build based on fe64f55

previews/PR186/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-10-31T06:14:31","documenter_version":"1.7.0"}}
+{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-10-31T06:54:56","documenter_version":"1.7.0"}}

previews/PR186/manual/counterterms/index.html

@@ -15,4 +15,4 @@
  \frac{(\delta\lambda)^n}{n!} \partial^{n}_\lambda V^m_\lambda = \left(\frac{\lambda}{8\pi}\right)^n \frac{m^{(n)}}{n!} V^{m+n}_\lambda = \binom{m + n - 1}{n} V^m_\lambda \delta^n_\lambda,
 \end{aligned}\]</p><p>where <span>$m^{(n)} = \prod_{i=0}^{n-1} (m + i)$</span> is a rising factorial and</p><p class="math-container">\[ \binom{m + n - 1}{n} = \frac{(m + n - 1)!}{n! (m-1)!} = \frac{m^{(n)}}{n!}.\]</p><p>Thus,</p><p class="math-container">\[\begin{aligned}
  V^m &amp; = \left(\sum^\infty_{n=0} \frac{(\delta\lambda)^n}{n!} \partial^n_\lambda V_\lambda\right)^m = V^m_\lambda \left(1 - \delta_\lambda\right)^{-m} = \sum^\infty_{n=0} \binom{m+n-1}{n} V^m_\lambda \delta^n_\lambda = \sum^\infty_{n=0} \frac{(\delta\lambda)^{n}}{n!} \partial^{n}_\lambda V^m_\lambda.
-\end{aligned}\]</p><p>Since the order of differentiation w.r.t. <span>$\mu$</span> and <span>$\lambda$</span> does not matter, it follows that a general diagram <span>$\mathcal{D}[g, V] \sim g^n V^m$</span> may be represented either by pre-expanding <span>$g[g_\mu]$</span> and <span>$V[V_\lambda]$</span> and collecting terms, or by directly evaluating terms in the Taylor series for <span>$\mathcal{D}[g_\mu, V_\lambda]$</span>; this codebase uses the latter approach.</p><h2 id="Evaluation-of-interaction-counterterms"><a class="docs-heading-anchor" href="#Evaluation-of-interaction-counterterms">Evaluation of interaction counterterms</a><a id="Evaluation-of-interaction-counterterms-1"></a><a class="docs-heading-anchor-permalink" href="#Evaluation-of-interaction-counterterms" title="Permalink"></a></h2><p>An example of the interaction counterterm evaluation for a diagram with <span>$n_\lambda = 3$</span> and <span>$m$</span> interaction lines. Since the Julia implementation evaluates the interaction counterterms of a given diagram as <span>$\frac{(-\lambda)^n}{n!}\partial^n_\lambda V^m_\lambda$</span>, we pick up an extra factor of <span>$l!$</span> on each <span>$l$</span>th-order derivative in the chain rule.</p><p><img src="../../assets/derivative_example.svg" alt="An example of the representation of interaction counterterm diagrams via differentiation."/></p><h2 id="Benchmark-of-counterterms-in-the-UEG"><a class="docs-heading-anchor" href="#Benchmark-of-counterterms-in-the-UEG">Benchmark of counterterms in the UEG</a><a id="Benchmark-of-counterterms-in-the-UEG-1"></a><a class="docs-heading-anchor-permalink" href="#Benchmark-of-counterterms-in-the-UEG" title="Permalink"></a></h2><p>As a concrete example, we have evaluated the individual diagrams and associated counterterms entering the RPT series for the total density <span>$n[g_\mu, V_\lambda]$</span> in units of the non-interacting density <span>$n_0$</span>. The diagrams/counterterms are denoted by partitions <span>$\mathcal{P} \equiv (n_{\text{loop}}, n_\mu, n_\lambda)$</span> indicating the total loop order and number of <span>$\mu$</span> and <span>$\lambda$</span> derivatives.</p><h3 id="3D-UEG"><a class="docs-heading-anchor" href="#3D-UEG">3D UEG</a><a id="3D-UEG-1"></a><a class="docs-heading-anchor-permalink" href="#3D-UEG" title="Permalink"></a></h3><p>For this benchmark, we take <span>$r_s = 1$</span>, <span>$\beta = 40 \epsilon_F$</span>, and <span>$\lambda = 0.6$</span>. All partitions contributing up to 4th order are included, as well as some selected partitions at 5th and 6th order.</p><table><tr><th style="text-align: center"><span>$(n_{\text{loop}}, n_\lambda, n_\mu)$</span></th><th style="text-align: center"><span>$n / n_0$</span></th></tr><tr><td style="text-align: center">(1, 0, 1)</td><td style="text-align: center">0.40814(16)</td></tr><tr><td style="text-align: center">(1, 0, 2)</td><td style="text-align: center">0.02778(21)</td></tr><tr><td style="text-align: center">(1, 0, 3)</td><td style="text-align: center">-0.00096(60)</td></tr><tr><td style="text-align: center">(2, 0, 0)</td><td style="text-align: center">0.28853(12)</td></tr><tr><td style="text-align: center">(2, 1, 0)</td><td style="text-align: center">0.07225(3)</td></tr><tr><td style="text-align: center">(2, 2, 0)</td><td style="text-align: center">0.02965(2)</td></tr><tr><td style="text-align: center">(2, 0, 1)</td><td style="text-align: center">0.09774(30)</td></tr><tr><td style="text-align: center">(2, 1, 1)</td><td style="text-align: center">0.01594(10)</td></tr><tr><td style="text-align: center">(2, 0, 2)</td><td style="text-align: center">0.00240(130)</td></tr><tr><td style="text-align: center">(3, 0, 0)</td><td style="text-align: center">0.10027(37)</td></tr><tr><td style="text-align: center">(3, 1, 0)</td><td style="text-align: center">0.04251(21)</td></tr><tr><td style="text-align: center">(3, 0, 1)</td><td style="text-align: center">0.02600(150)</td></tr><tr><td style="text-align: center">(4, 0, 0)</td><td style="text-align: center">0.00320(130)</td></tr><tr><td style="text-align: center">(2, 1, 2)</td><td style="text-align: center">-0.00111(18)</td></tr><tr><td style="text-align: center">(2, 0, 3)</td><td style="text-align: center">-0.00430(150)</td></tr><tr><td style="text-align: center">(3, 2, 0)</td><td style="text-align: center">0.02241(8)</td></tr><tr><td style="text-align: center">(3, 3, 0)</td><td style="text-align: center">0.01429(7)</td></tr></table></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../">« Home</a><a class="docs-footer-nextpage" href="../feynman_rule/">Feynman Rules »</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 1.7.0 on <span class="colophon-date" title="Thursday 31 October 2024 06:14">Thursday 31 October 2024</span>. Using Julia version 1.11.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{aligned}\]</p><p>Since the order of differentiation w.r.t. <span>$\mu$</span> and <span>$\lambda$</span> does not matter, it follows that a general diagram <span>$\mathcal{D}[g, V] \sim g^n V^m$</span> may be represented either by pre-expanding <span>$g[g_\mu]$</span> and <span>$V[V_\lambda]$</span> and collecting terms, or by directly evaluating terms in the Taylor series for <span>$\mathcal{D}[g_\mu, V_\lambda]$</span>; this codebase uses the latter approach.</p><h2 id="Evaluation-of-interaction-counterterms"><a class="docs-heading-anchor" href="#Evaluation-of-interaction-counterterms">Evaluation of interaction counterterms</a><a id="Evaluation-of-interaction-counterterms-1"></a><a class="docs-heading-anchor-permalink" href="#Evaluation-of-interaction-counterterms" title="Permalink"></a></h2><p>An example of the interaction counterterm evaluation for a diagram with <span>$n_\lambda = 3$</span> and <span>$m$</span> interaction lines. Since the Julia implementation evaluates the interaction counterterms of a given diagram as <span>$\frac{(-\lambda)^n}{n!}\partial^n_\lambda V^m_\lambda$</span>, we pick up an extra factor of <span>$l!$</span> on each <span>$l$</span>th-order derivative in the chain rule.</p><p><img src="../../assets/derivative_example.svg" alt="An example of the representation of interaction counterterm diagrams via differentiation."/></p><h2 id="Benchmark-of-counterterms-in-the-UEG"><a class="docs-heading-anchor" href="#Benchmark-of-counterterms-in-the-UEG">Benchmark of counterterms in the UEG</a><a id="Benchmark-of-counterterms-in-the-UEG-1"></a><a class="docs-heading-anchor-permalink" href="#Benchmark-of-counterterms-in-the-UEG" title="Permalink"></a></h2><p>As a concrete example, we have evaluated the individual diagrams and associated counterterms entering the RPT series for the total density <span>$n[g_\mu, V_\lambda]$</span> in units of the non-interacting density <span>$n_0$</span>. The diagrams/counterterms are denoted by partitions <span>$\mathcal{P} \equiv (n_{\text{loop}}, n_\mu, n_\lambda)$</span> indicating the total loop order and number of <span>$\mu$</span> and <span>$\lambda$</span> derivatives.</p><h3 id="3D-UEG"><a class="docs-heading-anchor" href="#3D-UEG">3D UEG</a><a id="3D-UEG-1"></a><a class="docs-heading-anchor-permalink" href="#3D-UEG" title="Permalink"></a></h3><p>For this benchmark, we take <span>$r_s = 1$</span>, <span>$\beta = 40 \epsilon_F$</span>, and <span>$\lambda = 0.6$</span>. All partitions contributing up to 4th order are included, as well as some selected partitions at 5th and 6th order.</p><table><tr><th style="text-align: center"><span>$(n_{\text{loop}}, n_\lambda, n_\mu)$</span></th><th style="text-align: center"><span>$n / n_0$</span></th></tr><tr><td style="text-align: center">(1, 0, 1)</td><td style="text-align: center">0.40814(16)</td></tr><tr><td style="text-align: center">(1, 0, 2)</td><td style="text-align: center">0.02778(21)</td></tr><tr><td style="text-align: center">(1, 0, 3)</td><td style="text-align: center">-0.00096(60)</td></tr><tr><td style="text-align: center">(2, 0, 0)</td><td style="text-align: center">0.28853(12)</td></tr><tr><td style="text-align: center">(2, 1, 0)</td><td style="text-align: center">0.07225(3)</td></tr><tr><td style="text-align: center">(2, 2, 0)</td><td style="text-align: center">0.02965(2)</td></tr><tr><td style="text-align: center">(2, 0, 1)</td><td style="text-align: center">0.09774(30)</td></tr><tr><td style="text-align: center">(2, 1, 1)</td><td style="text-align: center">0.01594(10)</td></tr><tr><td style="text-align: center">(2, 0, 2)</td><td style="text-align: center">0.00240(130)</td></tr><tr><td style="text-align: center">(3, 0, 0)</td><td style="text-align: center">0.10027(37)</td></tr><tr><td style="text-align: center">(3, 1, 0)</td><td style="text-align: center">0.04251(21)</td></tr><tr><td style="text-align: center">(3, 0, 1)</td><td style="text-align: center">0.02600(150)</td></tr><tr><td style="text-align: center">(4, 0, 0)</td><td style="text-align: center">0.00320(130)</td></tr><tr><td style="text-align: center">(2, 1, 2)</td><td style="text-align: center">-0.00111(18)</td></tr><tr><td style="text-align: center">(2, 0, 3)</td><td style="text-align: center">-0.00430(150)</td></tr><tr><td style="text-align: center">(3, 2, 0)</td><td style="text-align: center">0.02241(8)</td></tr><tr><td style="text-align: center">(3, 3, 0)</td><td style="text-align: center">0.01429(7)</td></tr></table></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../">« Home</a><a class="docs-footer-nextpage" href="../feynman_rule/">Feynman Rules »</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 1.7.0 on <span class="colophon-date" title="Thursday 31 October 2024 06:54">Thursday 31 October 2024</span>. Using Julia version 1.11.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

previews/PR186/manual/feynman_rule/index.html

@@ -8,4 +8,4 @@
 \end{aligned}\]</p><p>The sign of <span>$\Gamma^{(3)}$</span> diagram is given by <span>$(-1)^{n_v} \xi^{n_F}$</span>.</p><h2 id="Feynman-Rules-for-the-4-point-Vertex-Function"><a class="docs-heading-anchor" href="#Feynman-Rules-for-the-4-point-Vertex-Function">Feynman Rules for the 4-point Vertex Function</a><a id="Feynman-Rules-for-the-4-point-Vertex-Function-1"></a><a class="docs-heading-anchor-permalink" href="#Feynman-Rules-for-the-4-point-Vertex-Function" title="Permalink"></a></h2><p>The 4-point vertex function is related to the 3-point vertex function through an equation,</p><p class="math-container">\[\Gamma^{(3)}_{4,y,x} = \xi \cdot G_{4,s} \cdot G_{t, 4} \cdot \Gamma^{(4)}_{s, t, y, x},\]</p><p>where the indices <span>$x, y, s, t$</span> could be different from diagrams to diagrams.</p><p><img src="../../assets/diagrams/gamma4.svg" alt="Diagrammatic expansion of the 4-point vertex function."/></p><p>The diagram weights are given by,</p><p class="math-container">\[\begin{aligned}
 \Gamma^{(4)}= &amp; (-1) V_{56}^{\text{direct}} + (-1)\xi V_{56}^{exchange}\\
 +&amp;(-1)^2 \xi V_{56} V_{78} g_{58} g_{85}+(-1)^2 V_{56} V_{78}+\cdots,
-\end{aligned}\]</p><p>where we used the identity <span>$\xi^2 = 1$</span>.</p><p>The sign of <span>$\Gamma^{(4)}$</span> diagram is given by <span>$(-1)^{n_v} \xi^{n_F}$</span> multiplied with a sign from the permutation of the external legs.</p><h2 id="Feynman-Rules-for-the-Susceptibility"><a class="docs-heading-anchor" href="#Feynman-Rules-for-the-Susceptibility">Feynman Rules for the Susceptibility</a><a id="Feynman-Rules-for-the-Susceptibility-1"></a><a class="docs-heading-anchor-permalink" href="#Feynman-Rules-for-the-Susceptibility" title="Permalink"></a></h2><p>The susceptibility can be derived from <span>$\Gamma^{(4)}$</span>.</p><p class="math-container">\[\chi_{1,2} \equiv \left&lt;\mathcal{T} n_1 n_2\right&gt;_{\text{connected}} = \xi G_{1,2} G_{2, 1} + \xi G_{1,s} G_{t, 1} \Gamma^{(4)}_{s, t, y, x} G_{2,y} G_{x, 2}\]</p><p><img src="../../assets/diagrams/susceptibility.svg" alt="Diagrammatic expansion of the susceptibility."/></p><p>We define the polarization <span>$P$</span> as the one-interaction irreducible (or proper) vertex function,</p><p class="math-container">\[\chi^{-1} = P^{-1} + V,\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../counterterms/">« Evaluation of counterterms</a><a class="docs-footer-nextpage" href="../interaction/">Interaction/Scattering-Amplitude Convention »</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 1.7.0 on <span class="colophon-date" title="Thursday 31 October 2024 06:14">Thursday 31 October 2024</span>. Using Julia version 1.11.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{aligned}\]</p><p>where we used the identity <span>$\xi^2 = 1$</span>.</p><p>The sign of <span>$\Gamma^{(4)}$</span> diagram is given by <span>$(-1)^{n_v} \xi^{n_F}$</span> multiplied with a sign from the permutation of the external legs.</p><h2 id="Feynman-Rules-for-the-Susceptibility"><a class="docs-heading-anchor" href="#Feynman-Rules-for-the-Susceptibility">Feynman Rules for the Susceptibility</a><a id="Feynman-Rules-for-the-Susceptibility-1"></a><a class="docs-heading-anchor-permalink" href="#Feynman-Rules-for-the-Susceptibility" title="Permalink"></a></h2><p>The susceptibility can be derived from <span>$\Gamma^{(4)}$</span>.</p><p class="math-container">\[\chi_{1,2} \equiv \left&lt;\mathcal{T} n_1 n_2\right&gt;_{\text{connected}} = \xi G_{1,2} G_{2, 1} + \xi G_{1,s} G_{t, 1} \Gamma^{(4)}_{s, t, y, x} G_{2,y} G_{x, 2}\]</p><p><img src="../../assets/diagrams/susceptibility.svg" alt="Diagrammatic expansion of the susceptibility."/></p><p>We define the polarization <span>$P$</span> as the one-interaction irreducible (or proper) vertex function,</p><p class="math-container">\[\chi^{-1} = P^{-1} + V,\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../counterterms/">« Evaluation of counterterms</a><a class="docs-footer-nextpage" href="../interaction/">Interaction/Scattering-Amplitude Convention »</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 1.7.0 on <span class="colophon-date" title="Thursday 31 October 2024 06:54">Thursday 31 October 2024</span>. Using Julia version 1.11.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>