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88 lines
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HTML
88 lines
4.2 KiB
HTML
---
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title: Particle-Hole Symmetry
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excerpt:
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layout: none
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---
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<!DOCTYPE html>
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<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
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<head>
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<meta charset="utf-8" />
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<meta name="generator" content="pandoc" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
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<title>Particle-Hole Symmetry</title>
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<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
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<script src="/assets/js/thesis_scrollspy.js"></script>
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<link rel="stylesheet" href="/assets/css/styles.css">
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<body>
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<!--Capture the table of contents from pandoc as a jekyll variable -->
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{% capture tableOfContents %}
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<br>
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<nav aria-label="Table of Contents" class="page-table-of-contents">
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<ul>
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<li><a href="#chap:appendices" id="toc-chap:appendices">Appendices</a></li>
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<li><a href="#evaluation-of-the-fermion-free-energy" id="toc-evaluation-of-the-fermion-free-energy">Evaluation of the Fermion Free Energy</a></li>
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</ul>
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{% endcapture %}
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<!-- Give the table of contents to header as a variable so it can be put into the sidebar-->
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{% include header.html extra=tableOfContents %}
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<main>
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<!-- Table of Contents -->
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<!-- <nav id="TOC" role="doc-toc">
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<ul>
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<li><a href="#chap:appendices" id="toc-chap:appendices">Appendices</a></li>
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<li><a href="#evaluation-of-the-fermion-free-energy" id="toc-evaluation-of-the-fermion-free-energy">Evaluation of the Fermion Free Energy</a></li>
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</ul>
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</nav>
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-->
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<!-- Main Page Body -->
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<div id="page-header">
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<p>Appendices</p>
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<hr />
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</div>
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<section id="chap:appendices" class="level1">
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<h1>Appendices</h1>
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</section>
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<section id="evaluation-of-the-fermion-free-energy" class="level1">
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<h1>Evaluation of the Fermion Free Energy</h1>
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<p>There are <span class="math inline">\(2^N\)</span> possible ion configurations <span class="math inline">\(\{ n_i \}\)</span>, we define <span class="math inline">\(n^k_i\)</span> to be the occupation of the ith site of the kth configuration. The quantum part of the free energy can then be defined through the quantum partition function <span class="math inline">\(\mathcal{Z}^k\)</span> associated with each ionic state <span class="math inline">\(n^k_i\)</span>: <span class="math display">\[\begin{aligned}
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F^k &= -1/\beta \ln{\mathcal{Z}^k} \\
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\end{aligned}\]</span> % Such that the overall partition function is: <span class="math display">\[\begin{aligned}
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\mathcal{Z} &= \sum_k e^{- \beta H^k} Z^k \\
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&= \sum_k e^{-\beta (H^k + F^k)} \\
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\end{aligned}\]</span></p>
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<p>Because fermions are limited to occupation numbers of 0 or 1 <span class="math inline">\(Z^k\)</span> simplifies nicely. If <span class="math inline">\(m^j_i = \{0,1\}\)</span> is defined as the occupation of the level with energy <span class="math inline">\(\epsilon^k_i\)</span> then the partition function is a sum over all the occupation states labelled by j: <span class="math display">\[\begin{aligned}
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Z^k &= \mathrm{Tr} e^{-\beta F^k} = \sum_j e^{-\beta \sum_i m^j_i \epsilon^k_i}\\
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&= \sum_j \prod_i e^{- \beta m^j_i \epsilon^k_i}= \prod_i \sum_j e^{- \beta m^j_i \epsilon^k_i}\\
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&= \prod_i (1 + e^{- \beta \epsilon^k_i})\\
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F^k &= -1/\beta \sum_k \ln{(1 + e^{- \beta \epsilon^k_i})}
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\end{aligned}\]</span> % Observables can then be calculated from the partition function, for examples the occupation numbers:</p>
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<p><span class="math display">\[\begin{aligned}
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\langle N \rangle &= \frac{1}{\beta} \frac{1}{Z} \frac{\partial Z}{\partial \mu} = - \frac{\partial F}{\partial \mu}\\
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&= \frac{1}{\beta} \frac{1}{Z} \frac{\partial}{\partial \mu} \sum_k e^{-\beta (H^k + F^k)}\\
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&= 1/Z \sum_k (N^k_{\mathrm{ion}} + N^k_{\mathrm{electron}}) e^{-\beta (H^k + F^k)}\\
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\end{aligned}\]</span> % with the definitions:</p>
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<p><span class="math display">\[\begin{aligned}
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N^k_{\mathrm{ion}} &= - \frac{\partial H^k}{\partial \mu} = \sum_i n^k_i\\
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N^k_{\mathrm{electron}} &= - \frac{\partial F^k}{\partial \mu} = \sum_i \left(1 + e^{\beta \epsilon^k_i}\right)^{-1}\\
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\end{aligned}\]</span></p>
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<p>Next Section: <a href="../6_Appendices/A.1_Particle_Hole_Symmetry.html">Particle-Hole Symmetry</a></p>
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</section>
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</main>
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</body>
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</html>
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