontransfer/#Bibliography","page":"Proton Transfer Model","title":"Bibliography","text":"","category":"section"},{"location":"examples/protontransfer/","page":"Proton Transfer Model","title":"Proton Transfer Model","text":"[lede_ESIPT_2024]: Le Dé, B.; Huppert, S.; Spezia, R.; Chin, A.W Extending Non-Perturbative Simulation Techniques for Open-Quantum Systems to Excited-State Proton Transfer and Ultrafast Non-Adiabatic Dynamics https://arxiv.org/abs/2405.08693","category":"page"},{"location":"methods/#List-of-all-methods","page":"Methods","title":"List of all methods","text":"","category":"section"},{"location":"methods/#Matrix-Product-State-(MPS)","page":"Methods","title":"Matrix Product State (MPS)","text":"","category":"section"},{"location":"methods/","page":"Methods","title":"Methods","text":"Modules = [MPSDynamics]\nPages = [\"mpsBasics.jl\", \"reshape.jl\",\"tensorOps.jl\", \"fundamentals.jl\", \"switchmpo\"]","category":"page"},{"location":"methods/#MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Int64}","page":"Methods","title":"MPSDynamics.apply1siteoperator!","text":"apply1siteoperator!(A, O, sites::Int)\n\nApply an operator O on the MPS A. O is acting on only one site ::Int. The resulting MPS A is the MPS modified by the operator O.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Vector{Int64}}","page":"Methods","title":"MPSDynamics.apply1siteoperator!","text":"apply1siteoperator!(A, O, sites::Vector{Int})\n\nApply an operator O on the MPS A. O is acting on several sites ::Vector{Int}. The resulting MPS A is the MPS modified by the operator O.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.applympo!-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.applympo!","text":"applympo!(A, H)\n\nApply an MPO H on the MPS A. H must have the same number of site than A. The resulting MPS A is the MPS modified by the MPO H.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.chainmps-Tuple{Int64, Int64, Int64}","page":"Methods","title":"MPSDynamics.chainmps","text":"chainmps(N::Int, site::Int, numex::Int)\n\nGenerate an MPS with numex excitations on site\n\nThe returned MPS will have bond-dimensions and physical dimensions numex+1\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.chainmps-Tuple{Int64, Vector{Int64}, Int64}","page":"Methods","title":"MPSDynamics.chainmps","text":"chainmps(N::Int, sites::Vector{Int}, numex::Int)\n\nGenerate an MPS with numex excitations of an equal super-position over sites\n\nThe returned MPS will have bond-dimensions and physical dimensions numex+1\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.displacedchainmps-Tuple{Vector{Any}, Int64, Int64, Any}","page":"Methods","title":"MPSDynamics.displacedchainmps","text":"displacedchainmps(A::Vector{Any}, N::Int, Nm::Int, γ::Any)\n\nGiven a MPS A, return a MPS B where the Nm-long chain is displaced by γ without displacing the N-long system.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.electron2kmps","page":"Methods","title":"MPSDynamics.electron2kmps","text":"electronkmps(N::Int, k::Vector{Int}, spin=:Up, chainparams=[fill(1.0,N), fill(1.0,N-1)])\n\nGenerate an MPS with 2 electrons in k-states k1 and k2.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.electronkmps","page":"Methods","title":"MPSDynamics.electronkmps","text":"electronkmps(N::Int, k::Int, spin=:Up, chainparams=[fill(1.0,N), fill(1.0,N-1)])\n\nGenerate an MPS for an electron with momentum k.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.elementmpo-Tuple{Any, Vararg{Any}}","page":"Methods","title":"MPSDynamics.elementmpo","text":"elementmpo(M, el...)\n\nReturn the element of the MPO M for the set of physical states el...\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.elementmps-Tuple{Any, Vararg{Any}}","page":"Methods","title":"MPSDynamics.elementmps","text":"elementmps(A, el...)\n\nReturn the element of the MPS A for the set of physical states el...\n\nExamples\n\njulia> A = chainmps(6, [2,4], 1);\n\njulia> elementmps(A, 1, 2, 1, 1, 1, 1)\n0.7071067811865475\n\njulia> elementmps(A, 1, 1, 1, 2, 1, 1)\n0.7071067811865475\n\njulia> elementmps(A, 1, 2, 1, 2, 1, 1)\n0.0\n\njulia> elementmps(A, 1, 1, 1, 1, 1, 1)\n0.0\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.entanglemententropy-Tuple{Any}","page":"Methods","title":"MPSDynamics.entanglemententropy","text":"entanglemententropy(A)\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the entanglement entropy for a bipartite cut for every bond.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.modemps","page":"Methods","title":"MPSDynamics.modemps","text":"modemps(N::Int, k::Vector{Int}, numex::Int, chainparams=[fill(1.0,N), fill(1.0,N-1)])\n\nGenerate an MPS with numex excitations of an equal superposition of modes k of a bosonic tight-binding chain.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.modemps-2","page":"Methods","title":"MPSDynamics.modemps","text":"modemps(N::Int, k::Int, numex::Int, chainparams=[fill(1.0,N), fill(1.0,N-1)])\n\nGenerate an MPS with numex excitations of mode k of a bosonic tight-binding chain. \n\nchainparams takes the form [e::Vector, t::Vector] where e are the on-site energies and t are the hoppping parameters.\n\nThe returned MPS will have bond-dimensions and physical dimensions numex+1\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.mpsembed!-Tuple{Vector, Int64}","page":"Methods","title":"MPSDynamics.mpsembed!","text":"mpsembed!(A::Vector, Dmax::Int)\n\nEmbed MPS A in manifold of max bond-dimension Dmax\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsleftnorm!","page":"Methods","title":"MPSDynamics.mpsleftnorm!","text":"mpsleftnorm!(A::Vector, jq::Int=length(A))\n\nLeft orthoganalise MPS A up to site jq.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.mpsmixednorm!-Tuple{Vector, Int64}","page":"Methods","title":"MPSDynamics.mpsmixednorm!","text":"mpsmixednorm!(A::Vector, OC::Int)\n\nPut MPS A into mixed canonical form with orthogonality centre on site OC.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsrightnorm!","page":"Methods","title":"MPSDynamics.mpsrightnorm!","text":"mpsrightnorm!(A::Vector, jq::Int=1)\n\nRight orthoganalise MPS A up to site jq.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.multiply-Tuple{Vector, Vector}","page":"Methods","title":"MPSDynamics.multiply","text":"multiply(M1::Vector, M2::Vector)\n\nCalculates M1*M2 where M1 and M2 are MPOs\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.normmps-Tuple{Vector}","page":"Methods","title":"MPSDynamics.normmps","text":"normmps(A::Vector; mpsorthog=:None)\n\nCalculate norm of MPS A.\n\nSetting mpsorthog=:Right/:Left will calculate the norm assuming right/left canonical form. Setting mpsorthog=OC::Int will cause the norm to be calculated assuming the orthoganility center is on site OC. If mpsorthog is :None the norm will be calculated as an MPS-MPS product.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.orthcentersmps-Tuple{Vector}","page":"Methods","title":"MPSDynamics.orthcentersmps","text":"orthcentersmps(A::Vector)\n\nCompute the orthoganality centres of MPS A.\n\nReturn value is a list in which each element is the corresponding site tensor of A with the orthogonality centre on that site. Assumes A is right normalised.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.physdims-Tuple{Vector}","page":"Methods","title":"MPSDynamics.physdims","text":"physdims(M)\n\nReturn the physical dimensions of an MPS or MPO M.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.productstatemps","page":"Methods","title":"MPSDynamics.productstatemps","text":"productstatemps(physdims::Dims, Dmax=1; state=:Vacuum, mpsorthog=:Right)\n\nReturn an MPS representing a product state with local Hilbert space dimensions given by physdims.\n\nBy default all bond-dimensions will be 1 since the state is a product state. However, to embed the product state in a manifold of greater bond-dimension, Dmax can be set accordingly.\n\nThe indvidual states of the MPS sites can be provided by setting state to a list of column vectors. Setting state=:Vacuum will produce an MPS in the vacuum state (where the state of each site is represented by a column vector with a 1 in the first row and zeros elsewhere). Setting state=:FullOccupy will produce an MPS in which each site is fully occupied (ie. a column vector with a 1 in the last row and zeros elsewhere).\n\nThe argument mpsorthog can be used to set the gauge of the resulting MPS.\n\nExample\n\njulia> ψ = unitcol(1,2); d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1\n\njulia> cpars = chaincoeffs_ohmic(N, α, s)\n\njulia> H = spinbosonmpo(ω0, Δ, d, N, cpars)\n\njulia> A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.productstatemps-2","page":"Methods","title":"MPSDynamics.productstatemps","text":"productstatemps(N::Int, d::Int, Dmax=1; state=:Vacuum, mpsorthog=:Right)\n\nReturn an N-site MPS with all local Hilbert space dimensions given by d. \n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.randmps","page":"Methods","title":"MPSDynamics.randmps","text":"randmps(N::Int, d::Int, Dmax::Int, T=Float64)\n\nConstruct a random, N-site, right-normalised MPS with all local Hilbert space dimensions given by d.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.randmps-Union{Tuple{N}, Tuple{Tuple{Vararg{Int64, N}}, Int64}, Tuple{Tuple{Vararg{Int64, N}}, Int64, Type{<:Number}}} where N","page":"Methods","title":"MPSDynamics.randmps","text":"randmps(physdims::Dims{N}, Dmax::Int, T::Type{<:Number} = Float64) where {N}\n\nConstruct a random, right-normalised MPS with local Hilbert space dimensions given by physdims and max bond-dimension given by Dmax. \n\nT specifies the element type, eg. use T=ComplexF64 for a complex valued MPS.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.reversempo!-Tuple{Any}","page":"Methods","title":"MPSDynamics.reversempo!","text":"reversempo!(M)\n\nReverse the left and right dimensions of the MPO M. The resulting MPO M is the reversed MPO.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.reversempo-Tuple{Any}","page":"Methods","title":"MPSDynamics.reversempo","text":"reversempo(M)\n\nReverse the left and right dimensions of the MPO M.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.reversemps!-Tuple{Any}","page":"Methods","title":"MPSDynamics.reversemps!","text":"reversemps!(A)\n\nReverse the left and right dimensions of the MPS A. The resulting MPS A is the reversed MPS.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.reversemps-Tuple{Any}","page":"Methods","title":"MPSDynamics.reversemps","text":"reversemps(A)\n\nReverse the left and right bond-dimensions of the MPS A.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.svdmps-Tuple{Any}","page":"Methods","title":"MPSDynamics.svdmps","text":"svdmps(A)\n\nFor a right normalised mps A compute the full svd spectrum for a bipartition at every bond.\n\n\n\n\n\n","category":"method"},{"location":"methods/#Base.reshape-Tuple{Number, Vararg{Any}}","page":"Methods","title":"Base.reshape","text":"Base.reshape(x::Number, dims...)\n\nReshape any matrix with dimensions \"dims\"\n\nExample\n\njulia> A = rand(2,3,2)\n\njulia> B = reshape(A,4,3)\n\njulia> size(B) == (4,3) \n\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.svdtrunc-Tuple{Any}","page":"Methods","title":"MPSDynamics.svdtrunc","text":"U, S, Vd = svdtrunc(A; truncdim = max(size(A)...), truncerr = 0.)\n\nPerform a truncated SVD, with maximum number of singular values to keep equal to truncdim or truncating any singular values smaller than truncerr. If both options are provided, the smallest number of singular values will be kept. Unlike the SVD in Julia, this returns matrix U, a diagonal matrix (not a vector) S, and Vt such that A ≈ U * S * Vt\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.MPOtoVector-Tuple{ITensors.ITensorMPS.MPO}","page":"Methods","title":"MPSDynamics.MPOtoVector","text":"MPOtoVector(mpo::MPO)\n\nConvert an ITensors chain MPO into a form compatible with MPSDynamics\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.chainprop-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.chainprop","text":"chainprop(t, cparams)\n\nPropagate an excitation placed initially on the first site of a tight-binding chain with parameters given by cparams for a time t and return occupation expectation for each site.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.cosineh-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.cosineh","text":"cosineh(omega, bet)\n\nCalculates the hyperbolic cosine function function based on the input parameters,  for the Bogoliubov transformation necessary for the thermofield transformation.\n\nArguments\n\nomega::Float64: The frequency parameter.\nbet::Float64: The beta parameter.\n\nReturns\n\nFloat64: The result of the modified cosine function.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.disp-Tuple{Any, Any, Any}","page":"Methods","title":"MPSDynamics.disp","text":"disp(d,ωvib,m)\n\nDisplacement operator  X = fracsqrt22sqrtm omega_vib(a + a^dagger)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.disp-Tuple{Any}","page":"Methods","title":"MPSDynamics.disp","text":"disp(d)\n\nMass and frequency-weighted displacement operator  X = frac12(a + a^dagger)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.dynamap-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.dynamap","text":"dynamap(ps1,ps2,ps3,ps4)\n\nCalculate complete dynamical map to time step at which ps1, ps2, ps3 and ps4 are specified.\n\nArguments\n\nps1 : time evolved system density matrix starting from initial state up\nps2 : time evolved system density matrix starting from initial state down\nps3 : time evolved system density matrix starting from initial state (up + down)/sqrt(2)\nps4 : time evolved system density matrix starting from initial state (up - i*down)/sqrt(2)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.eigenchain-Tuple{Any}","page":"Methods","title":"MPSDynamics.eigenchain","text":"eigenchain(cparams; nummodes=nothing)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.endsiteocc-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.endsiteocc","text":"endsiteocc(t, cparams)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.entropy-Tuple{Any}","page":"Methods","title":"MPSDynamics.entropy","text":"entropy(rho)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.findchainlength","page":"Methods","title":"MPSDynamics.findchainlength","text":"findchainlength(T, ωc::Float64, β=Inf)\n\nEstimate length of chain using universal asymptotic properties of chain mapped environments given the simulation time T, the bath cut-off frequency ωc, and the inverse temperature β.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.findchainlength-Tuple{Any, Vector}","page":"Methods","title":"MPSDynamics.findchainlength","text":"findchainlength(T, cparams::Vector; eps=10^-6, verbose=false)\n\nEstimate length of chain required for a particular set of chain parameters by calculating how long an excitation on the first site takes to reach the end. The chain length is given as the length required for the excitation to have just reached the last site after time T. The initial number of sites in cparams has to be larger than the findchainlength result.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measurecorrs-Tuple{Any, Vector, Vector}","page":"Methods","title":"MPSDynamics.measurecorrs","text":"measurecorrs(oper, , e::Vector, t::Vector)\n\nParameters\n\noper: Square matrix (Matrix{Float64}) representing the operator to be transformed. e: Vector (Vector{Float64}) of diagonal (on-site energy) chain coefficients. t: Vector (Vector{Float64}) of off-diagonal (hopping terms) chain coefficients.\n\nReturns\n\nMatrix{Float64}: This matrix is the operator oper transformed back from the chain  representation to the representation corresponding to the extended bath. The resulting  operator represents quantities like mode occupations or other properties in the basis  of environmental modes associated with specific frequencies omega_i.\n\nDescription\n\nThis function performs a basis transformation of the operator oper. Specifically,  this transformation reverses the unitary transformation that maps the extended bath Hamiltonian into the chain representation. \n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measuremodes","page":"Methods","title":"MPSDynamics.measuremodes","text":"measuremodes(adaga, e=1.0, t=1.0)\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.measuremodes-Tuple{Any, Tuple{Int64, Int64}, AbstractArray}","page":"Methods","title":"MPSDynamics.measuremodes","text":"measuremodes(A, chainsection::Tuple{Int64,Int64}, U::AbstractArray)\n\nfor longer chains it can be worth calculating U in advance\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measuremodes-Tuple{Any, Tuple{Int64, Int64}, Vector{Float64}, Vector{Float64}}","page":"Methods","title":"MPSDynamics.measuremodes","text":"measuremodes(A, chainsection::Tuple{Int64,Int64}, e::Array{Float64,1}, t::Array{Float64,1})\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measuremodes-Tuple{Any, Vector, Vector}","page":"Methods","title":"MPSDynamics.measuremodes","text":"measuremodes(adaga, e::Vector, t::Vector)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.physical_occup-NTuple{6, Any}","page":"Methods","title":"MPSDynamics.physical_occup","text":"physical_occup(corr_constr, corr_destr, omega, occup, b, M)\n\nCalculates the physical occupation based on correlation matrices, omega values,  and other parameters. The physical occupation in the original frequency environment is computed by reverting the thermofield transformation.\n\nArguments\n\ncorr_constr::Matrix{ComplexF64}: The correlation construction matrix.\ncorr_destr::Matrix{ComplexF64}: The correlation destruction matrix.\nomega::Vector{Float64}: The omega values.\noccup::Matrix{Float64}: The occupation matrix.\nb::Float64: The beta parameter.\nM::Int: The number of points for interpolation.\n\nReturns\n\nVector{Float64}: The physical occupation values.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.randisometry-Tuple{Type, Int64, Int64}","page":"Methods","title":"MPSDynamics.randisometry","text":"randisometry([T=Float64], dims...)\n\nConstruct a random isometry\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.rmsd-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.rmsd","text":"rmsd(ob1, ob2)\n\nCalculate the root mean squared difference between two measurements of an observable over the same time period.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.sineh-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.sineh","text":"sineh(omega, bet)\n\nCalculates the hyperbolic sine function function based on the input parameters,  for the Bogoliubov transformation necessary for the thermofield transformation.\n\nArguments\n\nomega::Float64: The frequency parameter.\nbet::Float64: The beta parameter.\n\nReturns\n\nFloat64: The result of the modified cosine function.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.therHam-Tuple{Any, Any, Any}","page":"Methods","title":"MPSDynamics.therHam","text":" therHam(psi, site1, site2)\n\nCalculates Hβ such that ρ = e^(-βH) for some density matrix ρ obatined from tracing out everything outside the range [site1,site2] in the MPS psi\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.thermaloccupations-Tuple{Any, Vararg{Any}}","page":"Methods","title":"MPSDynamics.thermaloccupations","text":"thermaloccupations(β, cparams...)\n\n\n\n\n\n","category":"method"},{"location":"methods/#Tree-Tensor-Network-(TTN)","page":"Methods","title":"Tree Tensor Network (TTN)","text":"","category":"section"},{"location":"methods/","page":"Methods","title":"Methods","text":"Modules = [MPSDynamics]\nPages = [\"treeBasics.jl\", \"treeTDVP.jl\",\"treeMeasure.jl\", \"treeIterators.jl\", \"treeTDVP.jl\"]","category":"page"},{"location":"methods/#MPSDynamics.addchild!-Tuple{MPSDynamics.Tree, Int64}","page":"Methods","title":"MPSDynamics.addchild!","text":"addchild!(tree::Tree, id::Int)\n\nAdd child to node id of tree.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.addchildren!-Tuple{MPSDynamics.Tree, Int64, Int64}","page":"Methods","title":"MPSDynamics.addchildren!","text":"addchildren!(tree::Tree, id::Int, n::Int)\n\nAdd n children to node id of tree.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.findchild-Tuple{MPSDynamics.TreeNode, Int64}","page":"Methods","title":"MPSDynamics.findchild","text":"findchild(node::TreeNode, id::Int)\n\nReturn integer corresponding to the which number child site id is of node.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.randtree-Tuple{Int64, Int64}","page":"Methods","title":"MPSDynamics.randtree","text":"randtree(numnodes::Int, maxdegree::Int)\n\nConstruct a random tree with nummodes modes and max degree maxdegree.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure1siteoperator-Tuple{MPSDynamics.TreeNetwork, Any, Int64}","page":"Methods","title":"MPSDynamics.measure1siteoperator","text":"measure1siteoperator(net::TreeNetwork, O, id::Int)\n\nFor a Tree, compute the local expectation value of a one-site operator O for site id.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure1siteoperator-Tuple{MPSDynamics.TreeNetwork, Any, Tuple{Int64, Int64}}","page":"Methods","title":"MPSDynamics.measure1siteoperator","text":"measure1siteoperator(net::TreeNetwork, O, sites::Tuple{Int,Int})\n\nFor a Tree, compute the local expectation value of a one-site operator O for the specified site range.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure2siteoperator-Tuple{MPSDynamics.TreeNetwork, Any, Any, Tuple{Int64, Int64}}","page":"Methods","title":"MPSDynamics.measure2siteoperator","text":"measure2siteoperator(net::TreeNetwork, O1, O2, sites::Tuple{Int,Int})\n\nFor a Tree, compute the local expectation value of two one-site operators O1 and O2 for the specified site range.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.bonddims-Tuple{MPSDynamics.TreeNetwork}","page":"Methods","title":"MPSDynamics.bonddims","text":"bonddims(A::TreeNetwork)\n\nReturn the bon-dimension of a tree-MPS A.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsembed!-Tuple{MPSDynamics.TreeNetwork, Int64}","page":"Methods","title":"MPSDynamics.mpsembed!","text":"mpsembed(A::TreeNetwork, Dmax::Int)\n\nEmbed tree-MPS A in manifold of max bond-dimension Dmax.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsmixednorm!-Tuple{MPSDynamics.TreeNetwork, Int64}","page":"Methods","title":"MPSDynamics.mpsmixednorm!","text":"mpsmixednorm!(A::TreeNetwork, id::Int)\n\nNormalise tree-MPS A such that orthogonality centre is on site id.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsmoveoc!-Tuple{MPSDynamics.TreeNetwork, Int64}","page":"Methods","title":"MPSDynamics.mpsmoveoc!","text":"mpsmoveoc!(A::TreeNetwork, id::Int)\n\nMove the orthogonality centre of right normalised tree-MPS A to site id.\n\nThis function will be more efficient than using mpsmixednorm! if the tree-MPS is already right-normalised.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsrightnorm!-Tuple{MPSDynamics.TreeNetwork}","page":"Methods","title":"MPSDynamics.mpsrightnorm!","text":"mpsrightnorm!(A::TreeNetwork)\n\nWhen applied to a tree-MPS, right normalise towards head-node.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsshiftoc!-Tuple{MPSDynamics.TreeNetwork, Int64}","page":"Methods","title":"MPSDynamics.mpsshiftoc!","text":"mpsshiftoc!(A::TreeNetwork, newhd::Int)\n\nShift the orthogonality centre by one site, setting new head-node newhd.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.normmps-Tuple{MPSDynamics.TreeNetwork}","page":"Methods","title":"MPSDynamics.normmps","text":"normmps(net::TreeNetwork; mpsorthog=:None)\n\nWhen applied to a tree-MPS mpsorthog=:Left is not defined.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.physdims-Tuple{MPSDynamics.TreeNetwork}","page":"Methods","title":"MPSDynamics.physdims","text":"physdims(M::TreeNetwork)\n\nReturn the physical dimensions of a tree-MPS or tree-MPO M.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.productstatemps-3","page":"Methods","title":"MPSDynamics.productstatemps","text":"productstatemps(tree_::Tree, physdims::Dims, Dmax::Int=1; state=:Vacuum)\n\nReturn a tree-MPS representing a product state with local Hilbert space dimensions given by physdims.\n\nBy default all bond-dimensions will be 1 since the state is a product state. However, to embed the product state in a manifold of greater bond-dimension, Dmax can be set accordingly.\n\nThe indvidual states of the MPS sites can be provided by setting state to a list of column vectors. Setting state=:Vacuum will produce an MPS in the vacuum state (where the state of each site is represented by a column vector with a 1 in the first row and zeros elsewhere). Setting state=:FullOccupy will produce an MPS in which each site is fully occupied (ie. a column vector with a 1 in the last row and zeros elsewhere).\n\nExample\n\njulia> ψ = unitcol(1,2); d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1\n\njulia> cpars = chaincoeffs_ohmic(N, α, s)\n\njulia> H = spinbosonmpo(ω0, Δ, d, N, cpars, tree=true)\n\njulia> A = productstatemps(H.tree, physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # tree-MPS representation of |ψ>|Vacuum>\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.randmps-2","page":"Methods","title":"MPSDynamics.randmps","text":"randmps(tree::Tree, physdims, Dmax::Int, T::Type{<:Number} = Float64)\n\nConstruct a random, right-normalised, tree-MPS, with structure given by tree and max bond-dimension given by Dmax.\n\nThe local Hilbert space dimensions are specified by physdims which can either be of type Dims{length(tree)}, specifying the dimension of each site, or of type Int, in which case the same local dimension is used for every site.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.tdvp1sweep!-Tuple{Any, MPSDynamics.TreeNetwork, MPSDynamics.TreeNetwork, Vector, Int64}","page":"Methods","title":"MPSDynamics.tdvp1sweep!","text":"tdvp1sweep!(dt, A::TreeNetwork, M::TreeNetwork, F::Vector, id::Int; verbose=false, kwargs...)\n\nPropagates the tree-MPS A with the tree-MPO M following the 1-site TDVP method. The sweep is done back and forth with a time step dt/2. F represents the merged left and right parts of the site being propagated.  \n\n\n\n\n\n","category":"method"},{"location":"methods/#Measure-and-Obervables","page":"Methods","title":"Measure and Obervables","text":"","category":"section"},{"location":"methods/","page":"Methods","title":"Methods","text":"Modules = [MPSDynamics]\nPages = [\"measure.jl\", \"observables.jl\"]","category":"page"},{"location":"methods/#MPSDynamics.OneSiteObservable-Tuple{Any, Any, Any}","page":"Methods","title":"MPSDynamics.OneSiteObservable","text":"OneSiteObservable(name,op,sites)\n\nComputes the local expectation value of the one-site operator op on the specified sites. Used to define one-site observables that are obs and convobs parameters for the runsim function.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.OneSiteObservable-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.OneSiteObservable","text":"OneSiteObservable(name,op)\n\nComputes the local expectation value of the one-site operator op on the every site. Used to define one-site observables that are obs and convobs parameters for the runsim function.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.TwoSiteObservable","page":"Methods","title":"MPSDynamics.TwoSiteObservable","text":"TwoSiteObservable(name,op1,op2,sites1=nothing,sites2=nothing)\n\nComputes the local expectation value of operators op1 and op2 where op1 acts on sites1 and op2 acts on sites2. Used to define several-site observables that are obs and convobs parameters for the runsim function.\n\n\n\n\n\n","category":"type"},{"location":"methods/#MPSDynamics.measure-Tuple{Any, OneSiteObservable}","page":"Methods","title":"MPSDynamics.measure","text":"measure(A, O; kwargs...)\n\nMeasure observable O on mps state A\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure-Tuple{Vector, Vector}","page":"Methods","title":"MPSDynamics.measure","text":" measure(A::Vector, Os::Vector; kwargs...)\n\nCaculate expectation of Os on MPS A.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Int64}","page":"Methods","title":"MPSDynamics.measure1siteoperator","text":"measure1siteoperator(A::Vector, O, site::Int)\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for a single site.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Tuple{Int64, Int64}}","page":"Methods","title":"MPSDynamics.measure1siteoperator","text":"measure1siteoperator(A::Vector, O, chainsection::Tuple{Int64,Int64})\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for a chainsection.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Vector{Int64}}","page":"Methods","title":"MPSDynamics.measure1siteoperator","text":"measure1siteoperator(A::Vector, O, sites::Vector{Int})\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for every site or just one if it is specified.\n\nFor calculating operators on single sites this will be more efficient if the site is on the left of the mps.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure1siteoperator-Tuple{Vector, Any}","page":"Methods","title":"MPSDynamics.measure1siteoperator","text":"measure1siteoperator(A::Vector, O)\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for every site.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure2siteoperator-Tuple{Vector, Any, Any, Int64, Int64}","page":"Methods","title":"MPSDynamics.measure2siteoperator","text":" measure2siteoperator(A::Vector, M1, M2, j1, j2)\n\nCaculate expectation of M1*M2 where M1 acts on site j1 and M2 acts on site j2, assumes A is right normalised.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure2siteoperator-Tuple{Vector, Any, Any, Vector{Int64}, Vector{Int64}}","page":"Methods","title":"MPSDynamics.measure2siteoperator","text":" measure2siteoperator(A::Vector, M1, M2, sites1::Vector{Int}, sites2::Vector{Int})\n\nCaculate expectation of M1*M2 where M1 acts on sites1 and M2 acts on sites2, assumes A is right normalised.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measurempo-Tuple{Vector, Vector, Tuple{Int64, Int64}}","page":"Methods","title":"MPSDynamics.measurempo","text":"measurempo(A::Vector, M::Vector, sites::Tuples{Int,Int})\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the local expectation value of the MPO M on specified sites.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measurempo-Tuple{Vector, Vector}","page":"Methods","title":"MPSDynamics.measurempo","text":"measurempo(A::Vector, M::Vector)\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the local expectation value of the MPO M on every site.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.rhoreduced_1site","page":"Methods","title":"MPSDynamics.rhoreduced_1site","text":" rhoreduced_1site(A::Vector, site::Int=1)\n\nCaculate the reduced density matrix of the MPS A at the specified site.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.rhoreduced_2sites-Tuple{Vector, Tuple{Int64, Int64}}","page":"Methods","title":"MPSDynamics.rhoreduced_2sites","text":" rhoreduced_2sites(A::Vector, site::Tuple{Int, Int})\n\nCaculate the reduced density matrix of the MPS A of two neigbour sites. The resulting dimensions will be the four physical dimensions in total,  corresponding to the dimensions  of the two sites\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure-Tuple{Any, FockError}","page":"Methods","title":"MPSDynamics.measure","text":"measure(A, obs::FockError; t=0, kwargs...)\n\nReturn the measure of the observable obs on the MPS A. \n\n\n\n\n\n","category":"method"},{"location":"methods/#Models-and-Hamiltonians-(MPO)","page":"Methods","title":"Models and Hamiltonians (MPO)","text":"","category":"section"},{"location":"methods/","page":"Methods","title":"Methods","text":"Modules = [MPSDynamics]\nPages = [\"models.jl\", \"electronkmpo.jl\", \"excitonphononmpo.jl\"]","category":"page"},{"location":"methods/#MPSDynamics.chaincoeffs_ohmic-Tuple{Any, Any, Any}","page":"Methods","title":"MPSDynamics.chaincoeffs_ohmic","text":"chaincoeffs_ohmic(N, α, s; ωc=1, soft=false)\n\nGenerate chain coefficients ϵ_0ϵ_1t_0t_1c_0 for an Harmonic bath at zero temperature with a power law spectral density given by:\n\nsoft cutoff: J(ω) = 2αω_c (fracωω_c)^s exp(-ωω_c) \n\nhard cutoff: J(ω) = 2αω_c (fracωω_c)^s θ(ω-ω_c)\n\nThe coefficients parameterise the chain Hamiltonian\n\nH = H_S + c_0 A_SB_0+sum_i=0^N-1t_i (b_i+1^dagger b_i +hc) + sum_i=0^N ϵ_ib_i^dagger b_i\n\nwhich is unitarily equivalent (before the truncation to N sites) to\n\nH = H_S + A_Sint_0^dωsqrtJ(ω)B_ω + int_0^dωωb_ω^dagger b_ω.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.correlatedenvironmentmpo-Tuple{Vector, Int64, Int64}","page":"Methods","title":"MPSDynamics.correlatedenvironmentmpo","text":"correlatedenvironmentmpo(R::Vector, Nm::Int, d::Int; chainparams, fnamecc::String, s=1, α=1, ωc=1, c_phonon=1, β=\"inf\", issoft=false)\n\nGenerate a MPO for a one-dimensional bosonic bath spatially correlated to a multi-component system \n\nH_B + H_int = int_-^+ dk ω_k b_k^dagger b_k + _j int_-^+dk sqrtJ(k)(A_j b_k e^i k R_j + hc).\n\nThe interactions between the system and the chain-mapped bath are long range, i.e. each site interacts with all the chain modes. The spectral density is assumed to be Ohmic J(ω) = 2αωc(ωωc)^s.\n\nArguments\n\nR: List of system's components positions\nNm: Number of chain modes. The actual number of mode will be doubled to account for the left and right moving excitations.\nd: Local Hilbert space dimension of the bath modes\nchainparams: chain parameters, of the form chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature.\nfnamecc: Path to a file containing pre-computed long-range coupling coefficient. If not provided, the coupling coefficients will be computed and stored.\ns: Ohmicity\nα: Kondo parameter\nωc: Bath cut-off frequency\nc_phonon: Speed of sound in the bath\nβ: Inverse temperature \nissoft: Is the cut-off of the Ohmic SD soft or hard?\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.hbathchain-Tuple{Int64, Int64, Any, Vararg{Any}}","page":"Methods","title":"MPSDynamics.hbathchain","text":"hbathchain(N::Int, d::Int, chainparams, longrangecc...; tree=false, reverse=false, coupletox=false)\n\nGenerate MPO representing a tight-binding chain of N oscillators with d Fock states each. Chain parameters are supplied in the standard form: chainparams =ϵ_0ϵ_1t_0t_1c_0. The output does not itself represent a complete MPO but will possess an end which is open and should be attached to another tensor site, usually representing the system.\n\nArguments\n\nreverse: If reverse=true create a chain were the last (i.e. Nth) site is the site which couples to the system\ncoupletox: Used to choose the form of the system coupling. coupletox=true gives a non-number conserving coupling of the form H_textI= A_textS(b_0^dagger + b_0) where A_textS is a system operator, while coupletox=false gives the number-converving coupling H_textI=(A_textS b_0^dagger + A_textS^dagger b_0)\ntree: If true the resulting chain will be of type TreeNetwork; useful for construcing tree-MPOs \n\nExample\n\nOne can constuct a system site tensor to couple to a chain by using the function up to populate the tensor. For example, to construct a system site with Hamiltonian Hs and coupling operator As, the system tensor M is constructed as follows for a non-number conserving interaction:\n\nu = one(Hs) # system identity\nM = zeros(1,3,2,2)\nM[1, :, :, :] = up(Hs, As, u)\n\nThe full MPO can then be constructed with:\n\nHmpo = [M, hbathchain(N, d, chainparams, coupletox=true)...]\n\nSimilarly for a number conserving interaction the site tensor would look like:\n\nu = one(Hs) # system identity\nM = zeros(1,4,2,2)\nM[1, :, :, :] = up(Hs, As, As', u)\n\nAnd the full MPO would be\n\nHmpo = [M, hbathchain(N, d, chainparams; coupletox=false)...]\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.heisenbergmpo","page":"Methods","title":"MPSDynamics.heisenbergmpo","text":"heisenbergmpo(N::Int, J=1.0) = xyzmpo(N; Jx=J)\n\nGenerate MPO for the N-spin Heisenberg XXX model, defined by the Hamiltonian\n\nH = sum_n=1^N-1 -J σ_x^n σ_x^n+1 - J σ_y^n σ_y^n+1 - J σ_z^n σ_z^n+1\n\nwith σ_x^n σ_y^n σ_z^n the Pauli spin-1/2 matrices of the n^textth site.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.ibmmpo-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.ibmmpo","text":"ibmmpo(ω0, d, N, chainparams; tree=false)\n\nGenerate MPO for a spin-1/2 coupled to a chain of harmonic oscillators with the interacting boson model (IBM), defined by the Hamiltonian\n\nH = fracω_02σ_z +  c_0σ_z(b_0^dagger+b_0) + sum_i=0^N-1 t_i (b_i+1^dagger b_i +hc) + sum_i=0^N ϵ_ib_i^dagger b_i.\n\nThe spin is on site 1 of the MPS and the bath modes are to the right.\n\nThis Hamiltonain is unitarily equivalent (before the truncation to N sites) to the spin-boson Hamiltonian defined by\n\nH =  fracω_02σ_z + σ_zint_0^ dωsqrtJ(ω)(b_ω^dagger+b_ω) + int_0^ dω ωb_ω^dagger b_ω.\n\nThe chain parameters, supplied by chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.interleaved_tightbinding_mpo-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.interleaved_tightbinding_mpo","text":"interleaved_tightbinding_mpo(N, ϵd, chainparams1, chainparams2)\n\nGenerate MPO for a tight-binding chain of N fermionic sites with a single impurity site (fermionic as well) of energy ϵd. The impurity is coupled to two leads, each described by a set of chain parameters.  The interactions are next-nearest-neighbour, with the first site corresponding to the impurity, and the two chains organised in an interleaved fashion.\n\nArguments\n\nN::Int: number of sites in the chain\nϵd::Real: energy of the impurity site at the first site, as Ed - μ, where μ is the chemical potential\nchainparams1::Array{Real,1}: chain parameters for the first lead\nchainparams2::Array{Real,1}: chain parameters for the second lead\n\nThe chain parameters are given in the standard form: chainparams =ϵ_0ϵ_1t_0t_1c_0.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.isingmpo-Tuple{Int64}","page":"Methods","title":"MPSDynamics.isingmpo","text":"isingmpo(N; J=1.0, h=1.0)\n\nGenerate MPO for the N-spin 1D Ising model with external field vech = (00h), defined by the Hamiltonian\n\nH = sum_n=1^N-1 -J_x σ_x^n σ_x^n+1 + sum_n=1^N(- h_z σ_z^n)\n\nwith σ_x^n σ_y^n σ_z^n the Pauli spin-1/2 matrices of the n^textth site.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.longrange_isingmpo","page":"Methods","title":"MPSDynamics.longrange_isingmpo","text":"longrange_isingmpo(N::Int, α::Float64=0.; J=1.0, h=1.0) = longrange_xyzmpo(N, α; Jx=J, Jy=0., Jz=0., hz=h, hx=0.)\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.longrange_xyzmpo","page":"Methods","title":"MPSDynamics.longrange_xyzmpo","text":"longrange_xyzmpo(N::Int, α::Float64=0.; Jx=1.0, Jy=Jx, Jz=Jx, hx=0., hz=0.)\n\nGennerate MPO for the N-spin long-range XYZ model with external field vech=(h_x 0 h_z), , defined by the Hamiltonian\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.multisitempo","page":"Methods","title":"MPSDynamics.multisitempo","text":"multisitempo(N::Int, E=[]::Vector, J=[]::Vector, As=[]::Vector)\n\nGenerate a MPO for a N-site multi-component system to be coupled with a correlated environment.\n\nArguments:\n\nN: number of sites\nE: list of on-site energies\nJ: list of sites tunnelling energies\nAs: list of system operators coupling to the environment\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.nearestneighbourmpo","page":"Methods","title":"MPSDynamics.nearestneighbourmpo","text":"nearestneighbourmpo(N::Int, h0, A, Ad = A')\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.nearestneighbourmpo-2","page":"Methods","title":"MPSDynamics.nearestneighbourmpo","text":"nearestneighbourmpo(tree_::Tree, h0, A, Ad = A')\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.protontransfermpo-NTuple{11, Any}","page":"Methods","title":"MPSDynamics.protontransfermpo","text":"protontransfermpo(ω0e,ω0k,x0e,x0k, Δ, dRC, d, N, chainparams, RCparams, λreorg)\n\nGenerate a MPO for a system described in space with a reaction coordinate (RC) tensor. The Hamiltonian of the two-level system and of the reaction coordinate tensor reads \n\nH_S + H_RC + H_int^S-RC = omega^0_e erangle langle e + omega^0_k krangle langle k + Delta (erangle langle k + krangle langle e) + omega_RC (d^daggerd + frac12) + g_e erangle langle e( d + d^dagger)+ g_k k rangle langle k( d + d^dagger) The RC tensor is coupled to a bosonic bath, taking into account the induced reorganization energy H_B + H_int^RC-B = int_-^+ dk ω_k b_k^dagger b_k - (d + d^dagger)int_0^ dωsqrtJ(ω)(b_ω^dagger+b_ω) + lambda_reorg(d + d^dagger)^2 with lambda_reorg = int fracJ(omega)omegadomega\n\nArguments\n\nω0e: enol energy at reaction coordinate value x=0 \nω0k: keto energy at reaction coordinate value x=0\nx0e: enol equilibrium displacement\nx0k: keto equilibrium displacement \nΔ: direct coupling between enol and keto\ndRC: fock space of the RC tensor \nd: number of Fock states of the chain modes\nN: length of the bosonic chain\nchainparams: chain parameters, of the form chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature. \nRCparams: RC tensor parameter, of the form RCparams=ω_RC-gx \nλreorg: reorganization energy\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.puredephasingmpo-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.puredephasingmpo","text":"puredephasingmpo(ΔE, dchain, Nchain, chainparams; tree=false)\n\nGenerate MPO for a pure dephasing model, defined by the Hamiltonian H = fracΔE2 σ_z +  fracσ_z2 c_0 (b_0^dagger + b_0) + sum_i=0^N-1 t_i (b_i+1^dagger b_i +hc) + sum_i=0^N-1 ϵ_i b_i^dagger b_i\n\nThe spin is on site 1 of the MPS and the bath modes are to the right.\n\nArguments\n\nΔE::Real: energy splitting of the spin\ndchain::Int: physical dimension of the chain sites truncated Hilbert spaces\nNchain::Int: number of sites in the chain\nchainparams::Array{Real,1}: chain parameters for the bath chain. The chain parameters are given in the standard form: chainparams =ϵ_0ϵ_1t_0t_1c_0.\ntree::Bool: if true, return a TreeNetwork object, otherwise return a vector of MPO tensors\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.readchaincoeffs-Tuple{Any, Vararg{Any}}","page":"Methods","title":"MPSDynamics.readchaincoeffs","text":"readchaincoeffs(fdir, params...)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.spinbosonmpo-NTuple{5, Any}","page":"Methods","title":"MPSDynamics.spinbosonmpo","text":"spinbosonmpo(ω0, Δ, d, N, chainparams; rwa=false, tree=false)\n\nGenerate MPO for a spin-1/2 coupled to a chain of harmonic oscillators, defined by the Hamiltonian\n\nH = fracω_02σ_z + Δσ_x + c_0σ_x(b_0^dagger+b_0) + sum_i=0^N-1 t_i (b_i+1^dagger b_i +hc) + sum_i=0^N ϵ_ib_i^dagger b_i.\n\nThe spin is on site 1 of the MPS and the bath modes are to the right.\n\nThis Hamiltonain is unitarily equivalent (before the truncation to N sites) to the spin-boson Hamiltonian defined by\n\nH =  fracω_02σ_z + Δσ_x + σ_xint_0^ dωsqrtJ(ω)(b_ω^dagger+b_ω) + int_0^ dω ωb_ω^dagger b_ω.\n\nThe chain parameters, supplied by chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature.\n\nThe rotating wave approximation can be made by setting rwa=true.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.spinchainmpo-Tuple{Int64}","page":"Methods","title":"MPSDynamics.spinchainmpo","text":"spinchainmpo(N::Int; J=1.0, hz=1.0, hx=0.0, i=div(N,2))\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.tightbinding_mpo-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.tightbinding_mpo","text":"tightbinding_mpo(N, ϵd, chainparams1, chainparams2)\n\nGenerate MPO for a tight-binding chain of N fermionic sites with a single impurity site (fermionic as well)  of energy ϵd at the center. The impurity is coupled to two leads, each described by a set of chain parameters. The interactions are nearest-neighbour, with the first N/2-1 sites corresponding to the first lead, the Nth site corresponding to the impurity, and the rest of the sites corresponding to the second lead.\n\nArguments\n\nN::Int: number of sites in the chain\nϵd::Real: energy of the impurity site at the center, as Ed - μ, where μ is the chemical potential\nchainparams1::Array{Real,1}: chain parameters for the first lead\nchainparams2::Array{Real,1}: chain parameters for the second lead\n\nThe chain parameters are given in the standard form: chainparams =ϵ_0ϵ_1t_0t_1c_0.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.tightbindingmpo-Tuple{Int64, Int64}","page":"Methods","title":"MPSDynamics.tightbindingmpo","text":"tightbindingmpo(N::Int, d::Int; J=1.0, e=1.0)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.tunnelingmpo-Tuple{Any, Any, Any, Any, Any, Int64, Int64}","page":"Methods","title":"MPSDynamics.tunnelingmpo","text":"tunnelingmpo(ϵ, delta, α, s, β, d::Int, nummodes::Int; tree=false, ωc=1)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.twobathspinmpo","page":"Methods","title":"MPSDynamics.twobathspinmpo","text":"twobathspinmpo(ω0, Δ, Nl, Nr, dl, dr, chainparamsl=[fill(1.0,N),fill(1.0,N-1), 1.0], chainparamsr=chainparamsl; tree=false)\n\nGenerate MPO for a spin-1/2 coupled to two chains of harmonic oscillators, defined by the Hamiltonian\n\nH = fracω_02σ_z + Δσ_x + c_0^rσ_x(b_0^dagger+b_0) + sum_i=0^N_r-1 t_i^r (b_i+1^dagger b_i +hc) + sum_i=0^N_r ϵ_i^rb_i^dagger b_i + c_0^lσ_x(d_0^dagger+d_0) + sum_i=0^N_l-1 t_i^l (d_i+1^dagger d_i +hc) + sum_i=0^N_l ϵ_i^l d_i^dagger d_i.\n\nThe spin is on site N_l + 1 of the MPS, surrounded by the left chain modes and the right chain modes.\n\nThis Hamiltonain is unitarily equivalent (before the truncation to N sites) to the spin-boson Hamiltonian defined by\n\nH =  fracω_02σ_z + Δσ_x + σ_xint_0^ dωsqrtJ(ω)(b_ω^dagger+b_ω) + int_0^ dω ωb_ω^dagger b_ωi + σ_xint_0^ dωsqrtJ^l(ω)(d_ω^dagger+d_ω) + int_0^ dω ωd_ω^dagger d_ω.\n\nThe chain parameters, supplied by chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature. The two chains can have a different spectral density.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.xxzmpo","page":"Methods","title":"MPSDynamics.xxzmpo","text":"xxzmpo(N::Int, Δ = 1.0, J=1.0) = xyzmpo(N; Jx=J, Jy=J, Jz=J*Δ)\n\nGenerate MPO for the N-spin XXZ model, defined by the Hamiltonian\n\nH = sum_n=1^N-1 -J σ_x^n σ_x^n+1 - J σ_y^n σ_y^n+1 - Delta J σ_z^n σ_z^n+1\n\nwith σ_x^n σ_y^n σ_z^n the Pauli spin-1/2 matrices of the n^textth site.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.xyzmpo-Tuple{Int64}","page":"Methods","title":"MPSDynamics.xyzmpo","text":"xyzmpo(N::Int; Jx=1.0, Jy=Jx, Jz=Jx, hx=0., hz=0.)\n\nGenerate MPO for the N-spin XYZ model with external field vech=(h_x 0 h_z), , defined by the Hamiltonian\n\nH = sum_n=1^N-1 -J_x σ_x^n σ_x^n+1 - J_y σ_y^n σ_y^n+1 - J_z σ_z^n σ_z^n+1 + sum_n=1^N(- h_x σ_x^n - h_z σ_z^n)\n\nwith σ_x^n σ_y^n σ_z^n the Pauli spin-1/2 matrices of the n^textth site.\n\n\n\n\n\n","category":"method"},{"location":"methods/#Chain-Mapping","page":"Methods","title":"Chain-Mapping","text":"","category":"section"},{"location":"methods/","page":"Methods","title":"Methods","text":"Modules = [MPSDynamics]\nPages = [\"finitetemperature.jl\", \"chainOhmT.jl\", \"gauss.jl\", \"lanczos.jl\", \"mcdis2.jl\" , \"ohmicSpec.jl\" , \"quadohmT.jl\", \"r_jacobi.jl\", \"stieltjes.jl\", ]","category":"page"},{"location":"methods/#MPSDynamics.chaincoeffs_fermionic-Tuple{Any, Any, Any}","page":"Methods","title":"MPSDynamics.chaincoeffs_fermionic","text":"chaincoeffs_fermionic(nummodes, β, chain; ϵ=x, ωc=1, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=true)\n\nGenerate chain coefficients ϵ_0ϵ_1t_0t_1c_0 for a fermionic bath at the inverse temperature β.\n\nArguments\n\nnummodes: Number of bath modes\nβ: inverse temperature\nchain: 1 if the chain modes are empty, 2 if the chain modes are filled\nϵ: user-provided dispersion relation. Should be a function f(x) where x is the wavenumber\nJ: user-provided spectral density. Should be a function f(x) where x is the wavenumber\nωc: the maximum frequency allowwed in the spectral density\nmc: the number of component intervals\nmp: the number of points in the discrete part of the measure (mp=0 if there is none)\niq: a parameter to be set equal to 1, if the user provides his or her own quadrature routine, and different from 1 otherwise\nidelta: a parameter whose default value is 1, but is preferably set equal to 2, if iq=1 and the user provides Gauss-type quadrature routines\nprocedure: choice between the Stieltjes and the Lanczos procedure\nAB: component intervals\nMmax: maximum number of integration points\nsave: if true the coefficients are saved\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.chaincoeffs_finiteT","page":"Methods","title":"MPSDynamics.chaincoeffs_finiteT","text":"chaincoeffs_finiteT(nummodes, β, ohmic=true; α, s, J, ωc=1, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=true)\n\nGenerate chain coefficients ϵ_0ϵ_1t_0t_1c_0 for a harmonic bath at the inverse temperature β.\n\nBy default a Ohmic spectral density J(ω) = 2αω_c (fracωω_c)^s θ(ω-ω_c) is considered. Users can provide their own spectral density.\n\nArguments\n\nnummodes: Number of bath modes\nβ: inverse temperature\nohmic: true if the spectral density is Ohmic, false if the user provides its own spectral density\nα: Kondo parameter of the Ohmic spectral density \ns: ohmicity\nJ: user-provided spectral density. Should be a function f(x,i) where x is the frequency and i ∈ {1,...,mc} labels the intervals on which the SD is defined\nωc: the maximum frequency of the Ohmic spectral density\nmc: the number of component intervals\nmp: the number of points in the discrete part of the measure (mp=0 if there is none)\niq: a parameter to be set equal to 1, if the user provides his or her own quadrature routine, and different from 1 otherwise\nidelta: a parameter whose default value is 1, but is preferably set equal to 2, if iq=1 and the user provides Gauss-type quadrature routines\nprocedure: choice between the Stieltjes and the Lanczos procedure\nAB: component intervals\nMmax: maximum number of integration points\nsave: if true the coefficients are saved\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.gauss-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.gauss","text":"gauss(N,ab)\n\nGauss quadrature rule for N sites on an interval ab. Given a weight function w encoded by the Nx2 array ab of the first N recurrence coefficients for the associated orthogonal polynomials, the first column of ab containing the N alpha-coefficients and the second column the N beta-coefficients, the call xw = gauss(N,ab) generates the nodes and weights xw of the N-point Gauss quadrature rule for the weight function w. The nodes, in increasing order, are stored in the first column, the n corresponding weights in the second column, of the Nx2 array xw.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.lanczos-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.lanczos","text":"lanczos(N,xw)\n\nGiven the discrete inner product whose nodes are contained in the first column, and whose weights are contained in the second column, of the Nx2 array xw, the call ab=lanczos(N,xw) generates the first N recurrence coefficients ab of the corresponding discrete orthogonal polynomials. The N alpha-coefficients are stored in the first column, the N beta-coefficients in the second column, of the Nx2 array ab.\n\nThe script is adapted from the routine RKPW in W.B. Gragg and W.J. Harrod, ``The numerically stable reconstruction of Jacobi matrices from spectral data'', Numer. Math. 44 (1984), 317-335.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.stieltjes-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.stieltjes","text":"stieltjes(N,xw)\n\nDiscretized Stieltjes procedure. Given the discrete inner product whose nodes are contained in the first column, and whose weights are contained in the second column, of the Nx2 array xw, the call ab=stieltjes(N,xw) generates the first N recurrence coefficients ab of the corresponding discrete orthogonal polynomials. The N alpha- coefficients are stored in the first column, the N beta-coefficients in the second column, of the Nx2 array ab.\n\n\n\n\n\n","category":"method"},{"location":"methods/#Dynamics-propagation-function","page":"Methods","title":"Dynamics propagation function","text":"","category":"section"},{"location":"methods/","page":"Methods","title":"Methods","text":"Modules = [MPSDynamics]\nPages = [\"MPSDynamics.jl\", \"chain2TDVP.jl\", \"chainA1TDVP.jl\",\"chainDMRG.jl\",\"chainTDVP.jl\", \"run_1TDVP.jl\", \"run_1TDVPLC.jl\", \"run_2TDVP.jl\", \"run_all.jl\",\"run_DTDVP.jl\"]","category":"page"},{"location":"methods/#MPSDynamics.runsim-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.runsim","text":"runsim(dt, tmax, A, H; \n\tmethod=:TDVP1, \n\tmachine=LocalMachine(), \n\tparams=[], \n\tobs=[], \n\tconvobs=[],\n            convparams=error(\"Must specify convergence parameters\"),\n            save=false,\n            plot=save,\n            savedir=string(homedir(),\"/MPSDynamics/\"),\n            unid=randstring(5),\n            name=nothing,\n\tkwargs...\n\t)\n\nPropagate the MPS A with the MPO H up to time tmax in time steps of dt. The final MPS is returned to A and the measurement data is returned to dat \n\nArguments\n\nmethod: Several methods are implemented in MPSDynamics. :TDVP1 refers to 1-site TDVP on tree and chain MPS, :TDVP2 refers to 2-site TDVP on chain MPS, :DTDVP refers to a variant of 1-site TDVP with dynamics bond-dimensions on chain MPS\nmachine: LocalMachine() points local ressources, RemoteMachine() points distant ressources\nparams: list of parameters written in the log.txt file to describe the dynamics. Can be listed with @LogParams(). \nobs: list of observables that will be measured at every time step for the most accurate convergence parameter supplied to convparams \nconvobs: list of observables that will be measure at every time step for every convergence parameter supplied to convparams \nconvparams: list of convergence parameter with which the propagation will be calculated for every parameter. At each parameter, convobs are measured while obs are measured only for the most accurate dynamics\nsave: Used to choose whether the data will also be saved to a file  \nplot: Used to choose whether plots for 1D observables will be automatically generated and saved along with the data\nsavedir: Used to specify the path where resulting files are stored\nunid: Used to specify the name of the directory containing the resulting files\nname: Used to describe the calculation. This name will appear in the log.txt file\n\n\n\n\n\n","category":"method"},{"location":"methods/#Advanced","page":"Methods","title":"Advanced","text":"","category":"section"},{"location":"methods/","page":"Methods","title":"Methods","text":"Modules = [MPSDynamics]\nPages = [\"flattendict.jl\", \"logging.jl\", \"logiter.jl\",\"machines.jl\"]","category":"page"},{"location":"theory/#Theoretical-Background","page":"Theoretical Background","title":"Theoretical Background","text":"","category":"section"},{"location":"theory/#Chain-Mapping-of-bosonic-environments","page":"Theoretical Background","title":"Chain-Mapping of bosonic environments","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"We consider, in the Schrödinger picture, a general Hamiltonian where a non-specified system interacts linearly with a bosonic environments (hbar = 1)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"beginaligned\n    hatH = hatH_S + int_0^+infty omegahata^dagger_omegahata_omega mathrmdomega + hatA_S otimes int_0^+inftysqrtJ(omega)left(hata_omega + hata^dagger_omegaright)mathrmdomega\nendaligned","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where hat a_omega (hat a^dagger_omega) is a bosonic annihilation (creation) operator for a normal mode of the environment of energy omega, hatA_S is a system operator, and J(omega) = sum_k g_k^2delta(omega - omega_k) is the bath spectral density (SD), defined with the microscopic system-environment coupling strength g_k. The SD quantifies the coupling strengths of the different normal modes of the environment with the system. Any SD that is not flat corresponds to a non-Markovian environment.","category":"page"},{"location":"theory/#Zero-Temperature","page":"Theoretical Background","title":"Zero Temperature","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Let us consider the Hamiltonian presented in Eq.(1). To investigate its rich phenomenology, using the powerful tools provided by tensor network techniques, it is convenient to map the system and its environment from a star-like to a chain-like configuration, with local interactions. To do so, we introduce a unitary transformation of the continuous normal modes hata_omega to an infinite discrete set of interacting modes hatb_n[chin_exact_2010].","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hata_omega = sum_n=0^+infty U_n(omega)hatb_n = sum_n=0^+infty sqrtJ(omega)P_n(omega)hatb_n ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where P_n(omega) are orthonormal polynomials such that","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    int_0^+inftyP_n(omega)P_m(omega)J(omega)mathrmdomega = delta_nm ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"and the inverse transformation is","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hatb_n = int_0^+infty U^*_n(omega)hata_omegamathrmdomega ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Note that the orthonormality of the polynomials ensures the unitarity of the transformation defined in Eq.(2). The mapping from a continuous set of modes to a (still infinite) discrete set might seem counter-intuitive, however it is a direct consequence of the separability of the underlying Hilbert space.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Under this transformation, the Hamiltonian in Eq.(1) becomes","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hatH= hatH_S + sum_n=0^+inftyvarepsilon_nhatb_n^daggerhatb_n + t_n(hatb_n+1^daggerhatb_n + mathrmhc) + c_0 hatA_S(hatb_0 + hatb_0^dagger) ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Hence, this mapping transforms the normal bath Hamiltonian into a tight-binding Hamiltonian with on-site energies varepsilon_n and hopping energies t_n. Another important consequence of this mapping is that now the system only interacts with the first mode n = 0 of the chain-mapped environment. The chain coefficients varepsilon_n, t_n, and the coupling c_0 depend solely on the SD.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"This makes chain mapping a tool of choice for describing systems coupled to environment with highly structured SD (e.g. experimentally measured or calculated ab initio)[chin_role_2013][alvertis_nonequilibrium_2019][dunnett_influence_2021][caycedosoler_exact_2022]. In this new representation, the Hamiltonian in Eq.(5) has naturally a 1D chain topology. This makes its representation as a Matrix Product Operator (MPO) and the representation of the joint {System + Environment} wave-function as a Matrix Product State (MPS) suited [orus_practical_2014][paeckel_timeevolution_2019].","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The orthogonal polynomial-based chain mapping and the subsequent representation of the joint wave-function as a MPS (and the operators as MPO) are the building blocks of the Time-dependent Density operator with Orthonormal Polynomials Algorithm (TEDOPA) one of the state-of-the-art numerically exact method to simulate the dynamics of open quantum systems especially in the non-Markovian, non-perturbative regimes both at zero and finite temperatures [prior_efficient_2010][woods_simulating_2015][tamascelli_efficient_2019][dunnett_simulating_2021][lacroix_unveiling_2021].","category":"page"},{"location":"theory/#Finite-Temperature-with-T-TEDOPA","page":"Theoretical Background","title":"Finite Temperature with T-TEDOPA","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Assuming a unitary evolution for both the system and environment, the system's dynamics can be isolated by tracing out the environmental degrees of freedom. The density operator for the system at time t is described as","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"hatrho_S(t) = textTr_ElefthatU(t) hatrho_S(0) otimes hatrho_E(0) hatU^dagger(t)right","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Initially, the system state, hatrho_S(0), can be pure or mixed, and the environment is in a thermal state defined by the inverse temperature beta = (k_B T)^-1. This state is represented by a product of Gaussian states","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"hatrho_E(0) = bigotimes_omega frace^-beta omega hata_omega^dagger hata_omegaZ_omega(beta) ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The system's evolution is dictated by the environment's two-time correlation function S(t), which in turn is determined by the spectral density function J and the temperature beta via the Bose-Einstein distribution n_omega(beta)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"S(t) = int_0^infty domega J(omega)lefte^-iomega t(1 + n_omega(beta)) + e^iomega t n_omega(beta)right ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"To simulate finite temperature effects using a zero-temperature model with the T-TEDOPA method [tamascelli_efficient_2019], we extend the spectral density function to cover both positive and negative frequencies, allowing us to use a pure state description for the environment. This is achieved by defining a new spectral density function J(omega beta) that incorporates the Boltzmann factors, supporting the entire real axis","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"J(omega beta) = fractextsign(omega)J(leftomegaright)2 Big(1 + cothBig(fracbeta omega2Big)Big)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"This modified bath has the same correlation function S(t) and thus allows us to maintain a pure state description of the environment, represented as a vacuum state, and avoid the computational complexities of density matrices","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"lefttextvacrightrangle = bigotimes_omega left0_omegarightrangle ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The Hamiltonian of the system interacting with this extended bath now includes temperature-dependent interactions:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"hatH = hatH_S + int_-infty^+infty omega hata_omega^dagger hata_omega mathrmdomega + hatA_S otimes int_-infty^+infty sqrtJ(omegabeta)left(hata_omega^dagger+hata_omegaright) mathrmdomega","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"This method simplifies the simulation of finite temperature effects by treating them within an effective zero-temperature framework, thereby keeping the computational advantages of using pure states. In conclusion: the dynamics of the system resulting from the interaction with the original bath, starting in a thermal state at finite temperature, is exactly the same as the one resulting from the interaction with the extended environment, starting in the vacuum state at zero temperature. Once computed the chain coefficients at a given inverse temperature beta, the time evolution of the vacuum state interacting with the extended environment can be efficiently simulated using MPS time evolution methods.","category":"page"},{"location":"theory/#Finite-temperature-with-the-thermofield-transformation","page":"Theoretical Background","title":"Finite temperature with the thermofield transformation","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Finite temperature initial states of the system are challenging: in the previous section we have seen that a very efficient solution is to reabsorb the effect of the temperature inside of the spectral density function that characterizes the environment. De Vega and Bañuls present[devega_thermo_2015] another suggestive approach based on the idea of the thermofield transformation. The idea is to double the environmental degrees of freedom and then to apply a Bogoliubov transformation: the real environment in a thermal state is transformed into two virtual environments in the vacuum state, defined as the thermofield vacuum. For any operator of the real environment, the expectation values in the thermal state are equivalent to those calculated in the thermofield vacuum. Behind the thermofield approach there is the concept of purification: the initial mixed state of the environment can be represented as the partial trace of a pure state (the thermofield vacuum) defined on the larger Hilbert space of the modes of the two virtual environments at zero temperature. The interaction of the system and each one of the two environments is then mapped on two separate chains, using the TEDOPA chain mapping to define a unitary transformation that can be applied to the bath modes, in order to define new modes for the environment, mapping it from a star-like to a one dimensional chain-like configuration, which is very well suited for the application of tensor network techniques.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"In the thermofield approach, the first step is to introduce the auxiliary environment E^textaux of non-interacting bosonic modes of negative frequencies:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hat H^textaux = hat H - sum_k omega_k hat c_k^dagger hat c_k","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where the Hamiltonian hat H is the bosonic Hamiltonian defined in Eq. \\ref{eq:bosonic_ham}. The two environments, of positive and negative frequencies, are assumed to be in a thermal environment at inverse temperature beta; the second step is to apply a thermal Bogoliubov transformation to change the basis. The applied transformation produces two-modes squeezed states:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"         hat a_1k=e^-iG  hat b_k e^iG= cosh(theta_k) hat b_k -sinh(theta_k)  hat c_k^dagger \n         hat a_2k=e^-iG hat c_k e^iG= cosh(theta_k)  hat c_k -sinh(theta_k)  hat b_k^dagger","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where the exponent of the squeeze operator is G = i sum_k theta_k(hat b_k^dagger hat c_k^dagger-hat c_k hat b_k), and theta_k is dependent on the temperature as in the following relations, where the number of excitations in the k-th mode is n_k = 1(e^beta omega_k-1):","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"         cosh(theta_k) = sqrt1+n_k = sqrtfrac11-e^-beta omega_k \n         sinh(theta_k) =quadsqrtn_kquad= sqrtfrac1e^beta omega_k-1","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The Bogoliubov transformation defines a new squeezed vacuum state, which we write in terms of the vacuum state  Omega_0 rangle of the operators hat b_k, hat c_k:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    Omegarangle = e^iG Omega_0rangle quad textsuch that quad hat a_1k Omegarangle = 0  hat a_2k Omegarangle = 0","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"From the vacuum state, we can obtain the thermal state of the original environment:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hat rho_E = textTr_textaux OmegaranglelangleOmega ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"and it can be now used as pure an initial state for both of the environments. Moreover, the expectation value on  Omega rangle of the number of physical modes hat n_k does not vanish:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    n_k= langleOmega hat b_k^dagger hat b_k Omegarangle = sinh^2(theta_k)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Therefore, solving the dynamics given by the original Hamiltonian hat H, starting from the initial condition hat rho_S(0) otimes hat rho_E(beta), is equivalent to solving the dynamics given by the following Hamiltonian:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"       hat H = hat H_S +hat H_E +hat H_I   \n     hat H  = overbracehat A_S^hat H_S + overbracesum_k omega_k big(hat a_1k^dagger hat a_1k - hat a_2k^dagger hat a_2k big)^hat H_E + overbracehat L_S otimes sum_k g_1k(hat a_1k^dagger + hat a_1k)+hat L_S otimes sum_k g_2k(hat a_2k^dagger + hat a_2k)^hat H_I","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where hat L_S = hat L_S^dagger, considering hat rho_S(0) otimes Omega rangle langle Omega  as the initial state of system and environment. It is this Hamiltonian that is mapped on two chains with TEDOPA, to be able to perform the time evolution using tensor network techniques. ","category":"page"},{"location":"theory/#Computation-of-the-chain-coefficients","page":"Theoretical Background","title":"Computation of the chain coefficients","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"A useful property of the orthonormal polynomials is that they obey a recurrence relation","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    P_n(omega) = (C_n-1omega - A_n-1)P_n-1(omega) + B_n-1P_n-2(omega) ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where A_n is related to the first moment of P_n, B_n and C_n to the norms of P_n and P_n-1[appel_mathematics_2007]. This recurrence relation can be used to construct the polynomials with the conditions that P_0(omega) = p_0^-1 = left(int_mathbbR^+ J(omega)mathrmdomega right)^-frac12 and P_-1(omega) = 0, with bullet the norm of bullet with respect to the measure J(omega), and P_n(omega) = p_n(omega)p_n^-1 ; where the polynomials p_n_ninmathbbN are the so called monic polynomials where the factor a_n in front of omega^n is equal to 1.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The energy of the chain mode n is given by varepsilon_n = A_n C_n^-1 and t_n=C_n^-1 is the coupling between mode n and n+1[chin_exact_2010].","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The system couples only to the first mode with the coupling strength c_0 = p_0.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Explain that for some weight function/SD they are known analytically and that for others we can use the build-in routines inspired by Gautschi or the PolyChaos.jl package.","category":"page"},{"location":"theory/#Tensor-Networks","page":"Theoretical Background","title":"Tensor Networks","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"A multipartite quantum state psirangle, e.g. a N-site system where the sites can each be in a state phi_irangle belonging to a d-dimensional Hilbert space, can be written as follows","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    psirangle = sum_i_kc_i_1ldots i_Nphi_i_1rangleotimesldotsotimesphi_i_Nrangle ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where the complex numbers c_i_1ldots i_N are the amplitudes of each state phi_i_1rangleotimesldotsotimesphi_i_Nrangle whose superpositions form in full generality the state psirangle. Thus the state psirangle can be completely represented by a rank-N tensor c that is the collection of all possible amplitudes c_i_1ldots i_N. Here by the rank of a tensor, we simply mean the number of indices it has.","category":"page"},{"location":"theory/#MPS","page":"Theoretical Background","title":"MPS","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The tensor c of a quantum state psirangle corresponding to a one-dimensional system can be decomposed into a product of N smaller rank-3 tensors T_k (except for the first and last sites where the tensors will have a rank-2)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    c_i_1ldots i_N = sum_alpha T^alpha_1_i_1T^alpha_1alpha_2 _i_2T^alpha_2alpha_3 _i_3ldots T^alpha_N-1_i_N  ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"In this form, the local tensor T_k contains the information on the quantum state on site k and its relation (especially the entanglement) with the neighbouring sites.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The decomposition of the tensor of the amplitudes of a quantum state into a product of smaller rank tensors is called a Matrix Product State decomposition.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The contracted indices alpha_k between the tensors are called virtual indices and carry information about the correlations between bi-partitions of the state at bond k. The number of different values a virtual index can take is called the bond dimension and is denoted D. The free indices i_k associated with local quantum states are called physical indices. Thus, they can take d values (with d the dimension of the local Hilbert space).","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Any state in the Hilbert space of a one-dimensional many-body system can in principle be represented by a MPS by choosing a sufficiently large value for the bond dimension D [Orus]. On top of this intellectually satisfying property of MPSs being a dense set of states for a 1d-system, they can also be used as a practical Ansätze for a many-body quantum states by setting a maximal allowed value chi for the bond dimension D. In doing so, we restrict ourselves to a corner of the total Hilbert space. The rationale behind this Ansatz is the following: if the initial quantum state of a many-body system has a low bond dimension (typically if the initial state is a product state with D = 1), then in a finite time it will only be able to explore a region of the Hilbert space that is not to far away from its starting point. Thus, the bond dimension will not have the time to diverge exponentially [poulin_quantum_2011]. However, depending on the physical system at hand, this sub-manifold of the Hilbert space could still be \"too large\". There is an additional reason that explains why MPSs are good Ansätze for 1d physical systems. Most many-body Hamiltonians we (physicists) are interested in are local, meaning that the interactions they describe involve objects that are \"neighbours\". For such Hamiltonians, the ground states (outside of potential critical phases) follow the so called area law for the entanglement entropy [srednicki_entropy_1993][vidal_entanglement_2003][wolf_area_2008]. This law states that the entanglement entropy S_vN of a bi-partition of the system is proportional, not to the volume of the partition as one might expect, but to the hyper-surface of the partition's boundary; hence the name \"area law\". For a 3d system this corresponds to an actual surface area A, S_vN sim A; for a 2d system it corresponds to the length L of the partition's boundary, S_vN sim L; and in 1d the boundary reduces to a point, thus the entropy will be independent of the size of the system S_vN sim textconstant. The MPSs are states that satisfy this area law.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"An application of the Singular Value Decomposition is to create efficient approximations of quantum states to perform computations. The main idea is to reduce the content of the MPS to keep only the parts that contain the physics of interest. One method to realise this approximation is to do a SVD on each of the tensors of the MPS after each time step of the state time-evolution and to trim the smallest singular values in order to decrease the bond dimension of the MPS down to a chosen maximal value chi. The corresponding columns and rows of the unitary matrices U and V^dagger are also removed. Then, the trimmed matrices tildeU, tildeS and tildeV^dagger are contracted back to give an approximated tensor T with a smaller bond dimension. Another way to apply the restricted rank approximation is to restrict oneself into working in a manifold of fixed bond dimension D and to use methods that can enforce this constraint.","category":"page"},{"location":"theory/#MPO","page":"Theoretical Background","title":"MPO","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"In order to compute expectation values of observables or apply unitary transformations to a quantum state, we need a TN representation of operators. In the same fashion as a one-dimensional quantum state can be represented as a MPS, operators acting on those states can be represented as Matrix Product Operators (MPO). For an operator hatO, its MPO can be defined as follows","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hatO = sum_i_ki_k^ w W^i_1 i^_1_1 w_0w_1ldots  W^i_N i^_N_N w_N-1w_N phi_i_1^ldots phi_i_N^ranglelanglephi_i_1ldots phi_i_N ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The contracted indices between the tensors are called virtual indices. The free indices are called physical indices and correspond to the different input and output local quantum states.  They can take d values (with d the dimension of the local Hilbert space).","category":"page"},{"location":"theory/#TTN","page":"Theoretical Background","title":"TTN","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"A natural extension to the MPS is the (loop-free) tree tensor network.  A TTN is a generalisation of the MPS wherein each site, instead of being connected to only one other site to its right, may be connected to any arbitrary number of child sites. Provided the tree does not contain any loops, everything that one can do to an MPS/MPO can be extended straight-forwardly to TTN states and TTN operators.  The generalisation to trees introduces no new conceptual complexity (only implementational complexity). The sites of a TTN are usually referred to as nodes.  For our purposes, every node of a TTN state and operator has one parent leg, and any number (including zero) of child legs.  The first node is known as the head-node and has a dummy parent leg with dimension 1.","category":"page"},{"location":"theory/#Time-Dependent-Variational-Principle","page":"Theoretical Background","title":"Time-Dependent Variational Principle","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The original idea behind TDVP goes back to Dirac [dirac_note_1930] and Frenkel [frenkel_wave_1934]. The main point, in the modern tensor networks formulation, is that instead of solving the Schrödinger equation and then truncating the MPS representation of the quantum state, one can solve the equations of motion projected into a space of restricted bond dimension [haegeman_timedependent_2011][haegeman_unifying_2016].","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The general formulation of the Dirac-Frenkel Variational Principle [raab_diracfrenkelmclachlan_2000] is that one looks for a solution varphirangle in mathcalM of the Schrödinger equation where mathcalM subset mathcalH is a manifold of the total Hilbert space mathcalH in which we think that the relevant physical states `live'.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"We define T_varphiranglemathcalM the tangent space of mathcalM around the state varphirangle. The criterion to find varphirangle is that for every state chirangle in T_varphiranglemathcalM","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    langlechileft(fracmathrmdmathrmdt - frac1mathrmihbarhatHright)varphirangle =0 ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"which can be interpreted as saying that the time evolution procedure should keep varphirangle inside of the manifold mathcalM.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The term variational in the name of the method comes from the fact that in practice one aims at minimising the right-hand side of the hereinabove Eq. to find varphirangle.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Introducing hatP_T_varphiranglemathcalM the projector onto the tangent space T_varphiranglemathcalM, we can write the state chirangle = hatP_T_varphiranglemathcalMphirangle with phirangle a state in mathcalH. Leading to","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    forall phirangle in mathcalH  langlephihatP_T_varphiranglemathcalMleft(fracmathrmdmathrmdt - frac1mathrmihbarhatHright)varphirangle =0 ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Because the time derivation and the projector commute, we have","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    forall phirangle in mathcalH  langlephileft(fracmathrmdmathrmdt - frac1mathrmihbarhatP_T_varphiranglemathcalMhatHright)varphirangle =0 ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"This equation must be true for any phirangle in mathcalH, Eq.~(\\ref{eq:DiracFrenkel1}) can thus be written","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    left(fracmathrmdmathrmdt - frac1mathrmihbarhatP_T_varphiranglemathcalMhatHright)varphirangle =0 ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"In the context of MPS, the manifold mathcalM will correspond to the space of full-ranked MPS of a given bond dimension D, and the tangent space will be the space spanned by variations of single MPS tensors. The projector defines an effective Hamiltonian under which the dynamics are constrained on mathcalM. Constraining the dynamics on a manifold, introduces a projection error: the time evolution will obey to an effective Hamiltonian different from the starting one. After the introduction of TDVP as a time evolution method for MPS, Haegeman et al. pointed out [haegeman_unifying_2016] that there exist an analytical decomposition for the projector operator hatmathcalP that simplifies the resolution of the equation, turning the problem into one where each matrix A_i can be updated with an effective on site Hamiltonian hat H_texteff via a Schroedinger like equation. The effective Hamiltonian hat H_texteff is a contraction of the Hamiltonian MPO and the current state of the other matrices composing the MPS. This allows to do a sequential update.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"There exist different versions of the TDVP algorithm. In MPSDynamics.jl three methods have been so far implemented:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"the one-site TDVP (1TDVP)\nthe two-sites TDVP (2TDVP) \nthe adaptive TDVP (DTDVP) [dunnett_efficient_2021]","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The main advantage of the one-site 1TDVP algorithm is that it preserves the unitarity of the MPS during the time evolution. Its main problem, conversely, is that the time evolution is constrained to happen on a manifold constituted by tensors of fixed bond dimension, a quantity closely related to the amount of entanglement in the MPS, and such a bond dimension has therefore to be fixed before the beginning of the time evolution. This strategy will necessarily be non optimal: the growth of the bond dimensions required to describe the quantum state should ideally mirror the entanglement growth induced by the time evolution. 2TDVP allows for such a dynamical growth of the bond dimensions, and therefore better describes the entanglement in the MPS. It suffers however of other drawbacks: first of all, a truncation error is introduced (by the means of an SVD decomposition), which entails a loss of unitarity of the time-evolved MPS. Moreover, 2TDVP has bad scaling properties with the size of the local dimensions of the MPS: this is a major issue when dealing with bosons. The DTDVP algorithm combines the best features of 1TDVP and 2TDVP: it preserves unitarity, it has the same scaling properties of 1TDVP, and it adapts the bond dimensions to the entanglement evolution at each site and at each time-step. DTDVP does not suffer from a truncation error, but introduces only a projection error.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"________________","category":"page"},{"location":"theory/#Bibliography","page":"Theoretical Background","title":"Bibliography","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[chin_exact_2010]: Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. Journal of Mathematical Physics 2010, 51 (9), 092109. https://doi.org/10.1063/1.3490188.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[chin_role_2013]: Chin, A. W.; Prior, J.; Rosenbach, R.; Caycedo-Soler, F.; Huelga, S. F.; Plenio, M. B. The Role of Non-Equilibrium Vibrational Structures in Electronic Coherence and Recoherence in Pigment–Protein Complexes. Nature Phys 2013, 9 (2), 113–118. https://doi.org/10.1038/nphys2515.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[alvertis_nonequilibrium_2019]: Alvertis, A. M.; Schröder, F. A. Y. N.; Chin, A. W. Non-Equilibrium Relaxation of Hot States in Organic Semiconductors: Impact of Mode-Selective Excitation on Charge Transfer. J. Chem. Phys. 2019, 151 (8), 084104. https://doi.org/10.1063/1.5115239.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[dunnett_influence_2021]: Dunnett, A. J.; Gowland, D.; Isborn, C. M.; Chin, A. W.; Zuehlsdorff, T. J. Influence of Non-Adiabatic Effects on Linear Absorption Spectra in the Condensed Phase: Methylene Blue. J. Chem. Phys. 2021, 155 (14), 144112. https://doi.org/10.1063/5.0062950.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[caycedosoler_exact_2022]: Caycedo-Soler, F.; Mattioni, A.; Lim, J.; Renger, T.; Huelga, S. F.; Plenio, M. B. Exact Simulation of Pigment-Protein Complexes Unveils Vibronic Renormalization of Electronic Parameters in Ultrafast Spectroscopy. Nat Commun 2022, 13 (1), 2912. https://doi.org/10.1038/s41467-022-30565-4.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[orus_practical_2014]: Orus, R. A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States. Annals of Physics 2014, 349, 117–158. https://doi.org/10.1016/j.aop.2014.06.013.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[paeckel_timeevolution_2019]: Paeckel, S.; Köhler, T.; Swoboda, A.; Manmana, S. R.; Schollwöck, U.; Hubig, C. Time-Evolution Methods for Matrix-Product States. Annals of Physics 2019, 411, 167998. https://doi.org/10.1016/j.aop.2019.167998.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[prior_efficient_2010]: Prior, J.; Chin, A. W.; Huelga, S. F.; Plenio, M. B. Efficient Simulation of Strong System-Environment Interactions. Phys. Rev. Lett. 2010, 105 (5), 050404. https://doi.org/10.1103/PhysRevLett.105.050404.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[woods_simulating_2015]: Woods, M. P.; Cramer, M.; Plenio, M. B. Simulating Bosonic Baths with Error Bars. Phys. Rev. Lett. 2015, 115 (13), 130401. https://doi.org/10.1103/PhysRevLett.115.130401.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[tamascelli_efficient_2019]: Tamascelli, D.; Smirne, A.; Lim, J.; Huelga, S. F.; Plenio, M. B. Efficient Simulation of Finite-Temperature Open Quantum Systems. Phys. Rev. Lett. 2019, 123 (9), 090402. https://doi.org/10.1103/PhysRevLett.123.090402.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[dunnett_simulating_2021]: Dunnett, A. J.; Chin, A. W. Simulating Quantum Vibronic Dynamics at Finite Temperatures With Many Body Wave Functions at 0 K. Front. Chem. 2021, 8. https://doi.org/10.3389/fchem.2020.600731.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[lacroix_unveiling_2021]: Lacroix, T.; Dunnett, A.; Gribben, D.; Lovett, B. W.; Chin, A. Unveiling Non-Markovian Spacetime Signaling in Open Quantum Systems with Long-Range Tensor Network Dynamics. Phys. Rev. A 2021, 104 (5), 052204. https://doi.org/10.1103/PhysRevA.104.052204.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[appel_mathematics_2007]: Appel, W. Mathematics for Physics and Physicists; Princeton University Press, 2007.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[devega_thermo_2015]: de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[Orus]: Orus, R. A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States. Annals of Physics 2014, 349, 117–158. https://doi.org/10.1016/j.aop.2014.06.013.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[poulin_quantum_2011]: Poulin, D.; Qarry, A.; Somma, R.; Verstraete, F. Quantum Simulation of Time-Dependent Hamiltonians and the Convenient Illusion of Hilbert Space. Phys. Rev. Lett. 2011, 106 (17), 170501. https://doi.org/10.1103/PhysRevLett.106.170501.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[srednicki_entropy_1993]: Srednicki, M. Entropy and Area. Phys. Rev. Lett. 1993, 71 (5), 666–669. https://doi.org/10.1103/PhysRevLett.71.666.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[vidal_entanglement_2003]: Vidal, G.; Latorre, J. I.; Rico, E.; Kitaev, A. Entanglement in Quantum Critical Phenomena. Phys. Rev. Lett. 2003, 90 (22), 227902. https://doi.org/10.1103/PhysRevLett.90.227902.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[wolf_area_2008]: Wolf, M. M.; Verstraete, F.; Hastings, M. B.; Cirac, J. I. Area Laws in Quantum Systems: Mutual Information and Correlations. Phys. Rev. Lett. 2008, 100 (7), 070502. https://doi.org/10.1103/PhysRevLett.100.070502.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[dirac_note_1930]: Dirac, P. A. M. Note on Exchange Phenomena in the Thomas Atom. Mathematical Proceedings of the Cambridge Philosophical Society 1930, 26 (3), 376–385. https://doi.org/10.1017/S0305004100016108.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[frenkel_wave_1934]: Frenkel, Y. Wave Mechanics, Advanced General Theory; Oxford, 1934; Vol. 1.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[haegeman_timedependent_2011]: Haegeman, J.; Cirac, J. I.; Osborne, T. J.; Pižorn, I.; Verschelde, H.; Verstraete, F. Time-Dependent Variational Principle for Quantum Lattices. Phys. Rev. Lett. 2011, 107 (7), 070601. https://doi.org/10.1103/PhysRevLett.107.070601.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[haegeman_unifying_2016]: Haegeman, J.; Lubich, C.; Oseledets, I.; Vandereycken, B.; Verstraete, F. Unifying Time Evolution and Optimization with Matrix Product States. Phys. Rev. B 2016, 94 (16), 165116. https://doi.org/10.1103/PhysRevB.94.165116.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[raab_diracfrenkelmclachlan_2000]: Raab, A. On the Dirac–Frenkel/McLachlan Variational Principle. Chemical Physics Letters 2000, 319 (5), 674–678. https://doi.org/10.1016/S0009-2614(00)00200-1.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[dunnett_efficient_2021]: Dunnett, A. J.; Chin A. W. Efficient bond-adaptive approach for finite-temperature open quantum dynamics using the one-site time-dependent variational principle for matrix product states. Physical Review B, 104(21):214302, Dec 2021. doi:10.1103/PhysRevB.104.214302. ","category":"page"},{"location":"examples/bath-observables/#Inspecting-the-bath-by-undoing-the-chain-mapping","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"","category":"section"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"Here we give some context on the example script provided in MPSDynamics/example/bath-observables.jl. This example demonstrates the setup and execution of a simulation for a two-level system coupled to an Ohmic bath at finite temperature, where we exploit the access to the chain observables to:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"undo the chain mapping[chin_exact_2010], thus obtaining their representation in the extended bath of T-TEDOPA [tamascelli_efficient_2019], characterized by J(omega beta)\ninverting the thermofield transformation[devega_thermo_2015], thus obtaining the representation of the physical frequencies in the original environment, characterized by J(omega)","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"T-TEDOPA allows to substitute a thermally occupied bath by an extended one, in the pure state of the vacuum, extending the bath of frequencies to negative values: the creation from the system of a mode of negative frequency in the extended bath of frequencies corresponds to the absorption of energy for the system from the thermally occupied modes of the environment. However, by exploiting the thermofield transformation, it is possible to recover the picture of the occupations of the steady states in the physical bath of frequencies[riva_thermal_2023]. ","category":"page"},{"location":"examples/bath-observables/#The-code","page":"Inspecting the bath by undoing the chain mapping","title":"The code","text":"","category":"section"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We start by defining the parameters of the simulation:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"d = 10      # number of Fock states of the chain modes\nN = 60      # length of the chain\nα = 0.01    # coupling strength\nω0 = 0.2  # TLS gap\ns = 1       # ohmicity\nωc = 1.  # Cut-off of the spectral density J(ω)\nβ = 20    # Thermalized environment","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We set the specifics of the simulation:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"method = :TDVP1         # time-evolution method\nconv = 3                # bond dimension for the TDVP1\ndt = 0.5                # time step\ntfinal = 60.0           # simulation time","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"And then compute the chain coefficients, i.e. on-site energies epsilon_i, hopping energies t_i, and system-chain coupling c_0, that define the chain representation of the pure-dephasing model:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"cpars = chaincoeffs_finiteT(N, β; α=α, s=s, J=nothing, ωc=ωc, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=false)","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"with this, we can compute the corresponding MPO, and the initial state in MPS form (1sqrt2(0rangle + 1rangle)):","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"H = puredephasingmpo(ω0, d, N, cpars)\n\nψ = zeros(2)\nψ[1] = 1/sqrt(2)\nψ[2] = 1/sqrt(2)\n\nA = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...])","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We can now define the observables we are interested in computing. Importantly, we not only compute observables related to the system, but also to the chain modes, so that we can inspect the environment.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"ob1 = OneSiteObservable(\"sz\", sz, 1)\nob2 = OneSiteObservable(\"sx\", sx, 1)\nob3 = OneSiteObservable(\"chain_mode_occupation\", numb(d), (2,N+1))\nob4 = OneSiteObservable(\"c\", crea(d), collect(2:N+1))\nob5 = OneSiteObservable(\"cdag\", crea(d), collect(2:N+1))\nob6 = TwoSiteObservable(\"cdagc\", crea(d), anih(d), collect(2:N+1), collect(2:N+1))\nob7 = TwoSiteObservable(\"cdagcdag\", crea(d), crea(d), collect(2:N+1), collect(2:N+1))\nob8 = TwoSiteObservable(\"cc\", anih(d), anih(d), collect(2:N+1), collect(2:N+1))","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"with this we run the simulation:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"A, dat = runsim(dt, tfinal, A, H, prec=1E-4;\n                name = \"Bath observables in the pure dephasing model\",\n                method = method,\n                obs = [ob1, ob2, ob3, ob4, ob5, ob6, ob7, ob8],\n                convobs = [ob1],\n                params = @LogParams(ω0, N, d, α, s, ψ),\n                convparams = conv,\n                reduceddensity = true,\n                verbose = false,\n                save = false,\n                plot = true,\n                );","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"After the simulation, we need to post-process the data a bit, in order to map the correlations from the chain representation to the original environment. We start by defining the matrices associated to the mean values of c and c^dagger:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"T = length(dat[\"data/times\"])\n\nconstr = Array{ComplexF64}(undef, N, N, T)\ndestr = Array{ComplexF64}(undef, N, N, T)\nfor t in 1:T\n    constr[:,:,t] = diagm(0 => dat[\"data/cdag\"][:,t])\n    destr[:,:,t] = diagm(0 => dat[\"data/c\"][:,t])\nend","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"During a numerical simulation we work with chain of finite length N. This truncation on chain modes, introduces a sampling on the modes in the original star-like environment. To recover the frequency modes that are implicitly sampled, one has to diagonalize the tri-diagonal Ntimes N matrix H^textchain, where the diagonal is formed by the e_n coefficients, that is the chain's frequencies, and the upper and lower diagonals by the N-1 hopping coefficients t_n. The unitary matrix that defines the change of basis from the star-like to the chain-like environment is U_n, constituted by the eigenvectors of H^textchain. In the code, we use the MPSDynamics.eigenchain function:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"omeg = eigenchain(cpars, nummodes=N).values","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"At each time step of the simulation, a number of one-site and two-sites observables where evaluated on the chain. To obtain their value in the extended bath of T-tedopa, characterized by J(omegabeta), the unitary transformation that maps the extended bath Hamiltonian into the chain representation has to be reversed. For instance, when measuring the single site hat n^c_i=hat c_i^dagger hat c_i occupation number, we are not measuring the occupation number of the bosonic mode associated to the omega_i frequency, but the occupation number of the i-th chain mode. Therefore to calculate the number of modes of the environment associated to a specific frequency omega_i, the mapping must be reversed, to obtain the diagonal representation of the bosonic number operator:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"beginaligned\nhat n^b_i = hat b_i^dagger hat b_i = sum_kl U_ik^* hat c_k^dagger hat c_l U_li  \nendaligned","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"This is done in the code using the MPSDynamics.measuremodes function, which outputs the vector of the diagonal elements of the operators, in the following way:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"bath_occup = mapslices(X -> measuremodes(X, cpars[1], cpars[2]), dat[\"data/cdagc\"], dims = [1,2])\ncdag_average = mapslices(X -> measuremodes(X, cpars[1], cpars[2]), constr, dims = [1,2])\nc_average = mapslices(X -> measuremodes(X, cpars[1], cpars[2]), destr, dims = [1,2])","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"To compute the correlators, we need the full matrix in the original basis. We therefore use the measurecorrs function:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"cc_average = mapslices(X -> measurecorrs(X, cpars[1], cpars[2]), dat[\"data/cc\"], dims = [1,2])\ncdagcdag_average = mapslices(X -> measurecorrs(X, cpars[1], cpars[2]), dat[\"data/cdagcdag\"], dims = [1,2])\n\ncorrelations_c = [\n    cc_average[i, j, t] - c_average[i, 1, t] .* c_average[j, 1, t]\n    for i in 1:size(cc_average, 1), j in 1:size(cc_average, 2), t in 1:size(cc_average, 3)\n]\ncorrelations_cdag = [\n    cdagcdag_average[i, j, t] - cdag_average[i, 1, t] .* cdag_average[j, 1, t]\n    for i in 1:size(cdagcdag_average, 1), j in 1:size(cdagcdag_average, 2), t in 1:size(cdagcdag_average,3)\n]","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"It is possible to invert the thermofield transformation (details in [riva_thermal_2023]). The expression of the mean value of the number operator for the physical modes can be expressed as a function of mean values in the extended bath, which we denote langle hat a_2k^dagger hat a_2k rangle:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"beginaligned\n    langle hat b_k^dagger hat b_k rangle = coshtheta_ksinhtheta_k (langle hat a_2khat a_1krangle + langle hat a_1k^daggerhat a_2k^daggerrangle ) + sinh^2theta_k (1+ langle hat a_2k^dagger hat a_2k rangle ) ++ cosh^2theta_k langle hat a_1k^dagger hat a_1k rangle\nendaligned","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We remark that in the thermofield case, a negative frequency omega_2k is associated to each positive frequency omega_1k. The sampling is therefore symmetric around zero. This marks a difference with T-TEDOPA, where the sampling of frequencies was obtained through the thermalized measure dmu(beta) = sqrtJ(omega beta)domega, and was not symmetric. To recover the results for the physical bath of frequencies starting from the results of our simulations, that were conducted using the T-TEDOPA chain mapping, we need to do an extrapolation for all of the mean values, in order to have their values for each omega at -omega as well. This is done in the code with the MPSDynamics.physical_occup function:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"bath_occup_phys = physical_occup(cdagcdag_average[:,:,T], cc_average[:,:,T], omeg, bath_occup[:,:,T], β, N)","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"Finally, in the pure dephasing case, it is also possible to obtain the analytical prediction of the time evolution of the occupations of the bath's modes, so that we can compare our numerical results with the analytical ones, exploiting the Heisenberg time evolution relation:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"beginaligned\nfracd langle hat b_omega rangledt = -i langle hat b_omega hat H rangle = - i omega langlehat b_omega rangle - i fraclangle hat sigma_x rangle2 sqrtJ(omega beta)     \nfracd langle hat n_omega rangledt = -i langlehat b_omega^dagger hat b_omega hat H rangle= 2 fracJ(omegabeta)omega sin(omega t)\nendaligned","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"To this end, it is convenient to choose one of the eigenstates of hat sigma_z as the initial state, so that langle hat sigma_x rangle = pm 1. By solving these differential equations, one obtains the time evolved theoretical behavior of the bath. We define the function for the comparison with analytical predictions:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"Johmic(ω,s) = (2*α*ω^s)/(ωc^(s-1))\n\ntime_analytical = LinRange(0.0, tfinal, Int(tfinal))\n\nΓohmic(t) = - quadgk(x -> Johmic(x,s)*(1 - cos(x*t))*coth(β*x/2)/x^2, 0, ωc)[1]\n\nDecoherence_ohmic(t) = 0.5 * exp(Γohmic(t))\n\n\nα_theo = 0.25 * α\nfunction Jtherm(x)\n    if 1 >= x >= 0\n        return +α_theo * abs(x)^s * (1 + coth(β*0.5*x))\n    elseif -1 <= x <= 0\n        return -α_theo * abs(x)^s * (1 + coth(β*0.5*x))\n    else\n        return 0\n    end\nend\n\nbath_occup_analytical(ω, t) = abs(Jtherm(ω))/(ω^2)*2*(1-cos(ω*t)) ","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We conclude the example by plotting.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"ρ12 = abs.(dat[\"data/Reduced ρ\"][1,2,:])\n\np1 = plot(time_analytical, t->Decoherence_ohmic(t), label=\"Analytics\", title=L\"Pure Dephasing, Ohmic $s=%$s$, $\\beta = %$β ~\\mathrm{K}$\", linecolor=:black, xlabel=\"Time (arb. units)\", ylabel=L\"Coherence $|\\rho_{12}(t)|$\", linewidth=4, titlefontsize=16, legend=:best, legendfontsize=16, xguidefontsize=16, yguidefontsize=16, tickfontsize=10)\np1 = plot!(dat[\"data/times\"], ρ12, lw=4, ls=:dash, label=\"Numerics\")\n\ncumul = [bath_occup_analytical(omeg[i], tfinal)*(omeg[i+1]-omeg[i]) for i in 1:(length(omeg)-1)]\n\np2 = plot(omeg[1:length(omeg)-1], cumul, lw = 4, linecolor=:black,\n             xlabel=L\"\\omega\", ylabel=L\"\\langle n^b_\\omega \\rangle\", label=\"Analytics\",\n             title=\"Mode occupation in the extended bath\")\np2 = plot!(omeg, bath_occup[:, :, T], lw=4, ls=:dash, label=\"Numerics\")\n\np3 = heatmap(omeg, omeg, abs.(real.(correlations_cdag[:,:,T]) .+ im*imag.(correlations_cdag[:,:,T])), \n            xlabel=L\"\\omega\",\n            ylabel=L\"\\omega\", title=\"Environmental correlations\")\n\n\nMhalf = Int(length(omeg)*0.5)+1\nM = length(omeg)\n\np4 = plot(omeg[Mhalf:M], bath_occup_phys, lw=4,\n            xlabel=L\"\\omega\", ylabel=L\"\\langle n^b_\\omega \\rangle\",\n            title=\"Mode occupation in the physical bath\")\n\nplot(p1, p2, p3, p4, layout = (2, 2), size = (1400, 1200))","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"___________________","category":"page"},{"location":"examples/bath-observables/#Bibliography","page":"Inspecting the bath by undoing the chain mapping","title":"Bibliography","text":"","category":"section"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"[chin_exact_2010]: Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. Journal of Mathematical Physics 2010, 51 (9), 092109. https://doi.org/10.1063/1.3490188.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"[tamascelli_efficient_2019]: Tamascelli, D.; Smirne, A.; Lim, J.; Huelga, S. F.; Plenio, M. B. Efficient Simulation of Finite-Temperature Open Quantum Systems. Phys. Rev. Lett. 2019, 123 (9), 090402. https://doi.org/10.1103/PhysRevLett.123.090402.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"[devega_thermo_2015]: de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"[riva_thermal_2023]: Riva, A.; Tamascelli, D.; Dunnett, A. J.; Chin, A. W. Thermal cycle and polaron formation in structured bosonic environments. Phys. Rev. B 2023, 108, 195138, https://doi.org/10.1103/PhysRevB.108.195138.","category":"page"},{"location":"examples/puredephasing/#Pure-Dephasing","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"","category":"section"},{"location":"examples/puredephasing/#Context","page":"Pure-Dephasing","title":"Context","text":"","category":"section"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The Pure-Dephasing Model describes a two-level system interacting linearly with an environment characterised by a spectral density (SD) J(omega). The coupling only acts on diagonal terms through the sigma_z operator. The Hamiltonian reads (hbar = 1)","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"        hatH = fracomega_02hatsigma_z + int_0^+inftyomega hata^dagger_omegahata_omega mathrmdomega + frachatsigma_z2int_0^+inftysqrtJ(omega)(hata_omega + hata^dagger_omega)mathrmdomega","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Although this interaction will not change the population of the two-level system, the coherences between the two states will vary due to the environment. Introducing the two-level system reduced density matrix rho_ij(t) with ij in (01), the diagonal terms rho_ii(t) are the populations of the states and the anti-diagonal terms rho_ij(t) with i neq j are the coherences between the two states. The effect of the sigma_z bath interaction is to decouple the two states 0rangle and 1rangle. ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The density matrix can be calculated within the MPS formalism with the MPO psirangle langle psi. Tracing out the environment leads to the reduced density matrix. This can be done with the function MPSDynamics.rhoreduced_1site. ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"An analytical formula can be found for the decoherence function Gamma(t), taking into account the SD as well as the temperature of the environment [breuer]","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"    Gamma(t) = - int_0^omega_c mathrmd omega J(omega)frac(1 - cos(omega t))omega^2 coth(βomega2) ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"with beta = (k_B T)^-1. For the case where beta longrightarrow infty, the integral reads","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"    Gamma(t) = - int_0^omega_c mathrmd omega J(omega)frac(1 - cos(omega t))omega^2 ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The time-dependent anti-diagonal terms of the reduced density matrix are then expressed as","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"    rho_12(t) = rho_21(t)^* =rho_12(0) exp(Gamma(t)) ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Setting up the initial two-level system as a cat state psirangle_S(0) = frac0rangle pm 1ranglesqrt2, this leads to rho_12(0)=frac12.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Here we break out and comment the script in MPSDynamics/examples/puredephasing.jl and MPSDynamics/examples/puredephasing_temperature.jl to show how to simulate this model with an Ohmic SD (hard cut-off) using the T-TEDOPA method as implemented in MPSDynamics.jl.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The T-TEDOPA method relies on a truncated chain mapping that transform the initial Hamiltonian into","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"        hatH = fracomega_02 hatsigma_z +  c_0 frachatsigma_z2(hatb_0^dagger + hatb_0) + sum_i=0^N-1 t_i (hatb_i+1^dagger hatb_i + mathrmhc) + sum_i=0^N-1 epsilon_i hatb_i^dagger hatb_i","category":"page"},{"location":"examples/puredephasing/#The-code","page":"Pure-Dephasing","title":"The code","text":"","category":"section"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"First, we load the MPSdynamics.jl package to be able to perform the simulation, the Plots.jl one to plot the results, the LaTeXStrings.jl one to be able to use LaTeX in the plots and eventually QuadGK.jl to perform the analytical integral calculations. A notation is introduced for +∞.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"using MPSDynamics, Plots, LaTeXStrings, QuadGK\n\nconst ∞  = Inf","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"We then define variables for the physical parameters of the simulation. Among these, two are convergence parameters:","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"d is the number of states we retain for the truncated harmonic oscillators representation of environmental modes\nN is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The value of β determines whether the environment is thermalized or not. The example as it is is the zero-temperature case. For the finite-temperature case, 'β = ∞' has be commented instead of the line above that precises a β value.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#----------------------------\n# Physical parameters\n#----------------------------\n\nd = 10 # number of Fock states of the chain modes\n\nN = 30 # length of the chain\n\nα = 0.01 # coupling strength\n\nω0 = 0.008 # TLS gap\n\ns = 1 # ohmicity\n\nωc = 0.035 # Cut-off of the spectral density J(ω)\n\n#β = 100 # Thermalized environment\n\nβ = ∞ # Case zero temperature T=0, β → ∞","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The chain coefficient have then to be calculated. This part differs for the zero-temperature case and the thermalized bath. For β = , the chain coeffficients can be calculated with the function MPSDynamics.chaincoeffs_ohmic whereas the function MPSDynamics.chaincoeffs_finiteT is used when T neq 0  textK:","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"if β == ∞\n    cpars = chaincoeffs_ohmic(N, α, s; ωc=ωc)  # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0\nelse\n    cpars = chaincoeffs_finiteT(N, β; α=α, s=s, J=nothing, ωc=ωc, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=false)  # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0\n    #= #If cpars is stored in \"../ChainOhmT/ohmicT\" \n    curdir = @__DIR__\n    dir_chaincoeff = abspath(joinpath(curdir, \"../ChainOhmT/ohmicT\"))\n    cpars  = readchaincoeffs(\"$dir_chaincoeff/chaincoeffs.h5\",N,α,s,β) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0\n    =#\nend","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"An option is provided if the coefficient are saved to be reused.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is mathcalO(dt^3). In this example we present the regular one-site method with the keyword :TDVP1 where all the virtual bonds of the MPS have the same bond dimension D. It is worh pointing out that DTDVP performs poorly in this specific example due to the exact solution of the system.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Logically the constant bond dimension of the MPS for TDVP1 is the convergence parameter.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#-----------------------\n# Simulation parameters\n#-----------------------\n\ndt = 1.0 # time step\n\ntfinal = 300.0 # simulation time\n\nmethod = :TDVP1 # time-evolution method\n\nD = 2 # MPS bond dimension","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Using MPSDynamics.jl built-in methods we define the Pure Dephasing model MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the 'local state' of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#---------------------------\n# MPO and initial state MPS\n#---------------------------\n\nH = puredephasingmpo(ΔE, d, N, cpars)\n\n# Initial electronic system in a superposition of 1 and 2\nψ = zeros(2)\nψ[1] = 1/sqrt(2)\nψ[2] = 1/sqrt(2)\n\nA = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"We then chose the observables that will be stored in the data and the MPSDynamics.runsim arguments. This example relies on the storage of the reduced density matrix (called reduceddensity in runsim).","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"\n#---------------------------\n# Definition of observables\n#---------------------------\n\nob1 = OneSiteObservable(\"sz\", sz, 1)\n\n\n#-------------\n# Simulation\n#------------\n\nA, dat = runsim(dt, tfinal, A, H, prec=1E-4;\n                name = \"pure dephasing model with temperature\",\n                method = method,\n                obs = [ob1],\n                convobs = [ob1],\n                params = @LogParams(ΔE, N, d, α, s),\n                convparams = D,\n                reduceddensity=true,\n                verbose = false,\n                save = true,\n                plot = true,\n                );","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"After having performed the dynamics, the analytical decoherence function is numerically calculated with the help of the quadgk function (from the QuadGK.jl package). It reads","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#----------\n# Analytical results at specified temperature \n# (see The Theory of Open Quantum System, H.-P. Breuer & F. Petruccione 2002, Chapter 4)\n#----------\n\nJohmic(ω,s) = (2*α*ω^s)/(ωc^(s-1))\n\ntime_analytical = LinRange(0.0,tfinal,Int(tfinal))\n\nΓohmic(t) = - quadgk(x -> Johmic(x,s)*(1 - cos(x*t))*coth(β*x/2)/x^2, 0, ωc)[1]\n\nDecoherence_ohmic(t) = 0.5*exp(Γohmic(t))","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Eventually, the stored reduced density matrix is compared against the analytical formula","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#-------------\n# Plots\n#------------\n\nρ12 = abs.(dat[\"data/Reduced ρ\"][1,2,:])\n\nplot(time_analytical, t->Decoherence_ohmic(t), label=\"Analytics\", title=L\"Pure Dephasing, Ohmic $s=%$s$, $\\beta = %$β ~\\mathrm{K}$\", linecolor=:black, xlabel=\"Time (arb. units)\", ylabel=L\"Coherence $|\\rho_{12}(t)|$\", linewidth=4, titlefontsize=16, legend=:best, legendfontsize=16, xguidefontsize=16, yguidefontsize=16, tickfontsize=10)\n\nplot!(dat[\"data/times\"], ρ12, lw=4, ls=:dash, label=\"Numerics\")","category":"page"},{"location":"examples/puredephasing/#Bibliography","page":"Pure-Dephasing","title":"Bibliography","text":"","category":"section"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"[breuer]: Breuer, H.;Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press, 2002.","category":"page"},{"location":"examples/anderson-model/#The-Anderson-Impurity-Model","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"Here we give some context on the examples provided in MPSDynamics/examples/anderson_model_double.jl and MPSDynamics/examples/anderson_model_interleaved.jl. In these two examples, we use the fermionic chain mapping proposed in [khon_efficient_2021] to show how to perform tensor network simulations of the Single Impurity Anderson Model (SIAM) with the MPSDynamics.jl library. ","category":"page"},{"location":"examples/anderson-model/#Basics-of-the-fermionic-chain-mapping","page":"The Anderson Impurity Model","title":"Basics of the fermionic chain mapping","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"Before giving the code implementation, a short recap on the problem statement and on the fermionic mapping used to solved it. The SIAM Hamiltonian is defined as:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    hat H^textSIAM  = hat H_textloc + hat H_texthyb + hat H_textcond = overbraceepsilon_d hat d^dagger hat d^hat H_textloc + underbracesum_k V_k Big( hat d^dagger hat c_k + hat c_k^dagger hat d Big)_hat H_texthyb + underbracesum_k epsilon_k hat c_k^dagger hat c_k_hat H_textcond","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"This Hamiltonian represents a located impurity (hat H_textloc), conduction electrons (hat H_textcond) and a hybridization term between the impurity and the conduction electrons (hat H_texthyb). All of the operators obey to the usual fermionic anti-commutation relations: hat c_i hat c_j^dagger  = delta_ij, hat c_i hat c_j  =hat c_i^dagger hat c_j^dagger  =0 forall ij. The chain mapping is based on a thermofield-like transformation  [devega_thermo_2015], performed with fermions: ancillary fermionic operators hat c_2k are defined, one for each of the original fermionic modes hat c_1k. A Bogoliubov transformation is then applied, so that two new fermionic modes hat f_1k and hat f_2k are defined as a linear combination of hat c_1k and hat c_2k. Two chains are defined: the chain labelled 1 for the empty modes:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"hat f_1k=e^-iG  hat c_k e^iG= cosh(theta_k) hat c_1k -sinh(theta_k)  hat c_2k^dagger \nhat f_2k=e^-iG hat c_k e^iG= cosh(theta_k)  hat c_1k +sinh(theta_k)  hat c_2k^dagger","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We remark that this is the same Bogoliubov transformation used in the thermofield[devega_thermo_2015] for the bosonic case: the only thing that changes is a plus sign, that takes into account the fermionic anti-commutation relations. With these new fermionic operators we obtain the thermofield-transformed Hamiltonian, where the system interacts with two environments at zero temperature, allowing for pure state simulations (and thus the employement of MPS).","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The thermofield-transformed Hamiltonian is then mapped on two chains, defined and constructed using the TEDOPA chain mapping: the chain labelled 1 is for the empty modes, the chain labelled 2 for the filled modes. The following relations are used to define the functions equivalent to the spectral density of the bosonic case, one for each chain:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"beginaligned\n   V_1k = V_k sin theta_k = sqrtfrac1e^beta epsilon_k+1 \n   V_2k = V_k cos theta_k = sqrtfrac1e^-beta epsilon_k+1 \nendaligned","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where we choose the spectral function that characterizes the fermionic bath to be: V_k= sqrt1-k^2, and we define the dispersion relation as: e_k = k, that is, a linear dispersion relation with propagation speed equal to 1. This latter choice corresponds to a model of metals (gapless energy spectrum). We select a filled state as the initial state of the defect. Using the mapping proposed, the chain Hamiltonian becomes:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    beginaligned\n    hat H^textchain  = hat H_textloc + sum_i = 12bigg J_i0 Big(hat d^dagger hat a_i0 + hat d hat a_i0^dagger Big) +  + sum_n=1^infty  Big( J_in hat a_in^dagger hat a_in-1 +  J_in hat  a_in-1^dagger hat a_in Big) + sum_n=0^infty E_in hat  a_in^dagger hat a_in bigg\n    endaligned","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where the J_in coefficients are the couplings between the chain sites and the E_in coefficients are the energies associated to each chain site. Clearly, the interactions are between nearest neighbors. This, combined with the fact that the fermions in our model are spinless, enables a straightforward mapping into fermionic operators of the bosonic creation and annihilation operators, that on their part obey to the bosonic commutation relations: hat b_i hat b_j^dagger = delta_ij, hat b_i hat b_j =hat b_i^dagger hat b_j^dagger =0 forall ij. The mapping derived from Jordan-Wigner transformations for spinless fermions is:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    hat a_i^dagger hat a_i+1 + hat a_i+1^dagger hat a_i = hat b_i^dagger hat b_i+1 + hat b_i+1^dagger hat b_i  ","category":"page"},{"location":"examples/anderson-model/#Double-chain-MPO","page":"The Anderson Impurity Model","title":"Double chain MPO","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"(Image: image)","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We can represent the Hamiltonian hat H^textchain as a MPOs bond dimension is: chi = 4. The MPO has the structure of a double chain, with an impurty site at the center, and therefore can be seen as the product of the following matrices:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    H = W_1N cdotcdot W_1 0 cdot W_d cdot W_20 cdot  cdot W_2 N  ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where the matrices are defined as:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"beginaligned\n W_1N = \nbeginbmatrix\nhatmathbb I  J_2N hat b_2N^dagger  J_2N hat b_2N  E_2N hat b_2N^dagger hat b_2N \nendbmatrix quad   W_1 0 = \nbeginbmatrix\n hat mathbb I  J_20 hat b_20^dagger  J_20 hat b_20  E_20 hat b_20^dagger hat b_20\n0 0  0  hat b_20 \n0 0  0  hat b_20^dagger \n0 0  0  hatmathbb I\nendbmatrix \n W_d = \nbeginbmatrix\n hat mathbb I  hat d^dagger  hat d  epsilon_d hat d^dagger hat d\n0 0  0  hat d \n0 0  0  hat d^dagger \n0 0  0  hatmathbb I\nendbmatrix \n W_2 0 =\nbeginbmatrix\n hat mathbb I  hat b_10^dagger  hat b_10  E_10 hat b_10^dagger hat b_10\n0 0  0  hat J_10b_10 \n0 0  0  hat J_10b_10^dagger \n0 0  0  hatmathbb I\nendbmatrix\n quad W_2 N =\nbeginbmatrix\n E_2N hat b_2N^dagger hat b_2N  J_2N hat b_2N  J_2N hat b_2N^dagger  hatmathbb I\nendbmatrix\nendaligned","category":"page"},{"location":"examples/anderson-model/#Interleaved-chain-MPO","page":"The Anderson Impurity Model","title":"Interleaved chain MPO","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"(Image: image)","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The drawback of the double chain representation is that the particle-hole pairs are spatially separated in the MPS, creating correlations and therefore leading to a dramatic increase in the bond dimensions. This is why Kohn and Santoro propose an interleaved geometry, the advantages of which are thoroughly explained in [khon_eff_2022]. Exploiting the interleaved representation, the interaction comes to be between next-nearest neighbors: a string operator appears in the Jordan-Wigner transformation from bosons to fermions:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    hat a_i^dagger hat a_i+2 + hat a_i+2^dagger hat a_i = hat b_i^dagger hat F_i+1 hat b_i+2 + hat b_i hat F_i+1 hat b_i+2^dagger","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where the string operator hat F_i is defined as: hat F_i = (-1)^hat n_i = hatmathbb I -2 hat n_i = hatmathbb I-2 hat b_i^dagger hat b_i.  It is possible to find the analytical form also for MPOs with long range interaction. In the case of next-nearest neighbors interactions between spinless fermions, in the interleaved geometry of the figure above, the MPO representation will require a bond dimension chi=6. We explicitly write it as:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    H = W_d cdot W_2 0 cdot W_1 0 cdotcdot W_2N cdot W_1 N  ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where the matrices are defined as: ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"beginaligned\n W_d = \nbeginbmatrix\n hatmathbb I  hat d  hat d^dagger  0  0  E_d hat d^dagger hat d \nendbmatrix quad W_2 0 = \nbeginbmatrix\n hat mathbb I  hat b_20  hat b_20^dagger  0  0  E_20 hat b_20^dagger hat b_20\n0 0  0  hatF_20  0  J_20 hat b_20^dagger \n0 0  0  0  hatF_20  J_20 hat b_20 \n0 0  0  0  0   0\n0 0  0  0  0  0 \n0 0  0  0  0  hatmathbb I\nendbmatrix \n W_1 0 = \nbeginbmatrix\n hat mathbb I  hat b_10  hat b_10^dagger  0  0  E_10 hat b_10^dagger hat b_10\n0 0  0  hat F_10  0  0 \n0 0  0  0  hatF_10  0 \n0 0  0  0  0  J_10 hat b_10^dagger \n0 0  0  0  0  J_10 hat b_10 \n0 0  0  0  0  hatmathbb I\nendbmatrix \n W_2N = \nbeginbmatrix\n hat mathbb I  hat b_2N  hat b_2N^dagger  0  0  E_2N hat b_2N^dagger hat b_2N\n0 0  0  hatF_2N  0  0 \n0 0  0  0  hatF_2N  0 \n0 0  0  0  0  J_2N hat b_2N^dagger \n0 0  0  0  0  J_2N hat b_2N \n0 0  0  0  0  hatmathbb I\nendbmatrix \nquad W_1 N\nbeginbmatrix\n E_1N hat b_1N^dagger hat b_1N  0 0  J_1N hat b_1N^dagger  J_1N hat b_1N  hatmathbb I\nendbmatrix \nendaligned","category":"page"},{"location":"examples/anderson-model/#Code-implementation","page":"The Anderson Impurity Model","title":"Code implementation","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The fermionic mapping can be strightforwardly implemented with the methods of MPSDynamics.jl. We provide two examples: ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"examples/anderson_model_double, simulating the SIAM with the double-chain geometry\nexamples/anderson_model_interleaved, simulating the SIAM with the interleaved-chain geometry","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"They only differ on the way the Hamiltonian MPO is defined. We now briefly review the important bits of the code.","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"Both of the examples start with the definition of the physical parameters,","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"N = 40      # number of chain sites\nβ = 10.0     # inverse temperature\nμ = 0.      # chemical potential\nEd = 0.3    # energy of the impurity\nϵd = Ed - μ # energy of the impurity minus the chemical potential","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The MPSDynamics.chaincoeffs_fermionic function is needed to compute the chain coefficients. It requires as inputs the number of modes of each chain N, the inverse temperature β, a label to specify if the chain modes are empty (label is 1.0) or filled (label is 2.0), and both the dispersion relation epsilon_k and the fermionic spectral density funciton V_k.","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"function ϵ(x)\n    return x\nend\n\nfunction V(x)\n    return sqrt(1 - x^2) # semi-circular density of states\nend\n\nchainparams1 = chaincoeffs_fermionic(N, β, 1.0; ϵ, V, save=false) # empty\nchainparams2 = chaincoeffs_fermionic(N, β, 2.0; ϵ, V, save=false) # filled","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We then specify the simulation parameters","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"dt = 0.25           # time step\nT = 15.0            # simulation time\nmethod = :DTDVP     # time-evolution method\nDmax = 150          # MPS max bond dimension\nprec = 0.0001       # precision for the adaptive TDVP\n","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"and with this we are ready to construct the Hamiltonian MPO and specify the initial state, which will obviously differ depending on the chosen geometry.","category":"page"},{"location":"examples/anderson-model/#Double-chain-geometry","page":"The Anderson Impurity Model","title":"Double chain geometry","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The Hamiltonian is defined using the MPSDynamics.tightbinding_mpo function, which takes as an input the number of modes of each chain N, the defect's energy ϵd, and the chain coefficients of the first chainparams1 and second chainparams2 chain. The MPS for the initial state is a factorized state made of: N filled states, a filled impurity, and N empty states. ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"H = tightbinding_mpo(N, ϵd, chainparams1, chainparams2)\n\nψ =  unitcol(2,2) # (0,1) filled impurity state\nA = productstatemps(physdims(H), state=[fill(unitcol(2,2), N)..., ψ, fill(unitcol(1,2), N)...]) # MPS","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"To avoid the DTDVP algorithm from getting stuck in a local minimum, it is better to embed the MPS in a manifold of bond dimension 2 (or more):","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"mpsembed!(A, 2) # to embed the MPS in a manifold of bond dimension 2","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We can now define the observables for the two chains and for the impurity","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"ob1 = OneSiteObservable(\"chain1_filled_occup\", numb(2), (1,N))\nob2 = OneSiteObservable(\"chain2_empty_occup\", numb(2), (N+2, 2N+1))\nob3 = OneSiteObservable(\"system_occup\", numb(2), N+1)","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"and run the simulation","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"A, dat = runsim(dt, T, A, H;\n                name = \"Anderson impurity problem (folded chain)\",\n                method = method,\n                obs = [ob1, ob2, ob3], \n                convobs = [ob1],\n                params = @LogParams(N, ϵd, β),\n                convparams = [prec],   \n                Dlim = Dmax,          \n                savebonddims = true,   # we want to save the bond dimension\n                verbose = false,\n                save = false,\n                plot = true,\n                );","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"With very minimal post-processing of the data","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"# Reshaping the vector to a column matrix and horizontal concatenation\nsystem_occup_col = reshape(dat[\"data/system_occup\"], :, 1)\nocc = hcat(dat[\"data/chain1_filled_occup\"]', system_occup_col)\nocc = vcat(occ', dat[\"data/chain2_empty_occup\"])","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"we plot the results:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"# Plot the system occupation    \np1 = plot(\n    dat[\"data/times\"],\n    dat[\"data/system_occup\"],\n    xlabel = L\"$t$\",\n    ylabel = L\"$n_d$\",\n    title = \"System Occupation\",\n    size = (700, 500)\n)\n\n# Plot the occupation of the chain sites\np2 = heatmap(\n    collect(1:2*N+1),\n    dat[\"data/times\"],\n    transpose(occ),  # Use the matrix form\n    cmap = :coolwarm,\n    aspect_ratio = :auto,\n    xlabel = L\"$N_{i,j}$ chain sites\",\n    ylabel = L\"$t$\",\n    title = \"Chain Occupation\",\n    colorbar = true,\n    size = (700, 500)\n)\n\n# Plot the bond dimensions\np3 = heatmap(\n    collect(1:2*N+2),\n    dat[\"data/times\"],\n    transpose(dat[\"data/bonddims\"]),\n    cmap = :magma,\n    aspect_ratio = :auto,\n    xlabel = L\"$N_{i,j}$ chain sites\",\n    ylabel = L\"$t$\",\n    title = \"Bond Dimensions\",\n    colorbar = true,\n    size = (700, 500)\n)\n\n# Define indices for columns to be plotted\ncolumns_to_plot = [1, 5, 10, 15, 20]\n\n# Plot vertical slices for occupancy\np4 = plot(title = \"Chain occupation\")\nfor col in columns_to_plot\n    plot!(p4, occ[:, col], label = L\"$t =$\"*\"$col\", xlabel = L\"$N_{i,j}$ chain sites\", ylabel = \"chain occupation\")\nend\n\n# Plot vertical slices for bond dimensions\np5 = plot(title = \"Bond Dimensions\")\nfor col in columns_to_plot\n    plot!(p5, dat[\"data/bonddims\"][:, col], label = L\"$t =$\"*\"$col\", xlabel = L\"$N_{i,j}$ chain sites\", ylabel = L\"$\\chi$\")\nend\n\n# Display the plots\nplot(p2, p3, p4, p5, p1, layout = (3, 2), size = (1400, 1200))\n","category":"page"},{"location":"examples/anderson-model/#Interleaved-chain-geometry","page":"The Anderson Impurity Model","title":"Interleaved chain geometry","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The Hamiltonian is defined using the MPSDynamics.interleaved_tightbinding_mpo function, which takes as an input the number of modes of each chain N, the defect's energy ϵd, and the chain coefficients of the first chainparams1 and second chainparams2 chain. The MPS for the initial state is a factorized state (bond dimension 1) made of: a filled impurity, and 2N alternate filled-empty states. ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"H = interleaved_tightbinding_mpo(N, ϵd, chainparams1, chainparams2)\n\nψ =  unitcol(2,2) # (0,1) filled impurity state\nTot = Any[]\npush!(Tot, ψ)\nfor i in 1:(N)\n    push!(Tot, unitcol(2,2))\n    push!(Tot, unitcol(1,2))\nend\n\nA = productstatemps(physdims(H), state=Tot) # MPS","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"To avoid the DTDVP algorithm from getting stuck in a local minimum, it is better to embed the MPS in a manifold of bond dimension 2 (or more):","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"mpsembed!(A, 2) # to embed the MPS in a manifold of bond dimension 2","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We can finally define the observables for the interleaved chain","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"ob1 = OneSiteObservable(\"system_occup\", numb(2), 1)\nob2 = OneSiteObservable(\"folded_chain_occup\", numb(2), (2,2N+1))","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"and run the simulation","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"A, dat = runsim(dt, T, A, H;\n                name = \"Anderson impurity problem (folded chain)\",\n                method = method,\n                obs = [ob1, ob2], \n                convobs = [ob1],\n                params = @LogParams(N, ϵd, β),\n                convparams = [prec],   \n                Dlim = Dmax,          \n                savebonddims = true,   # we want to save the bond dimension\n                verbose = false,\n                save = false,\n                plot = true,\n                );","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"To show the data in a clear way, we do a bit of post-processing of the data to unfold the chain, and move back to the double chain representation:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"unfolded_occ = Vector{Vector{Float64}}()  # Assuming the elements are of type Float64\nunfolded_bonds = Vector{Vector{Float64}}()  # Adjust the type based on actual data\n\n# Populate unfolded_occ by iterating in the specific order mentioned\nfor i in 1:N  # Adjusted for 1-based indexing\n    push!(unfolded_occ, dat[ \"data/folded_chain_occup\"][2N + 1 - 2i, :])\nend\n\npush!(unfolded_occ, dat[\"data/folded_chain_occup\"][1,:])\n\nfor i in 2:N\n    push!(unfolded_occ, dat[\"data/folded_chain_occup\"][2i,:])\nend\n\n# Populate unfolded_bonds similarly\nfor i in 1:(N+1)  # Adjusted for 1-based indexing\n    push!(unfolded_bonds, dat[\"data/bonddims\"][2N + 3 - 2i,:])  # Assuming bonddims is directly accessible\nend\n\npush!(unfolded_bonds, dat[\"data/bonddims\"][1,:])\n\nfor i in 2:(N+1)\n    push!(unfolded_bonds, dat[\"data/bonddims\"][2i,:])\nend\n\nunfolded_bonds_matrix = hcat(unfolded_bonds...)'\nunfolded_occ_matrix = hcat(unfolded_occ...)'","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We conclude by plotting the data","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"# Plot the system occupation    \np1 = plot(\n    dat[\"data/times\"],\n    dat[\"data/system_occup\"],\n    xlabel = L\"$t$\",\n    ylabel = L\"$n_d$\",\n    title = \"System Occupation\",\n    size = (700, 500)\n)\n\n# Plot the occupation of the chain sites\np2 = heatmap(\n    collect(1:2*N),\n    dat[\"data/times\"],\n    transpose(unfolded_occ_matrix),  # Use the matrix form\n    cmap = :coolwarm,\n    aspect_ratio = :auto,\n    xlabel = L\"$N_{i,j}$ chain sites\",\n    ylabel = L\"$t$\",\n    title = \"Chain Occupation\",\n    colorbar = true,\n    size = (700, 500)\n)\n\n# Plot the bond dimensions\np3 = heatmap(\n    collect(1:2*N+2),\n    dat[\"data/times\"],\n    transpose(unfolded_bonds_matrix),\n    cmap = :magma,\n    aspect_ratio = :auto,\n    xlabel = L\"$N_{i,j}$ chain sites\",\n    ylabel = L\"$t$\",\n    title = \"Bond Dimensions\",\n    colorbar = true,\n    size = (700, 500)\n)\n\n# Define indices for columns to be plotted\ncolumns_to_plot = [1, 5, 10, 15, 20]\n\n\n# Plot vertical slices for occupancy\np4 = plot(title = \"Chain occupation\")\nfor col in columns_to_plot\n    plot!(p4, unfolded_occ_matrix[:, col], label = L\"$t =$\"*\"$col\", xlabel = L\"$N_{i,j}$ chain sites\", ylabel = \"chain occupation\")\nend\n\n# Plot vertical slices for bond dimensions\np5 = plot(title = \"Bond Dimensions\")\nfor col in columns_to_plot\n    plot!(p5, unfolded_bonds_matrix[:, col], label = L\"$t =$\"*\"$col\", xlabel = L\"$N_{i,j}$ chain sites\", ylabel = L\"$\\chi$\")\nend\n\n# Display the plots\nplot(p2, p3, p4, p5, p1, layout = (3, 2), size = (1400, 1200))","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"________________","category":"page"},{"location":"examples/anderson-model/#Bibliography","page":"The Anderson Impurity Model","title":"Bibliography","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"[khon_efficient_2021]: Kohn, L.; Santoro, G. E. Efficient mapping for anderson impurity problems with matrix product states. Phys. Rev. B 2021, 104 (1), 014303. https://doi.org/10.1103/PhysRevB.104.014303.","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"[devega_thermo_2015]: de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"[khon_eff_2022]: Kohn L.; Santoro G. E. J. Stat. Mech. (2022) 063102 DOI 10.1088/1742-5468/ac729b.","category":"page"},{"location":"examples/sbm/#The-Spin-Boson-Model","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"","category":"section"},{"location":"examples/sbm/#Context","page":"The Spin-Boson Model","title":"Context","text":"","category":"section"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"The Spin-Boson Model (SBM) is a prototypical model in the theory of open quantum systems where a two level system interacts linearly with a bosonic bath (hbar = 1)","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"\thatH = fracomega_02hatsigma_z + Deltahatsigma_x + int_0^+inftyomega hata^dagger_omegahata_omega mathrmdomega + hatsigma_xint_0^+inftysqrtJ(omega)(hata_omega + hata^dagger_omega)mathrmdomega","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Even though this model is fairly simple it is physically very rich and it is not analytically solvable. For these reason it has become a test-bed for numerical methods simulating open quantum systems dynamics in the non-perturbative non-Markovian regime.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"For instance when the SD is Ohmic, this model presents a phase transition between a so called localised and a delocalised phase for alpha approx 12.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Here we break out and comment the script in MPSDynamics/examples/sbm_zero_temperature.jl to show how to simulate this model with an Ohmic SD (hard cut-off) using the T-TEDOPA method as implemented in MPSDynamics.jl.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"The T-TEDOPA method relies on a truncated chain mapping that transform the initial Hamiltonian into","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"\thatH = fracomega_02 hatsigma_z + Delta hatsigma_x + c_0 hatsigma_x(hatb_0^dagger + hatb_0) + sum_i=0^N-1 t_i (hatb_i+1^dagger hatb_i + mathrmhc) + sum_i=0^N-1 epsilon_i hatb_i^dagger hatb_i ","category":"page"},{"location":"examples/sbm/#The-code","page":"The Spin-Boson Model","title":"The code","text":"","category":"section"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"First a multi-line comment introduces the model and the aim of the script.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#=\n    Example of a zero-temperature Spin-Boson Model with an hard cut-off Ohmic spectral density J(ω) = 2αω when ω < ωc and 0 otherwise\n\n    The dynamics is simulated using the T-TEDOPA method that maps the normal modes environment into a non-uniform tight-binding chain.\n\n    H = \\\\frac{ω_0}{2} σ_z + Δ σ_x + c_0 σ_x(b_0^\\\\dagger + b_0) + \\\\sum_{i=0}^{N-1} t_i (b_{i+1}^\\\\dagger b_i +h.c.) + \\\\sum_{i=0}^{N-1} ϵ_i b_i^\\\\dagger b_i \n\n    Two variants of the one-site Time Dependent Variational Principal (TDVP) are presented for the time evolution of the quantum state.\n=#","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We load the MPSdynamics.jl package to be able to perform the simulation, the Plots.jl one to plot the results, and the LaTeXStrings.jl one to be able to use LaTeX in the plots.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"using MPSDynamics, Plots, LaTeXStrings","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We then define variables for the physical parameters of the simulation. Among these, two are convergence parameters:","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"d is the number of states we retain for the truncated harmonic oscillators representation of environmental modes \nN is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#----------------------------\n# Physical parameters\n#----------------------------\n\nd = 6 # number of Fock states of the chain modes\n\nN = 30 # length of the chain\n\nα = 0.1 # coupling strength\n\nΔ = 0.0 # tunneling \n\nω0 = 0.2 # TLS gap\n\ns = 1 # ohmicity\n\ncpars = chaincoeffs_ohmic(N, α, s) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is mathcalO(dt^3). In this example we present two one-site implementation of TDVP that both preserves the unitarity of the evolution:","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"the regular one-site method with the keyword :TDVP1 where all the virtual bonds of the MPS have the same bond dimension D\nthe adaptive method with the keyword :DTDVP where the bond dimension is locally increased at each time step if the TDVP projection error crosses a threshold value","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Logically the constant bond dimension of the MPS for TDVP1 and the threshold of the projection error for DTDVP are their respective convergence parameter. ","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#-----------------------\n# Simulation parameters\n#-----------------------\n\ndt = 0.5 # time step\n\ntfinal = 30.0 # simulation time\n\nmethod = :TDVP1 # Regular one-site TDVP (fixed bond dimension)\n\n# method = :DTDVP # Adaptive one-site TDVP (dynamically updating bond dimension)\n\nconvparams = [2,4,6] # MPS bond dimension (1TDVP)\n\n# convparams = [1e-2, 1e-3, 1e-4] # threshold value of the projection error (DTDVP)","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Using MPSDynamics.jl built-in methods we define the SBM MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the 'local state' of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#---------------------------\n# MPO and initial state MPS\n#---------------------------\n\nH = spinbosonmpo(ω0, Δ, d, N, cpars) # MPO representation of the Hamiltonian\n\nψ = unitcol(1,2) # Initial up-z system state \n\nA = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We then chose the observables that will be measured during the time evolution after being passed as an argument of the MPSDynamics.runsim method.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#---------------------------\n# Definition of observables\n#---------------------------\n\nob1 = OneSiteObservable(\"sz\", sz, 1)\n\nob2 = OneSiteObservable(\"chain mode occupation\", numb(d), (2,N+1))\n\nob3 = TwoSiteObservable(\"SXdisp\", sx, disp(d), [1], collect(2:N+1))","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We run the simulation from t = 0 to tfinal with time-steps of size dt using the method method. The convobs observables will be measured and stored for the convergence parameters convparams, whereas the obs observables will only be measured and stored for the most precise convergence parameter.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#-------------\n# Simulation\n#------------\n\nA, dat = runsim(dt, tfinal, A, H;\n                name = \"ohmic spin boson model\",\n                method = method,\n                obs = [ob2,ob3],\n                convobs = [ob1],\n                params = @LogParams(N, d, α, Δ, ω0, s),\n                convparams = convparams,\n                verbose = false,\n                savebonddims = true, # this keyword argument enables the bond dimension at each time step to be saved when using DTDVP\n                save = true,\n                plot = true,\n                );","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Finally, we plot the simulation results that are stored in the dat dictionnary.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#----------\n# Plots\n#----------\n\nmethod == :TDVP1 && plot(dat[\"data/times\"], dat[\"convdata/sz\"], label=[\"Dmax = 2\" \"Dmax = 4\" \"Dmax = 6\"], xlabel=L\"t\",ylabel=L\"\\sigma_z\")\n\nmethod == :DTDVP && plot(dat[\"data/times\"], dat[\"convdata/sz\"], label=[\"p = 1e-2\" \"p = 1e-3\" \"p = 1e-4\"], xlabel=L\"t\",ylabel=L\"\\sigma_z\") \n\nmethod == :DTDVP && heatmap(dat[\"data/times\"], collect(0:N+1), dat[\"data/bonddims\"], xlabel=L\"t\",ylabel=\"bond index\")\n\nheatmap(dat[\"data/times\"], collect(1:N), abs.(dat[\"data/SXdisp\"][1,:,:]), xlabel=L\"t\",ylabel=\"chain mode\")","category":"page"},{"location":"#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"The MPSDynamics.jl package provides an easy to use interface for performing tensor network simulations on matrix product states (MPS) and tree tensor network (TTN) states. Written in the Julia programming language, MPSDynamics.jl is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle (TDVP) method for time evolution.  The package also provides strong support for the measurement of observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics.  The package has been developed with the aim of studying non-Markovian open system dynamics at finite temperature using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm (T-TEDOPA) based on environment chain mapping. However the methods implemented can equally be applied to other areas of physics.","category":"page"},{"location":"#Installation","page":"Introduction","title":"Installation","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"The package may be installed by typing the following into a Julia REPL","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"    ] add https://github.com/shareloqs/MPSDynamics.git","category":"page"},{"location":"#Table-of-Contents","page":"Introduction","title":"Table of Contents","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"Pages = [\"index.md\", \"user-guide.md\", \"examples/sbm.md\", \"examples/puredephasing.md\", \"examples/timedep.md\", \"examples/anderson-model.md\", \"examples/bath-observables.md\", \"examples/protontransfer.md\", \"theory.md\", \"methods.md\", \"dev.md\"]\nDepth = 3","category":"page"},{"location":"#Citation","page":"Introduction","title":"Citation","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"If you use the package in your research, please consider citing it. You can add the Zenodo record and the software paper to your BibTex file:","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"@misc{mpsdynamics_zenodo2021,\n\ttitle = {shareloqs/{MPSDynamics}},\n\tshorttitle = {{MPSDynamics.jl}},\n\turl = {https://zenodo.org/record/5106435},\n\tabstract = {Tensor network simulations for finite temperature, open quantum system dynamics},\n\tpublisher = {Zenodo},\n\tauthor = {Dunnett, Angus and Lacroix, Thibaut and Le Dé, Brieuc and Riva, Angela},\n\tyear = {2021},\n\tdoi = {10.5281/zenodo.5106435},\n}\n\n@article{mpsdynamicsjl_2024,\n\ttitle = {{MPSDynamics}.jl: {Tensor} network simulations for finite-temperature (non-{Markovian}) open quantum system dynamics},\n\tvolume = {161},\n\tissn = {0021-9606},\n\tshorttitle = {{MPSDynamics}.jl},\n\turl = {https://doi.org/10.1063/5.0223107},\n\tdoi = {10.1063/5.0223107},\n\tnumber = {8},\n\tjournal = {The Journal of Chemical Physics},\n\tauthor = {Lacroix, Thibaut and Le Dé, Brieuc and Riva, Angela and Dunnett, Angus J. and Chin, Alex W.},\n\tmonth = aug,\n\tyear = {2024},\n\tpages = {084116},\n}\n","category":"page"}]
+[{"location":"nutshell/#In-a-nutshell:-MPSDynamics.jl-for-open-quantum-systems","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"","category":"section"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"MPSDynamics.jl was originally developed to perform tensor network simulations of open quantum systems by solving the Schrödinger equation for the closed {System + Environment}.  The key idea to construct the simulations is to reformulate the open quantum system Hamiltonian as a one-dimensional many-body problem with nearest-neighbor interactions.  The dynamics are then efficiently simulated with Tensor Networks methods. ","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"(Image: Sketch of the different ingredients behind MPSDynamics)","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"The starting point is a system linearly coupled to a bosonic environment, where the coupling between system and environment is specified by the spectral density function J(omega): the prototypical example being The Spin-Boson Model. At finite temperature, the initial state of the environment will be a thermal state (mixed), requiring density matrix formalism and thus making the problem more complex. To circumvent this issue, we consider instead an extended environment, with coupling to the system characterized by a thermalized spectral density function J(omegabeta).  The two environments induce the same reduced dynamics on the system, enabling us to deal with pure states only[1]. The vacuum state (pure) of the extended environment corresponds to the thermal state of the original environment. ","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"To exploit the computational efficiency of the matrix product states (MPS) description of pure states, we apply the so called Chain-Mapping of bosonic environments —a unitary transformation dependent on J(omegabeta)— to the Spin-Boson Hamiltonian, mapping it on a chain Hamiltonian with nearest-neighbor interactions. This enables efficient representation of the full {System + Environment} state and its time evolution using the Time-Dependent Variational Principle.","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"To go further, you will find in this documentation:","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"a User Guide explaining how to use the package\nseveral examples illustrating different features of the package: adaptive time-evolution methods, Time-dependent Hamiltonian, or fermionic environments with The Anderson Impurity Model\na Theoretical Background covering the chain mapping procedure, flying over the basics of tensor networks, and summarizing the TDVP methods","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"If this package was useful in your work, do not forget Citation. And if you would like to get involved in its development, you can find out How to Contribute.","category":"page"},{"location":"nutshell/","page":"In a nutshell: MPSDynamics.jl for open quantum systems","title":"In a nutshell: MPSDynamics.jl for open quantum systems","text":"[1]: Moreover, the dynamics of the original environment can still be recovered, see Inspecting the bath by undoing the chain mapping","category":"page"},{"location":"user-guide/#User-Guide","page":"User Guide","title":"User Guide","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Here we explain the different steps to perform a simulation.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Examples with detailed explanations can be found in Examples.","category":"page"},{"location":"user-guide/#Initial-State","page":"User Guide","title":"Initial State","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The initial many-body state of the {System + Environment} can be described easily as Matrix Product States (MPS) or Tree Tensor Networks (TTN) state (e.g. when a system is coupled to several environments).","category":"page"},{"location":"user-guide/#Matrix-Product-States","page":"User Guide","title":"Matrix Product States","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"A MPS can be initialized with several methods.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The productstatemps method enables  to instantiate arbitrary MPS of fixed uniform bond dimension with non-uniform physical dimensions. The individual states of the MPS sites can be provided by setting state to a list of column vectors.  Setting state=:Vacuum will produce an MPS in the vacuum state.  Setting state=:FullOccupy will produce an MPS in which each site is fully occupied. The gauge of the MPS can also be set using a keyword argument.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"julia> ψ = unitcol(1,2) # system initial state\n\njulia> d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1 \n\njulia> cpars = chaincoeffs_ohmic(N, α, s) # chain coefficient for an Ohmic spectral density\n\njulia> H = spinbosonmpo(ω0, Δ, d, N, cpars)\n\njulia> A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Alternatively, a chain with a specified number of excitation localised on one site, or delocalized accross several sites can be generated with MPSDynamics.chainmps.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Random MPS can also be generated with the randmps method.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"For the case of fermionic states (which need to be anti-symmetrized), the MPSDynamics.electronkmps method generate an MPS for an electron with momentum k, and the MPSDynamics.electron2kmps generate an MPS with 2 electrons in k-states k1 and k2.","category":"page"},{"location":"user-guide/#Tree-Tensor-Networks","page":"User Guide","title":"Tree Tensor Networks","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Write a quick explanation of the how trees are structured: parents, child, nodes; and that most methods to initialize a MPS are overloaded for TreeNetwork. ","category":"page"},{"location":"user-guide/#Hamiltonian","page":"User Guide","title":"Hamiltonian","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"In order to perform time evolution and have access to the dynamics of the many-body state a Hamiltonian needs to be specified in the form of a Matrix Product Operator (MPO) of as a tree tensor network. Either way, this can be done by using a Build-in Hamiltonian, Convert a MPO from ITensor, or creating a Tailored MPO.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"In the context of Open Quantum Systems, custom chain coefficients for the environment can be generated for finite temperature simulations, and/or user provided spectral densities (SDs).","category":"page"},{"location":"user-guide/#Build-in-Hamiltonian","page":"User Guide","title":"Build-in Hamiltonian","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"MPSDynamics provides several topical Hamiltonians directly in the form of MPO or Tree Tensor Networks such as the Ising model MPSDynamics.isingmpo, the XYZ Hamiltonian MPSDynamics.xyzmpo, the Spin Boson Model MPSDynamics.spinbosonmpo, a spin coupled to two bosonic baths MPSDynamics.twobathspinmpo, nearest neighbour interactions Hamiltonian MPSDynamics.nearestneighbourmpo, the independent boson model MPSDynamics.ibmmpo, (non-)uniform tight-binding chain Hamiltonian MPSDynamics.tightbindingmpo.","category":"page"},{"location":"user-guide/#Convert-a-MPO-from-ITensor","page":"User Guide","title":"Convert a MPO from ITensor","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The method MPSDynamics.MPOtoVector converts an ITensors chain MPO into a form compatible with MPSDynamics.","category":"page"},{"location":"user-guide/#Tailored-MPO","page":"User Guide","title":"Tailored MPO","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"One can also construct MPO tailored to the problem one is interested in. MPOs are fundamentally a lists of rank-4 tensors such that the right bond dimension of the nth tensor must be equal to the left bond dimension of the n+1th tensor; and the dimension of the physical bonds of the nth tensor must be equal to the corresponding physical bond on the MPS.","category":"page"},{"location":"user-guide/#Finite-Temperature-and-Custom-SD","page":"User Guide","title":"Finite Temperature and Custom SD","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"In the T-TEDOPA framework (see Theoretical Background for more details) finite temperature simulations are done with an effective pure state description of the system and the environment where the coupling coefficients (or the SD) is temperature-dependent.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The corresponding chain coefficients for an Ohmic or a user provided spectral density (that can thus in pratice be either at zero or finite temperature) are computed with the [chaincoeffs_finiteT](@ref). This method is based on the ORTHOPOL routines[Gautschi]","category":"page"},{"location":"user-guide/#Observables","page":"User Guide","title":"Observables","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"System and environment observables can be computed, as well as system-and-environment 'non-local' observables.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Observables that will be passed to MPSDynamics.runsim(@ref) to have their expectation value computated at each time step are defined with the OneSiteObservable and [TwoSiteObservable](@ref).","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"One-site and two-site obsevables work similarly, they need to be given a name, an (pair of) operator(s) and the (list of) site(s) on which they are evaluated.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"For instance one can calculated the average number of excitation in each of the N environmental modes","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"    ob = OneSiteObservable(\"chain mode occupation\", numb(d), (2,N+1))","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"It is also possible to measure composite system/environment observables, for example","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"    ob = TwoSiteObservable(\"SXdisp\", sx, disp(d), [1], collect(2:N+1))","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"which measure the correlation between the spin in the x-direction and the displacement of the bath modes.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Purely environmental 'non-local' observables such as ","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"    ob = TwoSiteObservable(\"bath coherence\", crea(d), anih(d), collect(2:N+1), collect(2:N+1))","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"that computes all the chain mode coherences langlehata_n^daggerhata_mrangle (and the chain mode occupation when n = m).","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"We note that if one knows the coherences and populations of all the chain modes, it is then possible to reconstruct the populations of the normal mode environment.","category":"page"},{"location":"user-guide/#Time-Evolution","page":"User Guide","title":"Time-Evolution","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Simulation are performed when calling the MPSDynamics.runsim function where the time-evolution method should be specified.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The time-evolution methods currently implemented belong to the familly of Time-Dependent Variational Principle (TDVP). The central point of this method, in the modern tensor networks formulation, is that instead of solving the Schrödinger equation and then truncating the MPS representation of the quantum state, one can solve the equations of motion projected into a space of restricted bond dimension.  The major advantage of this method is that it naturally preserves the unitarity of the time evolution and conserves the energy (except in its two-site implementation). Three variants of TDVP are implemented in the MPSDynamics.jl package:","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"one-site TDVP (TDVP1): fixed bond dimension, preserves unitarity, conserves energy, scales as mathcalO(D^2 d^2 w^2 + D^3 dw + D^3 d^2 ) where D is the MPS (or TTN) bond dimension, d the local dimension, w the MPO bond dimension.\ntwo-site TDVP (TDVP2): adaptive bond dimension, breaks unitarity, scales as mathcalO(D^2d^3 w^2 + D^3 d^2 w + D^3 d^3 ).\nadaptive one-site TDVP (DTDVP): adaptive bond dimension, preserves unitarity, conserves energy.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Local one-site and two-site observables, as well as non-local two-site observables, can be efficiently computed for each time-step of the time-evolution with the three methods.","category":"page"},{"location":"user-guide/#One-site-TDVP","page":"User Guide","title":"One-site TDVP","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This implementation of the TDVP can be used for MPS and TTN. It performs one-site updates of the tensor network tensors while keeping the bond dimension fixed. Because no reduction of the dimensionality is performed (for instnce by throwing away some simgular values), it naturally preserves unitarity and hence conserves energy. The otherside of the coin is that simulation might be slower because the bond dimension is large at all time.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The convergence parameter of the simulation is thus the bond dimension D.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This method can be used with in MPSDynamics.runsim with the key word argument method=:TDVP1.","category":"page"},{"location":"user-guide/#Two-Site-TDVP","page":"User Guide","title":"Two-Site TDVP","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This version of the TDVP evolves two-site tensors than can be split back into two individual site tensors by applying an SVD. The bond dimension between two neighbouring sites can then be truncated to any value  r by throwing away any singular values that fall below some threshold.  In this way, the MPS bond dimension will grow dynamically throughout the course of the evolution to capture entanglement, as and when it emerges. The truncation entails a loss of unitarity.  Indeed, for a nearest-neighbour Hamiltonian, applying the two-site projector will not entail a projection error, leading to a scheme that is almost identical to TEBD,wherein the error arises solely from the truncation. Furthermore, 2TDVP scales poorly with the local physical dimension, and is known to have issues with long-range interactions.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The convergence parameter is the threshold of the SVD.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This method can be used with in MPSDynamics.runsim with the key word argument method=:TDVP2.","category":"page"},{"location":"user-guide/#Adaptive-One-Site-TDVP","page":"User Guide","title":"Adaptive One-Site TDVP","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Ahead of any time evolution, one computes new bond dimensions of the MPS if the relative rate of change of the TDVP projection error with respect to the bond dimension is larger than a chosen precision p. Then, TDVP1 is performed, using projectors with sub-spaces expanded accordingly, to produce an MPS evolved by one time step with the new,increased, bond dimensions. This version is faster than TDVP1 due to the acceleration gained from having more optimised bond dimensions; the bond update step is cheap and so its cost should not normally outweigh this advantage. However, two things need to be noticed:","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"DTDVP can get stuck into local minima from the initial time. If the bond dimension does not change during the time-evolution, consider embeding your MPS in a larger manifold with mpsembed! before time-evolving it.\nFor models with analytical solutions that are MPS of a given bond dimension (such as the independent boson model), DTDVP can overshoot the analytical bond dimension because the relative rate of change of the projection error becomes dominated by random numerical fluctuations.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The convergence parameter is a threshold value p for the rate of change of the projection error with respect to the bond dimension.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"This method can be used with in MPSDynamics.runsim with the key word argument method=:DTDVP.","category":"page"},{"location":"user-guide/#Data-Storage","page":"User Guide","title":"Data Storage","text":"","category":"section"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The data (i.e. observables time-series) is stored in the JLD format which is based on HDF5.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The HDF5 format is natively supported across many platforms and languages (e.g. Python, or Mathematica).","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"For the data to be saved to a file after a run, the keyword argument save=true needs to be used in runsim.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"The directory where the data should be saved can be chosen by setting a path with the savedir keyword argument.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"A plot keyword argument can also be used to choose whether plots for 1D observables will be automatically generated and saved along with the data. ","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"Loading the data in Julia using the JLD.jl package will recover the full type information of the Julia variables that were stored.","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"julia> using JLD\n\njulia> dat = load(\"filename.jld\")\nDict{String, Any} with 2 entries:\n  \"parameters\" => Dict{String, Any}(\"tmax\"=>0.2, \"method\"=>:DTDVP, \"dt\"=>0.0005…\n  \"data\"       => Dict{String, Any}(\"bonddims\"=>[1 1 … 1 1; 1 2 … 2 2; … ; 1 1 …","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"If the data is loaded in an variable dat the structure is the folowing:","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"the parameters entry contains a dictionary where all the simulation parameters are stored\nthe data entry contains a dictionary where are stored simulation time, the observables and (whenever relevent) the bond dimension of the state at each time steps. ","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"julia> dat[\"parameters\"]\nDict{String, Any} with 11 entries:\n  \"tmax\"       => 0.2\n  \"method\"     => :DTDVP\n  \"dt\"         => 0.0005\n  \"name\"       => \"my model\"\n  \"Δ\"          => 0.0\n  \"β\"          => 0.0186854\n  \"N\"          => 300.0\n  \"d\"          => 15.0\n  \"unid\"       => \"Ovzm6\"\n  \"ω0\"         => 0.0\n  \"convparams\" => 0.0005\n\njulia> dat[\"data\"]\nDict{String, Any} with 6 entries:\n  \"bonddims\" => [1 1 … 1 1; 1 2 … 2 2; … ; 1 1 … 7 7; 1 1 … 1 1]\n  \"sx\"       => [1.0, 0.912818, 0.741759, 0.605797, 0.528792, 0.492497, 0.47976…\n  \"sz\"       => [0.0, 0.0871825, 0.25824, 0.394201, 0.471207, 0.507503, 0.52023…\n  \"nchain\"   => [0.0 0.23466 … 1.84319 1.76098; 0.0 0.00231507 … 0.83105 0.9033…\n  \"sy\"       => [0.0, -0.0133489, -0.0588887, -0.0858181, -0.0759996, -0.048539…\n  \"times\"    => [0.0, 0.0005, 0.001, 0.0015, 0.002, 0.0025, 0.003, 0.0035, 0.00…\n","category":"page"},{"location":"user-guide/","page":"User Guide","title":"User Guide","text":"[Gautschi]: Gautschi, W. Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 20, 21–62 (1994).","category":"page"},{"location":"examples/timedep/#Time-dependent-Hamiltonian","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"","category":"section"},{"location":"examples/timedep/#Context","page":"Time-dependent Hamiltonian","title":"Context","text":"","category":"section"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"To simulate a drive or a laser pulse, a time-dependent Hamiltonian is often needed. Here we explain thanks to the script in MPSDynamics/examples/sbm_Htimedependent.jl how this type of Hamiltonian can be integrated within the method and applied on a wavefunction. We will take a Spin-Boson Model (SBM) Hamiltonian with an Ohmic spectral density. However, the time-dependency is not model specific and can be adapated for other models. For more information about the SBM, see the dedicated example. For this example, we simulate a drive of the form","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"        hatH_textdrive(t) = epsilon hatsigma_x sin(omega_textdrive t)","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"with epsilon = frac2 piT_textRabi the amplitude of the drive, T_textRabi the Rabi period, and omega_textdrive the frequency of the drive. The drive is set up to be on resonance with the two-level system.","category":"page"},{"location":"examples/timedep/#The-code","page":"Time-dependent Hamiltonian","title":"The code","text":"","category":"section"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"First we load the MPSdynamics.jl package to be able to perform the simulation, the Plots.jl one to plot the results, and the LaTeXStrings.jl one to be able to use LaTeX in the plots. The function MPSDynamics.disp is also imported.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"using MPSDynamics, Plots, LaTeXStrings\n\nimport MPSDynamics: disp","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"We then define variables for the physical parameters of the simulation. Among these, three are convergence parameters:","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"d is the number of states we retain for the truncated harmonic oscillators representation of environmental modes\nN is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"The variable Ndrive represents the site of the MPO where the operator of the time-dependent part acts. For this example, the two-level system is at the first site of the MPS. ","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#----------------------------\n# Physical parameters\n#----------------------------\n\nd = 4 # number of Fock states of the chain modes\n\nN = 60 # length of the chain\n\nα = 0.005 # coupling strength\n\nΔ = 0.0 # tunneling \n\nω0 = 0.8 # TLS gap\n\ns = 1 # ohmicity\n\ncpars = chaincoeffs_ohmic(N, α, s) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0\n\nTrabi = 30.0 # Rabi period of the drive\n\nϵ = 2*pi / Trabi # Intensity of the drive\n\nωdrive = ω0 # Frequency of the drive\n\nNdrive = 1 #Number of the site on which the drive is applied","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is mathcalO(dt^3). In this example we present only one-site implementation of TDVP that preserves the unitarity of the evolution:","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"the regular one-site method with the keyword :TDVP1 where all the virtual bonds of the MPS have the same bond dimension D","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"Logically the constant bond dimension of the MPS for TDVP1 is the respective convergence parameter.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#-----------------------\n# Simulation parameters\n#-----------------------\n\ndt = 0.5 # time step\n\ntfinal = 100.0 # simulation time\n\nmethod = :TDVP1 # time-evolution method\n\nD = [6] # MPS bond dimension","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"Using MPSDynamics.jl built-in methods we define the SBM MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the 'local state' of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"In this part, the time-dependent terms of the MPO are stored in a list. This enables to add these terms to the MPO elements at each timestep and avoid the calculation of an entire new MPO. This way is not cumbersome since it adds a matrix sum at each time step.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#---------------------------\n# MPO and initial state MPS\n#---------------------------\n\ntimelist = collect(0:dt:tfinal)\nnumsteps = length(timelist)-1\n\nHt = [ϵ*sx*sin(ωdrive*tstep) for tstep in timelist] # Time-dependent Hamiltonian term\n\nH = spinbosonmpo(ω0, Δ, d, N, cpars) # MPO representation of the Hamiltonian\n\nψ = unitcol(2,2) # Initial down-z system state \n\nA = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"We then choose the observables that will be stored in the data and the MPSDynamics.runsim arguments.","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#---------------------------\n# Definition of observables\n#---------------------------\n\nob1 = OneSiteObservable(\"sz\", sz, 1)\n\nob2 = OneSiteObservable(\"chain mode occupation\", numb(d), (2,N+1))\n\nob3 = TwoSiteObservable(\"SXdisp\", sx, disp(d), [1], collect(2:N+1))\n","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"MPSDynamics.runsim is called to perform the dynamics. The argument timedep is set up to true and Ndrive and Htime are provided in the kwargs. ","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"#-------------\n# Simulation\n#------------\n\nA, dat = runsim(dt, tfinal, A, H;\n                name = \"Driving field on ohmic spin boson model\",\n                method = method,\n                obs = [ob1],\n                convobs = [ob1],\n                params = @LogParams(N, d, α, Δ, ω0, s),\n                convparams = D,\n                timedep = true, # the Hamiltonian is time dependent\n                Ndrive = Ndrive, # the first site of the MPS/MPO (i.e. the system) is concerned\n                Htime = Ht, # list of time-dependent terms\n                verbose = false,\n                save = true,\n                plot = true,\n                );","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"Eventually, the stored observables can be represented","category":"page"},{"location":"examples/timedep/","page":"Time-dependent Hamiltonian","title":"Time-dependent Hamiltonian","text":"\n#----------\n# Plots\n#----------\n\nplot(dat[\"data/times\"], dat[\"data/sz\"], label=\"Dmax = $(D...)\", xlabel=L\"t\",ylabel=L\"\\sigma_z\", title=\"\")","category":"page"},{"location":"dev/#Developers","page":"Developers","title":"Developers","text":"","category":"section"},{"location":"dev/#Simulation-Workflow","page":"Developers","title":"Simulation Workflow","text":"","category":"section"},{"location":"dev/","page":"Developers","title":"Developers","text":"(Image: Flowchart of the simulation workflow)","category":"page"},{"location":"dev/#How-to-Contribute","page":"Developers","title":"How to Contribute","text":"","category":"section"},{"location":"dev/","page":"Developers","title":"Developers","text":"Contributions are welcome! Don't hesitate to contact us if you","category":"page"},{"location":"dev/","page":"Developers","title":"Developers","text":"found a bug;\nhave a suggestion on how to improve the code and/or documentation;\nwould like to get involved in writing code and/or documentation.","category":"page"},{"location":"dev/","page":"Developers","title":"Developers","text":"You can contact us by raising an issue on Github, or by writing to one of the developers.","category":"page"},{"location":"methods/#List-of-all-methods","page":"Methods","title":"List of all methods","text":"","category":"section"},{"location":"methods/","page":"Methods","title":"Methods","text":"Modules = [MPSDynamics]","category":"page"},{"location":"methods/#MPSDynamics.OneSiteObservable-Tuple{Any, Any, Any}","page":"Methods","title":"MPSDynamics.OneSiteObservable","text":"OneSiteObservable(name,op,sites)\n\nComputes the local expectation value of the one-site operator op on the specified sites. Used to define one-site observables that are obs and convobs parameters for the runsim function.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.MPOtoVector-Tuple{ITensors.MPO}","page":"Methods","title":"MPSDynamics.MPOtoVector","text":"MPOtoVector(mpo::MPO)\n\nConvert an ITensors chain MPO into a form compatible with MPSDynamics\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.addchild!-Tuple{MPSDynamics.Tree, Int64}","page":"Methods","title":"MPSDynamics.addchild!","text":"addchild!(tree::Tree, id::Int)\n\nAdd child to node id of tree.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.addchildren!-Tuple{MPSDynamics.Tree, Int64, Int64}","page":"Methods","title":"MPSDynamics.addchildren!","text":"addchildren!(tree::Tree, id::Int, n::Int)\n\nAdd n children to node id of tree.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Int64}","page":"Methods","title":"MPSDynamics.apply1siteoperator!","text":"apply1siteoperator!(A, O, sites::Int)\n\nApply an operator O on the MPS A. O is acting on only one site ::Int. The resulting MPS A is the MPS modified by the operator O.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.apply1siteoperator!-Tuple{Any, Any, Vector{Int64}}","page":"Methods","title":"MPSDynamics.apply1siteoperator!","text":"apply1siteoperator!(A, O, sites::Vector{Int})\n\nApply an operator O on the MPS A. O is acting on several sites ::Vector{Int}. The resulting MPS A is the MPS modified by the operator O.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.applympo!-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.applympo!","text":"applympo!(A, H)\n\nApply an MPO H on the MPS A. H must have the same number of site than A. The resulting MPS A is the MPS modified by the mpo H.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.chaincoeffs_finiteT","page":"Methods","title":"MPSDynamics.chaincoeffs_finiteT","text":"chaincoeffs_finiteT(nummodes, β, ohmic=true; α, s, J, ωc=1, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=true)\n\nGenerate chain coefficients ϵ_0ϵ_1t_0t_1c_0 for a harmonic bath at the inverse temperature β.\n\nBy default a Ohmic spectral density J(ω) = 2αω_c (fracωω_c)^s θ(ω-ω_c) is considered. Users can provide their own spectral density.\n\nArguments\n\nnummodes: Number of bath modes\nβ: inverse temperature\nohmic: true if the spectral density is Ohmic, false if the user provides its own spectral density\nα: Kondo parameter of the Ohmic spectral density \ns: ohmicity\nJ: user-provided spectral density. Should be a function f(x,i) where x is the frequency and i ∈ {1,...,mc} labels the intervals on which the SD is defined\nωc: the maximum frequency of the Ohmic spectral density\nmc: the number of component intervals\nmp: the number of points in the discrete part of the measure (mp=0 if there is none)\niq: a parameter to be set equal to 1, if the user provides his or her own quadrature routine, and different from 1 otherwise\nidelta: a parameter whose default value is 1, but is preferably set equal to 2, if iq=1 and the user provides Gauss-type quadrature routines\nprocedure: choice between the Stieltjes and the Lanczos procedure\nAB: component intervals\nMmax: maximum number of integration points\nsave: if true the coefficients are saved\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.chaincoeffs_ohmic-Tuple{Any, Any, Any}","page":"Methods","title":"MPSDynamics.chaincoeffs_ohmic","text":"chaincoeffs_ohmic(N, α, s; ωc=1, soft=false)\n\nGenerate chain coefficients ϵ_0ϵ_1t_0t_1c_0 for an Harmonic bath at zero temperature with a power law spectral density given by:\n\nsoft cutoff: J(ω) = 2αω_c (fracωω_c)^s exp(-ωω_c) \n\nhard cutoff: J(ω) = 2αω_c (fracωω_c)^s θ(ω-ω_c)\n\nThe coefficients parameterise the chain Hamiltonian\n\nH = H_S + c_0 A_SB_0+sum_i=0^N-1t_i (b_i+1^dagger b_i +hc) + sum_i=0^N ϵ_ib_i^dagger b_i\n\nwhich is unitarily equivalent (before the truncation to N sites) to\n\nH = H_S + A_Sint_0^dωsqrtJ(ω)B_ω + int_0^dωωb_ω^dagger b_ω.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.chainmps-Tuple{Int64, Int64, Int64}","page":"Methods","title":"MPSDynamics.chainmps","text":"chainmps(N::Int, site::Int, numex::Int)\n\nGenerate an MPS with numex excitations on site\n\nThe returned MPS will have bond-dimensions and physical dimensions numex+1\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.chainmps-Tuple{Int64, Vector{Int64}, Int64}","page":"Methods","title":"MPSDynamics.chainmps","text":"chainmps(N::Int, sites::Vector{Int}, numex::Int)\n\nGenerate an MPS with numex excitations of an equal super-position over sites\n\nThe returned MPS will have bond-dimensions and physical dimensions numex+1\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.chainprop-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.chainprop","text":"chainprop(t, cparams...)\n\nPropagate an excitation placed initially on the first site of a tight-binding chain with parameters given by cparams for a time t and return occupation expectation for each site.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.correlatedenvironmentmpo-Tuple{Vector, Int64, Int64}","page":"Methods","title":"MPSDynamics.correlatedenvironmentmpo","text":"correlatedenvironmentmpo(R::Vector, Nm::Int, d::Int; chainparams, fnamecc::String, s=1, α=1, ωc=1, c_phonon=1, β=\"inf\", issoft=false)\n\nGenerate a MPO for a one-dimensional bosonic bath spatially correlated to a multi-component system \n\nH_B + H_int = int_-^+ dk ω_k b_k^dagger b_k + _j int_-^+dk sqrtJ(k)(A_j b_k e^i k R_j + hc).\n\nThe interactions between the system and the chain-mapped bath are long range, i.e. each site interacts with all the chain modes. The spectral density is assumed to be Ohmic J(ω) = 2αωc(ωωc)^s.\n\nArguments\n\nR: List of system's components positions\nNm: Number of chain modes. The actual number of mode will be doubled to account for the left and right moving excitations.\nd: Local Hilbert space dimension of the bath modes\nchainparams: chain parameters, of the form chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature.\nfnamecc: Path to a file containing pre-computed long-range coupling coefficient. If not provided, the coupling coefficients will be computed and stored.\ns: Ohmicity\nα: Kondo parameter\nωc: Bath cut-off frequency\nc_phonon: Speed of sound in the bath\nβ: Inverse temperature \nissoft: Is the cut-off of the Ohmic SD soft or hard?\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.displacedchainmps-Tuple{Vector{Any}, Int64, Int64, Any}","page":"Methods","title":"MPSDynamics.displacedchainmps","text":"displacedchainmps(A::Vector{Any}, N::Int, Nm::Int, γ::Any)\n\nGiven a mps A, return a mps B where the `Nm`-long chain is displaced by `γ` without displacing the `N`-long system.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.dynamap-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.dynamap","text":"dynamap(ps1,ps2,ps3,ps4)\n\nCalulate complete dynamical map to time step at which ps1, ps2, ps3 and ps4 are specified.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.electron2kmps","page":"Methods","title":"MPSDynamics.electron2kmps","text":"electronkmps(N::Int, k::Vector{Int}, spin=:Up, chainparams=[fill(1.0,N), fill(1.0,N-1)])\n\nGenerate an MPS with 2 electrons in k-states k1 and k2.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.electronkmps","page":"Methods","title":"MPSDynamics.electronkmps","text":"electronkmps(N::Int, k::Int, spin=:Up, chainparams=[fill(1.0,N), fill(1.0,N-1)])\n\nGenerate an MPS for an electron with momentum k.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.elementmpo-Tuple{Any, Vararg{Any}}","page":"Methods","title":"MPSDynamics.elementmpo","text":"elementmpo(M, el...)\n\nReturn the element of the MPO M for the set of physical states el...\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.elementmps-Tuple{Any, Vararg{Any}}","page":"Methods","title":"MPSDynamics.elementmps","text":"elementmps(A, el...)\n\nReturn the element of the MPS A for the set of physical states el...\n\nExamples\n\njulia> A = chainmps(6, [2,4], 1);\n\njulia> elementmps(A, 1, 2, 1, 1, 1, 1)\n0.7071067811865475\n\njulia> elementmps(A, 1, 1, 1, 2, 1, 1)\n0.7071067811865475\n\njulia> elementmps(A, 1, 2, 1, 2, 1, 1)\n0.0\n\njulia> elementmps(A, 1, 1, 1, 1, 1, 1)\n0.0\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.entanglemententropy-Tuple{Any}","page":"Methods","title":"MPSDynamics.entanglemententropy","text":"entanglemententropy(A)\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the entanglement entropy for a bipartite cut for every bond.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.findchainlength-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.findchainlength","text":"findchainlength(T, cparams...; eps=10^-6)\n\nEstimate length of chain required for a particular set of chain parameters by calculating how long an excitation on the first site takes to reach the end. The chain length is given as the length required for the excitation to have just reached the last site after time T. The initial number of sites in cparams has to be larger than the findchainlength result.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.findchild-Tuple{MPSDynamics.TreeNode, Int64}","page":"Methods","title":"MPSDynamics.findchild","text":"findchild(node::TreeNode, id::Int)\n\nReturn integer corresponding to the which number child site id is of node.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.gauss-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.gauss","text":"GAUSS Gauss quadrature rule.\n\nGiven a weight function w encoded by the nx2 array ab of the\nfirst n recurrence coefficients for the associated orthogonal\npolynomials, the first column of ab containing the n alpha-\ncoefficients and the second column the n beta-coefficients,\nthe call xw = GAUSS(n,ab) generates the nodes and weights xw of\nthe n-point Gauss quadrature rule for the weight function w.\nThe nodes, in increasing order, are stored in the first\ncolumn, the n corresponding weights in the second column, of\nthe nx2 array xw.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.hbathchain-Tuple{Int64, Int64, Any, Vararg{Any}}","page":"Methods","title":"MPSDynamics.hbathchain","text":"hbathchain(N::Int, d::Int, chainparams, longrangecc...; tree=false, reverse=false, coupletox=false)\n\nGenerate MPO representing a tight-binding chain of N oscillators with d Fock states each. Chain parameters are supplied in the standard form: chainparams =ϵ_0ϵ_1t_0t_1c_0. The output does not itself represent a complete MPO but will possess an end which is open and should be attached to another tensor site, usually representing the system.\n\nArguments\n\nreverse: If reverse=true create a chain were the last (i.e. Nth) site is the site which couples to the system\ncoupletox: Used to choose the form of the system coupling. coupletox=true gives a non-number conserving coupling of the form H_textI= A_textS(b_0^dagger + b_0) where A_textS is a system operator, while coupletox=false gives the number-converving coupling H_textI=(A_textS b_0^dagger + A_textS^dagger b_0)\ntree: If true the resulting chain will be of type TreeNetwork; useful for construcing tree-MPOs \n\nExample\n\nOne can constuct a system site tensor to couple to a chain by using the function up to populate the tensor. For example, to construct a system site with Hamiltonian Hs and coupling operator As, the system tensor M is constructed as follows for a non-number conserving interaction:\n\nu = one(Hs) # system identity\nM = zeros(1,3,2,2)\nM[1, :, :, :] = up(Hs, As, u)\n\nThe full MPO can then be constructed with:\n\nHmpo = [M, hbathchain(N, d, chainparams, coupletox=true)...]\n\nSimilarly for a number conserving interaction the site tensor would look like:\n\nu = one(Hs) # system identity\nM = zeros(1,4,2,2)\nM[1, :, :, :] = up(Hs, As, As', u)\n\nAnd the full MPO would be\n\nHmpo = [M, hbathchain(N, d, chainparams; coupletox=false)...]\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.heisenbergmpo","page":"Methods","title":"MPSDynamics.heisenbergmpo","text":"heisenbergmpo(N::Int, J=1.0) = xyzmpo(N; Jx=J)\n\nGenerate MPO for the N-spin Heisenberg XXX model, defined by the Hamiltonian\n\nH = sum_n=1^N-1 -J σ_x^n σ_x^n+1 - J σ_y^n σ_y^n+1 - J σ_z^n σ_z^n+1\n\nwith σ_x^n σ_y^n σ_z^n the Pauli spin-1/2 matrices of the n^textth site.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.ibmmpo-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.ibmmpo","text":"ibmmpo(ω0, d, N, chainparams; tree=false)\n\nGenerate MPO for a spin-1/2 coupled to a chain of harmonic oscillators with the interacting boson model (IBM), defined by the Hamiltonian\n\nH = fracω_02σ_z +  c_0σ_z(b_0^dagger+b_0) + sum_i=0^N-1 t_i (b_i+1^dagger b_i +hc) + sum_i=0^N ϵ_ib_i^dagger b_i.\n\nThe spin is on site 1 of the MPS and the bath modes are to the right.\n\nThis Hamiltonain is unitarily equivalent (before the truncation to N sites) to the spin-boson Hamiltonian defined by\n\nH =  fracω_02σ_z + σ_zint_0^ dωsqrtfracJ(ω)π(b_ω^dagger+b_ω) + int_0^ dω ωb_ω^dagger b_ω.\n\nThe chain parameters, supplied by chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.isingmpo-Tuple{Int64}","page":"Methods","title":"MPSDynamics.isingmpo","text":"isingmpo(N; J=1.0, h=1.0)\n\nGenerate MPO for the N-spin 1D Ising model with external field vech = (00h), defined by the Hamiltonian\n\nH = sum_n=1^N-1 -J_x σ_x^n σ_x^n+1 + sum_n=1^N(- h_z σ_z^n)\n\nwith σ_x^n σ_y^n σ_z^n the Pauli spin-1/2 matrices of the n^textth site.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.lanczos-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.lanczos","text":"LANCZOS Lanczos algorithm.\n\nGiven the discrete inner product whose nodes are contained\nin the first column, and whose weights are contained in the\nsecond column, of the nx2 array xw, the call ab=LANCZOS(n,xw)\ngenerates the first n recurrence coefficients ab of the\ncorresponding discrete orthogonal polynomials. The n alpha-\ncoefficients are stored in the first column, the n beta-\ncoefficients in the second column, of the nx2 array ab.\n\nThe script is adapted from the routine RKPW in\nW.B. Gragg and W.J. Harrod, ``The numerically stable\nreconstruction of Jacobi matrices from spectral data'',\nNumer. Math. 44 (1984), 317-335.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.longrange_isingmpo","page":"Methods","title":"MPSDynamics.longrange_isingmpo","text":"longrange_isingmpo(N::Int, α::Float64=0.; J=1.0, h=1.0) = longrange_xyzmpo(N, α; Jx=J, Jy=0., Jz=0., hz=h, hx=0.)\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.longrange_xyzmpo","page":"Methods","title":"MPSDynamics.longrange_xyzmpo","text":"longrange_xyzmpo(N::Int, α::Float64=0.; Jx=1.0, Jy=Jx, Jz=Jx, hx=0., hz=0.)\n\nGennerate MPO for the N-spin long-range XYZ model with external field vech=(h_x 0 h_z), , defined by the Hamiltonian\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.measure-Tuple{Any, OneSiteObservable}","page":"Methods","title":"MPSDynamics.measure","text":"measure(A, O; kwargs...)\n\nMeasure observable O on mps state A\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure1siteoperator-Tuple{Vector, Any, Vector{Int64}}","page":"Methods","title":"MPSDynamics.measure1siteoperator","text":"measure1siteoperator(A::Vector, O, sites::Vector{Int})\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for every site or just one if it is specified.\n\nFor calculating operators on single sites this will be more efficient if the site is on the left of the mps.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure1siteoperator-Tuple{Vector, Any}","page":"Methods","title":"MPSDynamics.measure1siteoperator","text":"measure1siteoperator(A::Vector, O)\n\nFor a list of tensors A representing a right orthonormalized MPS, compute the local expectation value of a one-site operator O for every site.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.measure2siteoperator-Tuple{Vector, Any, Any, Int64, Int64}","page":"Methods","title":"MPSDynamics.measure2siteoperator","text":" measure2siteoperator(A::Vector, M1, M2, j1, j2)\n\nCaculate expectation of M1*M2 where M1 acts on site j1 and M2 acts on site j2, assumes A is right normalised.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.modemps","page":"Methods","title":"MPSDynamics.modemps","text":"modemps(N::Int, k::Vector{Int}, numex::Int, chainparams=[fill(1.0,N), fill(1.0,N-1)])\n\nGenerate an MPS with numex excitations of an equal superposition of modes k of a bosonic tight-binding chain.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.modemps-2","page":"Methods","title":"MPSDynamics.modemps","text":"modemps(N::Int, k::Int, numex::Int, chainparams=[fill(1.0,N), fill(1.0,N-1)])\n\nGenerate an MPS with numex excitations of mode k of a bosonic tight-binding chain. \n\nchainparams takes the form [e::Vector, t::Vector] where e are the on-site energies and t are the hoppping parameters.\n\nThe returned MPS will have bond-dimensions and physical dimensions numex+1\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.mpsembed!-Tuple{Vector, Int64}","page":"Methods","title":"MPSDynamics.mpsembed!","text":"mpsembed(A::Vector, Dmax::Int)\n\nEmbed MPS A in manifold of max bond-dimension Dmax\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsleftnorm!","page":"Methods","title":"MPSDynamics.mpsleftnorm!","text":"mpsleftnorm!(A::Vector, jq::Int=length(A))\n\nLeft orthoganalise MPS A up to site jq.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.mpsmixednorm!-Tuple{MPSDynamics.TreeNetwork, Int64}","page":"Methods","title":"MPSDynamics.mpsmixednorm!","text":"mpsmixednorm!(A::TreeNetwork, id::Int)\n\nNormalise tree-MPS A such that orthogonality centre is on site id.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsmixednorm!-Tuple{Vector, Int64}","page":"Methods","title":"MPSDynamics.mpsmixednorm!","text":"mpsmixednorm!(A::Vector, OC::Int)\n\nPut MPS A into mixed canonical form with orthogonality centre on site OC.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsmoveoc!-Tuple{MPSDynamics.TreeNetwork, Int64}","page":"Methods","title":"MPSDynamics.mpsmoveoc!","text":"mpsmoveoc!(A::TreeNetwork, id::Int)\n\nMove the orthogonality centre of right normalised tree-MPS A to site id.\n\nThis function will be more efficient than using mpsmixednorm! if the tree-MPS is already right-normalised.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsrightnorm!","page":"Methods","title":"MPSDynamics.mpsrightnorm!","text":"mpsrightnorm!(A::Vector, jq::Int=1)\n\nRight orthoganalise MPS A up to site jq.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.mpsrightnorm!-Tuple{MPSDynamics.TreeNetwork}","page":"Methods","title":"MPSDynamics.mpsrightnorm!","text":"mpsrightnorm!(A::TreeNetwork)\n\nWhen applied to a tree-MPS, right normalise towards head-node.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.mpsshiftoc!-Tuple{MPSDynamics.TreeNetwork, Int64}","page":"Methods","title":"MPSDynamics.mpsshiftoc!","text":"mpsshiftoc!(A::TreeNetwork, newhd::Int)\n\nShift the orthogonality centre by one site, setting new head-node newhd.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.multiply-Tuple{Vector, Vector}","page":"Methods","title":"MPSDynamics.multiply","text":"multiply(M1::Vector, M2::Vector)\n\nCalculates M1*M2 where M1 and M2 are MPOs\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.nearestneighbourmpo","page":"Methods","title":"MPSDynamics.nearestneighbourmpo","text":"nearestneighbourmpo(N::Int, h0, A, Ad = A')\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.nearestneighbourmpo-2","page":"Methods","title":"MPSDynamics.nearestneighbourmpo","text":"nearestneighbourmpo(tree_::Tree, h0, A, Ad = A')\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.normmps-Tuple{MPSDynamics.TreeNetwork}","page":"Methods","title":"MPSDynamics.normmps","text":"normmps(net::TreeNetwork; mpsorthog=:None)\n\nWhen applied to a tree-MPS mpsorthog=:Left is not defined.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.normmps-Tuple{Vector}","page":"Methods","title":"MPSDynamics.normmps","text":"normmps(A::Vector; mpsorthog=:None)\n\nCalculate norm of MPS A.\n\nSetting mpsorthog=:Right/:Left will calculate the norm assuming right/left canonical form. Setting mpsorthog=OC::Int will cause the norm to be calculated assuming the orthoganility center is on site OC. If mpsorthog is :None the norm will be calculated as an MPS-MPS product.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.orthcentersmps-Tuple{Vector}","page":"Methods","title":"MPSDynamics.orthcentersmps","text":"orthcentersmps(A)\n\nCompute the orthoganality centres of MPS A.\n\nReturn value is a list in which each element is the corresponding site tensor of A with the orthoganility centre on that site. Assumes A is right normalised.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.physdims-Tuple{Vector}","page":"Methods","title":"MPSDynamics.physdims","text":"physdims(M)\n\nReturn the physical dimensions of an MPS or MPO M.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.productstatemps","page":"Methods","title":"MPSDynamics.productstatemps","text":"productstatemps(physdims::Dims, Dmax=1; state=:Vacuum, mpsorthog=:Right)\n\nReturn an MPS representing a product state with local Hilbert space dimensions given by physdims.\n\nBy default all bond-dimensions will be 1 since the state is a product state. However, to embed the product state in a manifold of greater bond-dimension, Dmax can be set accordingly.\n\nThe indvidual states of the MPS sites can be provided by setting state to a list of column vectors. Setting state=:Vacuum will produce an MPS in the vacuum state (where the state of each site is represented by a column vector with a 1 in the first row and zeros elsewhere). Setting state=:FullOccupy will produce an MPS in which each site is fully occupied (ie. a column vector with a 1 in the last row and zeros elsewhere).\n\nThe argument mpsorthog can be used to set the gauge of the resulting MPS.\n\nExample\n\njulia> ψ = unitcol(1,2); d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1\n\njulia> cpars = chaincoeffs_ohmic(N, α, s)\n\njulia> H = spinbosonmpo(ω0, Δ, d, N, cpars)\n\njulia> A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.productstatemps-2","page":"Methods","title":"MPSDynamics.productstatemps","text":"productstatemps(N::Int, d::Int, Dmax=1; state=:Vacuum, mpsorthog=:Right)\n\nReturn an N-site MPS with all local Hilbert space dimensions given by d. \n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.protontransfermpo-NTuple{11, Any}","page":"Methods","title":"MPSDynamics.protontransfermpo","text":"protontransfermpo(ω0e,ω0k,x0e,x0k, Δ, dRC, d, N, chainparams, RCparams, λreorg)\n\nGenerate a MPO for a system described in space with a reaction coordinate (RC) tensor. The RC tensor is coupled to a bosonic bath, taking into account the induced reorganization energy. \n\nH_S + H_RC + H_int^S-RC = omega^0_e erangle langle e + omega^0_k krangle langle k + Delta (erangle langle k + krangle langle e) + omega_RC (d^daggerd + frac12) + g_e erangle langle e( d + d^dagger)+ g_k k rangle langle k( d + d^dagger) H_B + H_int^RC-B = int_-^+ dk ω_k b_k^dagger b_k - (d + d^dagger)int_0^ dωsqrtJ(ω)(b_ω^dagger+b_ω) + lambda_reorg(d + d^dagger)^2. lambda_reorg = int fracJ(omega)omegadomega.\n\nArguments\n\nω0e: enol energy at x=0 \nω0k: keto energy at x=0\nx0e: enol equilibrium displacement\nx0k: keto equilibrium displacement \nΔ: direct coupling between enol and keto\ndRC: fock space of the RC tensor \nd: number of Fock states of the chain modes\nN: length of the chain\nchainparams: chain parameters, of the form chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature. \nRCparams: RC tensor parameter, of the form RCparams=ωRC-gx \nλreorg: reorganization energy\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.puredephasingmpo-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.puredephasingmpo","text":"puredephasingmpo(ΔE, dchain, Nchain, chainparams; tree=false)\n\nGenerate MPO for a pure dephasing model, defined by the Hamiltonian\n``H = \\frac{ΔE}{2} σ_z +  \\frac{σ_z}{2} c_0 (b_0^\\dagger + b_0) + \\sum_{i=0}^{N-1} t_i (b_{i+1}^\\dagger b_i +h.c.) + \\sum_{i=0}^{N-1} ϵ_i b_i^\\dagger b_i  ``\n\nThe spin is on site 1 of the MPS and the bath modes are to the right.\n\n### Arguments\n* `ΔE::Real`: energy splitting of the spin\n* `dchain::Int`: physical dimension of the chain sites truncated Hilbert spaces\n* `Nchain::Int`: number of sites in the chain\n* `chainparams::Array{Real,1}`: chain parameters for the bath chain. The chain parameters are given in the standard form: `chainparams` ``=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]``.\n* `tree::Bool`: if true, return a `TreeNetwork` object, otherwise return a vector of MPO tensors\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.randisometry-Tuple{Type, Int64, Int64}","page":"Methods","title":"MPSDynamics.randisometry","text":"randisometry([T=Float64], dims...)\n\nConstruct a random isometry\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.randmps","page":"Methods","title":"MPSDynamics.randmps","text":"randmps(tree::Tree, physdims, Dmax::Int, T::Type{<:Number} = Float64)\n\nConstruct a random, right-normalised, tree-MPS, with structure given by tree and max bond-dimension given by Dmax.\n\nThe local Hilbert space dimensions are specified by physdims which can either be of type Dims{length(tree)}, specifying the dimension of each site, or of type Int, in which case the same local dimension is used for every site.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.randmps-2","page":"Methods","title":"MPSDynamics.randmps","text":"randmps(N::Int, d::Int, Dmax::Int, T=Float64)\n\nConstruct a random, N-site, right-normalised MPS with all local Hilbert space dimensions given by d.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.randmps-Union{Tuple{N}, Tuple{Tuple{Vararg{Int64, N}}, Int64}, Tuple{Tuple{Vararg{Int64, N}}, Int64, Type{<:Number}}} where N","page":"Methods","title":"MPSDynamics.randmps","text":"randmps(physdims::Dims{N}, Dmax::Int, T::Type{<:Number} = Float64) where {N}\n\nConstruct a random, right-normalised MPS with local Hilbert space dimensions given by physdims and max bond-dimension given by Dmax. \n\nT specifies the element type, eg. use T=ComplexF64 for a complex valued MPS.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.randtree-Tuple{Int64, Int64}","page":"Methods","title":"MPSDynamics.randtree","text":"randtree(numnodes::Int, maxdegree::Int)\n\nConstruct a random tree with nummodes modes and max degree maxdegree.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.readchaincoeffs-Tuple{Any, Vararg{Any}}","page":"Methods","title":"MPSDynamics.readchaincoeffs","text":"readchaincoeffs(fdir, params...)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.rmsd-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.rmsd","text":"rmsd(dat1::Vector{Float64}, dat2::Vector{Float64})\n\nCalculate the root mean squared difference between two measurements of an observable over the same time period.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.runsim-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.runsim","text":"runsim(dt, tmax, A, H; \n\tmethod=:TDVP1, \n\tmachine=LocalMachine(), \n\tparams=[], \n\tobs=[], \n\tconvobs=[],\n            convparams=error(\"Must specify convergence parameters\"),\n            save=false,\n            plot=save,\n            savedir=string(homedir(),\"/MPSDynamics/\"),\n            unid=randstring(5),\n            name=nothing,\n\tkwargs...\n\t)\n\nPropagate the MPS A with the MPO H up to time tmax in time steps of dt. The final MPS is returned to A and the measurement data is returned to dat \n\nArguments\n\nmethod: Several methods are implemented in MPSDynamics. :TDVP1 refers to 1-site TDVP on tree and chain MPS, :TDVP2 refers to 2-site TDVP on chain MPS, :DTDVP refers to a variant of 1-site TDVP with dynamics bond-dimensions on chain MPS\nmachine: LocalMachine() points local ressources, RemoteMachine() points distant ressources\nparams: list of parameters written in the log.txt file to describe the dynamics. Can be listed with @LogParams(). \nobs: list of observables that will be measured at every time step for the most accurate convergence parameter supplied to convparams \nconvobs: list of observables that will be measure at every time step for every convergence parameter supplied to convparams \nconvparams: list of convergence parameter with which the propagation will be calculated for every parameter. At each parameter, convobs are measured while obs are measured only for the most accurate dynamics\nsave: Used to choose whether the data will also be saved to a file  \nplot: Used to choose whether plots for 1D observables will be automatically generated and saved along with the data\nsavedir: Used to specify the path where resulting files are stored\nunid: Used to specify the name of the directory containing the resulting files\nname: Used to describe the calculation. This name will appear in the log.txt file\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.spinbosonmpo-NTuple{5, Any}","page":"Methods","title":"MPSDynamics.spinbosonmpo","text":"spinbosonmpo(ω0, Δ, d, N, chainparams; rwa=false, tree=false)\n\nGenerate MPO for a spin-1/2 coupled to a chain of harmonic oscillators, defined by the Hamiltonian\n\nH = fracω_02σ_z + Δσ_x + c_0σ_x(b_0^dagger+b_0) + sum_i=0^N-1 t_i (b_i+1^dagger b_i +hc) + sum_i=0^N ϵ_ib_i^dagger b_i.\n\nThe spin is on site 1 of the MPS and the bath modes are to the right.\n\nThis Hamiltonain is unitarily equivalent (before the truncation to N sites) to the spin-boson Hamiltonian defined by\n\nH =  fracω_02σ_z + Δσ_x + σ_xint_0^ dωsqrtJ(ω)(b_ω^dagger+b_ω) + int_0^ dω ωb_ω^dagger b_ω.\n\nThe chain parameters, supplied by chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature.\n\nThe rotating wave approximation can be made by setting rwa=true.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.spinchainmpo-Tuple{Int64}","page":"Methods","title":"MPSDynamics.spinchainmpo","text":"spinchainmpo(N::Int; J=1.0, hz=1.0, hx=0.0, i=div(N,2))\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.stieltjes-Tuple{Any, Any}","page":"Methods","title":"MPSDynamics.stieltjes","text":"STIELTJES Discretized Stieltjes procedure.\n\nGiven the discrete inner product whose nodes are contained\nin the first column, and whose weights are contained in the\nsecond column, of the nx2 array xw, the call ab=STIELTJES(n,xw)\ngenerates the first n recurrence coefficients ab of the\ncorresponding discrete orthogonal polynomials. The n alpha-\ncoefficients are stored in the first column, the n beta-\ncoefficients in the second column, of the nx2 array ab.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.svdmps-Tuple{Any}","page":"Methods","title":"MPSDynamics.svdmps","text":"svdmps(A)\n\nFor a right normalised mps A compute the full svd spectrum for a bipartition at every bond.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.svdtrunc-Tuple{Any}","page":"Methods","title":"MPSDynamics.svdtrunc","text":"U, S, Vd = svdtrunc(A; truncdim = max(size(A)...), truncerr = 0.)\n\nPerform a truncated SVD, with maximum number of singular values to keep equal to truncdim or truncating any singular values smaller than truncerr. If both options are provided, the smallest number of singular values will be kept. Unlike the SVD in Julia, this returns matrix U, a diagonal matrix (not a vector) S, and Vt such that A ≈ U * S * Vt\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.tightbinding_mpo-NTuple{4, Any}","page":"Methods","title":"MPSDynamics.tightbinding_mpo","text":"tightbinding_mpo(N, ϵd, chainparams1, chainparams2)\n\nGenerate MPO for a tight-binding chain of N fermionic sites with a single impurity site (fermionic as well) \nof energy ϵd at the center. The impurity is coupled to two leads, each described by a set of chain parameters.\nThe interactions are nearest-neighbour, with the first N/2-1 sites corresponding to the first lead,\nthe Nth site corresponding to the impurity, and the rest of the sites corresponding to the second\nlead.\n\n# Arguments\n\n* `N::Int`: number of sites in the chain\n* `ϵd::Real`: energy of the impurity site at the center, as Ed - μ, where μ is the chemical potential\n* chainparams1::Array{Real,1}: chain parameters for the first lead\n* chainparams2::Array{Real,1}: chain parameters for the second lead\n\nThe chain parameters are given in the standard form: `chainparams` ``=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]``.\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.tightbindingmpo-Tuple{Int64, Int64}","page":"Methods","title":"MPSDynamics.tightbindingmpo","text":"tightbindingmpo(N::Int, d::Int; J=1.0, e=1.0)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.tunnelingmpo-Tuple{Any, Any, Any, Any, Any, Int64, Int64}","page":"Methods","title":"MPSDynamics.tunnelingmpo","text":"tunnelingmpo(ϵ, delta, α, s, β, d::Int, nummodes::Int; tree=false, ωc=1)\n\n\n\n\n\n","category":"method"},{"location":"methods/#MPSDynamics.twobathspinmpo","page":"Methods","title":"MPSDynamics.twobathspinmpo","text":"twobathspinmpo(ω0, Δ, Nl, Nr, dl, dr, chainparamsl=[fill(1.0,N),fill(1.0,N-1), 1.0], chainparamsr=chainparamsl; tree=false)\n\nGenerate MPO for a spin-1/2 coupled to two chains of harmonic oscillators, defined by the Hamiltonian\n\nH = fracω_02σ_z + Δσ_x + c_0^rσ_x(b_0^dagger+b_0) + sum_i=0^N_r-1 t_i^r (b_i+1^dagger b_i +hc) + sum_i=0^N_r ϵ_i^rb_i^dagger b_i + c_0^lσ_x(d_0^dagger+d_0) + sum_i=0^N_l-1 t_i^l (d_i+1^dagger d_i +hc) + sum_i=0^N_l ϵ_i^l d_i^dagger d_i.\n\nThe spin is on site N_l + 1 of the MPS, surrounded by the left chain modes and the right chain modes.\n\nThis Hamiltonain is unitarily equivalent (before the truncation to N sites) to the spin-boson Hamiltonian defined by\n\nH =  fracω_02σ_z + Δσ_x + σ_xint_0^ dωsqrtfracJ(ω)π(b_ω^dagger+b_ω) + int_0^ dω ωb_ω^dagger b_ωi + σ_xint_0^ dωsqrtfracJ^l(ω)π(d_ω^dagger+d_ω) + int_0^ dω ωd_ω^dagger d_ω.\n\nThe chain parameters, supplied by chainparams=ϵ_0ϵ_1t_0t_1c_0, can be chosen to represent any arbitrary spectral density J(ω) at any temperature. The two chains can have a different spectral density.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.xxzmpo","page":"Methods","title":"MPSDynamics.xxzmpo","text":"xxzmpo(N::Int, Δ = 1.0, J=1.0) = xyzmpo(N; Jx=J, Jy=J, Jz=J*Δ)\n\nGenerate MPO for the N-spin XXZ model, defined by the Hamiltonian\n\nH = sum_n=1^N-1 -J σ_x^n σ_x^n+1 - J σ_y^n σ_y^n+1 - Delta J σ_z^n σ_z^n+1\n\nwith σ_x^n σ_y^n σ_z^n the Pauli spin-1/2 matrices of the n^textth site.\n\n\n\n\n\n","category":"function"},{"location":"methods/#MPSDynamics.xyzmpo-Tuple{Int64}","page":"Methods","title":"MPSDynamics.xyzmpo","text":"xyzmpo(N::Int; Jx=1.0, Jy=Jx, Jz=Jx, hx=0., hz=0.)\n\nGenerate MPO for the N-spin XYZ model with external field vech=(h_x 0 h_z), , defined by the Hamiltonian\n\nH = sum_n=1^N-1 -J_x σ_x^n σ_x^n+1 - J_y σ_y^n σ_y^n+1 - J_z σ_z^n σ_z^n+1 + sum_n=1^N(- h_x σ_x^n - h_z σ_z^n)\n\nwith σ_x^n σ_y^n σ_z^n the Pauli spin-1/2 matrices of the n^textth site.\n\n\n\n\n\n","category":"method"},{"location":"theory/#Theoretical-Background","page":"Theoretical Background","title":"Theoretical Background","text":"","category":"section"},{"location":"theory/#Chain-Mapping-of-bosonic-environments","page":"Theoretical Background","title":"Chain-Mapping of bosonic environments","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"We consider, in the Schrödinger picture, a general Hamiltonian where a non-specified system interacts linearly with a bosonic environments","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"beginaligned\n    hatH = hatH_S + int_0^+infty hbaromegahata^dagger_omegahata_omega mathrmdomega + hatA_Sint_0^+inftysqrtJ(omega)left(hata_omega + hata^dagger_omegaright)mathrmdomega\nendaligned","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where hat a_omega (hat a^dagger_omega) is a bosonic annihilation (creation) operator for a normal mode of the environment of energy hbaromega, hatA_S is a system operator, and J(omega) = sum_k g_k^2delta(omega - omega_k) is the bath spectral density (SD), defined with the microscopic system-environment coupling strength g_k. The SD quantifies the coupling strengths of the different normal modes of the environment with the system. Any SD that is not flat corresponds to a non-Markovian environment.","category":"page"},{"location":"theory/#Zero-Temperature","page":"Theoretical Background","title":"Zero Temperature","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Let us consider the Hamiltonian presented in Eq.(1). To investigate its rich phenomenology, using the powerful tools provided by tensor network techniques, it is convenient to map the system and its environment from a star-like to a chain-like configuration, with local interactions. To do so, we introduce a unitary transformation of the continuous normal modes hata_omega to an infinite discrete set of interacting modes hatb_n[chin_exact_2010].","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hata_omega = sum_n=0^+infty U_n(omega)hatb_n = sum_n=0^+infty sqrtJ(omega)P_n(omega)hatb_n ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where P_n(omega) are orthonormal polynomials such that","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    int_0^+inftyP_n(omega)P_m(omega)J(omega)mathrmdomega = delta_nm ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"and the inverse transformation is","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hatb_n = int_0^+infty U_n(omega)hata_omegamathrmdomega ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Note that the orthonormality of the polynomials ensures the unitarity of the transformation defined in Eq.(2). The mapping from a continuous set of modes to a (still infinite) discrete set might seem counter-intuitive, however it is a direct consequence of the separability of the underlying Hilbert space.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Under this transformation, the Hamiltonian in Eq.(1) becomes","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hatH= hatH_S + sum_n=0^+inftyvarepsilon_nhatb_n^daggerhatb_n + t_n(hatb_n+1^daggerhatb_n + mathrmhc) + kappahatA_S(hatb_0 + hatb_0^dagger) ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Hence, this mapping transforms the normal bath Hamiltonian into a tight-binding Hamiltonian with on-site energies varepsilon_n and hopping energies t_n. Another important consequence of this mapping is that now the system only interacts with the first mode n = 0 of the chain-mapped environment. The chain coefficients varepsilon_n, t_n, and the coupling kappa depend solely on the SD.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"This makes chain mapping a tool of choice for describing systems coupled to environment with highly structured SD (e.g. experimentally measured or calculated ab initio)[chin_role_2013][alvertis_nonequilibrium_2019][dunnett_influence_2021][caycedosoler_exact_2022]. In this new representation, the Hamiltonian in Eq.(5) has naturally a 1D chain topology. This makes its representation as a Matrix Product Operator (MPO) and the representation of the joint {System + Environment} wave-function as a Matrix Product State (MPS) suited [orus_practical_2014][paeckel_timeevolution_2019].","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The orthogonal polynomial-based chain mapping and the subsequent representation of the joint wave-function as a MPS (and the operators as MPO) are the building blocks of the Time-dependent Density operator with Orthonormal Polynomials Algorithm (TEDOPA) one of the state-of-the-art numerically exact method to simulate the dynamics of open quantum systems especially in the non-Markovian, non-perturbative regimes both at zero and finite temperatures [prior_efficient_2010][woods_simulating_2015][tamascelli_efficient_2019][dunnett_simulating_2021][lacroix_unveiling_2021].","category":"page"},{"location":"theory/#Finite-Temperature-with-T-TEDOPA","page":"Theoretical Background","title":"Finite Temperature with T-TEDOPA","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Assuming a unitary evolution for both the system and environment, the system's dynamics can be isolated by tracing out the environmental degrees of freedom. The density operator for the system at time t is described as","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"hatrho_S(t) = textTr_ElefthatU(t) hatrho_S(0) otimes hatrho_E(0) hatU^dagger(t)right","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Initially, the system state, hatrho_S(0), can be pure or mixed, and the environment is in a thermal state defined by the inverse temperature beta = (k_B T)^-1. This state is represented by a product of Gaussian states","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"hatrho_E(0) = bigotimes_omega frace^-beta omega hata_omega^dagger hata_omegaZ_omega(beta) ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The system's evolution is dictated by the environment's two-time correlation function S(t), which in turn is determined by the spectral density function J and the temperature beta via the Bose-Einstein distribution n_omega(beta)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"S(t) = int_0^infty domega J(omega)lefte^-iomega t(1 + n_omega(beta)) + e^iomega t n_omega(beta)right ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"To simulate finite temperature effects using a zero-temperature model with the T-TEDOPA method [tamascelli_efficient_2019], we extend the spectral density function to cover both positive and negative frequencies, allowing us to use a pure state description for the environment. This is achieved by defining a new spectral density function J(omega beta) that incorporates the Boltzmann factors, supporting the entire real axis","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"J(omega beta) = fractextsign(omega)J(leftomegaright)2 Big(1 + cothBig(fracbeta omega2Big)Big)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"This modified bath has the same correlation function S(t) and thus allows us to maintain a pure state description of the environment, represented as a vacuum state, and avoid the computational complexities of density matrices","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"lefttextvacrightrangle = bigotimes_omega left0_omegarightrangle ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The Hamiltonian of the system interacting with this extended bath now includes temperature-dependent interactions:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"hatH = hatH_S + int_-infty^+infty domega omega hata_omega^dagger hata_omega + hatA_S otimes int_-infty^+infty domega sqrtJ(omegabeta)left(hata_omega^dagger+hata_omegaright)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"This method simplifies the simulation of finite temperature effects by treating them within an effective zero-temperature framework, thereby keeping the computational advantages of using pure states. In conclusion: the dynamics of the system resulting from the interaction with the original bath, starting in a thermal state at finite temperature, is exactly the same as the one resulting from the interaction with the extended environment, starting in the vacuum state at zero temperature. Once computed the chain coefficients at a given inverse temperature beta, the time evolution of the vacuum state interacting with the extended environment can be efficiently simulated using MPS time evolution methods.","category":"page"},{"location":"theory/#Finite-temperature-with-the-thermofield-transformation","page":"Theoretical Background","title":"Finite temperature with the thermofield transformation","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Finite temperature initial states of the system are challenging: in the previous section we have seen that a very efficient solution is to reabsorb the effect of the temperature inside of the spectral density function that characterizes the environment. De Vega and Bañuls present[devega_thermo_2015] another suggestive approach based on the idea of the thermofield transformation. The idea is to double the environmental degrees of freedom and then to apply a Bogoliubov transformation: the real environment in a thermal state is transformed into two virtual environments in the vacuum state, defined as the thermofield vacuum. For any operator of the real environment, the expectation values in the thermal state are equivalent to those calculated in the thermofield vacuum. Behind the thermofield approach there is the concept of purification: the initial mixed state of the environment can be represented as the partial trace of a pure state (the thermofield vacuum) defined on the larger Hilbert space of the modes of the two virtual environments at zero temperature. The interaction of the system and each one of the two environments is then mapped on two separate chains, using the TEDOPA chain mapping to define a unitary transformation that can be applied to the bath modes, in order to define new modes for the environment, mapping it from a star-like to a one dimensional chain-like configuration, which is very well suited for the application of tensor network techniques.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"In the thermofield approach, the first step is to introduce the auxiliary environment E^textaux of non-interacting bosonic modes of negative frequencies:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hat H^textaux = hat H - sum_k omega_k hat c_k^dagger hat c_k","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where the Hamiltonian hat H is the bosonic Hamiltonian defined in Eq. \\ref{eq:bosonic_ham}. The two environments, of positive and negative frequencies, are assumed to be in a thermal environment at inverse temperature beta; the second step is to apply a thermal Bogoliubov transformation to change the basis. The applied transformation produces two-modes squeezed states:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"         hat a_1k=e^-iG  hat b_k e^iG= cosh(theta_k) hat b_k -sinh(theta_k)  hat c_k^dagger \n         hat a_2k=e^-iG hat c_k e^iG= cosh(theta_k)  hat c_k -sinh(theta_k)  hat b_k^dagger","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where the exponent of the squeeze operator is G = i sum_k theta_k(hat b_k^dagger hat c_k^dagger-hat c_k hat b_k), and theta_k is dependent on the temperature as in the following relations, where the number of excitations in the k-th mode is n_k = 1(e^beta omega_k-1):","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"         cosh(theta_k) = sqrt1+n_k = sqrtfrac11-e^-beta omega_k \n         sinh(theta_k) =quadsqrtn_kquad= sqrtfrac1e^beta omega_k-1","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The Bogoliubov transformation defines a new squeezed vacuum state, which we write in terms of the vacuum state ketOmega_0 of the operators hat b_k, hat c_k:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    Omegarangle = e^iG Omega_0rangle quad textsuch that quad hat a_1k Omegarangle = 0  hat a_2k Omegarangle = 0","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"From the vacuum state, we can obtain the thermal state of the original environment:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hat rho_E = textTr_textaux OmegaranglelangleOmega ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"and it can be now used as pure an initial state for both of the environments. Moreover, the expectation value on ketOmega of the number of physical modes hat n_k does not vanish:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    n_k= langleOmega hat b_k^dagger hat b_k Omegarangle = sinh^2(theta_k)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Therefore, solving the dynamics given by the original Hamiltonian hat H, starting from the initial condition hat rho_S(0) otimes hat rho_E(beta), is equivalent to solving the dynamics given by the following Hamiltonian:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"       hat H = hat H_S +hat H_E +hat H_I  = \n       = overbracehat A_S^hat H_S + overbracesum_k omega_k big(hat a_1k^dagger hat a_1k - hat a_2k^dagger hat a_2k big)^H_E + overbracehat L_S otimes sum_k g_1k(hat a_1k^dagger + hat a_1k)+hat L_S otimes sum_k g_2k(hat a_2k^dagger + hat a_2k)^H_I","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where hat L_S = hat L_S^dagger, considering hat rho_S(0) otimes ketOmegabraOmega as the initial state of system and environment. It is this Hamiltonian that is mapped on two chains with TEDOPA, to be able to perform the time evolution using tensor network techniques. ","category":"page"},{"location":"theory/#Computation-of-the-chain-coefficients","page":"Theoretical Background","title":"Computation of the chain coefficients","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"A useful property of the orthonormal polynomials is that they obey a recurrence relation","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    P_n(omega) = (C_n-1omega - A_n-1)P_n-1(omega) + B_n-1P_n-2(omega) ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where A_n is related to the first moment of P_n, B_n and C_n to the norms of P_n and P_n-1[appel_mathematics_2007]. This recurrence relation can be used to construct the polynomials with the conditions that P_0(omega) = p_0^-1 = left(int_mathbbR^+ J(omega)mathrmdomega right)^-frac12 and P_-1(omega) = 0, with bullet the norm of bullet with respect to the measure J(omega), and P_n(omega) = p_n(omega)p_n^-1 ; where the polynomials p_n_ninmathbbN are the so called monic polynomials where the factor a_n in front of omega^n is equal to 1.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The energy of the chain mode n is given by varepsilon_n = A_n C_n^-1 and t_n=C_n^-1 is the coupling between mode n and n+1[chin_exact_2010].","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The system couples only to the first mode with the coupling strength kappa = p_0.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Explain that for some weight function/SD they are known analytically and that for others we can use the build-in routines inspired by Gautschi or the PolyChaos.jl package.","category":"page"},{"location":"theory/#Tensor-Networks","page":"Theoretical Background","title":"Tensor Networks","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"A multipartite quantum state psirangle, e.g. a N-site system where the sites can each be in a state phi_irangle belonging to a d-dimensional Hilbert space, can be written as follows","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    psirangle = sum_i_kc_i_1ldots i_Nphi_i_1rangleotimesldotsotimesphi_i_Nrangle ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"where the complex numbers c_i_1ldots i_N are the amplitudes of each state phi_i_1rangleotimesldotsotimesphi_i_Nrangle whose superpositions form in full generality the state psirangle. Thus the state psirangle can be completely represented by a rank-N tensor c that is the collection of all possible amplitudes c_i_1ldots i_N. Here by the rank of a tensor, we simply mean the number of indices it has.","category":"page"},{"location":"theory/#MPS","page":"Theoretical Background","title":"MPS","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The tensor c of a quantum state psirangle corresponding to a one-dimensional system can be decomposed into a product of N smaller rank-3 tensors T_k (except for the first and last sites where the tensors will have a rank-2)","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    c_i_1ldots i_N = sum_alpha T^alpha_1_i_1T^alpha_1alpha_2 _i_2T^alpha_2alpha_3 _i_3ldots T^alpha_N-1_i_N  ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"In this form, the local tensor T_k contains the information on the quantum state on site k and its relation (especially the entanglement) with the neighbouring sites.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The decomposition of the tensor of the amplitudes of a quantum state into a product of smaller rank tensors is called a Matrix Product State decomposition.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The contracted indices alpha_k between the tensors are called virtual indices and carry information about the correlations between bi-partitions of the state at bond k. The number of different values a virtual index can take is called the bond dimension and is denoted D. The free indices i_k associated with local quantum states are called physical indices. Thus, they can take d values (with d the dimension of the local Hilbert space).","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Any state in the Hilbert space of a one-dimensional many-body system can in principle be represented by a MPS by choosing a sufficiently large value for the bond dimension D [Orus]. On top of this intellectually satisfying property of MPSs being a dense set of states for a 1d-system, they can also be used as a practical Ansätze for a many-body quantum states by setting a maximal allowed value chi for the bond dimension D. In doing so, we restrict ourselves to a corner of the total Hilbert space. The rationale behind this Ansatz is the following: if the initial quantum state of a many-body system has a low bond dimension (typically if the initial state is a product state with D = 1), then in a finite time it will only be able to explore a region of the Hilbert space that is not to far away from its starting point. Thus, the bond dimension will not have the time to diverge exponentially [poulin_quantum_2011]. However, depending on the physical system at hand, this sub-manifold of the Hilbert space could still be \"too large\". There is an additional reason that explains why MPSs are good Ansätze for 1d physical systems. Most many-body Hamiltonians we (physicists) are interested in are local, meaning that the interactions they describe involve objects that are \"neighbours\". For such Hamiltonians, the ground states (outside of potential critical phases) follow the so called area law for the entanglement entropy [srednicki_entropy_1993][vidal_entanglement_2003][wolf_area_2008]. This law states that the entanglement entropy S_vN of a bi-partition of the system is proportional, not to the volume of the partition as one might expect, but to the hyper-surface of the partition's boundary; hence the name \"area law\". For a 3d system this corresponds to an actual surface area A, S_vN sim A; for a 2d system it corresponds to the length L of the partition's boundary, S_vN sim L; and in 1d the boundary reduces to a point, thus the entropy will be independent of the size of the system S_vN sim textconstant. The MPSs are states that satisfy this area law.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"An application of the Singular Value Decomposition is to create efficient approximations of quantum states to perform computations. The main idea is to reduce the content of the MPS to keep only the parts that contain the physics of interest. One method to realise this approximation is to do a SVD on each of the tensors of the MPS after each time step of the state time-evolution and to trim the smallest singular values in order to decrease the bond dimension of the MPS down to a chosen maximal value chi. The corresponding columns and rows of the unitary matrices U and V^dagger are also removed. Then, the trimmed matrices tildeU, tildeS and tildeV^dagger are contracted back to give an approximated tensor T with a smaller bond dimension. Another way to apply the restricted rank approximation is to restrict oneself into working in a manifold of fixed bond dimension D and to use methods that can enforce this constraint.","category":"page"},{"location":"theory/#MPO","page":"Theoretical Background","title":"MPO","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"In order to compute expectation values of observables or apply unitary transformations to a quantum state, we need a TN representation of operators. In the same fashion as a one-dimensional quantum state can be represented as a MPS, operators acting on those states can be represented as Matrix Product Operators (MPO). For an operator hatO, its MPO can be defined as follows","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    hatO = sum_i_ki_k^ w W^i_1 i^_1_1 w_0w_1ldots  W^i_N i^_N_N w_N-1w_N phi_i_1^ldots phi_i_N^ranglelanglephi_i_1ldots phi_i_N ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The contracted indices between the tensors are called virtual indices. The free indices are called physical indices and correspond to the different input and output local quantum states.  They can take d values (with d the dimension of the local Hilbert space).","category":"page"},{"location":"theory/#TTN","page":"Theoretical Background","title":"TTN","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"A natural extension to the MPS is the (loop-free) tree tensor network.  A TTN is a generalisation of the MPS wherein each site, instead of being connected to only one other site to its right, may be connected to any arbitrary number of child sites. Provided the tree does not contain any loops, everything that one can do to an MPS/MPO can be extended straight-forwardly to TTN states and TTN operators.  The generalisation to trees introduces no new conceptual complexity (only implementational complexity). The sites of a TTN are usually referred to as nodes.  For our purposes, every node of a TTN state and operator has one parent leg, and any number (including zero) of child legs.  The first node is known as the head-node and has a dummy parent leg with dimension 1.","category":"page"},{"location":"theory/#Time-Dependent-Variational-Principle","page":"Theoretical Background","title":"Time-Dependent Variational Principle","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The original idea behind TDVP goes back to Dirac [dirac_note_1930] and Frenkel [frenkel_wave_1934]. The main point, in the modern tensor networks formulation, is that instead of solving the Schrödinger equation and then truncating the MPS representation of the quantum state, one can solve the equations of motion projected into a space of restricted bond dimension [haegeman_timedependent_2011][haegeman_unifying_2016].","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The general formulation of the Dirac-Frenkel Variational Principle [raab_diracfrenkelmclachlan_2000] is that one looks for a solution varphirangle in mathcalM of the Schrödinger equation where mathcalM subset mathcalH is a manifold of the total Hilbert space mathcalH in which we think that the relevant physical states `live'.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"We define T_varphiranglemathcalM the tangent space of mathcalM around the state varphirangle. The criterion to find varphirangle is that for every state chirangle in T_varphiranglemathcalM","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    langlechileft(fracmathrmdmathrmdt - frac1mathrmihbarhatHright)varphirangle =0 ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"which can be interpreted as saying that the time evolution procedure should keep varphirangle inside of the manifold mathcalM.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The term variational in the name of the method comes from the fact that in practice one aims at minimising the right-hand side of the hereinabove Eq. to find varphirangle.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Introducing hatP_T_varphiranglemathcalM the projector onto the tangent space T_varphiranglemathcalM, we can write the state chirangle = hatP_T_varphiranglemathcalMphirangle with phirangle a state in mathcalH. Leading to","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    forall phirangle in mathcalH  langlephihatP_T_varphiranglemathcalMleft(fracmathrmdmathrmdt - frac1mathrmihbarhatHright)varphirangle =0 ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"Because the time derivation and the projector commute, we have","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    forall phirangle in mathcalH  langlephileft(fracmathrmdmathrmdt - frac1mathrmihbarhatP_T_varphiranglemathcalMhatHright)varphirangle =0 ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"This equation must be true for any phirangle in mathcalH, Eq.~(\\ref{eq:DiracFrenkel1}) can thus be written","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"    left(fracmathrmdmathrmdt - frac1mathrmihbarhatP_T_varphiranglemathcalMhatHright)varphirangle =0 ","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"In the context of MPS, the manifold mathcalM will correspond to the space of full-ranked MPS of a given bond dimension D, and the tangent space will be the space spanned by variations of single MPS tensors. The projector defines an effective Hamiltonian under which the dynamics are constrained on mathcalM. Constraining the dynamics on a manifold, introduces a projection error: the time evolution will obey to an effective Hamiltonian different from the starting one. After the introduction of TDVP as a time evolution method for MPS, Haegeman et al. pointed out [haegeman_unifying_2016] that there exist an analytical decomposition for the projector operator hatmathcalP that simplifies the resolution of the equation, turning the problem into one where each matrix A_i can be updated with an effective on site Hamiltonian hat H_texteff via a Schroedinger like equation. The effective Hamiltonian hat H_texteff is a contraction of the Hamiltonian MPO and the current state of the other matrices composing the MPS. This allows to do a sequential update.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"There exist different versions of the TDVP algorithm. In MPSDynamics.jl three methods have been so far implemented:","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"the one-site TDVP (1TDVP)\nthe two-sites TDVP (2TDVP) \nthe adaptive TDVP (DTDVP) [dunnett_efficient_2021]","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"The main advantage of the one-site 1TDVP algorithm is that it preserves the unitarity of the MPS during the time evolution. Its main problem, conversely, is that the time evolution is constrained to happen on a manifold constituted by tensors of fixed bond dimension, a quantity closely related to the amount of entanglement in the MPS, and such a bond dimension has therefore to be fixed before the beginning of the time evolution. This strategy will necessarily be non optimal: the growth of the bond dimensions required to describe the quantum state should ideally mirror the entanglement growth induced by the time evolution. 2TDVP allows for such a dynamical growth of the bond dimensions, and therefore better describes the entanglement in the MPS. It suffers however of other drawbacks: first of all, a truncation error is introduced (by the means of an SVD decomposition), which entails a loss of unitarity of the time-evolved MPS. Moreover, 2TDVP has bad scaling properties with the size of the local dimensions of the MPS: this is a major issue when dealing with bosons. The DTDVP algorithm combines the best features of 1TDVP and 2TDVP: it preserves unitarity, it has the same scaling properties of 1TDVP, and it adapts the bond dimensions to the entanglement evolution at each site and at each time-step. DTDVP does not suffer from a truncation error, but introduces only a projection error.","category":"page"},{"location":"theory/#Bibliography","page":"Theoretical Background","title":"Bibliography","text":"","category":"section"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[chin_exact_2010]: Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. Journal of Mathematical Physics 2010, 51 (9), 092109. https://doi.org/10.1063/1.3490188.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[chin_role_2013]: Chin, A. W.; Prior, J.; Rosenbach, R.; Caycedo-Soler, F.; Huelga, S. F.; Plenio, M. B. The Role of Non-Equilibrium Vibrational Structures in Electronic Coherence and Recoherence in Pigment–Protein Complexes. Nature Phys 2013, 9 (2), 113–118. https://doi.org/10.1038/nphys2515.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[alvertis_nonequilibrium_2019]: Alvertis, A. M.; Schröder, F. A. Y. N.; Chin, A. W. Non-Equilibrium Relaxation of Hot States in Organic Semiconductors: Impact of Mode-Selective Excitation on Charge Transfer. J. Chem. Phys. 2019, 151 (8), 084104. https://doi.org/10.1063/1.5115239.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[dunnett_influence_2021]: Dunnett, A. J.; Gowland, D.; Isborn, C. M.; Chin, A. W.; Zuehlsdorff, T. J. Influence of Non-Adiabatic Effects on Linear Absorption Spectra in the Condensed Phase: Methylene Blue. J. Chem. Phys. 2021, 155 (14), 144112. https://doi.org/10.1063/5.0062950.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[caycedosoler_exact_2022]: Caycedo-Soler, F.; Mattioni, A.; Lim, J.; Renger, T.; Huelga, S. F.; Plenio, M. B. Exact Simulation of Pigment-Protein Complexes Unveils Vibronic Renormalization of Electronic Parameters in Ultrafast Spectroscopy. Nat Commun 2022, 13 (1), 2912. https://doi.org/10.1038/s41467-022-30565-4.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[orus_practical_2014]: Orus, R. A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States. Annals of Physics 2014, 349, 117–158. https://doi.org/10.1016/j.aop.2014.06.013.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[paeckel_timeevolution_2019]: Paeckel, S.; Köhler, T.; Swoboda, A.; Manmana, S. R.; Schollwöck, U.; Hubig, C. Time-Evolution Methods for Matrix-Product States. Annals of Physics 2019, 411, 167998. https://doi.org/10.1016/j.aop.2019.167998.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[prior_efficient_2010]: Prior, J.; Chin, A. W.; Huelga, S. F.; Plenio, M. B. Efficient Simulation of Strong System-Environment Interactions. Phys. Rev. Lett. 2010, 105 (5), 050404. https://doi.org/10.1103/PhysRevLett.105.050404.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[woods_simulating_2015]: Woods, M. P.; Cramer, M.; Plenio, M. B. Simulating Bosonic Baths with Error Bars. Phys. Rev. Lett. 2015, 115 (13), 130401. https://doi.org/10.1103/PhysRevLett.115.130401.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[tamascelli_efficient_2019]: Tamascelli, D.; Smirne, A.; Lim, J.; Huelga, S. F.; Plenio, M. B. Efficient Simulation of Finite-Temperature Open Quantum Systems. Phys. Rev. Lett. 2019, 123 (9), 090402. https://doi.org/10.1103/PhysRevLett.123.090402.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[dunnett_simulating_2021]: Dunnett, A. J.; Chin, A. W. Simulating Quantum Vibronic Dynamics at Finite Temperatures With Many Body Wave Functions at 0 K. Front. Chem. 2021, 8. https://doi.org/10.3389/fchem.2020.600731.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[lacroix_unveiling_2021]: Lacroix, T.; Dunnett, A.; Gribben, D.; Lovett, B. W.; Chin, A. Unveiling Non-Markovian Spacetime Signaling in Open Quantum Systems with Long-Range Tensor Network Dynamics. Phys. Rev. A 2021, 104 (5), 052204. https://doi.org/10.1103/PhysRevA.104.052204.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[appel_mathematics_2007]: Appel, W. Mathematics for Physics and Physicists; Princeton University Press, 2007.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[devega_thermo_2015]: de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[Orus]: Orus, R. A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States. Annals of Physics 2014, 349, 117–158. https://doi.org/10.1016/j.aop.2014.06.013.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[poulin_quantum_2011]: Poulin, D.; Qarry, A.; Somma, R.; Verstraete, F. Quantum Simulation of Time-Dependent Hamiltonians and the Convenient Illusion of Hilbert Space. Phys. Rev. Lett. 2011, 106 (17), 170501. https://doi.org/10.1103/PhysRevLett.106.170501.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[srednicki_entropy_1993]: Srednicki, M. Entropy and Area. Phys. Rev. Lett. 1993, 71 (5), 666–669. https://doi.org/10.1103/PhysRevLett.71.666.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[vidal_entanglement_2003]: Vidal, G.; Latorre, J. I.; Rico, E.; Kitaev, A. Entanglement in Quantum Critical Phenomena. Phys. Rev. Lett. 2003, 90 (22), 227902. https://doi.org/10.1103/PhysRevLett.90.227902.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[wolf_area_2008]: Wolf, M. M.; Verstraete, F.; Hastings, M. B.; Cirac, J. I. Area Laws in Quantum Systems: Mutual Information and Correlations. Phys. Rev. Lett. 2008, 100 (7), 070502. https://doi.org/10.1103/PhysRevLett.100.070502.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[dirac_note_1930]: Dirac, P. A. M. Note on Exchange Phenomena in the Thomas Atom. Mathematical Proceedings of the Cambridge Philosophical Society 1930, 26 (3), 376–385. https://doi.org/10.1017/S0305004100016108.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[frenkel_wave_1934]: Frenkel, Y. Wave Mechanics, Advanced General Theory; Oxford, 1934; Vol. 1.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[haegeman_timedependent_2011]: Haegeman, J.; Cirac, J. I.; Osborne, T. J.; Pižorn, I.; Verschelde, H.; Verstraete, F. Time-Dependent Variational Principle for Quantum Lattices. Phys. Rev. Lett. 2011, 107 (7), 070601. https://doi.org/10.1103/PhysRevLett.107.070601.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[haegeman_unifying_2016]: Haegeman, J.; Lubich, C.; Oseledets, I.; Vandereycken, B.; Verstraete, F. Unifying Time Evolution and Optimization with Matrix Product States. Phys. Rev. B 2016, 94 (16), 165116. https://doi.org/10.1103/PhysRevB.94.165116.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[raab_diracfrenkelmclachlan_2000]: Raab, A. On the Dirac–Frenkel/McLachlan Variational Principle. Chemical Physics Letters 2000, 319 (5), 674–678. https://doi.org/10.1016/S0009-2614(00)00200-1.","category":"page"},{"location":"theory/","page":"Theoretical Background","title":"Theoretical Background","text":"[dunnett_efficient_2021]: Dunnett, A. J.; Chin A. W. Efficient bond-adaptive approach for finite-temperature open quantum dynamics using the one-site time-dependent variational principle for matrix product states. Physical Review B, 104(21):214302, Dec 2021. doi:10.1103/PhysRevB.104.214302. ","category":"page"},{"location":"examples/bath-observables/#Inspecting-the-bath-by-undoing-the-chain-mapping","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"","category":"section"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"Here we give some context on the example script provided in MPSDynamics/example/bath-observables.jl. This example demonstrates the setup and execution of a simulation for a two-level system coupled to an Ohmic bath at finite temperature, where we exploit the access to the chain observables to:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"undo the chain mapping[chin_exact_2010], thus obtaining their representation in the extended bath of T-TEDOPA [tamascelli_efficient_2019], characterized by J(omega beta)\ninverting the thermofield transformation[devega_thermo_2015], thus obtaining the representation of the physical frequencies in the original environment, characterized by J(omega)","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"T-TEDOPA allows to substitute a thermally occupied bath by an extended one, in the pure state of the vacuum, extending the bath of frequencies to negative values: the creation from the system of a mode of negative frequency in the extended bath of frequencies corresponds to the absorption of energy for the system from the thermally occupied modes of the environment. However, by exploiting the thermofield transformation, it is possible to recover the picture of the occupations of the steady states in the physical bath of frequencies[riva_thermal_2023]. ","category":"page"},{"location":"examples/bath-observables/#The-code","page":"Inspecting the bath by undoing the chain mapping","title":"The code","text":"","category":"section"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We start by defining the parameters of the simulation:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"d = 10      # number of Fock states of the chain modes\nN = 60      # length of the chain\nα = 0.01    # coupling strength\nω0 = 0.2  # TLS gap\ns = 1       # ohmicity\nωc = 1.  # Cut-off of the spectral density J(ω)\nβ = 20    # Thermalized environment","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We set the specifics of the simulation:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"method = :TDVP1         # time-evolution method\nconv = 3                # bond dimension for the TDVP1\ndt = 0.5                # time step\ntfinal = 60.0           # simulation time","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"And then compute the chain coefficients, i.e. on-site energies epsilon_i, hopping energies t_i, and system-chain coupling c_0, that define the chain representation of the pure-dephasing model:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"cpars = chaincoeffs_finiteT(N, β; α=α, s=s, J=nothing, ωc=ωc, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=false)","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"with this, we can compute the corresponding MPO, and the initial state in MPS form (1sqrt2(0rangle + 1rangle)):","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"H = puredephasingmpo(ω0, d, N, cpars)\n\nψ = zeros(2)\nψ[1] = 1/sqrt(2)\nψ[2] = 1/sqrt(2)\n\nA = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...])","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We can now define the observables we are interested in computing. Importantly, we not only compute observables related to the system, but also to the chain modes, so that we can inspect the environment.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"ob1 = OneSiteObservable(\"sz\", sz, 1)\nob2 = OneSiteObservable(\"sx\", sx, 1)\nob3 = OneSiteObservable(\"chain_mode_occupation\", numb(d), (2,N+1))\nob4 = OneSiteObservable(\"c\", crea(d), collect(2:N+1))\nob5 = OneSiteObservable(\"cdag\", crea(d), collect(2:N+1))\nob6 = TwoSiteObservable(\"cdagc\", crea(d), anih(d), collect(2:N+1), collect(2:N+1))\nob7 = TwoSiteObservable(\"cdagcdag\", crea(d), crea(d), collect(2:N+1), collect(2:N+1))\nob8 = TwoSiteObservable(\"cc\", anih(d), anih(d), collect(2:N+1), collect(2:N+1))","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"with this we run the simulation:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"A, dat = runsim(dt, tfinal, A, H, prec=1E-4;\n                name = \"Bath observables in the pure dephasing model\",\n                method = method,\n                obs = [ob1, ob2, ob3, ob4, ob5, ob6, ob7, ob8],\n                convobs = [ob1],\n                params = @LogParams(ω0, N, d, α, s, ψ),\n                convparams = conv,\n                reduceddensity = true,\n                verbose = false,\n                save = false,\n                plot = true,\n                );","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"After the simulation, we need to post-process the data a bit, in order to map the correlations from the chain representation to the original environment. We start by defining the matrices associated to the mean values of c and c^dagger:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"T = length(dat[\"data/times\"])\n\nconstr = Array{ComplexF64}(undef, N, N, T)\ndestr = Array{ComplexF64}(undef, N, N, T)\nfor t in 1:T\n    constr[:,:,t] = diagm(0 => dat[\"data/cdag\"][:,t])\n    destr[:,:,t] = diagm(0 => dat[\"data/c\"][:,t])\nend","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"During a numerical simulation we work with chain of finite length N. This truncation on chain modes, introduces a sampling on the modes in the original star-like environment. To recover the frequency modes that are implicitly sampled, one has to diagonalize the tri-diagonal Ntimes N matrix H^textchain, where the diagonal is formed by the e_n coefficients, that is the chain's frequencies, and the upper and lower diagonals by the N-1 hopping coefficients t_n. The unitary matrix that defines the change of basis from the star-like to the chain-like environment is U_n, constituted by the eigenvectors of H^textchain. In the code, we use the eigenchain function:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"omeg = eigenchain(cpars, nummodes=N).values","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"At each time step of the simulation, a number of one-site and two-sites observables where evaluated on the chain. To obtain their value in the extended bath of T-tedopa, characterized by J(omegabeta), the unitary transformation that maps the extended bath Hamiltonian into the chain representation has to be reversed. For instance, when measuring the single site hat n^c_i=hat c_i^dagger hat c_i occupation number, we are not measuring the occupation number of the bosonic mode associated to the omega_i frequency, but the occupation number of the i-th chain mode. Therefore to calculate the number of modes of the environment associated to a specific frequency omega_i, the mapping must be reversed, to obtain the diagonal representation of the bosonic number operator:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"beginaligned\nhat n^b_i = hat b_i^dagger hat b_i = sum_kl U_ik^* hat c_k^dagger hat c_l U_li  \nendaligned","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"This is done in the code using the measuremodes(X, cpars[1], cpars[2]) function, which outputs the vector of the diagonal elements of the operators, in the following way:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"bath_occup = mapslices(X -> measuremodes(X, cpars[1], cpars[2]), dat[\"data/cdagc\"], dims = [1,2])\ncdag_average = mapslices(X -> measuremodes(X, cpars[1], cpars[2]), constr, dims = [1,2])\nc_average = mapslices(X -> measuremodes(X, cpars[1], cpars[2]), destr, dims = [1,2])","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"To compute the correlators, we need the full matrix in the original basis. We therefore use the measurecorrs function:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"cc_average = mapslices(X -> measurecorrs(X, cpars[1], cpars[2]), dat[\"data/cc\"], dims = [1,2])\ncdagcdag_average = mapslices(X -> measurecorrs(X, cpars[1], cpars[2]), dat[\"data/cdagcdag\"], dims = [1,2])\n\ncorrelations_c = [\n    cc_average[i, j, t] - c_average[i, 1, t] .* c_average[j, 1, t]\n    for i in 1:size(cc_average, 1), j in 1:size(cc_average, 2), t in 1:size(cc_average, 3)\n]\ncorrelations_cdag = [\n    cdagcdag_average[i, j, t] - cdag_average[i, 1, t] .* cdag_average[j, 1, t]\n    for i in 1:size(cdagcdag_average, 1), j in 1:size(cdagcdag_average, 2), t in 1:size(cdagcdag_average,3)\n]","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"It is possible to invert the thermofield transformation (details in [riva_thermal_2023]). The expression of the mean value of the number operator for the physical modes can be expressed as a function of mean values in the extended bath, which we denote langle hat a_2k^dagger hat a_2k rangle:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"beginaligned\n    langle hat b_k^dagger hat b_k rangle = coshtheta_ksinhtheta_k (langle hat a_2khat a_1krangle + langle hat a_1k^daggerhat a_2k^daggerrangle ) + sinh^2theta_k (1+ langle hat a_2k^dagger hat a_2k rangle ) ++ cosh^2theta_k langle hat a_1k^dagger hat a_1k rangle\nendaligned","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We remark that in the thermofield case, a negative frequency omega_2k is associated to each positive frequency omega_1k. The sampling is therefore symmetric around zero. This marks a difference with T-TEDOPA, where the sampling of frequencies was obtained through the thermalized measure dmu(beta) = sqrtJ(omega beta)domega, and was not symmetric. To recover the results for the physical bath of frequencies starting from the results of our simulations, that were conducted using the T-TEDOPA chain mapping, we need to do an extrapolation for all of the mean values, in order to have their values for each omega at -omega as well. This is done in the code with the physical_occup function:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"bath_occup_phys = physical_occup(cdagcdag_average[:,:,T], cc_average[:,:,T], omeg, bath_occup[:,:,T], β, N)","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"Finally, in the pure dephasing case, it is also possible to obtain the analytical prediction of the time evolution of the occupations of the bath's modes, so that we can compare our numerical results with the analytical ones, exploiting the Heisenberg time evolution relation:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"beginaligned\nfracd langle hat b_omega rangledt = -i langle hat b_omega hat H rangle = - i omega langlehat b_omega rangle - i fraclangle hat sigma_x rangle2 sqrtJ(omega beta)     \nfracd langle hat n_omega rangledt = -i langlehat b_omega^dagger hat b_omega hat H rangle= 2 fracJ(omegabeta)omega sin(omega t)\nendaligned","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"To this end, it is convenient to choose one of the eigenstates of hat sigma_z as the initial state, so that langle hat sigma_x rangle = pm 1. By solving these differential equations, one obtains the time evolved theoretical behavior of the bath. We define the function for the comparison with analytical predictions:","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"Johmic(ω,s) = (2*α*ω^s)/(ωc^(s-1))\n\ntime_analytical = LinRange(0.0, tfinal, Int(tfinal))\n\nΓohmic(t) = - quadgk(x -> Johmic(x,s)*(1 - cos(x*t))*coth(β*x/2)/x^2, 0, ωc)[1]\n\nDecoherence_ohmic(t) = 0.5 * exp(Γohmic(t))\n\n\nα_theo = 0.25 * α\nfunction Jtherm(x)\n    if 1 >= x >= 0\n        return +α_theo * abs(x)^s * (1 + coth(β*0.5*x))\n    elseif -1 <= x <= 0\n        return -α_theo * abs(x)^s * (1 + coth(β*0.5*x))\n    else\n        return 0\n    end\nend\n\nbath_occup_analytical(ω, t) = abs(Jtherm(ω))/(ω^2)*2*(1-cos(ω*t)) ","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"We conclude the example by plotting.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"ρ12 = abs.(dat[\"data/Reduced ρ\"][1,2,:])\n\np1 = plot(time_analytical, t->Decoherence_ohmic(t), label=\"Analytics\", title=L\"Pure Dephasing, Ohmic $s=%$s$, $\\beta = %$β ~\\mathrm{K}$\", linecolor=:black, xlabel=\"Time (arb. units)\", ylabel=L\"Coherence $|\\rho_{12}(t)|$\", linewidth=4, titlefontsize=16, legend=:best, legendfontsize=16, xguidefontsize=16, yguidefontsize=16, tickfontsize=10)\np1 = plot!(dat[\"data/times\"], ρ12, lw=4, ls=:dash, label=\"Numerics\")\n\ncumul = [bath_occup_analytical(omeg[i], tfinal)*(omeg[i+1]-omeg[i]) for i in 1:(length(omeg)-1)]\n\np2 = plot(omeg[1:length(omeg)-1], cumul, lw = 4, linecolor=:black,\n             xlabel=L\"\\omega\", ylabel=L\"\\langle n^b_\\omega \\rangle\", label=\"Analytics\",\n             title=\"Mode occupation in the extended bath\")\np2 = plot!(omeg, bath_occup[:, :, T], lw=4, ls=:dash, label=\"Numerics\")\n\np3 = heatmap(omeg, omeg, abs.(real.(correlations_cdag[:,:,T]) .+ im*imag.(correlations_cdag[:,:,T])), \n            xlabel=L\"\\omega\",\n            ylabel=L\"\\omega\", title=\"Environmental correlations\")\n\n\nMhalf = Int(length(omeg)*0.5)+1\nM = length(omeg)\n\np4 = plot(omeg[Mhalf:M], bath_occup_phys, lw=4,\n            xlabel=L\"\\omega\", ylabel=L\"\\langle n^b_\\omega \\rangle\",\n            title=\"Mode occupation in the physical bath\")\n\nplot(p1, p2, p3, p4, layout = (2, 2), size = (1400, 1200))","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"___________________","category":"page"},{"location":"examples/bath-observables/#References","page":"Inspecting the bath by undoing the chain mapping","title":"References","text":"","category":"section"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"[chin_exact_2010]: Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. Journal of Mathematical Physics 2010, 51 (9), 092109. https://doi.org/10.1063/1.3490188.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"[tamascelli_efficient_2019]: Tamascelli, D.; Smirne, A.; Lim, J.; Huelga, S. F.; Plenio, M. B. Efficient Simulation of Finite-Temperature Open Quantum Systems. Phys. Rev. Lett. 2019, 123 (9), 090402. https://doi.org/10.1103/PhysRevLett.123.090402.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"[devega_thermo_2015]: de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.","category":"page"},{"location":"examples/bath-observables/","page":"Inspecting the bath by undoing the chain mapping","title":"Inspecting the bath by undoing the chain mapping","text":"[riva_thermal_2023]: Riva, A.; Tamascelli, D.; Dunnett, A. J.; Chin, A. W. Thermal cycle and polaron formation in structured bosonic environments. Phys. Rev. B 2023, 108, 195138, https://doi.org/10.1103/PhysRevB.108.195138.","category":"page"},{"location":"examples/puredephasing/#Pure-Dephasing","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"","category":"section"},{"location":"examples/puredephasing/#Context","page":"Pure-Dephasing","title":"Context","text":"","category":"section"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The Pure-Dephasing Model describes a two-level system interacting linearly with an environment characterised by a spectral density (SD) J(omega). The coupling only acts on diagonal terms through the sigma_z operator. The Hamiltonian reads","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"        hatH = fracomega_02hatsigma_z + int_0^+inftyomega hata^dagger_omegahata_omega mathrmdomega + frachatsigma_z2int_0^+inftysqrtJ(omega)(hata_omega + hata^dagger_omega)mathrmdomega","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Although this interaction will not change the population of the two-level system, the coherences between the two states will vary due to the environment. Introducing the two-level system reduced density matrix rho_ij(t) with ij in (01), the diagonal terms rho_ii(t) are the populations of the states and the anti-diagonal terms rho_ij(t) with i neq j are the coherences between the two states. The effect of the sigma_z bath interaction is to decouple the two states 0rangle and 1rangle. ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The density matrix can be calculated within the MPS formalism with the MPO psirangle langle psi. Tracing out the environment leads to the reduced density matrix. This can be done with the function MPSDynamics.rhoreduced_1site. ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"An analytical formula can be found for the decoherence function Gamma(t), taking into account the SD as well as the temperature of the environment [breuer]","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"    Gamma(t) = - int_0^omega_c mathrmd omega J(omega)frac(1 - cos(omega t))omega^2 coth(βomega2) ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"with beta = (k_B T)^-1. For the case where beta longrightarrow infty, the integral reads","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"    Gamma(t) = - int_0^omega_c mathrmd omega J(omega)frac(1 - cos(omega t))omega^2 ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The time-dependent anti-diagonal terms of the reduced density matrix are then expressed as","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"    rho_12(t) = rho_21(t)^* =rho_12(0) exp(Gamma(t)) ","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Setting up the initial two-level system as a cat state psirangle_S(0) = frac0rangle pm 1ranglesqrt2, this leads to rho_12(0)=frac12.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Here we break out and comment the script in MPSDynamics/examples/puredephasing.jl and MPSDynamics/examples/puredephasing_temperature.jl to show how to simulate this model with an Ohmic SD (hard cut-off) using the T-TEDOPA method as implemented in MPSDynamics.jl.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The T-TEDOPA method relies on a truncated chain mapping that transform the initial Hamiltonian into","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"        hatH = fracomega_02 hatsigma_z +  c_0 frachatsigma_z2(hatb_0^dagger + hatb_0) + sum_i=0^N-1 t_i (hatb_i+1^dagger hatb_i + mathrmhc) + sum_i=0^N-1 epsilon_i hatb_i^dagger hatb_i","category":"page"},{"location":"examples/puredephasing/#The-code","page":"Pure-Dephasing","title":"The code","text":"","category":"section"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"First, we load the MPSdynamics.jl package to be able to perform the simulation, the Plots.jl one to plot the results, the LaTeXStrings.jl one to be able to use LaTeX in the plots and eventually QuadGK.jl to perform the analytical integral calculations. A notation is introduced for +∞.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"using MPSDynamics, Plots, LaTeXStrings, QuadGK\n\nconst ∞  = Inf","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"We then define variables for the physical parameters of the simulation. Among these, two are convergence parameters:","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"d is the number of states we retain for the truncated harmonic oscillators representation of environmental modes\nN is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The value of β determines whether the environment is thermalized or not. The example as it is is the zero-temperature case. For the finite-temperature case, 'β = ∞' has be commented instead of the line above that precises a β value.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#----------------------------\n# Physical parameters\n#----------------------------\n\nd = 10 # number of Fock states of the chain modes\n\nN = 30 # length of the chain\n\nα = 0.01 # coupling strength\n\nω0 = 0.008 # TLS gap\n\ns = 1 # ohmicity\n\nωc = 0.035 # Cut-off of the spectral density J(ω)\n\n#β = 100 # Thermalized environment\n\nβ = ∞ # Case zero temperature T=0, β → ∞","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"The chain coefficient have then to be calculated. This part differs for the zero-temperature case and the thermalized bath. For β = , the chain coeffficients can be calculated with the function MPSDynamics.chaincoeffs_ohmic whereas the function MPSDynamics.chaincoeffs_finiteT is used when T neq 0  textK:","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"if β == ∞\n    cpars = chaincoeffs_ohmic(N, α, s; ωc=ωc)  # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0\nelse\n    cpars = chaincoeffs_finiteT(N, β; α=α, s=s, J=nothing, ωc=ωc, mc=4, mp=0, AB=nothing, iq=1, idelta=2, procedure=:Lanczos, Mmax=5000, save=false)  # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0\n    #= #If cpars is stored in \"../ChainOhmT/ohmicT\" \n    curdir = @__DIR__\n    dir_chaincoeff = abspath(joinpath(curdir, \"../ChainOhmT/ohmicT\"))\n    cpars  = readchaincoeffs(\"$dir_chaincoeff/chaincoeffs.h5\",N,α,s,β) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0\n    =#\nend","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"An option is provided if the coefficient are saved to be reused.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is mathcalO(dt^3). In this example we present two one-site implementation of TDVP that both preserves the unitarity of the evolution:","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"the regular one-site method with the keyword :TDVP1 where all the virtual bonds of the MPS have the same bond dimension D\nthe adaptive method with the keyword :DTDVP where the bond dimension is locally increased at each time step if the TDVP projection error crosses a threshold value","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Logically the constant bond dimension of the MPS for TDVP1 and the threshold of the projection error for DTDVP are their respective convergence parameter.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#-----------------------\n# Simulation parameters\n#-----------------------\n\ndt = 1.0 # time step\n\ntfinal = 300.0 # simulation time\n\nmethod = :TDVP1 # time-evolution method\n\n#method = :DTDVP # time-evolution method\n\nD = 2 # MPS bond dimension","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Using MPSDynamics.jl built-in methods we define the Pure Dephasing model MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the 'local state' of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#---------------------------\n# MPO and initial state MPS\n#---------------------------\n\nH = puredephasingmpo(ΔE, d, N, cpars)\n\n# Initial electronic system in a superposition of 1 and 2\nψ = zeros(2)\nψ[1] = 1/sqrt(2)\nψ[2] = 1/sqrt(2)\n\nA = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"We then chose the observables that will be stored in the data and the MPSDynamics.runsim arguments. This example relies on the storage of the reduced density matrix (called reduceddensity in runsim).","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"\n#---------------------------\n# Definition of observables\n#---------------------------\n\nob1 = OneSiteObservable(\"sz\", sz, 1)\n\n\n#-------------\n# Simulation\n#------------\n\nA, dat = runsim(dt, tfinal, A, H, prec=1E-4;\n                name = \"pure dephasing model with temperature\",\n                method = method,\n                obs = [ob1],\n                convobs = [ob1],\n                params = @LogParams(ΔE, N, d, α, s),\n                convparams = D,\n                reduceddensity=true,\n                verbose = false,\n                save = true,\n                plot = true,\n                );","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"After having performed the dynamics, the analytical decoherence function is numerically calculated with the help of the quadgk function (from the QuadGK.jl package). It reads","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#----------\n# Analytical results at specified temperature \n# (see The Theory of Open Quantum System, H.-P. Breuer & F. Petruccione 2002, Chapter 4)\n#----------\n\nJohmic(ω,s) = (2*α*ω^s)/(ωc^(s-1))\n\ntime_analytical = LinRange(0.0,tfinal,Int(tfinal))\n\nΓohmic(t) = - quadgk(x -> Johmic(x,s)*(1 - cos(x*t))*coth(β*x/2)/x^2, 0, ωc)[1]\n\nDecoherence_ohmic(t) = 0.5*exp(Γohmic(t))","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"Eventually, the stored reduced density matrix is compared against the analytical formula","category":"page"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"#-------------\n# Plots\n#------------\n\nρ12 = abs.(dat[\"data/Reduced ρ\"][1,2,:])\n\nplot(time_analytical, t->Decoherence_ohmic(t), label=\"Analytics\", title=L\"Pure Dephasing, Ohmic $s=%$s$, $\\beta = %$β ~\\mathrm{K}$\", linecolor=:black, xlabel=\"Time (arb. units)\", ylabel=L\"Coherence $|\\rho_{12}(t)|$\", linewidth=4, titlefontsize=16, legend=:best, legendfontsize=16, xguidefontsize=16, yguidefontsize=16, tickfontsize=10)\n\nplot!(dat[\"data/times\"], ρ12, lw=4, ls=:dash, label=\"Numerics\")","category":"page"},{"location":"examples/puredephasing/#Bibliography","page":"Pure-Dephasing","title":"Bibliography","text":"","category":"section"},{"location":"examples/puredephasing/","page":"Pure-Dephasing","title":"Pure-Dephasing","text":"[breuer]: Breuer, H.;Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press, 2002.","category":"page"},{"location":"examples/anderson-model/#The-Anderson-Impurity-Model","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"Here we give some context on the examples provided in MPSDynamics/examples/anderson_model_double.jl and MPSDynamics/examples/anderson_model_interleaved.jl. In these two examples, we use the fermionic chain mapping proposed in [khon_efficient_2021] to show how to perform tensor network simulations of the Single Impurity Anderson Model (SIAM) with the MPSDynamics.jl library. ","category":"page"},{"location":"examples/anderson-model/#Basics-of-the-fermionic-chain-mapping","page":"The Anderson Impurity Model","title":"Basics of the fermionic chain mapping","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"Before giving the code implementation, a short recap on the problem statement and on the fermionic mapping used to solved it. The SIAM Hamiltonian is defined as:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    hat H^textSIAM  = hat H_textloc + hat H_texthyb + hat H_textcond = overbraceepsilon_d hat d^dagger hat d^hat H_textloc + underbracesum_k V_k Big( hat d^dagger hat c_k + hat c_k^dagger hat d Big)_H_texthyb + underbracesum_k epsilon_k hat c_k^dagger hat c_k_H_I^textchain","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"All of the operators obey to the usual fermionic anti-commutation relations: hat c_i hat c_j^dagger  = delta_ij, hat c_i hat c_j  =hat c_i^dagger hat c_j^dagger  =0 forall ij. The chain mapping is based on a thermofield-like transformation  [devega_thermo_2015], performed with fermions: ancillary fermionic operators hat c_2k are defined, one for each of the original fermionic modes hat c_1k. A Bogoliubov transformation is then applied, so that two new fermionic modes hat f_1k and hat f_2k are defined as a linear combination of hat c_1k and hat c_2k. Two chains are defined: the chain labelled 1 for the empty modes:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"hat f_1k=e^-iG  hat c_k e^iG= cosh(theta_k) hat c_1k -sinh(theta_k)  hat c_2k^dagger \nhat f_2k=e^-iG hat c_k e^iG= cosh(theta_k)  hat c_1k +sinh(theta_k)  hat c_2k^dagger","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We remark that this is the same Bogoliubov transformation used in the thermofield[devega_thermo_2015] for the bosonic case: the only thing that changes is a plus sign, that takes into account the fermionic anti-commutation relations. With these new fermionic operators we obtain the thermofield-transformed Hamiltonian, where the system interacts with two environments at zero temperature, allowing for pure state simulations (and thus the employement of MPS).","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The thermofield-transformed Hamiltonian is then mapped on two chains, defined and constructed using the TEDOPA chain mapping: the chain labelled 1 is for the empty modes, the chain labelled 2 for the filled modes. The following relations are used to define the functions equivalent to the spectral density of the bosonic case, one for each chain:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"beginaligned\n   V_1k = V_k sin theta_k = sqrtfrac1e^beta epsilon_k+1 \n   V_2k = V_k cos theta_k = sqrtfrac1e^-beta epsilon_k+1 \nendaligned","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where we choose the spectral function that characterizes the fermionic bath to be: V_k= sqrt1-k^2, and we define the dispersion relation as: e_k = k, that is, a linear dispersion relation with propagation speed equal to 1. This latter choice corresponds to a model of metals (gapless energy spectrum). We select a filled state as the initial state of the defect. Using the mapping proposed in [1], the chain Hamiltonian becomes:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    beginaligned\n    hat H^textchain  = hat H_textloc + sum_i = 12bigg J_i0 Big(hat d^dagger hat a_i0 + hat d hat a_i0^dagger Big) +  + sum_n=1^infty  Big( J_in hat a_in^dagger hat a_in-1 +  J_in hat  a_in-1^dagger hat a_in Big) + sum_n=0^infty E_in hat  a_in^dagger hat a_in bigg\n    endaligned","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where the J_in coefficients are the couplings between the chain sites and the E_in coefficients are the energies associated to each chain site. Clearly, the interactions are between nearest neighbors. This, combined with the fact that the fermions in our model are spinless, enables a straightforward mapping into fermionic operators of the bosonic creation and annihilation operators, that on their part obey to the bosonic commutation relations: hat b_i hat b_j^dagger = delta_ij, hat b_i hat b_j =hat b_i^dagger hat b_j^dagger =0 forall ij. The mapping derived from Jordan-Wigner transformations for spinless fermions is:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    hat a_i^dagger hat a_i+1 + hat a_i+1^dagger hat a_i = hat b_i^dagger hat b_i+1 + hat b_i+1^dagger hat b_i  ","category":"page"},{"location":"examples/anderson-model/#Double-chain-MPO","page":"The Anderson Impurity Model","title":"Double chain MPO","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"(Image: image)","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We can represent the Hamiltonian hat H^textchain as a MPOs bond dimension is: chi = 4. The MPO has the structure of a double chain, with an impurty site at the center, and therefore can be seen as the product of the following matrices:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    H = W_1N cdotcdot W_1 0 cdot W_d cdot W_20 cdot  cdot W_2 N  ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where the matrices are defined as:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"beginaligned\n W_1N = \nbeginbmatrix\nhatmathbb I  J_2N hat b_2N^dagger  J_2N hat b_2N  E_2N hat b_2N^dagger hat b_2N \nendbmatrix quad   W_1 0 = \nbeginbmatrix\n hat mathbb I  J_20 hat b_20^dagger  J_20 hat b_20  E_20 hat b_20^dagger hat b_20\n0 0  0  hat b_20 \n0 0  0  hat b_20^dagger \n0 0  0  hatmathbb I\nendbmatrix \n W_d = \nbeginbmatrix\n hat mathbb I  hat d^dagger  hat d  epsilon_d hat d^dagger hat d\n0 0  0  hat d \n0 0  0  hat d^dagger \n0 0  0  hatmathbb I\nendbmatrix \n W_2 0 =\nbeginbmatrix\n hat mathbb I  hat b_10^dagger  hat b_10  E_10 hat b_10^dagger hat b_10\n0 0  0  hat J_10b_10 \n0 0  0  hat J_10b_10^dagger \n0 0  0  hatmathbb I\nendbmatrix\n quad W_2 N =\nbeginbmatrix\n E_2N hat b_2N^dagger hat b_2N  J_2N hat b_2N  J_2N hat b_2N^dagger  hatmathbb I\nendbmatrix\nendaligned","category":"page"},{"location":"examples/anderson-model/#Interleaved-chain-MPO","page":"The Anderson Impurity Model","title":"Interleaved chain MPO","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"(Image: image)","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The drawback of the double chain representation is that the particle-hole pairs are spatially separated in the MPS, creating correlations and therefore leading to a dramatic increase in the bond dimensions. This is why Kohn and Santoro propose an interleaved geometry, the advantages of which are thoroughly explained in [khon_eff_2022]. Exploiting the interleaved representation, the interaction comes to be between next-nearest neighbors: a string operator appears in the Jordan-Wigner transformation from bosons to fermions:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    hat a_i^dagger hat a_i+2 + hat a_i+2^dagger hat a_i = hat b_i^dagger hat F_i+1 hat b_i+2 + hat b_i hat F_i+1 hat b_i+2^dagger","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where the string operator hat F_i is defined as: hat F_i = (-1)^hat n_i = hatmathbb I -2 hat n_i = hatmathbb I-2 hat b_i^dagger hat b_i.  It is possible to find the analytical form also for MPOs with long range interaction \\cite{mpo}. In the case of next-nearest neighbors interactions between spinless fermions, in the interleaved geometry of Fig. \\ref{subfig:folded}, the MPO representation will require a bond dimension chi=6. We explicitly write it as:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"    H = W_d cdot W_2 0 cdot W_1 0 cdotcdot W_2N cdot W_1 N  ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"where the matrices are defined as: ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"beginaligned\n W_d = \nbeginbmatrix\n hatmathbb I  hat d  hat d^dagger  0  0  E_d hat d^dagger hat d \nendbmatrix quad W_2 0 = \nbeginbmatrix\n hat mathbb I  hat b_20  hat b_20^dagger  0  0  E_20 hat b_20^dagger hat b_20\n0 0  0  hatF_20  0  J_20 hat b_20^dagger \n0 0  0  0  hatF_20  J_20 hat b_20 \n0 0  0  0  0   0\n0 0  0  0  0  0 \n0 0  0  0  0  hatmathbb I\nendbmatrix \n W_1 0 = \nbeginbmatrix\n hat mathbb I  hat b_10  hat b_10^dagger  0  0  E_10 hat b_10^dagger hat b_10\n0 0  0  hat F_10  0  0 \n0 0  0  0  hatF_10  0 \n0 0  0  0  0  J_10 hat b_10^dagger \n0 0  0  0  0  J_10 hat b_10 \n0 0  0  0  0  hatmathbb I\nendbmatrix \n W_2N = \nbeginbmatrix\n hat mathbb I  hat b_2N  hat b_2N^dagger  0  0  E_2N hat b_2N^dagger hat b_2N\n0 0  0  hatF_2N  0  0 \n0 0  0  0  hatF_2N  0 \n0 0  0  0  0  J_2N hat b_2N^dagger \n0 0  0  0  0  J_2N hat b_2N \n0 0  0  0  0  hatmathbb I\nendbmatrix \nquad W_1 N\nbeginbmatrix\n E_1N hat b_1N^dagger hat b_1N  0 0  J_1N hat b_1N^dagger  J_1N hat b_1N  hatmathbb I\nendbmatrix \nendaligned","category":"page"},{"location":"examples/anderson-model/#Code-implementation","page":"The Anderson Impurity Model","title":"Code implementation","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The fermionic mapping can be strightforwardly implemented with the methods of MPSDyanmics.jl. We provide two examples: ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"examples/anderson_model_double, simulating the SIAM with the double-chain geometry\nexamples/anderson_model_interleaved, simulating the SIAM with the interleaved-chain geometry","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"They only differ on the way the Hamiltonian MPO is defined. We now briefly review the important bits of the code.","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"Both of the examples start with the definition of the physical parameters,","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"N = 40      # number of chain sites\nβ = 10.0     # inverse temperature\nμ = 0.      # chemical potential\nEd = 0.3    # energy of the impurity\nϵd = Ed - μ # energy of the impurity minus the chemical potential","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The chaincoeffs_fermionic function is needed to compute the chain coefficients. It requires as inputs the number of modes of each chain N, the inverse temperature \\beta, a label to specify if the chain modes are empty (label is 1.0) or filled (label is 2.0), and both the dispersion relation epsilon_k and the fermionic spectral density funciton V_k.","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"function ϵ(x)\n    return x\nend\n\nfunction J(x)\n    return sqrt(1 - x^2) # semi-circular density of states\nend\n\nchainparams1 = chaincoeffs_fermionic(N, β, 1.0; ϵ, J, save=false) # empty\nchainparams2 = chaincoeffs_fermionic(N, β, 2.0; ϵ, J, save=false) # filled","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We then specify the simulation parameters","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"dt = 0.25           # time step\nT = 15.0            # simulation time\nmethod = :DTDVP     # time-evolution method\nDmax = 150          # MPS max bond dimension\nprec = 0.0001       # precision for the adaptive TDVP\n","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"and with this we are ready to construct the Hamiltonian MPO and specify the initial state, which will obviously differ depending on the chosen geometry.","category":"page"},{"location":"examples/anderson-model/#Double-chain-geometry","page":"The Anderson Impurity Model","title":"Double chain geometry","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The Hamiltonian is defined using the tightbinding_mpo function, which takes as an input the number of modes of each chain N, the defect's energy \\epsilon_d, and the chain coefficients of the first chainparams1 and second chainparams2 chain. The MPS for the initial state is a factorized state made of: N filled states, a filled impurity, and N empty states. ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"H = tightbinding_mpo(N, ϵd, chainparams1, chainparams2)\n\nψ =  unitcol(2,2) # (0,1) filled impurity state\nA = productstatemps(physdims(H), state=[fill(unitcol(2,2), N)..., ψ, fill(unitcol(1,2), N)...]) # MPS","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"To avoid the DTDVP algorithm from getting stuck in a local minimum, it is better to embed the MPS in a manifold of bond dimension 2 (or more):","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"mpsembed!(A, 2) # to embed the MPS in a manifold of bond dimension 2","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We can now define the observables for the two chains and for the impurity","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"ob1 = OneSiteObservable(\"chain1_filled_occup\", numb(2), (1,N))\nob2 = OneSiteObservable(\"chain2_empty_occup\", numb(2), (N+2, 2N+1))\nob3 = OneSiteObservable(\"system_occup\", numb(2), N+1)","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"and run the simulation","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"A, dat = runsim(dt, T, A, H;\n                name = \"Anderson impurity problem (folded chain)\",\n                method = method,\n                obs = [ob1, ob2, ob3], \n                convobs = [ob1],\n                params = @LogParams(N, ϵd, β, c1, c2),\n                convparams = [prec],   \n                Dlim = Dmax,          \n                savebonddims = true,   # we want to save the bond dimension\n                verbose = false,\n                save = false,\n                plot = true,\n                );","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"With very minimal post-processing of the data","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"# Reshaping the vector to a column matrix and horizontal concatenation\nsystem_occup_col = reshape(dat[\"data/system_occup\"], :, 1)\nocc = hcat(dat[\"data/chain1_filled_occup\"]', system_occup_col)\nocc = vcat(occ', dat[\"data/chain2_empty_occup\"])","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"we plot the results:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"# Plot the system occupation    \np1 = plot(\n    dat[\"data/times\"],\n    dat[\"data/system_occup\"],\n    xlabel = L\"$t$\",\n    ylabel = L\"$n_d$\",\n    title = \"System Occupation\",\n    size = (700, 500)\n)\n\n# Plot the occupation of the chain sites\np2 = heatmap(\n    collect(1:2*N+1),\n    dat[\"data/times\"],\n    transpose(occ),  # Use the matrix form\n    cmap = :coolwarm,\n    aspect_ratio = :auto,\n    xlabel = L\"$N_{i,j}$ chain sites\",\n    ylabel = L\"$t$\",\n    title = \"Chain Occupation\",\n    colorbar = true,\n    size = (700, 500)\n)\n\n# Plot the bond dimensions\np3 = heatmap(\n    collect(1:2*N+2),\n    dat[\"data/times\"],\n    transpose(dat[\"data/bonddims\"]),\n    cmap = :magma,\n    aspect_ratio = :auto,\n    xlabel = L\"$N_{i,j}$ chain sites\",\n    ylabel = L\"$t$\",\n    title = \"Bond Dimensions\",\n    colorbar = true,\n    size = (700, 500)\n)\n\n# Define indices for columns to be plotted\ncolumns_to_plot = [1, 5, 10, 15, 20]\n\n# Plot vertical slices for occupancy\np4 = plot(title = \"Chain occupation\")\nfor col in columns_to_plot\n    plot!(p4, occ[:, col], label = L\"$t =$\"*\"$col\", xlabel = L\"$N_{i,j}$ chain sites\", ylabel = \"chain occupation\")\nend\n\n# Plot vertical slices for bond dimensions\np5 = plot(title = \"Bond Dimensions\")\nfor col in columns_to_plot\n    plot!(p5, dat[\"data/bonddims\"][:, col], label = L\"$t =$\"*\"$col\", xlabel = L\"$N_{i,j}$ chain sites\", ylabel = L\"$\\chi$\")\nend\n\n# Display the plots\nplot(p2, p3, p4, p5, p1, layout = (3, 2), size = (1400, 1200))\n","category":"page"},{"location":"examples/anderson-model/#Interleaved-chain-geometry","page":"The Anderson Impurity Model","title":"Interleaved chain geometry","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"The Hamiltonian is defined using the interleaved_tightbinding_mpo function, which takes as an input the number of modes of each chain N, the defect's energy \\epsilon_d, and the chain coefficients of the first chainparams1 and second chainparams2 chain. The MPS for the initial state is a factorized state (bond dimension 1) made of: a filled impurity, and 2N alternate filled-empty states. ","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"H = interleaved_tightbinding_mpo(N, ϵd, chainparams1, chainparams2)\n\nψ =  unitcol(2,2) # (0,1) filled impurity state\nTot = Any[]\npush!(Tot, ψ)\nfor i in 1:(N)\n    push!(Tot, unitcol(2,2))\n    push!(Tot, unitcol(1,2))\nend\n\nA = productstatemps(physdims(H), state=Tot) # MPS","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"To avoid the DTDVP algorithm from getting stuck in a local minimum, it is better to embed the MPS in a manifold of bond dimension 2 (or more):","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"mpsembed!(A, 2) # to embed the MPS in a manifold of bond dimension 2","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We can finally define the observables for the interleaved chain","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"ob1 = OneSiteObservable(\"system_occup\", numb(2), 1)\nob2 = OneSiteObservable(\"folded_chain_occup\", numb(2), (2,2N+1))","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"and run the simulation","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"A, dat = runsim(dt, T, A, H;\n                name = \"Anderson impurity problem (folded chain)\",\n                method = method,\n                obs = [ob1, ob2], \n                convobs = [ob1],\n                params = @LogParams(N, ϵd, β, c1, c2),\n                convparams = [prec],   \n                Dlim = Dmax,          \n                savebonddims = true,   # we want to save the bond dimension\n                verbose = false,\n                save = false,\n                plot = true,\n                );","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"To show the data in a clear way, we do a bit of post-processing of the data to unfold the chain, and move back to the double chain representation:","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"unfolded_occ = Vector{Vector{Float64}}()  # Assuming the elements are of type Float64\nunfolded_bonds = Vector{Vector{Float64}}()  # Adjust the type based on actual data\n\n# Populate unfolded_occ by iterating in the specific order mentioned\nfor i in 1:N  # Adjusted for 1-based indexing\n    push!(unfolded_occ, dat[ \"data/folded_chain_occup\"][2N + 1 - 2i, :])\nend\n\npush!(unfolded_occ, dat[\"data/folded_chain_occup\"][1,:])\n\nfor i in 2:N\n    push!(unfolded_occ, dat[\"data/folded_chain_occup\"][2i,:])\nend\n\n# Populate unfolded_bonds similarly\nfor i in 1:(N+1)  # Adjusted for 1-based indexing\n    push!(unfolded_bonds, dat[\"data/bonddims\"][2N + 3 - 2i,:])  # Assuming bonddims is directly accessible\nend\n\npush!(unfolded_bonds, dat[\"data/bonddims\"][1,:])\n\nfor i in 2:(N+1)\n    push!(unfolded_bonds, dat[\"data/bonddims\"][2i,:])\nend\n\nunfolded_bonds_matrix = hcat(unfolded_bonds...)'\nunfolded_occ_matrix = hcat(unfolded_occ...)'","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"We conclude by plotting the data","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"# Plot the system occupation    \np1 = plot(\n    dat[\"data/times\"],\n    dat[\"data/system_occup\"],\n    xlabel = L\"$t$\",\n    ylabel = L\"$n_d$\",\n    title = \"System Occupation\",\n    size = (700, 500)\n)\n\n# Plot the occupation of the chain sites\np2 = heatmap(\n    collect(1:2*N),\n    dat[\"data/times\"],\n    transpose(unfolded_occ_matrix),  # Use the matrix form\n    cmap = :coolwarm,\n    aspect_ratio = :auto,\n    xlabel = L\"$N_{i,j}$ chain sites\",\n    ylabel = L\"$t$\",\n    title = \"Chain Occupation\",\n    colorbar = true,\n    size = (700, 500)\n)\n\n# Plot the bond dimensions\np3 = heatmap(\n    collect(1:2*N+2),\n    dat[\"data/times\"],\n    transpose(unfolded_bonds_matrix),\n    cmap = :magma,\n    aspect_ratio = :auto,\n    xlabel = L\"$N_{i,j}$ chain sites\",\n    ylabel = L\"$t$\",\n    title = \"Bond Dimensions\",\n    colorbar = true,\n    size = (700, 500)\n)\n\n# Define indices for columns to be plotted\ncolumns_to_plot = [1, 5, 10, 15, 20]\n\n\n# Plot vertical slices for occupancy\np4 = plot(title = \"Chain occupation\")\nfor col in columns_to_plot\n    plot!(p4, unfolded_occ_matrix[:, col], label = L\"$t =$\"*\"$col\", xlabel = L\"$N_{i,j}$ chain sites\", ylabel = \"chain occupation\")\nend\n\n# Plot vertical slices for bond dimensions\np5 = plot(title = \"Bond Dimensions\")\nfor col in columns_to_plot\n    plot!(p5, unfolded_bonds_matrix[:, col], label = L\"$t =$\"*\"$col\", xlabel = L\"$N_{i,j}$ chain sites\", ylabel = L\"$\\chi$\")\nend\n\n# Display the plots\nplot(p2, p3, p4, p5, p1, layout = (3, 2), size = (1400, 1200))","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"________________","category":"page"},{"location":"examples/anderson-model/#References","page":"The Anderson Impurity Model","title":"References","text":"","category":"section"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"[khon_efficient_2021]: Kohn, L.; Santoro, G. E. Efficient mapping for anderson impurity problems with matrix product states. Phys. Rev. B 2021, 104 (1), 014303. https://doi.org/10.1103/PhysRevB.104.014303.","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"[devega_thermo_2015]: de Vega, I.; Banuls, M-.C. Thermofield-based chain-mapping approach for open quantum systems. Phys. Rev. A 2015, 92 (5), 052116. https://doi.org/10.1103/PhysRevA.92.052116.","category":"page"},{"location":"examples/anderson-model/","page":"The Anderson Impurity Model","title":"The Anderson Impurity Model","text":"[khon_eff_2022]: Kohn L.; Santoro G. E. J. Stat. Mech. (2022) 063102 DOI 10.1088/1742-5468/ac729b.","category":"page"},{"location":"examples/sbm/#The-Spin-Boson-Model","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"","category":"section"},{"location":"examples/sbm/#Context","page":"The Spin-Boson Model","title":"Context","text":"","category":"section"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"The Spin-Boson Model (SBM) is a prototypical model in the theory of open quantum systems where a two level system interacts linearly with a bosonic bath","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"\thatH = fracomega_02hatsigma_z + Deltahatsigma_x + int_0^+inftyomega hata^dagger_omegahata_omega mathrmdomega + hatsigma_xint_0^+inftysqrtJ(omega)(hata_omega + hata^dagger_omega)mathrmdomega","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Even though this model is fairly simple it is physically very rich and it is not analytically solvable. For these reason it has become a test-bed for numerical methods simulating open quantum systems dynamics in the non-perturbative non-Markovian regime.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"For instance when the SD is Ohmic, this model presents a phase transition between a so called localised and a delocalised phase for alpha approx 12.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Here we break out and comment the script in MPSDynamics/examples/sbm_zero_temperature.jl to show how to simulate this model with an Ohmic SD (hard cut-off) using the T-TEDOPA method as implemented in MPSDynamics.jl.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"The T-TEDOPA method relies on a truncated chain mapping that transform the initial Hamiltonian into","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"\thatH = fracomega_02 hatsigma_z + Delta hatsigma_x + c_0 hatsigma_x(hatb_0^dagger + hatb_0) + sum_i=0^N-1 t_i (hatb_i+1^dagger hatb_i + mathrmhc) + sum_i=0^N-1 epsilon_i hatb_i^dagger hatb_i ","category":"page"},{"location":"examples/sbm/#The-code","page":"The Spin-Boson Model","title":"The code","text":"","category":"section"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"First a multi-line comment introduces the model and the aim of the script.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#=\n    Example of a zero-temperature Spin-Boson Model with an hard cut-off Ohmic spectral density J(ω) = 2αω when ω < ωc and 0 otherwise\n\n    The dynamics is simulated using the T-TEDOPA method that maps the normal modes environment into a non-uniform tight-binding chain.\n\n    H = \\\\frac{ω_0}{2} σ_z + Δ σ_x + c_0 σ_x(b_0^\\\\dagger + b_0) + \\\\sum_{i=0}^{N-1} t_i (b_{i+1}^\\\\dagger b_i +h.c.) + \\\\sum_{i=0}^{N-1} ϵ_i b_i^\\\\dagger b_i \n\n    Two variants of the one-site Time Dependent Variational Principal (TDVP) are presented for the time evolution of the quantum state.\n=#","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We load the MPSdynamics.jl package to be able to perform the simulation, the Plots.jl one to plot the results, and the LaTeXStrings.jl one to be able to use LaTeX in the plots.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"using MPSDynamics, Plots, LaTeXStrings","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We then define variables for the physical parameters of the simulation. Among these, two are convergence parameters:","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"d is the number of states we retain for the truncated harmonic oscillators representation of environmental modes \nN is the number of chain (environmental) modes we keep. This parameters determines the maximum simulation time of the simulation: indeed excitations that arrive at the end of the chain are reflected towards the system and can lead to unphysical results","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#----------------------------\n# Physical parameters\n#----------------------------\n\nd = 6 # number of Fock states of the chain modes\n\nN = 30 # length of the chain\n\nα = 0.1 # coupling strength\n\nΔ = 0.0 # tunneling \n\nω0 = 0.2 # TLS gap\n\ns = 1 # ohmicity\n\ncpars = chaincoeffs_ohmic(N, α, s) # chain parameters, i.e. on-site energies ϵ_i, hopping energies t_i, and system-chain coupling c_0","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We set the simulation parameters and choose a time evolution method. As always for simulations of dynamics, the time step must be chosen wisely. The error of the TDVP methods is mathcalO(dt^3). In this example we present two one-site implementation of TDVP that both preserves the unitarity of the evolution:","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"the regular one-site method with the keyword :TDVP1 where all the virtual bonds of the MPS have the same bond dimension D\nthe adaptive method with the keyword :DTDVP where the bond dimension is locally increased at each time step if the TDVP projection error crosses a threshold value","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Logically the constant bond dimension of the MPS for TDVP1 and the threshold of the projection error for DTDVP are their respective convergence parameter. ","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#-----------------------\n# Simulation parameters\n#-----------------------\n\ndt = 0.5 # time step\n\ntfinal = 30.0 # simulation time\n\nmethod = :TDVP1 # Regular one-site TDVP (fixed bond dimension)\n\n# method = :DTDVP # Adaptive one-site TDVP (dynamically updating bond dimension)\n\nconvparams = [2,4,6] # MPS bond dimension (1TDVP)\n\n# convparams = [1e-2, 1e-3, 1e-4] # threshold value of the projection error (DTDVP)","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Using MPSDynamics.jl built-in methods we define the SBM MPO and the MPS representing the initial state. This initial state is a product state between the system and the chain. It is constructed using a list of the 'local state' of each site of the MPS, and the dimensions of the physical legs of the MPS are set to be the same as the ones of the MPO.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#---------------------------\n# MPO and initial state MPS\n#---------------------------\n\nH = spinbosonmpo(ω0, Δ, d, N, cpars) # MPO representation of the Hamiltonian\n\nψ = unitcol(1,2) # Initial up-z system state \n\nA = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ>|Vacuum>","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We then chose the observables that will be measured during the time evolution after being passed as an argument of the MPSDynamics.runsim method.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#---------------------------\n# Definition of observables\n#---------------------------\n\nob1 = OneSiteObservable(\"sz\", sz, 1)\n\nob2 = OneSiteObservable(\"chain mode occupation\", numb(d), (2,N+1))\n\nob3 = TwoSiteObservable(\"SXdisp\", sx, disp(d), [1], collect(2:N+1))","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"We run the simulation from t = 0 to tfinal with time-steps of size dt using the method method. The convobs observables will be measured and stored for the convergence parameters convparams, whereas the obs observables will only be measured and stored for the most precise convergence parameter.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#-------------\n# Simulation\n#------------\n\nA, dat = runsim(dt, tfinal, A, H;\n                name = \"ohmic spin boson model\",\n                method = method,\n                obs = [ob2,ob3],\n                convobs = [ob1],\n                params = @LogParams(N, d, α, Δ, ω0, s),\n                convparams = convparams,\n                verbose = false,\n                savebonddims = true, # this keyword argument enables the bond dimension at each time step to be saved when using DTDVP\n                save = true,\n                plot = true,\n                );","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"Finally, we plot the simulation results that are stored in the dat dictionnary.","category":"page"},{"location":"examples/sbm/","page":"The Spin-Boson Model","title":"The Spin-Boson Model","text":"#----------\n# Plots\n#----------\n\nmethod == :TDVP1 && plot(dat[\"data/times\"], dat[\"convdata/sz\"], label=[\"Dmax = 2\" \"Dmax = 4\" \"Dmax = 6\"], xlabel=L\"t\",ylabel=L\"\\sigma_z\")\n\nmethod == :DTDVP && plot(dat[\"data/times\"], dat[\"convdata/sz\"], label=[\"p = 1e-2\" \"p = 1e-3\" \"p = 1e-4\"], xlabel=L\"t\",ylabel=L\"\\sigma_z\") \n\nmethod == :DTDVP && heatmap(dat[\"data/times\"], collect(0:N+1), dat[\"data/bonddims\"], xlabel=L\"t\",ylabel=\"bond index\")\n\nheatmap(dat[\"data/times\"], collect(1:N), abs.(dat[\"data/SXdisp\"][1,:,:]), xlabel=L\"t\",ylabel=\"chain mode\")","category":"page"},{"location":"#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"The MPSDynamics.jl package provides an easy to use interface for performing tensor network simulations on matrix product states (MPS) and tree tensor network (TTN) states. Written in the Julia programming language, MPSDynamics.jl is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle (TDVP) method for time evolution.  The package also provides strong support for the measurement of observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics.  The package has been developed with the aim of studying non-Markovian open system dynamics at finite temperature using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm (T-TEDOPA) based on environment chain mapping. However the methods implemented can equally be applied to other areas of physics.","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"warning: Warning\nThe documentation is currently undergoing massive restructurations/improvement. It's a work in progress until the next release scheduled for May, 2024.","category":"page"},{"location":"#Installation","page":"Introduction","title":"Installation","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"The package may be installed by typing the following into a Julia REPL","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"    ] add https://github.com/shareloqs/MPSDynamics.git","category":"page"},{"location":"#Table-of-Contents","page":"Introduction","title":"Table of Contents","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"Pages = [\"index.md\", \"user-guide.md\", \"examples/sbm.md\", \"examples/puredephasing.md\", \"examples/timedep.md\", \"examples/anderson-model.md\", \"examples/bath-observables.md\", \"theory.md\", \"methods.md\", \"dev.md\"]\nDepth = 3","category":"page"},{"location":"#Citation","page":"Introduction","title":"Citation","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"If you use the package in your research, please consider citing it. You can add the Zenodo record to your BibTex file:","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"@misc{mpsdynamics_zenodo2021,\n\ttitle = {shareloqs/{MPSDynamics}},\n\tshorttitle = {{MPSDynamics.jl}},\n\turl = {https://zenodo.org/record/5106435},\n\tabstract = {Tensor network simulations for finite temperature, open quantum system dynamics},\n\tpublisher = {Zenodo},\n\tauthor = {Dunnett, Angus and Lacroix, Thibaut and Le Dé, Brieuc and Riva, Angela},\n\tyear = {2021},\n\tdoi = {10.5281/zenodo.5106435},\n}","category":"page"}]
 }
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href="../examples/protontransfer/">Proton Transfer Model</a></li></ul></li><li class="is-active"><a class="tocitem" href>Theoretical Background</a><ul class="internal"><li><a class="tocitem" href="#Chain-Mapping-of-bosonic-environments"><span>Chain-Mapping of bosonic environments</span></a></li><li><a class="tocitem" href="#Computation-of-the-chain-coefficients"><span>Computation of the chain coefficients</span></a></li><li><a class="tocitem" href="#Tensor-Networks"><span>Tensor Networks</span></a></li><li><a class="tocitem" href="#Time-Dependent-Variational-Principle"><span>Time-Dependent Variational Principle</span></a></li><li><a class="tocitem" href="#Bibliography"><span>Bibliography</span></a></li></ul></li><li><a class="tocitem" href="../methods/">Methods</a></li><li><a class="tocitem" href="../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Theoretical Background</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Theoretical Background</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Theoretical-Background"><a class="docs-heading-anchor" href="#Theoretical-Background">Theoretical Background</a><a id="Theoretical-Background-1"></a><a class="docs-heading-anchor-permalink" href="#Theoretical-Background" title="Permalink"></a></h1><h2 id="Chain-Mapping-of-bosonic-environments"><a class="docs-heading-anchor" href="#Chain-Mapping-of-bosonic-environments">Chain-Mapping of bosonic environments</a><a id="Chain-Mapping-of-bosonic-environments-1"></a><a class="docs-heading-anchor-permalink" href="#Chain-Mapping-of-bosonic-environments" title="Permalink"></a></h2><p>We consider, in the Schrödinger picture, a general Hamiltonian where a non-specified system interacts linearly with a bosonic environments (<span>$\hbar = 1$</span>)</p><p class="math-container">\[\begin{aligned}
-    \hat{H} =&amp; \hat{H}_S + \int_0^{+\infty} \omega\hat{a}^\dagger_\omega\hat{a}_\omega\ \mathrm{d}\omega + \hat{A}_S \otimes \int_0^{+\infty}\sqrt{J(\omega)}\left(\hat{a}_\omega + \hat{a}^\dagger_\omega\right)\mathrm{d}\omega
-\end{aligned}\]</p><p>where <span>$\hat a_\omega$</span> (<span>$\hat a^\dagger_\omega$</span>) is a bosonic annihilation (creation) operator for a normal mode of the environment of energy <span>$\omega$</span>, <span>$\hat{A}_S$</span> is a system operator, and <span>$J(\omega) = \sum_k |g_k|^2\delta(\omega - \omega_k)$</span> is the bath spectral density (SD), defined with the microscopic system-environment coupling strength <span>$g_k$</span>. The SD quantifies the coupling strengths of the different normal modes of the environment with the system. Any SD that is not flat corresponds to a non-Markovian environment.</p><h3 id="Zero-Temperature"><a class="docs-heading-anchor" href="#Zero-Temperature">Zero Temperature</a><a id="Zero-Temperature-1"></a><a class="docs-heading-anchor-permalink" href="#Zero-Temperature" title="Permalink"></a></h3><p>Let us consider the Hamiltonian presented in Eq.(1). To investigate its rich phenomenology, using the powerful tools provided by tensor network techniques, it is convenient to map the system and its environment from a star-like to a chain-like configuration, with local interactions. To do so, we introduce a unitary transformation of the continuous normal modes <span>$\hat{a}_\omega$</span> to an infinite discrete set of interacting modes <span>$\hat{b}_n$</span><sup class="footnote-reference"><a id="citeref-chin_exact_2010" href="#footnote-chin_exact_2010">[chin_exact_2010]</a></sup>.</p><p class="math-container">\[    \hat{a}_\omega = \sum_{n=0}^{+\infty} U_n(\omega)\hat{b}_n = \sum_{n=0}^{+\infty} \sqrt{J(\omega)}P_n(\omega)\hat{b}_n\ ,\]</p><p>where <span>$P_n(\omega)$</span> are orthonormal polynomials such that</p><p class="math-container">\[    \int_{0}^{+\infty}P_n(\omega)P_m(\omega)J(\omega)\mathrm{d}\omega = \delta_{n,m}\ ;\]</p><p>and the inverse transformation is</p><p class="math-container">\[    \hat{b}_n = \int_0^{+\infty} U^*_n(\omega)\hat{a}_\omega\mathrm{d}\omega\ .\]</p><p>Note that the orthonormality of the polynomials ensures the unitarity of the transformation defined in Eq.(2). The mapping from a continuous set of modes to a (still infinite) discrete set might seem counter-intuitive, however it is a direct consequence of the separability of the underlying Hilbert space.</p><p>Under this transformation, the Hamiltonian in Eq.(1) becomes</p><p class="math-container">\[    \hat{H}= \hat{H}_S + \sum_{n=0}^{+\infty}\varepsilon_n\hat{b}_n^\dagger\hat{b}_n + t_n(\hat{b}_{n+1}^\dagger\hat{b}_n + \mathrm{h.c.}) + c_0 \hat{A}_S(\hat{b}_0 + \hat{b}_0^\dagger)\ .\]</p><p>Hence, this mapping transforms the normal bath Hamiltonian into a tight-binding Hamiltonian with on-site energies <span>$\varepsilon_n$</span> and hopping energies <span>$t_n$</span>. Another important consequence of this mapping is that now the system only interacts with the first mode <span>$n = 0$</span> of the chain-mapped environment. The chain coefficients <span>$\varepsilon_n$</span>, <span>$t_n$</span>, and the coupling <span>$c_0$</span> depend solely on the SD.</p><p>This makes chain mapping a tool of choice for describing systems coupled to environment with highly structured SD (e.g. experimentally measured or calculated <em>ab initio</em>)<sup class="footnote-reference"><a id="citeref-chin_role_2013" href="#footnote-chin_role_2013">[chin_role_2013]</a></sup><sup class="footnote-reference"><a id="citeref-alvertis_nonequilibrium_2019" href="#footnote-alvertis_nonequilibrium_2019">[alvertis_nonequilibrium_2019]</a></sup><sup class="footnote-reference"><a id="citeref-dunnett_influence_2021" href="#footnote-dunnett_influence_2021">[dunnett_influence_2021]</a></sup><sup class="footnote-reference"><a id="citeref-caycedosoler_exact_2022" href="#footnote-caycedosoler_exact_2022">[caycedosoler_exact_2022]</a></sup>. In this new representation, the Hamiltonian in Eq.(5) has naturally a 1D chain topology. This makes its representation as a Matrix Product Operator (MPO) and the representation of the joint {System + Environment} wave-function as a Matrix Product State (MPS) suited <sup class="footnote-reference"><a id="citeref-orus_practical_2014" href="#footnote-orus_practical_2014">[orus_practical_2014]</a></sup><sup class="footnote-reference"><a id="citeref-paeckel_timeevolution_2019" href="#footnote-paeckel_timeevolution_2019">[paeckel_timeevolution_2019]</a></sup>.</p><p>The orthogonal polynomial-based chain mapping and the subsequent representation of the joint wave-function as a MPS (and the operators as MPO) are the building blocks of the Time-dependent Density operator with Orthonormal Polynomials Algorithm (TEDOPA) one of the state-of-the-art numerically exact method to simulate the dynamics of open quantum systems especially in the non-Markovian, non-perturbative regimes both at zero and finite temperatures <sup class="footnote-reference"><a id="citeref-prior_efficient_2010" href="#footnote-prior_efficient_2010">[prior_efficient_2010]</a></sup><sup class="footnote-reference"><a id="citeref-woods_simulating_2015" href="#footnote-woods_simulating_2015">[woods_simulating_2015]</a></sup><sup class="footnote-reference"><a id="citeref-tamascelli_efficient_2019" href="#footnote-tamascelli_efficient_2019">[tamascelli_efficient_2019]</a></sup><sup class="footnote-reference"><a id="citeref-dunnett_simulating_2021" href="#footnote-dunnett_simulating_2021">[dunnett_simulating_2021]</a></sup><sup class="footnote-reference"><a id="citeref-lacroix_unveiling_2021" href="#footnote-lacroix_unveiling_2021">[lacroix_unveiling_2021]</a></sup>.</p><h3 id="Finite-Temperature-with-T-TEDOPA"><a class="docs-heading-anchor" href="#Finite-Temperature-with-T-TEDOPA">Finite Temperature with T-TEDOPA</a><a id="Finite-Temperature-with-T-TEDOPA-1"></a><a class="docs-heading-anchor-permalink" href="#Finite-Temperature-with-T-TEDOPA" title="Permalink"></a></h3><p>Assuming a unitary evolution for both the system and environment, the system&#39;s dynamics can be isolated by tracing out the environmental degrees of freedom. The density operator for the system at time <span>$t$</span> is described as</p><p class="math-container">\[\hat{\rho}_S(t) = \text{T}r_E\left\{\hat{U}(t) \hat{\rho}_S(0) \otimes \hat{\rho}_E(0) \hat{U}^\dagger(t)\right\}.\]</p><p>Initially, the system state, <span>$\hat{\rho}_S(0)$</span>, can be pure or mixed, and the environment is in a thermal state defined by the inverse temperature <span>$\beta = (k_B T)^{-1}$</span>. This state is represented by a product of Gaussian states</p><p class="math-container">\[\hat{\rho}_E(0) = \bigotimes_\omega \frac{e^{-\beta \omega \hat{a}_\omega^\dagger \hat{a}_\omega}}{Z_\omega(\beta)} .\]</p><p>The system&#39;s evolution is dictated by the environment&#39;s two-time correlation function <span>$S(t)$</span>, which in turn is determined by the spectral density function <span>$J$</span> and the temperature <span>$\beta$</span> via the Bose-Einstein distribution <span>$n_\omega(\beta)$</span></p><p class="math-container">\[S(t) = \int_0^\infty d\omega J(\omega)\left[e^{-i\omega t}(1 + n_\omega(\beta)) + e^{i\omega t} n_\omega(\beta)\right] .\]</p><p>To simulate finite temperature effects using a zero-temperature model with the T-TEDOPA method <sup class="footnote-reference"><a id="citeref-tamascelli_efficient_2019" href="#footnote-tamascelli_efficient_2019">[tamascelli_efficient_2019]</a></sup>, we extend the spectral density function to cover both positive and negative frequencies, allowing us to use a pure state description for the environment. This is achieved by defining a new spectral density function <span>$J(\omega, \beta)$</span> that incorporates the Boltzmann factors, supporting the entire real axis</p><p class="math-container">\[J(\omega, \beta) = \frac{\text{sign}(\omega)J(\left|\omega\right|)}{2} \Big(1 + \coth\Big(\frac{\beta \omega}{2}\Big)\Big).\]</p><p>This modified bath has the same correlation function <span>$S(t)$</span> and thus allows us to maintain a pure state description of the environment, represented as a vacuum state, and avoid the computational complexities of density matrices</p><p class="math-container">\[\left|\text{vac}\right\rangle = \bigotimes_\omega \left|0_\omega\right\rangle .\]</p><p>The Hamiltonian of the system interacting with this extended bath now includes temperature-dependent interactions:</p><p class="math-container">\[\hat{H} = \hat{H}_S + \int_{-\infty}^{+\infty} \omega \hat{a}_\omega^\dagger \hat{a}_\omega \mathrm{d}\omega + \hat{A}_S \otimes \int_{-\infty}^{+\infty} \sqrt{J(\omega,\beta)}\left(\hat{a}_\omega^\dagger+\hat{a}_\omega\right) \mathrm{d}\omega,\]</p><p>This method simplifies the simulation of finite temperature effects by treating them within an effective zero-temperature framework, thereby keeping the computational advantages of using pure states. In conclusion: the dynamics of the system resulting from the interaction with the original bath, starting in a thermal state at finite temperature, is exactly the same as the one resulting from the interaction with the extended environment, starting in the vacuum state at zero temperature. Once computed the chain coefficients at a given inverse temperature <span>$\beta$</span>, the time evolution of the vacuum state interacting with the extended environment can be efficiently simulated using MPS time evolution methods.</p><h3 id="Finite-temperature-with-the-thermofield-transformation"><a class="docs-heading-anchor" href="#Finite-temperature-with-the-thermofield-transformation">Finite temperature with the thermofield transformation</a><a id="Finite-temperature-with-the-thermofield-transformation-1"></a><a class="docs-heading-anchor-permalink" href="#Finite-temperature-with-the-thermofield-transformation" title="Permalink"></a></h3><p>Finite temperature initial states of the system are challenging: in the previous section we have seen that a very efficient solution is to reabsorb the effect of the temperature inside of the spectral density function that characterizes the environment. De Vega and Bañuls present<sup class="footnote-reference"><a id="citeref-devega_thermo_2015" href="#footnote-devega_thermo_2015">[devega_thermo_2015]</a></sup> another suggestive approach based on the idea of the thermofield transformation. The idea is to double the environmental degrees of freedom and then to apply a Bogoliubov transformation: the real environment in a thermal state is transformed into two virtual environments in the vacuum state, defined as the <em>thermofield vacuum</em>. For any operator of the real environment, the expectation values in the thermal state are equivalent to those calculated in the thermofield vacuum. Behind the thermofield approach there is the concept of purification: the initial mixed state of the environment can be represented as the partial trace of a pure state (the thermofield vacuum) defined on the larger Hilbert space of the modes of the two virtual environments at zero temperature. The interaction of the system and each one of the two environments is then mapped on two separate chains, using the TEDOPA chain mapping to define a unitary transformation that can be applied to the bath modes, in order to define new modes for the environment, mapping it from a star-like to a one dimensional chain-like configuration, which is very well suited for the application of tensor network techniques.</p><p>In the thermofield approach, the first step is to introduce the auxiliary environment <span>$E^{\text{aux}}$</span> of non-interacting bosonic modes of negative frequencies:</p><p class="math-container">\[    \hat H^\text{aux} = \hat H - \sum_k \omega_k \hat c_k^\dagger \hat c_k,\]</p><p>where the Hamiltonian <span>$\hat H$</span> is the bosonic Hamiltonian defined in Eq. \ref{eq:bosonic_ham}. The two environments, of positive and negative frequencies, are assumed to be in a thermal environment at inverse temperature <span>$\beta$</span>; the second step is to apply a thermal Bogoliubov transformation to change the basis. The applied transformation produces two-modes squeezed states:</p><p class="math-container">\[         \hat a_{1k}=e^{-iG}  \hat b_k e^{iG}= \cosh(\theta_k) \hat b_k -\sinh(\theta_k)  \hat c_k^\dagger \\
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Theoretical Background · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script 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nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li class="is-active"><a class="tocitem" href>Theoretical Background</a><ul class="internal"><li><a class="tocitem" href="#Chain-Mapping-of-bosonic-environments"><span>Chain-Mapping of bosonic environments</span></a></li><li><a class="tocitem" href="#Computation-of-the-chain-coefficients"><span>Computation of the chain coefficients</span></a></li><li><a class="tocitem" href="#Tensor-Networks"><span>Tensor Networks</span></a></li><li><a class="tocitem" href="#Time-Dependent-Variational-Principle"><span>Time-Dependent Variational Principle</span></a></li><li><a class="tocitem" href="#Bibliography"><span>Bibliography</span></a></li></ul></li><li><a class="tocitem" href="../methods/">Methods</a></li><li><a class="tocitem" href="../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Theoretical Background</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Theoretical Background</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/theory.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Theoretical-Background"><a class="docs-heading-anchor" href="#Theoretical-Background">Theoretical Background</a><a id="Theoretical-Background-1"></a><a class="docs-heading-anchor-permalink" href="#Theoretical-Background" title="Permalink"></a></h1><h2 id="Chain-Mapping-of-bosonic-environments"><a class="docs-heading-anchor" href="#Chain-Mapping-of-bosonic-environments">Chain-Mapping of bosonic environments</a><a id="Chain-Mapping-of-bosonic-environments-1"></a><a class="docs-heading-anchor-permalink" href="#Chain-Mapping-of-bosonic-environments" title="Permalink"></a></h2><p>We consider, in the Schrödinger picture, a general Hamiltonian where a non-specified system interacts linearly with a bosonic environments</p><p class="math-container">\[\begin{aligned}
+    \hat{H} =&amp; \hat{H}_S + \int_0^{+\infty} \hbar\omega\hat{a}^\dagger_\omega\hat{a}_\omega\ \mathrm{d}\omega + \hat{A}_S\int_0^{+\infty}\sqrt{J(\omega)}\left(\hat{a}_\omega + \hat{a}^\dagger_\omega\right)\mathrm{d}\omega
+\end{aligned}\]</p><p>where <span>$\hat a_\omega$</span> (<span>$\hat a^\dagger_\omega$</span>) is a bosonic annihilation (creation) operator for a normal mode of the environment of energy <span>$\hbar\omega$</span>, <span>$\hat{A}_S$</span> is a system operator, and <span>$J(\omega) = \sum_k |g_k|^2\delta(\omega - \omega_k)$</span> is the bath spectral density (SD), defined with the microscopic system-environment coupling strength <span>$g_k$</span>. The SD quantifies the coupling strengths of the different normal modes of the environment with the system. Any SD that is not flat corresponds to a non-Markovian environment.</p><h3 id="Zero-Temperature"><a class="docs-heading-anchor" href="#Zero-Temperature">Zero Temperature</a><a id="Zero-Temperature-1"></a><a class="docs-heading-anchor-permalink" href="#Zero-Temperature" title="Permalink"></a></h3><p>Let us consider the Hamiltonian presented in Eq.(1). To investigate its rich phenomenology, using the powerful tools provided by tensor network techniques, it is convenient to map the system and its environment from a star-like to a chain-like configuration, with local interactions. To do so, we introduce a unitary transformation of the continuous normal modes <span>$\hat{a}_\omega$</span> to an infinite discrete set of interacting modes <span>$\hat{b}_n$</span><sup class="footnote-reference"><a id="citeref-chin_exact_2010" href="#footnote-chin_exact_2010">[chin_exact_2010]</a></sup>.</p><p class="math-container">\[    \hat{a}_\omega = \sum_{n=0}^{+\infty} U_n(\omega)\hat{b}_n = \sum_{n=0}^{+\infty} \sqrt{J(\omega)}P_n(\omega)\hat{b}_n\ ,\]</p><p>where <span>$P_n(\omega)$</span> are orthonormal polynomials such that</p><p class="math-container">\[    \int_{0}^{+\infty}P_n(\omega)P_m(\omega)J(\omega)\mathrm{d}\omega = \delta_{n,m}\ ;\]</p><p>and the inverse transformation is</p><p class="math-container">\[    \hat{b}_n = \int_0^{+\infty} U_n(\omega)\hat{a}_\omega\mathrm{d}\omega\ .\]</p><p>Note that the orthonormality of the polynomials ensures the unitarity of the transformation defined in Eq.(2). The mapping from a continuous set of modes to a (still infinite) discrete set might seem counter-intuitive, however it is a direct consequence of the separability of the underlying Hilbert space.</p><p>Under this transformation, the Hamiltonian in Eq.(1) becomes</p><p class="math-container">\[    \hat{H}= \hat{H}_S + \sum_{n=0}^{+\infty}\varepsilon_n\hat{b}_n^\dagger\hat{b}_n + t_n(\hat{b}_{n+1}^\dagger\hat{b}_n + \mathrm{h.c.}) + \kappa\hat{A}_S(\hat{b}_0 + \hat{b}_0^\dagger)\ .\]</p><p>Hence, this mapping transforms the normal bath Hamiltonian into a tight-binding Hamiltonian with on-site energies <span>$\varepsilon_n$</span> and hopping energies <span>$t_n$</span>. Another important consequence of this mapping is that now the system only interacts with the first mode <span>$n = 0$</span> of the chain-mapped environment. The chain coefficients <span>$\varepsilon_n$</span>, <span>$t_n$</span>, and the coupling <span>$\kappa$</span> depend solely on the SD.</p><p>This makes chain mapping a tool of choice for describing systems coupled to environment with highly structured SD (e.g. experimentally measured or calculated <em>ab initio</em>)<sup class="footnote-reference"><a id="citeref-chin_role_2013" href="#footnote-chin_role_2013">[chin_role_2013]</a></sup><sup class="footnote-reference"><a id="citeref-alvertis_nonequilibrium_2019" href="#footnote-alvertis_nonequilibrium_2019">[alvertis_nonequilibrium_2019]</a></sup><sup class="footnote-reference"><a id="citeref-dunnett_influence_2021" href="#footnote-dunnett_influence_2021">[dunnett_influence_2021]</a></sup><sup class="footnote-reference"><a id="citeref-caycedosoler_exact_2022" href="#footnote-caycedosoler_exact_2022">[caycedosoler_exact_2022]</a></sup>. In this new representation, the Hamiltonian in Eq.(5) has naturally a 1D chain topology. This makes its representation as a Matrix Product Operator (MPO) and the representation of the joint {System + Environment} wave-function as a Matrix Product State (MPS) suited <sup class="footnote-reference"><a id="citeref-orus_practical_2014" href="#footnote-orus_practical_2014">[orus_practical_2014]</a></sup><sup class="footnote-reference"><a id="citeref-paeckel_timeevolution_2019" href="#footnote-paeckel_timeevolution_2019">[paeckel_timeevolution_2019]</a></sup>.</p><p>The orthogonal polynomial-based chain mapping and the subsequent representation of the joint wave-function as a MPS (and the operators as MPO) are the building blocks of the Time-dependent Density operator with Orthonormal Polynomials Algorithm (TEDOPA) one of the state-of-the-art numerically exact method to simulate the dynamics of open quantum systems especially in the non-Markovian, non-perturbative regimes both at zero and finite temperatures <sup class="footnote-reference"><a id="citeref-prior_efficient_2010" href="#footnote-prior_efficient_2010">[prior_efficient_2010]</a></sup><sup class="footnote-reference"><a id="citeref-woods_simulating_2015" href="#footnote-woods_simulating_2015">[woods_simulating_2015]</a></sup><sup class="footnote-reference"><a id="citeref-tamascelli_efficient_2019" href="#footnote-tamascelli_efficient_2019">[tamascelli_efficient_2019]</a></sup><sup class="footnote-reference"><a id="citeref-dunnett_simulating_2021" href="#footnote-dunnett_simulating_2021">[dunnett_simulating_2021]</a></sup><sup class="footnote-reference"><a id="citeref-lacroix_unveiling_2021" href="#footnote-lacroix_unveiling_2021">[lacroix_unveiling_2021]</a></sup>.</p><h3 id="Finite-Temperature-with-T-TEDOPA"><a class="docs-heading-anchor" href="#Finite-Temperature-with-T-TEDOPA">Finite Temperature with T-TEDOPA</a><a id="Finite-Temperature-with-T-TEDOPA-1"></a><a class="docs-heading-anchor-permalink" href="#Finite-Temperature-with-T-TEDOPA" title="Permalink"></a></h3><p>Assuming a unitary evolution for both the system and environment, the system&#39;s dynamics can be isolated by tracing out the environmental degrees of freedom. The density operator for the system at time <span>$t$</span> is described as</p><p class="math-container">\[\hat{\rho}_S(t) = \text{T}r_E\left\{\hat{U}(t) \hat{\rho}_S(0) \otimes \hat{\rho}_E(0) \hat{U}^\dagger(t)\right\}.\]</p><p>Initially, the system state, <span>$\hat{\rho}_S(0)$</span>, can be pure or mixed, and the environment is in a thermal state defined by the inverse temperature <span>$\beta = (k_B T)^{-1}$</span>. This state is represented by a product of Gaussian states</p><p class="math-container">\[\hat{\rho}_E(0) = \bigotimes_\omega \frac{e^{-\beta \omega \hat{a}_\omega^\dagger \hat{a}_\omega}}{Z_\omega(\beta)} .\]</p><p>The system&#39;s evolution is dictated by the environment&#39;s two-time correlation function <span>$S(t)$</span>, which in turn is determined by the spectral density function <span>$J$</span> and the temperature <span>$\beta$</span> via the Bose-Einstein distribution <span>$n_\omega(\beta)$</span></p><p class="math-container">\[S(t) = \int_0^\infty d\omega J(\omega)\left[e^{-i\omega t}(1 + n_\omega(\beta)) + e^{i\omega t} n_\omega(\beta)\right] .\]</p><p>To simulate finite temperature effects using a zero-temperature model with the T-TEDOPA method <sup class="footnote-reference"><a id="citeref-tamascelli_efficient_2019" href="#footnote-tamascelli_efficient_2019">[tamascelli_efficient_2019]</a></sup>, we extend the spectral density function to cover both positive and negative frequencies, allowing us to use a pure state description for the environment. This is achieved by defining a new spectral density function <span>$J(\omega, \beta)$</span> that incorporates the Boltzmann factors, supporting the entire real axis</p><p class="math-container">\[J(\omega, \beta) = \frac{\text{sign}(\omega)J(\left|\omega\right|)}{2} \Big(1 + \coth\Big(\frac{\beta \omega}{2}\Big)\Big).\]</p><p>This modified bath has the same correlation function <span>$S(t)$</span> and thus allows us to maintain a pure state description of the environment, represented as a vacuum state, and avoid the computational complexities of density matrices</p><p class="math-container">\[\left|\text{vac}\right\rangle = \bigotimes_\omega \left|0_\omega\right\rangle .\]</p><p>The Hamiltonian of the system interacting with this extended bath now includes temperature-dependent interactions:</p><p class="math-container">\[\hat{H} = \hat{H}_S + \int_{-\infty}^{+\infty} d\omega \omega \hat{a}_\omega^\dagger \hat{a}_\omega + \hat{A}_S \otimes \int_{-\infty}^{+\infty} d\omega \sqrt{J(\omega,\beta)}\left(\hat{a}_\omega^\dagger+\hat{a}_\omega\right),\]</p><p>This method simplifies the simulation of finite temperature effects by treating them within an effective zero-temperature framework, thereby keeping the computational advantages of using pure states. In conclusion: the dynamics of the system resulting from the interaction with the original bath, starting in a thermal state at finite temperature, is exactly the same as the one resulting from the interaction with the extended environment, starting in the vacuum state at zero temperature. Once computed the chain coefficients at a given inverse temperature <span>$\beta$</span>, the time evolution of the vacuum state interacting with the extended environment can be efficiently simulated using MPS time evolution methods.</p><h3 id="Finite-temperature-with-the-thermofield-transformation"><a class="docs-heading-anchor" href="#Finite-temperature-with-the-thermofield-transformation">Finite temperature with the thermofield transformation</a><a id="Finite-temperature-with-the-thermofield-transformation-1"></a><a class="docs-heading-anchor-permalink" href="#Finite-temperature-with-the-thermofield-transformation" title="Permalink"></a></h3><p>Finite temperature initial states of the system are challenging: in the previous section we have seen that a very efficient solution is to reabsorb the effect of the temperature inside of the spectral density function that characterizes the environment. De Vega and Bañuls present<sup class="footnote-reference"><a id="citeref-devega_thermo_2015" href="#footnote-devega_thermo_2015">[devega_thermo_2015]</a></sup> another suggestive approach based on the idea of the thermofield transformation. The idea is to double the environmental degrees of freedom and then to apply a Bogoliubov transformation: the real environment in a thermal state is transformed into two virtual environments in the vacuum state, defined as the <em>thermofield vacuum</em>. For any operator of the real environment, the expectation values in the thermal state are equivalent to those calculated in the thermofield vacuum. Behind the thermofield approach there is the concept of purification: the initial mixed state of the environment can be represented as the partial trace of a pure state (the thermofield vacuum) defined on the larger Hilbert space of the modes of the two virtual environments at zero temperature. The interaction of the system and each one of the two environments is then mapped on two separate chains, using the TEDOPA chain mapping to define a unitary transformation that can be applied to the bath modes, in order to define new modes for the environment, mapping it from a star-like to a one dimensional chain-like configuration, which is very well suited for the application of tensor network techniques.</p><p>In the thermofield approach, the first step is to introduce the auxiliary environment <span>$E^{\text{aux}}$</span> of non-interacting bosonic modes of negative frequencies:</p><p class="math-container">\[    \hat H^\text{aux} = \hat H - \sum_k \omega_k \hat c_k^\dagger \hat c_k,\]</p><p>where the Hamiltonian <span>$\hat H$</span> is the bosonic Hamiltonian defined in Eq. \ref{eq:bosonic_ham}. The two environments, of positive and negative frequencies, are assumed to be in a thermal environment at inverse temperature <span>$\beta$</span>; the second step is to apply a thermal Bogoliubov transformation to change the basis. The applied transformation produces two-modes squeezed states:</p><p class="math-container">\[         \hat a_{1k}=e^{-iG}  \hat b_k e^{iG}= \cosh(\theta_k) \hat b_k -\sinh(\theta_k)  \hat c_k^\dagger \\
          \hat a_{2k}=e^{-iG} \hat c_k e^{iG}= \cosh(\theta_k)  \hat c_k -\sinh(\theta_k)  \hat b_k^\dagger,\]</p><p>where the exponent of the squeeze operator is <span>$G = i \sum_k \theta_k(\hat b_k^\dagger \hat c_k^\dagger-\hat c_k \hat b_k)$</span>, and <span>$\theta_k$</span> is dependent on the temperature as in the following relations, where the number of excitations in the <span>$k$</span>-th mode is <span>$n_k = 1/(e^{\beta \omega_k}-1)$</span>:</p><p class="math-container">\[         \cosh(\theta_k) = \sqrt{1+n_k} = \sqrt{\frac{1}{1-e^{-\beta \omega_k}}} \\
-         \sinh(\theta_k) =\quad\sqrt{n_k}\quad= \sqrt{\frac{1}{e^{\beta \omega_k}-1}}.\]</p><p>The Bogoliubov transformation defines a new squeezed vacuum state, which we write in terms of the vacuum state <span>$| \Omega_0 \rangle$</span> of the operators <span>$\hat b_k$</span>, <span>$\hat c_k$</span>:</p><p class="math-container">\[    |\Omega\rangle = e^{iG} |\Omega_0\rangle, \quad \text{such that: }\quad \hat a_{1k} |\Omega\rangle = 0,  \hat a_{2k} |\Omega\rangle = 0.\]</p><p>From the vacuum state, we can obtain the thermal state of the original environment:</p><p class="math-container">\[    \hat \rho_E = \text{Tr}_{\text{aux}}\{ |\Omega\rangle\langle\Omega| \},\]</p><p>and it can be now used as pure an initial state for both of the environments. Moreover, the expectation value on <span>$| \Omega \rangle$</span> of the number of physical modes <span>$\hat n_k$</span> does not vanish:</p><p class="math-container">\[    n_k= \langle\Omega| \hat b_k^\dagger \hat b_k |\Omega\rangle = \sinh^2(\theta_k).\]</p><p>Therefore, solving the dynamics given by the original Hamiltonian <span>$\hat H$</span>, starting from the initial condition <span>$\hat \rho_S(0) \otimes \hat \rho_E(\beta)$</span>, is equivalent to solving the dynamics given by the following Hamiltonian:</p><p class="math-container">\[       \hat H = \hat H_S +\hat H_E +\hat H_I   \\
-     \hat H  = \overbrace{\hat A_S}^{\hat H_S} + \overbrace{\sum_k \omega_k \big(\hat a_{1k}^\dagger \hat a_{1k} - \hat a_{2k}^\dagger \hat a_{2k} \big)}^{\hat H_E} + \overbrace{\hat L_S \otimes \sum_k g_{1k}(\hat a_{1k}^\dagger + \hat a_{1k})+\hat L_S \otimes \sum_k g_{2k}(\hat a_{2k}^\dagger + \hat a_{2k})}^{\hat H_I}.\]</p><p>where <span>$\hat L_S = \hat L_S^\dagger$</span>, considering <span>$\hat \rho_S(0) \otimes |\Omega \rangle \langle \Omega |$</span> as the initial state of system and environment. It is this Hamiltonian that is mapped on two chains with TEDOPA, to be able to perform the time evolution using tensor network techniques. </p><h2 id="Computation-of-the-chain-coefficients"><a class="docs-heading-anchor" href="#Computation-of-the-chain-coefficients">Computation of the chain coefficients</a><a id="Computation-of-the-chain-coefficients-1"></a><a class="docs-heading-anchor-permalink" href="#Computation-of-the-chain-coefficients" title="Permalink"></a></h2><p>A useful property of the orthonormal polynomials is that they obey a recurrence relation</p><p class="math-container">\[    P_n(\omega) = (C_{n-1}\omega - A_{n-1})P_{n-1}(\omega) + B_{n-1}P_{n-2}(\omega)\ ,\]</p><p>where <span>$A_n$</span> is related to the first moment of <span>$P_n$</span>, <span>$B_n$</span> and <span>$C_n$</span> to the norms of <span>$P_n$</span> and <span>$P_{n-1}$</span><sup class="footnote-reference"><a id="citeref-appel_mathematics_2007" href="#footnote-appel_mathematics_2007">[appel_mathematics_2007]</a></sup>. This recurrence relation can be used to construct the polynomials with the conditions that <span>$P_0(\omega) = ||p_0||^{-1} = \left(\int_{\mathbb{R}^{+}} J(\omega)\mathrm{d}\omega \right)^{-\frac{1}{2}}$</span> and <span>$P_{-1}(\omega) = 0$</span>, with <span>$||\bullet||$</span> the norm of <span>$\bullet$</span> with respect to the measure <span>$J(\omega)$</span>, and <span>$P_n(\omega) = p_n(\omega)||p_n||^{-1}$</span> ; where the polynomials <span>$\{p_n\}_{n\in\mathbb{N}}$</span> are the so called <em>monic polynomials</em> where the factor <span>$a_n$</span> in front of <span>$\omega^{n}$</span> is equal to 1.</p><p>The energy of the chain mode <span>$n$</span> is given by <span>$\varepsilon_n = A_n C_n^{-1}$</span> and <span>$t_n=C_n^{-1}$</span> is the coupling between mode <span>$n$</span> and <span>$n+1$</span><sup class="footnote-reference"><a id="citeref-chin_exact_2010" href="#footnote-chin_exact_2010">[chin_exact_2010]</a></sup>.</p><p>The system couples <em>only</em> to the first mode with the coupling strength <span>$c_0 = ||p_0||$</span>.</p><p>Explain that for some weight function/SD they are known analytically and that for others we can use the build-in routines inspired by Gautschi or the PolyChaos.jl package.</p><h2 id="Tensor-Networks"><a class="docs-heading-anchor" href="#Tensor-Networks">Tensor Networks</a><a id="Tensor-Networks-1"></a><a class="docs-heading-anchor-permalink" href="#Tensor-Networks" title="Permalink"></a></h2><p>A multipartite quantum state <span>$|\psi\rangle$</span>, e.g. a <span>$N$</span>-site system where the sites can each be in a state <span>$|\phi_i\rangle$</span> belonging to a <span>$d$</span>-dimensional Hilbert space, can be written as follows</p><p class="math-container">\[    |\psi\rangle = \sum_{\{i_k\}}c_{i_1\ldots i_N}|\phi_{i_1}\rangle\otimes\ldots\otimes|\phi_{i_N}\rangle\ ,\]</p><p>where the complex numbers <span>$c_{i_1\ldots i_N}$</span> are the amplitudes of each state <span>$|\phi_{i_1}\rangle\otimes\ldots\otimes|\phi_{i_N}\rangle$</span> whose superpositions form in full generality the state <span>$|\psi\rangle$</span>. Thus the state <span>$|\psi\rangle$</span> can be completely represented by a rank-<span>$N$</span> tensor <span>$c$</span> that is the collection of all possible amplitudes <span>$c_{i_1\ldots i_N}$</span>. Here by the rank of a tensor, we simply mean the number of indices it has.</p><h3 id="MPS"><a class="docs-heading-anchor" href="#MPS">MPS</a><a id="MPS-1"></a><a class="docs-heading-anchor-permalink" href="#MPS" title="Permalink"></a></h3><p>The tensor <span>$c$</span> of a quantum state <span>$|\psi\rangle$</span> corresponding to a one-dimensional system can be decomposed into a product of <span>$N$</span> smaller rank-3 tensors <span>$T_{k}$</span> (except for the first and last sites where the tensors will have a rank-2)</p><p class="math-container">\[    c_{i_1\ldots i_N} = \sum_{\{\alpha\}} T^{\alpha_1}_{i_1}T^{\alpha_1\alpha_2\ }_{i_2}T^{\alpha_2\alpha_3\ }_{i_3}\ldots T^{\alpha_{N-1}}_{i_N} \ .\]</p><p>In this form, the local tensor <span>$T_k$</span> contains the information on the quantum state on site <span>$k$</span> and its relation (especially the entanglement) with the neighbouring sites.</p><p>The decomposition of the tensor of the amplitudes of a quantum state into a product of smaller rank tensors is called a <strong>Matrix Product State</strong> decomposition.</p><p>The contracted indices <span>$\alpha_k$</span> between the tensors are called <em>virtual indices</em> and carry information about the correlations between bi-partitions of the state at bond <span>$k$</span>. The number of different values a virtual index can take is called the <em>bond dimension</em> and is denoted <span>$D$</span>. The free indices <span>$i_k$</span> associated with local quantum states are called <em>physical indices</em>. Thus, they can take <span>$d$</span> values (with <span>$d$</span> the dimension of the local Hilbert space).</p><p>Any state in the Hilbert space of a one-dimensional many-body system can in principle be represented by a MPS by choosing a sufficiently large value for the bond dimension <span>$D$</span> <sup class="footnote-reference"><a id="citeref-Orus" href="#footnote-Orus">[Orus]</a></sup>. On top of this intellectually satisfying property of MPSs being a dense set of states for a 1d-system, they can also be used as a practical Ansätze for a many-body quantum states by setting a maximal allowed value <span>$\chi$</span> for the bond dimension <span>$D$</span>. In doing so, we restrict ourselves to a corner of the total Hilbert space. The rationale behind this Ansatz is the following: if the initial quantum state of a many-body system has a low bond dimension (typically if the initial state is a product state with <span>$D = 1$</span>), then in a finite time it will only be able to explore a region of the Hilbert space that is not to far away from its starting point. Thus, the bond dimension will not have the time to diverge exponentially <sup class="footnote-reference"><a id="citeref-poulin_quantum_2011" href="#footnote-poulin_quantum_2011">[poulin_quantum_2011]</a></sup>. However, depending on the physical system at hand, this sub-manifold of the Hilbert space could still be &quot;too large&quot;. There is an additional reason that explains why MPSs are good Ansätze for 1d physical systems. Most many-body Hamiltonians we (physicists) are interested in are local, meaning that the interactions they describe involve objects that are &quot;neighbours&quot;. For such Hamiltonians, the ground states (outside of potential critical phases) follow the so called <em>area law</em> for the entanglement entropy <sup class="footnote-reference"><a id="citeref-srednicki_entropy_1993" href="#footnote-srednicki_entropy_1993">[srednicki_entropy_1993]</a></sup><sup class="footnote-reference"><a id="citeref-vidal_entanglement_2003" href="#footnote-vidal_entanglement_2003">[vidal_entanglement_2003]</a></sup><sup class="footnote-reference"><a id="citeref-wolf_area_2008" href="#footnote-wolf_area_2008">[wolf_area_2008]</a></sup>. This law states that the entanglement entropy <span>$S_{vN}$</span> of a bi-partition of the system is proportional, not to the volume of the partition as one might expect, but to the hyper-surface of the partition&#39;s boundary; hence the name &quot;area law&quot;. For a 3d system this corresponds to an actual surface area <span>$A$</span>, <span>$S_{vN} \sim A$</span>; for a 2d system it corresponds to the length <span>$L$</span> of the partition&#39;s boundary, <span>$S_{vN} \sim L$</span>; and in 1d the boundary reduces to a point, thus the entropy will be independent of the size of the system <span>$S_{vN} \sim \text{constant}$</span>. The MPSs are states that satisfy this area law.</p><p>An application of the <a href="https://en.wikipedia.org/wiki/Singular_value_decomposition">Singular Value Decomposition</a> is to create efficient approximations of quantum states to perform computations. The main idea is to reduce the content of the MPS to keep only the parts that contain the physics of interest. One method to realise this approximation is to do a SVD on each of the tensors of the MPS after each time step of the state time-evolution and to trim the smallest singular values in order to decrease the bond dimension of the MPS down to a chosen maximal value <span>$\chi$</span>. The corresponding columns and rows of the unitary matrices <span>$U$</span> and <span>$V^\dagger$</span> are also removed. Then, the trimmed matrices <span>$\tilde{U}$</span>, <span>$\tilde{S}$</span> and <span>$\tilde{V}^\dagger$</span> are contracted back to give an approximated tensor <span>$T$</span> with a smaller bond dimension. Another way to apply the restricted rank approximation is to restrict oneself into working in a manifold of fixed bond dimension <span>$D$</span> and to use methods that can enforce this constraint.</p><h3 id="MPO"><a class="docs-heading-anchor" href="#MPO">MPO</a><a id="MPO-1"></a><a class="docs-heading-anchor-permalink" href="#MPO" title="Permalink"></a></h3><p>In order to compute expectation values of observables or apply unitary transformations to a quantum state, we need a TN representation of operators. In the same fashion as a one-dimensional quantum state can be represented as a MPS, operators acting on those states can be represented as <strong>Matrix Product Operators</strong> (MPO). For an operator <span>$\hat{O}$</span>, its MPO can be defined as follows</p><p class="math-container">\[    \hat{O} = \sum_{\{i_k\}\{i_k^{&#39;}\} \{w\}} W^{i_1\ i^{&#39;}_1}_{1\ w_0w_1}\ldots  W^{i_N\ i^{&#39;}_N}_{N\ w_{N-1}w_N} |\phi_{i_1^{&#39;}}\ldots \phi_{i_N^{&#39;}}\rangle\langle\phi_{i_1}\ldots \phi_{i_N}| \]</p><p>The contracted indices between the tensors are called <em>virtual indices</em>. The free indices are called <em>physical indices</em> and correspond to the different input and output local quantum states.  They can take <span>$d$</span> values (with <span>$d$</span> the dimension of the local Hilbert space).</p><h3 id="TTN"><a class="docs-heading-anchor" href="#TTN">TTN</a><a id="TTN-1"></a><a class="docs-heading-anchor-permalink" href="#TTN" title="Permalink"></a></h3><p>A natural extension to the MPS is the (loop-free) tree tensor network.  A TTN is a generalisation of the MPS wherein each site, instead of being connected to only one other site to its right, may be connected to any arbitrary number of <em>child</em> sites. Provided the tree does not contain any loops, everything that one can do to an MPS/MPO can be extended straight-forwardly to TTN states and TTN operators.  The generalisation to trees introduces no new conceptual complexity (only implementational complexity). The sites of a TTN are usually referred to as <em>nodes</em>.  For our purposes, every node of a TTN state and operator has one <em>parent</em> leg, and any number (including zero) of child legs.  The first node is known as the head-node and has a dummy parent leg with dimension 1.</p><h2 id="Time-Dependent-Variational-Principle"><a class="docs-heading-anchor" href="#Time-Dependent-Variational-Principle">Time-Dependent Variational Principle</a><a id="Time-Dependent-Variational-Principle-1"></a><a class="docs-heading-anchor-permalink" href="#Time-Dependent-Variational-Principle" title="Permalink"></a></h2><p>The original idea behind TDVP goes back to Dirac <sup class="footnote-reference"><a id="citeref-dirac_note_1930" href="#footnote-dirac_note_1930">[dirac_note_1930]</a></sup> and Frenkel <sup class="footnote-reference"><a id="citeref-frenkel_wave_1934" href="#footnote-frenkel_wave_1934">[frenkel_wave_1934]</a></sup>. The main point, in the modern tensor networks formulation, is that instead of solving the Schrödinger equation and then truncating the MPS representation of the quantum state, one can solve the equations of motion projected into a space of restricted bond dimension <sup class="footnote-reference"><a id="citeref-haegeman_timedependent_2011" href="#footnote-haegeman_timedependent_2011">[haegeman_timedependent_2011]</a></sup><sup class="footnote-reference"><a id="citeref-haegeman_unifying_2016" href="#footnote-haegeman_unifying_2016">[haegeman_unifying_2016]</a></sup>.</p><p>The general formulation of the Dirac-Frenkel Variational Principle <sup class="footnote-reference"><a id="citeref-raab_diracfrenkelmclachlan_2000" href="#footnote-raab_diracfrenkelmclachlan_2000">[raab_diracfrenkelmclachlan_2000]</a></sup> is that one looks for a solution <span>$|\varphi\rangle \in \mathcal{M}$</span> of the Schrödinger equation where <span>$\mathcal{M} \subset \mathcal{H}$</span> is a manifold of the total Hilbert space <span>$\mathcal{H}$</span> in which we think that the relevant physical states `live&#39;.</p><p>We define <span>$T_{|\varphi\rangle}\mathcal{M}$</span> the tangent space of <span>$\mathcal{M}$</span> around the state <span>$|\varphi\rangle$</span>. The criterion to find <span>$|\varphi\rangle$</span> is that for every state <span>$|\chi\rangle \in T_{|\varphi\rangle}\mathcal{M}$</span></p><p class="math-container">\[    \langle\chi|\left(\frac{\mathrm{d}}{\mathrm{d}t} - \frac{1}{\mathrm{i}\hbar}\hat{H}\right)|\varphi\rangle =0\ ,\]</p><p>which can be interpreted as saying that the time evolution procedure should keep <span>$|\varphi\rangle$</span> inside of the manifold <span>$\mathcal{M}$</span>.</p><p>The term <em>variational</em> in the name of the method comes from the fact that in practice one aims at minimising the right-hand side of the hereinabove Eq. to find <span>$|\varphi\rangle$</span>.</p><p>Introducing <span>$\hat{P}_{T_{|\varphi\rangle}\mathcal{M}}$</span> the projector onto the tangent space <span>$T_{|\varphi\rangle}\mathcal{M}$</span>, we can write the state <span>$|\chi\rangle = \hat{P}_{T_{|\varphi\rangle}\mathcal{M}}|\phi\rangle$</span> with <span>$|\phi\rangle$</span> a state in <span>$\mathcal{H}$</span>. Leading to</p><p class="math-container">\[    \forall |\phi\rangle \in \mathcal{H}, \ \langle\phi|\hat{P}_{T_{|\varphi\rangle}\mathcal{M}}\left(\frac{\mathrm{d}}{\mathrm{d}t} - \frac{1}{\mathrm{i}\hbar}\hat{H}\right)|\varphi\rangle =0\ .\]</p><p>Because the time derivation and the projector commute, we have</p><p class="math-container">\[    \forall |\phi\rangle \in \mathcal{H}, \ \langle\phi|\left(\frac{\mathrm{d}}{\mathrm{d}t} - \frac{1}{\mathrm{i}\hbar}\hat{P}_{T_{|\varphi\rangle}\mathcal{M}}\hat{H}\right)|\varphi\rangle =0\ .\]</p><p>This equation must be true for any <span>$|\phi\rangle \in \mathcal{H}$</span>, Eq.~(\ref{eq:DiracFrenkel1}) can thus be written</p><p class="math-container">\[    \left(\frac{\mathrm{d}}{\mathrm{d}t} - \frac{1}{\mathrm{i}\hbar}\hat{P}_{T_{|\varphi\rangle}\mathcal{M}}\hat{H}\right)|\varphi\rangle =0\ .\]</p><p>In the context of MPS, the manifold <span>$\mathcal{M}$</span> will correspond to the space of full-ranked MPS of a given bond dimension <span>$D$</span>, and the tangent space will be the space spanned by variations of single MPS tensors. The projector defines an effective Hamiltonian under which the dynamics are constrained on <span>$\mathcal{M}$</span>. Constraining the dynamics on a manifold, introduces a projection error: the time evolution will obey to an effective Hamiltonian different from the starting one. After the introduction of TDVP as a time evolution method for MPS, Haegeman <em>et al.</em> pointed out <sup class="footnote-reference"><a id="citeref-haegeman_unifying_2016" href="#footnote-haegeman_unifying_2016">[haegeman_unifying_2016]</a></sup> that there exist an analytical decomposition for the projector operator <span>$\hat{\mathcal{P}}$</span> that simplifies the resolution of the equation, turning the problem into one where each matrix <span>$A_i$</span> can be updated with an effective <em>on site</em> Hamiltonian <span>$\hat H_\text{eff}$</span> via a Schroedinger like equation. The effective Hamiltonian <span>$\hat H_\text{eff}$</span> is a contraction of the Hamiltonian MPO and the current state of the other matrices composing the MPS. This allows to do a sequential update.</p><p>There exist different versions of the TDVP algorithm. In <code>MPSDynamics.jl</code> three methods have been so far implemented:</p><ul><li>the one-site TDVP (1TDVP)</li><li>the two-sites TDVP (2TDVP) </li><li>the adaptive TDVP (DTDVP) <sup class="footnote-reference"><a id="citeref-dunnett_efficient_2021" href="#footnote-dunnett_efficient_2021">[dunnett_efficient_2021]</a></sup></li></ul><p>The main advantage of the one-site 1TDVP algorithm is that it preserves the unitarity of the MPS during the time evolution. Its main problem, conversely, is that the time evolution is constrained to happen on a manifold constituted by tensors of fixed bond dimension, a quantity closely related to the amount of entanglement in the MPS, and such a bond dimension has therefore to be fixed before the beginning of the time evolution. This strategy will necessarily be non optimal: the growth of the bond dimensions required to describe the quantum state should ideally mirror the entanglement growth induced by the time evolution. 2TDVP allows for such a dynamical growth of the bond dimensions, and therefore better describes the entanglement in the MPS. It suffers however of other drawbacks: first of all, a truncation error is introduced (by the means of an SVD decomposition), which entails a loss of unitarity of the time-evolved MPS. Moreover, 2TDVP has bad scaling properties with the size of the local dimensions of the MPS: this is a major issue when dealing with bosons. The DTDVP algorithm combines the best features of 1TDVP and 2TDVP: it preserves unitarity, it has the same scaling properties of 1TDVP, and it adapts the bond dimensions to the entanglement evolution at each site and at each time-step. DTDVP does not suffer from a truncation error, but introduces only a projection error.</p><p>________________</p><h2 id="Bibliography"><a class="docs-heading-anchor" href="#Bibliography">Bibliography</a><a id="Bibliography-1"></a><a class="docs-heading-anchor-permalink" href="#Bibliography" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-chin_exact_2010"><a class="tag is-link" href="#citeref-chin_exact_2010">chin_exact_2010</a><blockquote><p>Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. Journal of Mathematical Physics 2010, 51 (9), 092109. https://doi.org/10.1063/1.3490188.</p></blockquote></li><li class="footnote" id="footnote-chin_role_2013"><a class="tag is-link" href="#citeref-chin_role_2013">chin_role_2013</a><blockquote><p>Chin, A. W.; Prior, J.; Rosenbach, R.; Caycedo-Soler, F.; Huelga, S. F.; Plenio, M. B. The Role of Non-Equilibrium Vibrational Structures in Electronic Coherence and Recoherence in Pigment–Protein Complexes. Nature Phys 2013, 9 (2), 113–118. https://doi.org/10.1038/nphys2515.</p></blockquote></li><li class="footnote" id="footnote-alvertis_nonequilibrium_2019"><a class="tag is-link" href="#citeref-alvertis_nonequilibrium_2019">alvertis_nonequilibrium_2019</a><blockquote><p>Alvertis, A. M.; Schröder, F. A. Y. N.; Chin, A. W. Non-Equilibrium Relaxation of Hot States in Organic Semiconductors: Impact of Mode-Selective Excitation on Charge Transfer. J. Chem. Phys. 2019, 151 (8), 084104. https://doi.org/10.1063/1.5115239.</p></blockquote></li><li class="footnote" id="footnote-dunnett_influence_2021"><a class="tag is-link" href="#citeref-dunnett_influence_2021">dunnett_influence_2021</a><blockquote><p>Dunnett, A. J.; Gowland, D.; Isborn, C. M.; Chin, A. W.; Zuehlsdorff, T. J. Influence of Non-Adiabatic Effects on Linear Absorption Spectra in the Condensed Phase: Methylene Blue. J. Chem. Phys. 2021, 155 (14), 144112. https://doi.org/10.1063/5.0062950.</p></blockquote></li><li class="footnote" id="footnote-caycedosoler_exact_2022"><a class="tag is-link" href="#citeref-caycedosoler_exact_2022">caycedosoler_exact_2022</a><blockquote><p>Caycedo-Soler, F.; Mattioni, A.; Lim, J.; Renger, T.; Huelga, S. F.; Plenio, M. B. Exact Simulation of Pigment-Protein Complexes Unveils Vibronic Renormalization of Electronic Parameters in Ultrafast Spectroscopy. 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J.; Pižorn, I.; Verschelde, H.; Verstraete, F. Time-Dependent Variational Principle for Quantum Lattices. Phys. Rev. Lett. 2011, 107 (7), 070601. https://doi.org/10.1103/PhysRevLett.107.070601.</p></blockquote></li><li class="footnote" id="footnote-haegeman_unifying_2016"><a class="tag is-link" href="#citeref-haegeman_unifying_2016">haegeman_unifying_2016</a><blockquote><p>Haegeman, J.; Lubich, C.; Oseledets, I.; Vandereycken, B.; Verstraete, F. Unifying Time Evolution and Optimization with Matrix Product States. Phys. Rev. B 2016, 94 (16), 165116. https://doi.org/10.1103/PhysRevB.94.165116.</p></blockquote></li><li class="footnote" id="footnote-raab_diracfrenkelmclachlan_2000"><a class="tag is-link" href="#citeref-raab_diracfrenkelmclachlan_2000">raab_diracfrenkelmclachlan_2000</a><blockquote><p>Raab, A. On the Dirac–Frenkel/McLachlan Variational Principle. 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Physical Review B, 104(21):214302, Dec 2021. doi:10.1103/PhysRevB.104.214302. </p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../examples/protontransfer/">« Proton Transfer Model</a><a class="docs-footer-nextpage" href="../methods/">Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. 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+         \sinh(\theta_k) =\quad\sqrt{n_k}\quad= \sqrt{\frac{1}{e^{\beta \omega_k}-1}}.\]</p><p>The Bogoliubov transformation defines a new squeezed vacuum state, which we write in terms of the vacuum state <span>$\ket{\Omega_0}$</span> of the operators <span>$\hat b_k$</span>, <span>$\hat c_k$</span>:</p><p class="math-container">\[    |\Omega\rangle = e^{iG} |\Omega_0\rangle, \quad \text{such that: }\quad \hat a_{1k} |\Omega\rangle = 0,  \hat a_{2k} |\Omega\rangle = 0.\]</p><p>From the vacuum state, we can obtain the thermal state of the original environment:</p><p class="math-container">\[    \hat \rho_E = \text{Tr}_{\text{aux}}\{ |\Omega\rangle\langle\Omega| \},\]</p><p>and it can be now used as pure an initial state for both of the environments. Moreover, the expectation value on <span>$\ket{\Omega}$</span> of the number of physical modes <span>$\hat n_k$</span> does not vanish:</p><p class="math-container">\[    n_k= \langle\Omega| \hat b_k^\dagger \hat b_k |\Omega\rangle = \sinh^2(\theta_k).\]</p><p>Therefore, solving the dynamics given by the original Hamiltonian <span>$\hat H$</span>, starting from the initial condition <span>$\hat \rho_S(0) \otimes \hat \rho_E(\beta)$</span>, is equivalent to solving the dynamics given by the following Hamiltonian:</p><p class="math-container">\[       \hat H = \hat H_S +\hat H_E +\hat H_I  = \\
+       = \overbrace{\hat A_S}^{\hat H_S} + \overbrace{\sum_k \omega_k \big(\hat a_{1k}^\dagger \hat a_{1k} - \hat a_{2k}^\dagger \hat a_{2k} \big)}^{H_E} + \overbrace{\hat L_S \otimes \sum_k g_{1k}(\hat a_{1k}^\dagger + \hat a_{1k})+\hat L_S \otimes \sum_k g_{2k}(\hat a_{2k}^\dagger + \hat a_{2k})}^{H_I}.\]</p><p>where <span>$\hat L_S = \hat L_S^\dagger$</span>, considering <span>$\hat \rho_S(0) \otimes \ket{\Omega}\bra{\Omega}$</span> as the initial state of system and environment. It is this Hamiltonian that is mapped on two chains with TEDOPA, to be able to perform the time evolution using tensor network techniques. </p><h2 id="Computation-of-the-chain-coefficients"><a class="docs-heading-anchor" href="#Computation-of-the-chain-coefficients">Computation of the chain coefficients</a><a id="Computation-of-the-chain-coefficients-1"></a><a class="docs-heading-anchor-permalink" href="#Computation-of-the-chain-coefficients" title="Permalink"></a></h2><p>A useful property of the orthonormal polynomials is that they obey a recurrence relation</p><p class="math-container">\[    P_n(\omega) = (C_{n-1}\omega - A_{n-1})P_{n-1}(\omega) + B_{n-1}P_{n-2}(\omega)\ ,\]</p><p>where <span>$A_n$</span> is related to the first moment of <span>$P_n$</span>, <span>$B_n$</span> and <span>$C_n$</span> to the norms of <span>$P_n$</span> and <span>$P_{n-1}$</span><sup class="footnote-reference"><a id="citeref-appel_mathematics_2007" href="#footnote-appel_mathematics_2007">[appel_mathematics_2007]</a></sup>. This recurrence relation can be used to construct the polynomials with the conditions that <span>$P_0(\omega) = ||p_0||^{-1} = \left(\int_{\mathbb{R}^{+}} J(\omega)\mathrm{d}\omega \right)^{-\frac{1}{2}}$</span> and <span>$P_{-1}(\omega) = 0$</span>, with <span>$||\bullet||$</span> the norm of <span>$\bullet$</span> with respect to the measure <span>$J(\omega)$</span>, and <span>$P_n(\omega) = p_n(\omega)||p_n||^{-1}$</span> ; where the polynomials <span>$\{p_n\}_{n\in\mathbb{N}}$</span> are the so called <em>monic polynomials</em> where the factor <span>$a_n$</span> in front of <span>$\omega^{n}$</span> is equal to 1.</p><p>The energy of the chain mode <span>$n$</span> is given by <span>$\varepsilon_n = A_n C_n^{-1}$</span> and <span>$t_n=C_n^{-1}$</span> is the coupling between mode <span>$n$</span> and <span>$n+1$</span><sup class="footnote-reference"><a id="citeref-chin_exact_2010" href="#footnote-chin_exact_2010">[chin_exact_2010]</a></sup>.</p><p>The system couples <em>only</em> to the first mode with the coupling strength <span>$\kappa = ||p_0||$</span>.</p><p>Explain that for some weight function/SD they are known analytically and that for others we can use the build-in routines inspired by Gautschi or the PolyChaos.jl package.</p><h2 id="Tensor-Networks"><a class="docs-heading-anchor" href="#Tensor-Networks">Tensor Networks</a><a id="Tensor-Networks-1"></a><a class="docs-heading-anchor-permalink" href="#Tensor-Networks" title="Permalink"></a></h2><p>A multipartite quantum state <span>$|\psi\rangle$</span>, e.g. a <span>$N$</span>-site system where the sites can each be in a state <span>$|\phi_i\rangle$</span> belonging to a <span>$d$</span>-dimensional Hilbert space, can be written as follows</p><p class="math-container">\[    |\psi\rangle = \sum_{\{i_k\}}c_{i_1\ldots i_N}|\phi_{i_1}\rangle\otimes\ldots\otimes|\phi_{i_N}\rangle\ ,\]</p><p>where the complex numbers <span>$c_{i_1\ldots i_N}$</span> are the amplitudes of each state <span>$|\phi_{i_1}\rangle\otimes\ldots\otimes|\phi_{i_N}\rangle$</span> whose superpositions form in full generality the state <span>$|\psi\rangle$</span>. Thus the state <span>$|\psi\rangle$</span> can be completely represented by a rank-<span>$N$</span> tensor <span>$c$</span> that is the collection of all possible amplitudes <span>$c_{i_1\ldots i_N}$</span>. Here by the rank of a tensor, we simply mean the number of indices it has.</p><h3 id="MPS"><a class="docs-heading-anchor" href="#MPS">MPS</a><a id="MPS-1"></a><a class="docs-heading-anchor-permalink" href="#MPS" title="Permalink"></a></h3><p>The tensor <span>$c$</span> of a quantum state <span>$|\psi\rangle$</span> corresponding to a one-dimensional system can be decomposed into a product of <span>$N$</span> smaller rank-3 tensors <span>$T_{k}$</span> (except for the first and last sites where the tensors will have a rank-2)</p><p class="math-container">\[    c_{i_1\ldots i_N} = \sum_{\{\alpha\}} T^{\alpha_1}_{i_1}T^{\alpha_1\alpha_2\ }_{i_2}T^{\alpha_2\alpha_3\ }_{i_3}\ldots T^{\alpha_{N-1}}_{i_N} \ .\]</p><p>In this form, the local tensor <span>$T_k$</span> contains the information on the quantum state on site <span>$k$</span> and its relation (especially the entanglement) with the neighbouring sites.</p><p>The decomposition of the tensor of the amplitudes of a quantum state into a product of smaller rank tensors is called a <strong>Matrix Product State</strong> decomposition.</p><p>The contracted indices <span>$\alpha_k$</span> between the tensors are called <em>virtual indices</em> and carry information about the correlations between bi-partitions of the state at bond <span>$k$</span>. The number of different values a virtual index can take is called the <em>bond dimension</em> and is denoted <span>$D$</span>. The free indices <span>$i_k$</span> associated with local quantum states are called <em>physical indices</em>. Thus, they can take <span>$d$</span> values (with <span>$d$</span> the dimension of the local Hilbert space).</p><p>Any state in the Hilbert space of a one-dimensional many-body system can in principle be represented by a MPS by choosing a sufficiently large value for the bond dimension <span>$D$</span> <sup class="footnote-reference"><a id="citeref-Orus" href="#footnote-Orus">[Orus]</a></sup>. On top of this intellectually satisfying property of MPSs being a dense set of states for a 1d-system, they can also be used as a practical Ansätze for a many-body quantum states by setting a maximal allowed value <span>$\chi$</span> for the bond dimension <span>$D$</span>. In doing so, we restrict ourselves to a corner of the total Hilbert space. The rationale behind this Ansatz is the following: if the initial quantum state of a many-body system has a low bond dimension (typically if the initial state is a product state with <span>$D = 1$</span>), then in a finite time it will only be able to explore a region of the Hilbert space that is not to far away from its starting point. Thus, the bond dimension will not have the time to diverge exponentially <sup class="footnote-reference"><a id="citeref-poulin_quantum_2011" href="#footnote-poulin_quantum_2011">[poulin_quantum_2011]</a></sup>. However, depending on the physical system at hand, this sub-manifold of the Hilbert space could still be &quot;too large&quot;. There is an additional reason that explains why MPSs are good Ansätze for 1d physical systems. Most many-body Hamiltonians we (physicists) are interested in are local, meaning that the interactions they describe involve objects that are &quot;neighbours&quot;. For such Hamiltonians, the ground states (outside of potential critical phases) follow the so called <em>area law</em> for the entanglement entropy <sup class="footnote-reference"><a id="citeref-srednicki_entropy_1993" href="#footnote-srednicki_entropy_1993">[srednicki_entropy_1993]</a></sup><sup class="footnote-reference"><a id="citeref-vidal_entanglement_2003" href="#footnote-vidal_entanglement_2003">[vidal_entanglement_2003]</a></sup><sup class="footnote-reference"><a id="citeref-wolf_area_2008" href="#footnote-wolf_area_2008">[wolf_area_2008]</a></sup>. This law states that the entanglement entropy <span>$S_{vN}$</span> of a bi-partition of the system is proportional, not to the volume of the partition as one might expect, but to the hyper-surface of the partition&#39;s boundary; hence the name &quot;area law&quot;. For a 3d system this corresponds to an actual surface area <span>$A$</span>, <span>$S_{vN} \sim A$</span>; for a 2d system it corresponds to the length <span>$L$</span> of the partition&#39;s boundary, <span>$S_{vN} \sim L$</span>; and in 1d the boundary reduces to a point, thus the entropy will be independent of the size of the system <span>$S_{vN} \sim \text{constant}$</span>. The MPSs are states that satisfy this area law.</p><p>An application of the <a href="https://en.wikipedia.org/wiki/Singular_value_decomposition">Singular Value Decomposition</a> is to create efficient approximations of quantum states to perform computations. The main idea is to reduce the content of the MPS to keep only the parts that contain the physics of interest. One method to realise this approximation is to do a SVD on each of the tensors of the MPS after each time step of the state time-evolution and to trim the smallest singular values in order to decrease the bond dimension of the MPS down to a chosen maximal value <span>$\chi$</span>. The corresponding columns and rows of the unitary matrices <span>$U$</span> and <span>$V^\dagger$</span> are also removed. Then, the trimmed matrices <span>$\tilde{U}$</span>, <span>$\tilde{S}$</span> and <span>$\tilde{V}^\dagger$</span> are contracted back to give an approximated tensor <span>$T$</span> with a smaller bond dimension. Another way to apply the restricted rank approximation is to restrict oneself into working in a manifold of fixed bond dimension <span>$D$</span> and to use methods that can enforce this constraint.</p><h3 id="MPO"><a class="docs-heading-anchor" href="#MPO">MPO</a><a id="MPO-1"></a><a class="docs-heading-anchor-permalink" href="#MPO" title="Permalink"></a></h3><p>In order to compute expectation values of observables or apply unitary transformations to a quantum state, we need a TN representation of operators. In the same fashion as a one-dimensional quantum state can be represented as a MPS, operators acting on those states can be represented as <strong>Matrix Product Operators</strong> (MPO). For an operator <span>$\hat{O}$</span>, its MPO can be defined as follows</p><p class="math-container">\[    \hat{O} = \sum_{\{i_k\}\{i_k^{&#39;}\} \{w\}} W^{i_1\ i^{&#39;}_1}_{1\ w_0w_1}\ldots  W^{i_N\ i^{&#39;}_N}_{N\ w_{N-1}w_N} |\phi_{i_1^{&#39;}}\ldots \phi_{i_N^{&#39;}}\rangle\langle\phi_{i_1}\ldots \phi_{i_N}| \]</p><p>The contracted indices between the tensors are called <em>virtual indices</em>. The free indices are called <em>physical indices</em> and correspond to the different input and output local quantum states.  They can take <span>$d$</span> values (with <span>$d$</span> the dimension of the local Hilbert space).</p><h3 id="TTN"><a class="docs-heading-anchor" href="#TTN">TTN</a><a id="TTN-1"></a><a class="docs-heading-anchor-permalink" href="#TTN" title="Permalink"></a></h3><p>A natural extension to the MPS is the (loop-free) tree tensor network.  A TTN is a generalisation of the MPS wherein each site, instead of being connected to only one other site to its right, may be connected to any arbitrary number of <em>child</em> sites. Provided the tree does not contain any loops, everything that one can do to an MPS/MPO can be extended straight-forwardly to TTN states and TTN operators.  The generalisation to trees introduces no new conceptual complexity (only implementational complexity). The sites of a TTN are usually referred to as <em>nodes</em>.  For our purposes, every node of a TTN state and operator has one <em>parent</em> leg, and any number (including zero) of child legs.  The first node is known as the head-node and has a dummy parent leg with dimension 1.</p><h2 id="Time-Dependent-Variational-Principle"><a class="docs-heading-anchor" href="#Time-Dependent-Variational-Principle">Time-Dependent Variational Principle</a><a id="Time-Dependent-Variational-Principle-1"></a><a class="docs-heading-anchor-permalink" href="#Time-Dependent-Variational-Principle" title="Permalink"></a></h2><p>The original idea behind TDVP goes back to Dirac <sup class="footnote-reference"><a id="citeref-dirac_note_1930" href="#footnote-dirac_note_1930">[dirac_note_1930]</a></sup> and Frenkel <sup class="footnote-reference"><a id="citeref-frenkel_wave_1934" href="#footnote-frenkel_wave_1934">[frenkel_wave_1934]</a></sup>. The main point, in the modern tensor networks formulation, is that instead of solving the Schrödinger equation and then truncating the MPS representation of the quantum state, one can solve the equations of motion projected into a space of restricted bond dimension <sup class="footnote-reference"><a id="citeref-haegeman_timedependent_2011" href="#footnote-haegeman_timedependent_2011">[haegeman_timedependent_2011]</a></sup><sup class="footnote-reference"><a id="citeref-haegeman_unifying_2016" href="#footnote-haegeman_unifying_2016">[haegeman_unifying_2016]</a></sup>.</p><p>The general formulation of the Dirac-Frenkel Variational Principle <sup class="footnote-reference"><a id="citeref-raab_diracfrenkelmclachlan_2000" href="#footnote-raab_diracfrenkelmclachlan_2000">[raab_diracfrenkelmclachlan_2000]</a></sup> is that one looks for a solution <span>$|\varphi\rangle \in \mathcal{M}$</span> of the Schrödinger equation where <span>$\mathcal{M} \subset \mathcal{H}$</span> is a manifold of the total Hilbert space <span>$\mathcal{H}$</span> in which we think that the relevant physical states `live&#39;.</p><p>We define <span>$T_{|\varphi\rangle}\mathcal{M}$</span> the tangent space of <span>$\mathcal{M}$</span> around the state <span>$|\varphi\rangle$</span>. The criterion to find <span>$|\varphi\rangle$</span> is that for every state <span>$|\chi\rangle \in T_{|\varphi\rangle}\mathcal{M}$</span></p><p class="math-container">\[    \langle\chi|\left(\frac{\mathrm{d}}{\mathrm{d}t} - \frac{1}{\mathrm{i}\hbar}\hat{H}\right)|\varphi\rangle =0\ ,\]</p><p>which can be interpreted as saying that the time evolution procedure should keep <span>$|\varphi\rangle$</span> inside of the manifold <span>$\mathcal{M}$</span>.</p><p>The term <em>variational</em> in the name of the method comes from the fact that in practice one aims at minimising the right-hand side of the hereinabove Eq. to find <span>$|\varphi\rangle$</span>.</p><p>Introducing <span>$\hat{P}_{T_{|\varphi\rangle}\mathcal{M}}$</span> the projector onto the tangent space <span>$T_{|\varphi\rangle}\mathcal{M}$</span>, we can write the state <span>$|\chi\rangle = \hat{P}_{T_{|\varphi\rangle}\mathcal{M}}|\phi\rangle$</span> with <span>$|\phi\rangle$</span> a state in <span>$\mathcal{H}$</span>. Leading to</p><p class="math-container">\[    \forall |\phi\rangle \in \mathcal{H}, \ \langle\phi|\hat{P}_{T_{|\varphi\rangle}\mathcal{M}}\left(\frac{\mathrm{d}}{\mathrm{d}t} - \frac{1}{\mathrm{i}\hbar}\hat{H}\right)|\varphi\rangle =0\ .\]</p><p>Because the time derivation and the projector commute, we have</p><p class="math-container">\[    \forall |\phi\rangle \in \mathcal{H}, \ \langle\phi|\left(\frac{\mathrm{d}}{\mathrm{d}t} - \frac{1}{\mathrm{i}\hbar}\hat{P}_{T_{|\varphi\rangle}\mathcal{M}}\hat{H}\right)|\varphi\rangle =0\ .\]</p><p>This equation must be true for any <span>$|\phi\rangle \in \mathcal{H}$</span>, Eq.~(\ref{eq:DiracFrenkel1}) can thus be written</p><p class="math-container">\[    \left(\frac{\mathrm{d}}{\mathrm{d}t} - \frac{1}{\mathrm{i}\hbar}\hat{P}_{T_{|\varphi\rangle}\mathcal{M}}\hat{H}\right)|\varphi\rangle =0\ .\]</p><p>In the context of MPS, the manifold <span>$\mathcal{M}$</span> will correspond to the space of full-ranked MPS of a given bond dimension <span>$D$</span>, and the tangent space will be the space spanned by variations of single MPS tensors. The projector defines an effective Hamiltonian under which the dynamics are constrained on <span>$\mathcal{M}$</span>. Constraining the dynamics on a manifold, introduces a projection error: the time evolution will obey to an effective Hamiltonian different from the starting one. After the introduction of TDVP as a time evolution method for MPS, Haegeman <em>et al.</em> pointed out <sup class="footnote-reference"><a id="citeref-haegeman_unifying_2016" href="#footnote-haegeman_unifying_2016">[haegeman_unifying_2016]</a></sup> that there exist an analytical decomposition for the projector operator <span>$\hat{\mathcal{P}}$</span> that simplifies the resolution of the equation, turning the problem into one where each matrix <span>$A_i$</span> can be updated with an effective <em>on site</em> Hamiltonian <span>$\hat H_\text{eff}$</span> via a Schroedinger like equation. The effective Hamiltonian <span>$\hat H_\text{eff}$</span> is a contraction of the Hamiltonian MPO and the current state of the other matrices composing the MPS. This allows to do a sequential update.</p><p>There exist different versions of the TDVP algorithm. In <code>MPSDynamics.jl</code> three methods have been so far implemented:</p><ul><li>the one-site TDVP (1TDVP)</li><li>the two-sites TDVP (2TDVP) </li><li>the adaptive TDVP (DTDVP) <sup class="footnote-reference"><a id="citeref-dunnett_efficient_2021" href="#footnote-dunnett_efficient_2021">[dunnett_efficient_2021]</a></sup></li></ul><p>The main advantage of the one-site 1TDVP algorithm is that it preserves the unitarity of the MPS during the time evolution. Its main problem, conversely, is that the time evolution is constrained to happen on a manifold constituted by tensors of fixed bond dimension, a quantity closely related to the amount of entanglement in the MPS, and such a bond dimension has therefore to be fixed before the beginning of the time evolution. This strategy will necessarily be non optimal: the growth of the bond dimensions required to describe the quantum state should ideally mirror the entanglement growth induced by the time evolution. 2TDVP allows for such a dynamical growth of the bond dimensions, and therefore better describes the entanglement in the MPS. It suffers however of other drawbacks: first of all, a truncation error is introduced (by the means of an SVD decomposition), which entails a loss of unitarity of the time-evolved MPS. Moreover, 2TDVP has bad scaling properties with the size of the local dimensions of the MPS: this is a major issue when dealing with bosons. The DTDVP algorithm combines the best features of 1TDVP and 2TDVP: it preserves unitarity, it has the same scaling properties of 1TDVP, and it adapts the bond dimensions to the entanglement evolution at each site and at each time-step. DTDVP does not suffer from a truncation error, but introduces only a projection error.</p><h2 id="Bibliography"><a class="docs-heading-anchor" href="#Bibliography">Bibliography</a><a id="Bibliography-1"></a><a class="docs-heading-anchor-permalink" href="#Bibliography" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-chin_exact_2010"><a class="tag is-link" href="#citeref-chin_exact_2010">chin_exact_2010</a><blockquote><p>Chin, A. W.; Rivas, Á.; Huelga, S. F.; Plenio, M. B. Exact Mapping between System-Reservoir Quantum Models and Semi-Infinite Discrete Chains Using Orthogonal Polynomials. 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J.; Pižorn, I.; Verschelde, H.; Verstraete, F. Time-Dependent Variational Principle for Quantum Lattices. Phys. Rev. Lett. 2011, 107 (7), 070601. https://doi.org/10.1103/PhysRevLett.107.070601.</p></blockquote></li><li class="footnote" id="footnote-haegeman_unifying_2016"><a class="tag is-link" href="#citeref-haegeman_unifying_2016">haegeman_unifying_2016</a><blockquote><p>Haegeman, J.; Lubich, C.; Oseledets, I.; Vandereycken, B.; Verstraete, F. Unifying Time Evolution and Optimization with Matrix Product States. Phys. Rev. B 2016, 94 (16), 165116. https://doi.org/10.1103/PhysRevB.94.165116.</p></blockquote></li><li class="footnote" id="footnote-raab_diracfrenkelmclachlan_2000"><a class="tag is-link" href="#citeref-raab_diracfrenkelmclachlan_2000">raab_diracfrenkelmclachlan_2000</a><blockquote><p>Raab, A. On the Dirac–Frenkel/McLachlan Variational Principle. 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Physical Review B, 104(21):214302, Dec 2021. doi:10.1103/PhysRevB.104.214302. </p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../examples/bath-observables/">« Inspecting the bath by undoing the chain mapping</a><a class="docs-footer-nextpage" href="../methods/">Methods »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. 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id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Introduction</a></li><li class="is-active"><a class="tocitem" href>User Guide</a><ul class="internal"><li><a class="tocitem" href="#Initial-State"><span>Initial State</span></a></li><li><a class="tocitem" href="#Hamiltonian"><span>Hamiltonian</span></a></li><li><a class="tocitem" href="#Observables"><span>Observables</span></a></li><li><a class="tocitem" href="#Time-Evolution"><span>Time-Evolution</span></a></li><li><a class="tocitem" href="#Data-Storage"><span>Data Storage</span></a></li></ul></li><li><a class="tocitem" href="../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li><li><a class="tocitem" href="../examples/protontransfer/">Proton Transfer Model</a></li></ul></li><li><a class="tocitem" href="../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../methods/">Methods</a></li><li><a class="tocitem" href="../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>User Guide</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>User Guide</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="User-Guide"><a class="docs-heading-anchor" href="#User-Guide">User Guide</a><a id="User-Guide-1"></a><a class="docs-heading-anchor-permalink" href="#User-Guide" title="Permalink"></a></h1><p>Here we explain the different steps to perform a simulation.</p><p>Examples with detailed explanations can be found in the <code>Examples</code> section.</p><h2 id="Initial-State"><a class="docs-heading-anchor" href="#Initial-State">Initial State</a><a id="Initial-State-1"></a><a class="docs-heading-anchor-permalink" href="#Initial-State" title="Permalink"></a></h2><p>The initial many-body state of the {System + Environment} can be described easily as <a href="#Matrix-Product-States">Matrix Product States</a> (MPS) or <a href="#Tree-Tensor-Networks">Tree Tensor Networks</a> (TTN) state (e.g. when a system is coupled to several environments).</p><h3 id="Matrix-Product-States"><a class="docs-heading-anchor" href="#Matrix-Product-States">Matrix Product States</a><a id="Matrix-Product-States-1"></a><a class="docs-heading-anchor-permalink" href="#Matrix-Product-States" title="Permalink"></a></h3><p>A MPS can be initialized with several methods.</p><p>The <a href="../methods/#MPSDynamics.productstatemps"><code>productstatemps</code></a> method enables  to instantiate arbitrary MPS of fixed uniform bond dimension with non-uniform physical dimensions. The individual states of the MPS sites can be provided by setting state to a list of column vectors.  Setting <code>state=:Vacuum</code> will produce an MPS in the vacuum state.  Setting <code>state=:FullOccupy</code> will produce an MPS in which each site is fully occupied. The gauge of the MPS can also be set using a keyword argument.</p><pre><code class="language-julia hljs">julia&gt; ψ = unitcol(1,2) # system initial state
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>User Guide · MPSDynamics.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img src="../assets/logo.png" alt="MPSDynamics.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">MPSDynamics.jl</span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Introduction</a></li><li class="is-active"><a class="tocitem" href>User Guide</a><ul class="internal"><li><a class="tocitem" href="#Initial-State"><span>Initial State</span></a></li><li><a class="tocitem" href="#Hamiltonian"><span>Hamiltonian</span></a></li><li><a class="tocitem" href="#Observables"><span>Observables</span></a></li><li><a class="tocitem" href="#Time-Evolution"><span>Time-Evolution</span></a></li><li><a class="tocitem" href="#Data-Storage"><span>Data Storage</span></a></li></ul></li><li><a class="tocitem" href="../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/sbm/">The Spin-Boson Model</a></li><li><a class="tocitem" href="../examples/puredephasing/">Pure-Dephasing</a></li><li><a class="tocitem" href="../examples/timedep/">Time-dependent Hamiltonian</a></li><li><a class="tocitem" href="../examples/anderson-model/">The Anderson Impurity Model</a></li><li><a class="tocitem" href="../examples/bath-observables/">Inspecting the bath by undoing the chain mapping</a></li></ul></li><li><a class="tocitem" href="../theory/">Theoretical Background</a></li><li><a class="tocitem" href="../methods/">Methods</a></li><li><a class="tocitem" href="../dev/">Developers</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>User Guide</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>User Guide</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/user-guide.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="User-Guide"><a class="docs-heading-anchor" href="#User-Guide">User Guide</a><a id="User-Guide-1"></a><a class="docs-heading-anchor-permalink" href="#User-Guide" title="Permalink"></a></h1><p>Here we explain the different steps to perform a simulation.</p><p>Examples with detailed explanations can be found in <a href="@ref">Examples</a>.</p><h2 id="Initial-State"><a class="docs-heading-anchor" href="#Initial-State">Initial State</a><a id="Initial-State-1"></a><a class="docs-heading-anchor-permalink" href="#Initial-State" title="Permalink"></a></h2><p>The initial many-body state of the {System + Environment} can be described easily as <a href="#Matrix-Product-States">Matrix Product States</a> (MPS) or <a href="#Tree-Tensor-Networks">Tree Tensor Networks</a> (TTN) state (e.g. when a system is coupled to several environments).</p><h3 id="Matrix-Product-States"><a class="docs-heading-anchor" href="#Matrix-Product-States">Matrix Product States</a><a id="Matrix-Product-States-1"></a><a class="docs-heading-anchor-permalink" href="#Matrix-Product-States" title="Permalink"></a></h3><p>A MPS can be initialized with several methods.</p><p>The <a href="../methods/#MPSDynamics.productstatemps"><code>productstatemps</code></a> method enables  to instantiate arbitrary MPS of fixed uniform bond dimension with non-uniform physical dimensions. The individual states of the MPS sites can be provided by setting state to a list of column vectors.  Setting <code>state=:Vacuum</code> will produce an MPS in the vacuum state.  Setting <code>state=:FullOccupy</code> will produce an MPS in which each site is fully occupied. The gauge of the MPS can also be set using a keyword argument.</p><pre><code class="language-julia">julia&gt; ψ = unitcol(1,2) # system initial state
 
 julia&gt; d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1 
 
@@ -7,20 +7,12 @@
 
 julia&gt; H = spinbosonmpo(ω0, Δ, d, N, cpars)
 
-julia&gt; A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre><p>Alternatively, a chain with a specified number of excitation localised on one site, or delocalized accross several sites can be generated with <a href="../methods/#MPSDynamics.chainmps-Tuple{Int64, Int64, Int64}"><code>MPSDynamics.chainmps</code></a>.</p><p>Random MPS can also be generated with the <a href="../methods/#MPSDynamics.randmps"><code>randmps</code></a> method.</p><p>For the case of fermionic states (which need to be anti-symmetrized), the <a href="../methods/#MPSDynamics.electronkmps"><code>MPSDynamics.electronkmps</code></a> method generate an MPS for an electron with momentum <code>k</code>, and the <a href="../methods/#MPSDynamics.electron2kmps"><code>MPSDynamics.electron2kmps</code></a> generate an MPS with 2 electrons in k-states <code>k1</code> and <code>k2</code>.</p><h3 id="Tree-Tensor-Networks"><a class="docs-heading-anchor" href="#Tree-Tensor-Networks">Tree Tensor Networks</a><a id="Tree-Tensor-Networks-1"></a><a class="docs-heading-anchor-permalink" href="#Tree-Tensor-Networks" title="Permalink"></a></h3><p>TTN are a generalization of MPS that allows more ramifications to the one dimensional shape of the MSP. This is useful when, for instance, a system is interacting with more than two environments. Tensors of <code>TreeNetwork</code> are usually referred as nodes, while the first node, represented as the leftmost node is called the head node. The Tree structure is built such that a node has always one parent leg pointing to the left and its child legs pointing to the right. Most of the methods for MPS states are overloaded for <code>TreeNetwork</code>. For example, the <a href="../methods/#MPSDynamics.productstatemps"><code>productstatemps</code></a> method tolerates also TTN state and a simple <code>TreeNetwork</code> with only one branch can be initialized in the same way than with MPS.</p><pre><code class="language-julia hljs">julia&gt; ψ = unitcol(1,2)
-
-julia&gt; d = 6; N = 30; α = 0.1; Δ = 0.0; ω0 = 0.2; s = 1
-
-julia&gt; cpars = chaincoeffs_ohmic(N, α, s)
-
-julia&gt; H = spinbosonmpo(ω0, Δ, d, N, cpars, tree=true)
-
-julia&gt; A = productstatemps(H.tree, physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # tree-MPS representation of |ψ&gt;|Vacuum&gt;</code></pre><p>Building a specific tree-MPO such as <a href="../methods/#MPSDynamics.nearestneighbourmpo"><code>MPSDynamics.nearestneighbourmpo</code></a> adds no conceptual complexity and MPS methods have the option to built it as a <code>TreeNetwork</code>.</p><h2 id="Hamiltonian"><a class="docs-heading-anchor" href="#Hamiltonian">Hamiltonian</a><a id="Hamiltonian-1"></a><a class="docs-heading-anchor-permalink" href="#Hamiltonian" title="Permalink"></a></h2><p>In order to perform time evolution and have access to the dynamics of the many-body state a Hamiltonian needs to be specified in the form of a Matrix Product Operator (MPO) of as a tree tensor network. Either way, this can be done by using a <a href="#Build-in-Hamiltonian">Build-in Hamiltonian</a>, <a href="#Convert-a-MPO-from-ITensor">Convert a MPO from ITensor</a>, or creating a <a href="#Tailored-MPO">Tailored MPO</a>.</p><p>In the context of Open Quantum Systems, custom chain coefficients for the environment can be generated for finite temperature simulations, and/or user provided spectral densities (SDs).</p><h3 id="Build-in-Hamiltonian"><a class="docs-heading-anchor" href="#Build-in-Hamiltonian">Build-in Hamiltonian</a><a id="Build-in-Hamiltonian-1"></a><a class="docs-heading-anchor-permalink" href="#Build-in-Hamiltonian" title="Permalink"></a></h3><p>MPSDynamics provides several topical Hamiltonians directly in the form of MPO or Tree Tensor Networks such as the Ising model <a href="../methods/#MPSDynamics.isingmpo-Tuple{Int64}"><code>MPSDynamics.isingmpo</code></a>, the XYZ Hamiltonian <a href="../methods/#MPSDynamics.xyzmpo-Tuple{Int64}"><code>MPSDynamics.xyzmpo</code></a>, the Spin Boson Model <a href="../methods/#MPSDynamics.spinbosonmpo-NTuple{5, Any}"><code>MPSDynamics.spinbosonmpo</code></a>, a spin coupled to two bosonic baths <a href="../methods/#MPSDynamics.twobathspinmpo"><code>MPSDynamics.twobathspinmpo</code></a>, nearest neighbour interactions Hamiltonian <a href="../methods/#MPSDynamics.nearestneighbourmpo"><code>MPSDynamics.nearestneighbourmpo</code></a>, the independent boson model <a href="../methods/#MPSDynamics.ibmmpo-NTuple{4, Any}"><code>MPSDynamics.ibmmpo</code></a>, (non-)uniform tight-binding chain Hamiltonian <a href="../methods/#MPSDynamics.tightbindingmpo-Tuple{Int64, Int64}"><code>MPSDynamics.tightbindingmpo</code></a>.</p><h3 id="Convert-a-MPO-from-ITensor"><a class="docs-heading-anchor" href="#Convert-a-MPO-from-ITensor">Convert a MPO from ITensor</a><a id="Convert-a-MPO-from-ITensor-1"></a><a class="docs-heading-anchor-permalink" href="#Convert-a-MPO-from-ITensor" title="Permalink"></a></h3><p>The method <a href="../methods/#MPSDynamics.MPOtoVector-Tuple{ITensors.ITensorMPS.MPO}"><code>MPSDynamics.MPOtoVector</code></a> converts an ITensors chain MPO into a form compatible with MPSDynamics.</p><h3 id="Tailored-MPO"><a class="docs-heading-anchor" href="#Tailored-MPO">Tailored MPO</a><a id="Tailored-MPO-1"></a><a class="docs-heading-anchor-permalink" href="#Tailored-MPO" title="Permalink"></a></h3><p>One can also construct MPO tailored to the problem one is interested in. MPOs are fundamentally a lists of rank-4 tensors such that the right bond dimension of the nth tensor must be equal to the left bond dimension of the n+1th tensor; and the dimension of the physical bonds of the nth tensor must be equal to the corresponding physical bond on the MPS.</p><h3 id="Finite-Temperature-and-Custom-SD"><a class="docs-heading-anchor" href="#Finite-Temperature-and-Custom-SD">Finite Temperature and Custom SD</a><a id="Finite-Temperature-and-Custom-SD-1"></a><a class="docs-heading-anchor-permalink" href="#Finite-Temperature-and-Custom-SD" title="Permalink"></a></h3><p>In the T-TEDOPA framework (see <a href="../theory/#Theoretical-Background">Theoretical Background</a> for more details) finite temperature simulations are done with an effective pure state description of the system and the environment where the coupling coefficients (or the SD) is temperature-dependent.</p><p>The corresponding chain coefficients for an Ohmic or a user provided spectral density (that can thus in pratice be either at zero or finite temperature) are computed with the <a href="../methods/#MPSDynamics.chaincoeffs_finiteT"><code>chaincoeffs_finiteT</code></a>. This method is based on the <code>ORTHOPOL</code> routines<sup class="footnote-reference"><a id="citeref-Gautschi" href="#footnote-Gautschi">[Gautschi]</a></sup></p><h2 id="Observables"><a class="docs-heading-anchor" href="#Observables">Observables</a><a id="Observables-1"></a><a class="docs-heading-anchor-permalink" href="#Observables" title="Permalink"></a></h2><p>System and environment observables can be computed, as well as system-and-environment &#39;non-local&#39; observables.</p><p>Observables that will be passed to <a href="../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> to have their expectation value computated at each time step are defined with the <a href="../methods/#MPSDynamics.OneSiteObservable-Tuple{Any, Any, Any}"><code>OneSiteObservable</code></a> and <a href="../methods/#MPSDynamics.TwoSiteObservable"><code>TwoSiteObservable</code></a>.</p><p>One-site and two-site obsevables work similarly, they need to be given a name, an (pair of) operator(s) and the (list of) site(s) on which they are evaluated.</p><p>For instance one can calculated the average number of excitation in each of the <span>$N$</span> environmental modes</p><pre><code class="language-julia hljs">    ob = OneSiteObservable(&quot;chain mode occupation&quot;, numb(d), (2,N+1))</code></pre><p>It is also possible to measure composite system/environment observables, for example</p><pre><code class="language-julia hljs">    ob = TwoSiteObservable(&quot;SXdisp&quot;, sx, disp(d), [1], collect(2:N+1))</code></pre><p>which measure the correlation between the spin in the x-direction and the displacement of the bath modes.</p><p>Purely environmental &#39;non-local&#39; observables such as </p><pre><code class="language-julia hljs">    ob = TwoSiteObservable(&quot;bath coherence&quot;, crea(d), anih(d), collect(2:N+1), collect(2:N+1))</code></pre><p>that computes all the chain mode coherences <span>$\langle\hat{a}_n^\dagger\hat{a}_{m}\rangle$</span> (and the chain mode occupation when <span>$n = m$</span>).</p><p>We note that if one knows the coherences and populations of all the chain modes, it is then possible to reconstruct the populations of the normal mode environment.</p><h2 id="Time-Evolution"><a class="docs-heading-anchor" href="#Time-Evolution">Time-Evolution</a><a id="Time-Evolution-1"></a><a class="docs-heading-anchor-permalink" href="#Time-Evolution" title="Permalink"></a></h2><p>Simulation are performed when calling the <a href="../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> function where the time-evolution method should be specified.</p><p>The time-evolution methods currently implemented belong to the familly of Time-Dependent Variational Principle (TDVP). The central point of this method, in the modern tensor networks formulation, is that instead of solving the Schrödinger equation and then truncating the MPS representation of the quantum state, one can solve the equations of motion projected into a space of restricted bond dimension.  The major advantage of this method is that it naturally preserves the unitarity of the time evolution and conserves the energy (except in its two-site implementation). Three variants of TDVP are implemented in the <code>MPSDynamics.jl</code> package:</p><ul><li>one-site TDVP (TDVP1): fixed bond dimension, preserves unitarity, conserves energy, scales as <span>$\mathcal{O}(D^2 d^2 w^2 + D^3 dw + D^3 d^2 )$</span> where <span>$D$</span> is the MPS (or TTN) bond dimension, <span>$d$</span> the local dimension, <span>$w$</span> the MPO bond dimension.</li><li>two-site TDVP (TDVP2): adaptive bond dimension, breaks unitarity, scales as <span>$\mathcal{O}(D^2d^3 w^2 + D^3 d^2 w + D^3 d^3 )$</span>.</li><li>adaptive one-site TDVP (DTDVP): adaptive bond dimension, preserves unitarity, conserves energy.</li></ul><p>Local one-site and two-site observables, as well as non-local two-site observables, can be efficiently computed for each time-step of the time-evolution with the three methods.</p><h3 id="One-site-TDVP"><a class="docs-heading-anchor" href="#One-site-TDVP">One-site TDVP</a><a id="One-site-TDVP-1"></a><a class="docs-heading-anchor-permalink" href="#One-site-TDVP" title="Permalink"></a></h3><p>This implementation of the TDVP can be used for MPS and TTN. It performs one-site updates of the tensor network tensors while keeping the bond dimension fixed. Because no reduction of the dimensionality is performed (for instnce by throwing away some simgular values), it naturally preserves unitarity and hence conserves energy. The otherside of the coin is that simulation might be slower because the bond dimension is large at <em>all time</em>.</p><p>The convergence parameter of the simulation is thus the bond dimension <code>D</code>.</p><p>This method can be used with in <a href="../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> with the key word argument <code>method=:TDVP1</code>.</p><h3 id="Two-Site-TDVP"><a class="docs-heading-anchor" href="#Two-Site-TDVP">Two-Site TDVP</a><a id="Two-Site-TDVP-1"></a><a class="docs-heading-anchor-permalink" href="#Two-Site-TDVP" title="Permalink"></a></h3><p>This version of the TDVP evolves two-site tensors than can be split back into two individual site tensors by applying an SVD. The bond dimension between two neighbouring sites can then be truncated to any value <span>$≤ r$</span> by throwing away any singular values that fall below some threshold.  In this way, the MPS bond dimension will grow dynamically throughout the course of the evolution to capture entanglement, as and when it emerges. The truncation entails a loss of unitarity.  Indeed, for a nearest-neighbour Hamiltonian, applying the two-site projector will not entail a projection error, leading to a scheme that is almost identical to TEBD,wherein the error arises solely from the truncation. Furthermore, 2TDVP scales poorly with the local physical dimension, and is known to have issues with long-range interactions.</p><p>The convergence parameter is the threshold of the SVD.</p><p>This method can be used with in <a href="../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> with the key word argument <code>method=:TDVP2</code>.</p><h3 id="Adaptive-One-Site-TDVP"><a class="docs-heading-anchor" href="#Adaptive-One-Site-TDVP">Adaptive One-Site TDVP</a><a id="Adaptive-One-Site-TDVP-1"></a><a class="docs-heading-anchor-permalink" href="#Adaptive-One-Site-TDVP" title="Permalink"></a></h3><p>Ahead of any time evolution, one computes new bond dimensions of the MPS if the relative rate of change of the TDVP projection error with respect to the bond dimension is larger than a chosen precision <code>p</code>. Then, TDVP1 is performed, using projectors with sub-spaces expanded accordingly, to produce an MPS evolved by one time step with the new,increased, bond dimensions. This version is faster than TDVP1 due to the acceleration gained from having more optimised bond dimensions; the bond update step is cheap and so its cost should not normally outweigh this advantage. However, two things need to be noticed:</p><ul><li>DTDVP can get stuck into local minima from the initial time. If the bond dimension does not change during the time-evolution, consider embeding your MPS in a larger manifold with <a href="../methods/#MPSDynamics.mpsembed!-Tuple{Vector, Int64}"><code>MPSDynamics.mpsembed!</code></a> before time-evolving it.</li><li>For models with analytical solutions that are MPS of a given bond dimension (such as the independent boson model), DTDVP can overshoot the analytical bond dimension because the relative rate of change of the projection error becomes dominated by random numerical fluctuations.</li></ul><p>The convergence parameter is a threshold value <code>p</code> for the rate of change of the projection error with respect to the bond dimension.</p><p>This method can be used with in <a href="../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> with the key word argument <code>method=:DTDVP</code>.</p><h2 id="Data-Storage"><a class="docs-heading-anchor" href="#Data-Storage">Data Storage</a><a id="Data-Storage-1"></a><a class="docs-heading-anchor-permalink" href="#Data-Storage" title="Permalink"></a></h2><p>The data (i.e. observables time-series) is stored in the JLD format which is based on HDF5.</p><p>The HDF5 format is natively supported across many platforms and languages (e.g. <code>Python</code>, or <code>Mathematica</code>).</p><p>For the data to be saved to a file after a run, the keyword argument <code>save=true</code> needs to be used in <code>runsim</code>.</p><p>The directory where the data should be saved can be chosen by setting a path with the <code>savedir</code> keyword argument.</p><p>A <code>plot</code> keyword argument can also be used to choose whether plots for 1D observables will be automatically generated and saved along with the data. </p><p>Loading the data in Julia using the <a href="https://github.com/JuliaIO/JLD.jl"><code>JLD.jl</code></a> package will recover the full type information of the Julia variables that were stored.</p><pre><code class="language-julia hljs">julia&gt; using JLD
+julia&gt; A = productstatemps(physdims(H), state=[ψ, fill(unitcol(1,d), N)...]) # MPS representation of |ψ&gt;|Vacuum&gt;</code></pre><p>Alternatively, a chain with a specified number of excitation localised on one site, or delocalized accross several sites can be generated with <a href="../methods/#MPSDynamics.chainmps-Tuple{Int64, Int64, Int64}"><code>MPSDynamics.chainmps</code></a>.</p><p>Random MPS can also be generated with the <a href="../methods/#MPSDynamics.randmps"><code>randmps</code></a> method.</p><p>For the case of fermionic states (which need to be anti-symmetrized), the <a href="../methods/#MPSDynamics.electronkmps"><code>MPSDynamics.electronkmps</code></a> method generate an MPS for an electron with momentum <code>k</code>, and the <a href="../methods/#MPSDynamics.electron2kmps"><code>MPSDynamics.electron2kmps</code></a> generate an MPS with 2 electrons in k-states <code>k1</code> and <code>k2</code>.</p><h3 id="Tree-Tensor-Networks"><a class="docs-heading-anchor" href="#Tree-Tensor-Networks">Tree Tensor Networks</a><a id="Tree-Tensor-Networks-1"></a><a class="docs-heading-anchor-permalink" href="#Tree-Tensor-Networks" title="Permalink"></a></h3><p>Write a quick explanation of the how trees are structured: parents, child, nodes; and that most methods to initialize a MPS are overloaded for <code>TreeNetwork</code>. </p><h2 id="Hamiltonian"><a class="docs-heading-anchor" href="#Hamiltonian">Hamiltonian</a><a id="Hamiltonian-1"></a><a class="docs-heading-anchor-permalink" href="#Hamiltonian" title="Permalink"></a></h2><p>In order to perform time evolution and have access to the dynamics of the many-body state a Hamiltonian needs to be specified in the form of a Matrix Product Operator (MPO) of as a tree tensor network. Either way, this can be done by using a <a href="#Build-in-Hamiltonian">Build-in Hamiltonian</a>, <a href="#Convert-a-MPO-from-ITensor">Convert a MPO from ITensor</a>, or creating a <a href="#Tailored-MPO">Tailored MPO</a>.</p><p>In the context of Open Quantum Systems, custom chain coefficients for the environment can be generated for finite temperature simulations, and/or user provided spectral densities (SDs).</p><h3 id="Build-in-Hamiltonian"><a class="docs-heading-anchor" href="#Build-in-Hamiltonian">Build-in Hamiltonian</a><a id="Build-in-Hamiltonian-1"></a><a class="docs-heading-anchor-permalink" href="#Build-in-Hamiltonian" title="Permalink"></a></h3><p>MPSDynamics provides several topical Hamiltonians directly in the form of MPO or Tree Tensor Networks such as the Ising model <a href="../methods/#MPSDynamics.isingmpo-Tuple{Int64}"><code>MPSDynamics.isingmpo</code></a>, the XYZ Hamiltonian <a href="../methods/#MPSDynamics.xyzmpo-Tuple{Int64}"><code>MPSDynamics.xyzmpo</code></a>, the Spin Boson Model <a href="../methods/#MPSDynamics.spinbosonmpo-NTuple{5, Any}"><code>MPSDynamics.spinbosonmpo</code></a>, a spin coupled to two bosonic baths <code>MPSDynamics.twobathspinmpo</code>, nearest neighbour interactions Hamiltonian <code>MPSDynamics.nearestneighbourmpo</code>, the independent boson model <code>MPSDynamics.ibmmpo</code>, (non-)uniform tight-binding chain Hamiltonian <a href="../methods/#MPSDynamics.tightbindingmpo-Tuple{Int64, Int64}"><code>MPSDynamics.tightbindingmpo</code></a>.</p><h3 id="Convert-a-MPO-from-ITensor"><a class="docs-heading-anchor" href="#Convert-a-MPO-from-ITensor">Convert a MPO from ITensor</a><a id="Convert-a-MPO-from-ITensor-1"></a><a class="docs-heading-anchor-permalink" href="#Convert-a-MPO-from-ITensor" title="Permalink"></a></h3><p>The method <a href="../methods/#MPSDynamics.MPOtoVector-Tuple{ITensors.MPO}"><code>MPSDynamics.MPOtoVector</code></a> converts an ITensors chain MPO into a form compatible with MPSDynamics.</p><h3 id="Tailored-MPO"><a class="docs-heading-anchor" href="#Tailored-MPO">Tailored MPO</a><a id="Tailored-MPO-1"></a><a class="docs-heading-anchor-permalink" href="#Tailored-MPO" title="Permalink"></a></h3><p>One can also construct MPO tailored to the problem one is interested in. MPOs are fundamentally a lists of rank-4 tensors such that the right bond dimension of the nth tensor must be equal to the left bond dimension of the n+1th tensor; and the dimension of the physical bonds of the nth tensor must be equal to the corresponding physical bond on the MPS.</p><h3 id="Finite-Temperature-and-Custom-SD"><a class="docs-heading-anchor" href="#Finite-Temperature-and-Custom-SD">Finite Temperature and Custom SD</a><a id="Finite-Temperature-and-Custom-SD-1"></a><a class="docs-heading-anchor-permalink" href="#Finite-Temperature-and-Custom-SD" title="Permalink"></a></h3><p>In the T-TEDOPA framework (see <a href="../theory/#Theoretical-Background">Theoretical Background</a> for more details) finite temperature simulations are done with an effective pure state description of the system and the environment where the coupling coefficients (or the SD) is temperature-dependent.</p><p>The corresponding chain coefficients for an Ohmic or a user provided spectral density (that can thus in pratice be either at zero or finite temperature) are computed with the <code>[</code>chaincoeffs_finiteT<code>](@ref)</code>. This method is based on the <code>ORTHOPOL</code> routines<sup class="footnote-reference"><a id="citeref-Gautschi" href="#footnote-Gautschi">[Gautschi]</a></sup></p><h2 id="Observables"><a class="docs-heading-anchor" href="#Observables">Observables</a><a id="Observables-1"></a><a class="docs-heading-anchor-permalink" href="#Observables" title="Permalink"></a></h2><p>System and environment observables can be computed, as well as system-and-environment &#39;non-local&#39; observables.</p><p>Observables that will be passed to <code>MPSDynamics.runsim</code>(@ref) to have their expectation value computated at each time step are defined with the <a href="../methods/#MPSDynamics.OneSiteObservable-Tuple{Any, Any, Any}"><code>OneSiteObservable</code></a> and <code>[</code>TwoSiteObservable<code>](@ref)</code>.</p><p>One-site and two-site obsevables work similarly, they need to be given a name, an (pair of) operator(s) and the (list of) site(s) on which they are evaluated.</p><p>For instance one can calculated the average number of excitation in each of the <span>$N$</span> environmental modes</p><pre><code class="language-julia">    ob = OneSiteObservable(&quot;chain mode occupation&quot;, numb(d), (2,N+1))</code></pre><p>It is also possible to measure composite system/environment observables, for example</p><pre><code class="language-julia">    ob = TwoSiteObservable(&quot;SXdisp&quot;, sx, disp(d), [1], collect(2:N+1))</code></pre><p>which measure the correlation between the spin in the x-direction and the displacement of the bath modes.</p><p>Purely environmental &#39;non-local&#39; observables such as </p><pre><code class="language-julia">    ob = TwoSiteObservable(&quot;bath coherence&quot;, crea(d), anih(d), collect(2:N+1), collect(2:N+1))</code></pre><p>that computes all the chain mode coherences <span>$\langle\hat{a}_n^\dagger\hat{a}_{m}\rangle$</span> (and the chain mode occupation when <span>$n = m$</span>).</p><p>We note that if one knows the coherences and populations of all the chain modes, it is then possible to reconstruct the populations of the normal mode environment.</p><h2 id="Time-Evolution"><a class="docs-heading-anchor" href="#Time-Evolution">Time-Evolution</a><a id="Time-Evolution-1"></a><a class="docs-heading-anchor-permalink" href="#Time-Evolution" title="Permalink"></a></h2><p>Simulation are performed when calling the <a href="../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> function where the time-evolution method should be specified.</p><p>The time-evolution methods currently implemented belong to the familly of Time-Dependent Variational Principle (TDVP). The central point of this method, in the modern tensor networks formulation, is that instead of solving the Schrödinger equation and then truncating the MPS representation of the quantum state, one can solve the equations of motion projected into a space of restricted bond dimension.  The major advantage of this method is that it naturally preserves the unitarity of the time evolution and conserves the energy (except in its two-site implementation). Three variants of TDVP are implemented in the <code>MPSDynamics.jl</code> package:</p><ul><li>one-site TDVP (TDVP1): fixed bond dimension, preserves unitarity, conserves energy, scales as <span>$\mathcal{O}(D^2 d^2 w^2 + D^3 dw + D^3 d^2 )$</span> where <span>$D$</span> is the MPS (or TTN) bond dimension, <span>$d$</span> the local dimension, <span>$w$</span> the MPO bond dimension.</li><li>two-site TDVP (TDVP2): adaptive bond dimension, breaks unitarity, scales as <span>$\mathcal{O}(D^2d^3 w^2 + D^3 d^2 w + D^3 d^3 )$</span>.</li><li>adaptive one-site TDVP (DTDVP): adaptive bond dimension, preserves unitarity, conserves energy.</li></ul><p>Local one-site and two-site observables, as well as non-local two-site observables, can be efficiently computed for each time-step of the time-evolution with the three methods.</p><h3 id="One-site-TDVP"><a class="docs-heading-anchor" href="#One-site-TDVP">One-site TDVP</a><a id="One-site-TDVP-1"></a><a class="docs-heading-anchor-permalink" href="#One-site-TDVP" title="Permalink"></a></h3><p>This implementation of the TDVP can be used for MPS and TTN. It performs one-site updates of the tensor network tensors while keeping the bond dimension fixed. Because no reduction of the dimensionality is performed (for instnce by throwing away some simgular values), it naturally preserves unitarity and hence conserves energy. The otherside of the coin is that simulation might be slower because the bond dimension is large at <em>all time</em>.</p><p>The convergence parameter of the simulation is thus the bond dimension <code>D</code>.</p><p>This method can be used with in <a href="../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> with the key word argument <code>method=:TDVP1</code>.</p><h3 id="Two-Site-TDVP"><a class="docs-heading-anchor" href="#Two-Site-TDVP">Two-Site TDVP</a><a id="Two-Site-TDVP-1"></a><a class="docs-heading-anchor-permalink" href="#Two-Site-TDVP" title="Permalink"></a></h3><p>This version of the TDVP evolves two-site tensors than can be split back into two individual site tensors by applying an SVD. The bond dimension between two neighbouring sites can then be truncated to any value <span>$≤ r$</span> by throwing away any singular values that fall below some threshold.  In this way, the MPS bond dimension will grow dynamically throughout the course of the evolution to capture entanglement, as and when it emerges. The truncation entails a loss of unitarity.  Indeed, for a nearest-neighbour Hamiltonian, applying the two-site projector will not entail a projection error, leading to a scheme that is almost identical to TEBD,wherein the error arises solely from the truncation. Furthermore, 2TDVP scales poorly with the local physical dimension, and is known to have issues with long-range interactions.</p><p>The convergence parameter is the threshold of the SVD.</p><p>This method can be used with in <a href="../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> with the key word argument <code>method=:TDVP2</code>.</p><h3 id="Adaptive-One-Site-TDVP"><a class="docs-heading-anchor" href="#Adaptive-One-Site-TDVP">Adaptive One-Site TDVP</a><a id="Adaptive-One-Site-TDVP-1"></a><a class="docs-heading-anchor-permalink" href="#Adaptive-One-Site-TDVP" title="Permalink"></a></h3><p>Ahead of any time evolution, one computes new bond dimensions of the MPS if the relative rate of change of the TDVP projection error with respect to the bond dimension is larger than a chosen precision <code>p</code>. Then, TDVP1 is performed, using projectors with sub-spaces expanded accordingly, to produce an MPS evolved by one time step with the new,increased, bond dimensions. This version is faster than TDVP1 due to the acceleration gained from having more optimised bond dimensions; the bond update step is cheap and so its cost should not normally outweigh this advantage. However, two things need to be noticed:</p><ul><li>DTDVP can get stuck into local minima from the initial time. If the bond dimension does not change during the time-evolution, consider embeding your MPS in a larger manifold with <a href="@ref"><code>mpsembed!</code></a> before time-evolving it.</li><li>For models with analytical solutions that are MPS of a given bond dimension (such as the independent boson model), DTDVP can overshoot the analytical bond dimension because the relative rate of change of the projection error becomes dominated by random numerical fluctuations.</li></ul><p>The convergence parameter is a threshold value <code>p</code> for the rate of change of the projection error with respect to the bond dimension.</p><p>This method can be used with in <a href="../methods/#MPSDynamics.runsim-NTuple{4, Any}"><code>MPSDynamics.runsim</code></a> with the key word argument <code>method=:DTDVP</code>.</p><h2 id="Data-Storage"><a class="docs-heading-anchor" href="#Data-Storage">Data Storage</a><a id="Data-Storage-1"></a><a class="docs-heading-anchor-permalink" href="#Data-Storage" title="Permalink"></a></h2><p>The data (i.e. observables time-series) is stored in the JLD format which is based on HDF5.</p><p>The HDF5 format is natively supported across many platforms and languages (e.g. <code>Python</code>, or <code>Mathematica</code>).</p><p>For the data to be saved to a file after a run, the keyword argument <code>save=true</code> needs to be used in <code>runsim</code>.</p><p>The directory where the data should be saved can be chosen by setting a path with the <code>savedir</code> keyword argument.</p><p>A <code>plot</code> keyword argument can also be used to choose whether plots for 1D observables will be automatically generated and saved along with the data. </p><p>Loading the data in Julia using the <a href="https://github.com/JuliaIO/JLD.jl"><code>JLD.jl</code></a> package will recover the full type information of the Julia variables that were stored.</p><pre><code class="language-julia">julia&gt; using JLD
 
 julia&gt; dat = load(&quot;filename.jld&quot;)
 Dict{String, Any} with 2 entries:
   &quot;parameters&quot; =&gt; Dict{String, Any}(&quot;tmax&quot;=&gt;0.2, &quot;method&quot;=&gt;:DTDVP, &quot;dt&quot;=&gt;0.0005…
-  &quot;data&quot;       =&gt; Dict{String, Any}(&quot;bonddims&quot;=&gt;[1 1 … 1 1; 1 2 … 2 2; … ; 1 1 …</code></pre><p>If the data is loaded in an variable <code>dat</code> the structure is the folowing:</p><ul><li>the <code>parameters</code> entry contains a dictionary where all the simulation parameters are stored</li><li>the <code>data</code> entry contains a dictionary where are stored simulation time, the observables and (whenever relevent) the bond dimension of the state at each time steps. </li></ul><pre><code class="language-julia hljs">julia&gt; dat[&quot;parameters&quot;]
+  &quot;data&quot;       =&gt; Dict{String, Any}(&quot;bonddims&quot;=&gt;[1 1 … 1 1; 1 2 … 2 2; … ; 1 1 …</code></pre><p>If the data is loaded in an variable <code>dat</code> the structure is the folowing:</p><ul><li>the <code>parameters</code> entry contains a dictionary where all the simulation parameters are stored</li><li>the <code>data</code> entry contains a dictionary where are stored simulation time, the observables and (whenever relevent) the bond dimension of the state at each time steps. </li></ul><pre><code class="language-julia">julia&gt; dat[&quot;parameters&quot;]
 Dict{String, Any} with 11 entries:
   &quot;tmax&quot;       =&gt; 0.2
   &quot;method&quot;     =&gt; :DTDVP
@@ -42,4 +34,4 @@
   &quot;nchain&quot;   =&gt; [0.0 0.23466 … 1.84319 1.76098; 0.0 0.00231507 … 0.83105 0.9033…
   &quot;sy&quot;       =&gt; [0.0, -0.0133489, -0.0588887, -0.0858181, -0.0759996, -0.048539…
   &quot;times&quot;    =&gt; [0.0, 0.0005, 0.001, 0.0015, 0.002, 0.0025, 0.003, 0.0035, 0.00…
-</code></pre><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-Gautschi"><a class="tag is-link" href="#citeref-Gautschi">Gautschi</a><blockquote><p>Gautschi, W. Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 20, 21–62 (1994).</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Introduction</a><a class="docs-footer-nextpage" href="../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 22 November 2024 16:28">Friday 22 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+</code></pre><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-Gautschi"><a class="tag is-link" href="#citeref-Gautschi">Gautschi</a><blockquote><p>Gautschi, W. Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 20, 21–62 (1994).</p></blockquote></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Introduction</a><a class="docs-footer-nextpage" href="../nutshell/">In a nutshell: MPSDynamics.jl for open quantum systems »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 25 November 2024 10:51">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/versions.js b/versions.js
index 1f19bed..6c297c1 100644
--- a/versions.js
+++ b/versions.js
@@ -4,5 +4,3 @@ var DOC_VERSIONS = [
   "v1.0",
   "docs",
 ];
-var DOCUMENTER_NEWEST = "v1.1";
-var DOCUMENTER_STABLE = "stable";