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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 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>
+<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><h3 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></h3><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>, corresponds to a sampling on the modes in the original environment. </p><p>The length chain is connected to the simulation time in the following way: excitations injected from the system form a wave-front that travels along the chain. Any perturbation to the initial state outside of this wave-front are exponentially suppressed. It is thus natural to truncate in the chain basis, making sure that this truncation happens <strong>beyond</strong> the expanding wave-front, as in this way the sampling error is also exponentially small. </p><p>To make sure that, for a given simulation time, and for a given set of chain coefficients, the chain-length is the optimal one, we can do as follows. We set the total simulation time, and we compute the chain coefficients for a non-optimal chain, much longer than needed.</p><pre><code class="language-none">T = 100 
+N_huge = 1000
+cpars = chaincoeffs_flat(N_huge, αchain; ωc = 1.0)</code></pre><p>We can then use the built-in function <code>findchainlength</code>, that computes the required chain length <code>N_opt</code> from the propagation speed on the chain sites (given by the hopping coefficients <span>$t_n$</span>):</p><pre><code class="language-none">N_opt = findchainlength(T, cpars; eps=10^-4, verbose=false)</code></pre><h3 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></h3><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 <span>$d$</span> 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><h3 id><a class="docs-heading-anchor" href="#"></a><a id="-1"></a><a class="docs-heading-anchor-permalink" href="#" title="Permalink"></a></h3></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 14:49">Monday 25 November 2024</span>. Using Julia version 1.7.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>