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105 lines
4.6 KiB
HTML
105 lines
4.6 KiB
HTML
---
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title: Particle-Hole Symmetry
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excerpt:
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<title>Particle-Hole Symmetry</title>
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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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<!-- 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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<!-- 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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<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 configurations of the spins in the LRFK model. In the language of ions and electrons (immobile and mobile species), we define <span class="math inline">\(n^k_i\)</span> to be the occupation of the <span class="math inline">\(i\)</span>th site of the <span class="math inline">\(k\)</span>th 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 state <span class="math inline">\(n^k_i\)</span>:</p>
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<p><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></p>
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<p>such that the overall partition function is:</p>
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<p><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>Fermions are limited to occupation numbers of 0 or 1, so <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 <span class="math inline">\(j\)</span>:</p>
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<p><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></p>
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<p>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></p>
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<p>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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