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-<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Convergence checks · 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><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>Convergence checks</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Convergence checks</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/convergence.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="Convergence-checks"><a class="docs-heading-anchor" href="#Convergence-checks">Convergence checks</a><a id="Convergence-checks-1"></a><a class="docs-heading-anchor-permalink" href="#Convergence-checks" title="Permalink"></a></h1><p>In this page we give some tips on how to obtain converged simulations. </p><h2 id="Chain-length-N"><a class="docs-heading-anchor" href="#Chain-length-N">Chain length <span>$N$</span></a><a id="Chain-length-N-1"></a><a class="docs-heading-anchor-permalink" href="#Chain-length-N" title="Permalink"></a></h2><p>During a numerical simulation, a truncation on the number of chain modes (and therefore chain length) will be introduced, in order to work with a chain of finite length instead of a semi-infinite one. This truncation on chain modes, let us say <span>$N$</span>, introduces a sampling on the modes in the original environment.</p><h2 id="Local-dimension-d"><a class="docs-heading-anchor" href="#Local-dimension-d">Local dimension <span>$d$</span></a><a id="Local-dimension-d-1"></a><a class="docs-heading-anchor-permalink" href="#Local-dimension-d" title="Permalink"></a></h2><p>A second truncation has to be introduced on the local dimension <span>$d$</span> of each tensor comprising the MPS that will be dynamically evolved. The local dimension coincides with the truncated number of Fock&#39;s states of the chain mode&#39;s Hilbert space: each site of the MPS, can be filled up by at most <span>$d$</span> excitations. Since in this case the excitations are bosonic, the Pauli exclusion principle does not hold, and this number should in principle be infinite.</p></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 12:59">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Convergence checks · 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><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>Convergence checks</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Convergence checks</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/shareloqs/MPSDynamics/blob/master/docs/src/convergence.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="Convergence-checks"><a class="docs-heading-anchor" href="#Convergence-checks">Convergence checks</a><a id="Convergence-checks-1"></a><a class="docs-heading-anchor-permalink" href="#Convergence-checks" title="Permalink"></a></h1><p>In this section, we analyze the numerical accuracy of our simulations by outlining the necessary approximations and providing practical guidelines to ensure convergence of results.</p><h2 id="Chain-length-N"><a class="docs-heading-anchor" href="#Chain-length-N">Chain length <span>$N$</span></a><a id="Chain-length-N-1"></a><a class="docs-heading-anchor-permalink" href="#Chain-length-N" title="Permalink"></a></h2><p>During a numerical simulation, a truncation on the number of chain modes (and therefore chain length) will be introduced, in order to work with a chain of finite length instead of a semi-infinite one. This truncation on chain modes, let us say <span>$N$</span>, introduces a sampling on the modes in the original environment. </p><h2 id="Local-dimension-d"><a class="docs-heading-anchor" href="#Local-dimension-d">Local dimension <span>$d$</span></a><a id="Local-dimension-d-1"></a><a class="docs-heading-anchor-permalink" href="#Local-dimension-d" title="Permalink"></a></h2><p>Another critical truncation is imposed on the local dimension <span>$d$</span> of each tensor in the MPS that undergoes dynamic evolution. The local dimension dd corresponds to the number of Fock states retained in the Hilbert space of each chain mode. Since harmonic oscillators are, in principle, infinite-dimensional systems, truncating their Hilbert space to a finite <span>$d$</span> is necessary for numerical computations. The choice of dd determines the maximum number of excitations per site in the MPS and must be carefully tuned to capture the relevant physics while ensuring numerical convergence.</p></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 13:23">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>