updates

_sass/thesis.scss

@@ -88,5 +88,11 @@ nav.overall-table-of-contents > ul {
 
 // Page header
 div#page-header {
+    //make the header sticky, I don't really like how this looks but it's fun to play with
+    // position: sticky;
+    // top: 0px;
+    // background: white;
+    // z-index: 10;
+    // width: 100%;
     p { margin-block-end: 0px;}
 }

_thesis/1_Introduction/1_Intro.html

@@ -595,8 +595,7 @@ to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and
 localisation. Then Chapter 3 introduces and studies the Long Range
 Falikov-Kimball Model in one dimension while Chapter 4 focusses on the
 Amorphous Kitaev Model.</p>
-<p>Next Chapter: <a
+<p>Next Chapter: <a href="../2_Background/2.1_FK_Model.html">2
-href="../2_Background/2.1_FK_Model.html#the-falikov-kimball-model">2
 Background</a></p>
 </section>
 <section id="bibliography" class="level1 unnumbered">

_thesis/2_Background/2.1_FK_Model.html

@@ -138,7 +138,7 @@ href="#ref-gruberFalicovKimballModel2005"
 role="doc-biblioref">9</a>]</span>. The absence of a hopping term for
 the heavy electrons means they do not need the factor of <span
 class="math inline">\(\epsilon_i\)</span>. See appendix <a
-href="../6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">A.1</a>
+href="../6_Appendices/A.1_Particle_Hole_Symmetry-Copy1.html#particle-hole-symmetry">A.1</a>
 for a full derivation of the PH symmetry.</p>
 <div id="fig:simple_DOS" class="fignos">
 <figure>
@@ -419,9 +419,8 @@ j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span
 class="math inline">\(\alpha\)</span> is varied.</figcaption>
 </figure>
 </div>
-<p>Next Section: <a
+<p>Next Section: <a href="../2_Background/2.2_HKM_Model.html">The Kitaev
-href="../2_Background/2.2_HKM_Model.html#the-kitaev-honeycomb-model">The
+Honeycomb Model</a></p>
-Kitaev Honeycomb Model</a></p>
 </section>
 </section>
 <section id="bibliography" class="level1 unnumbered">

_thesis/2_Background/2.2_HKM_Model.html

@@ -140,8 +140,7 @@ role="doc-biblioref">1</a>]</span></li>
 <h2>Phase Diagram</h2>
 <div class="sourceCode" id="cb1"><pre
 class="sourceCode python"><code class="sourceCode python"></code></pre></div>
-<p>Next Section: <a
+<p>Next Section: <a href="../2_Background/2.3_Disorder.html">Disorder
-href="../2_Background/2.3_Disorder.html#bg-disorder-and-localisation">Disorder
 and Localisation</a></p>
 </section>
 </section>

_thesis/2_Background/2.3_Disorder.html

@@ -174,8 +174,8 @@ timescales to the infinite limit.</p>
 <p>-link to the Kitaev Model</p>
 <p>-link to the physics of amorphous systems</p>
 <p>Next Chapter: <a
-href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#fk-model">3
+href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3 The Long
-The Long Range Falikov-Kimball Model</a></p>
+Range Falikov-Kimball Model</a></p>
 </section>
 </section>
 <section id="bibliography" class="level1 unnumbered">

_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html

@@ -272,7 +272,7 @@ href="#ref-fukuiOrderNClusterMonte2009"
 role="doc-biblioref">30</a>]</span>. We only consider even system sizes
 given that odd system sizes are not commensurate with a CDW state.</p>
 <p>Next Section: <a
-href="../3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html#fk-methods">Methods</a></p>
+href="../3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html">Methods</a></p>
 </section>
 <section id="bibliography" class="level1 unnumbered">
 <h1 class="unnumbered">Bibliography</h1>

_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html

@@ -470,8 +470,8 @@ H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} -
 <div class="sourceCode" id="cb1"><pre
 class="sourceCode python"><code class="sourceCode python"></code></pre></div>
 <p>Next Chapter: <a
-href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html#vortices-and-their-movements">4
+href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">4 The Amorphous
-The Amorphous Kitaev Model</a></p>
+Kitaev Model</a></p>
 </section>
 <section id="bibliography" class="level1 unnumbered">
 <h1 class="unnumbered">Bibliography</h1>

_thesis/4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html

@@ -1070,8 +1070,7 @@ role="doc-biblioref">16</a>,<a
 href="#ref-kitaevFaulttolerantQuantumComputation2003"
 role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p>
 <p>Next Section: <a
-href="../4_Amorphous_Kitaev_Model/4.1_AMK_Model.html#amk-Model">The
+href="../4_Amorphous_Kitaev_Model/4.1_AMK_Model.html">The Model</a></p>
-Model</a></p>
 </section>
 </section>
 <section id="bibliography" class="level1 unnumbered">

_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html

@@ -769,7 +769,7 @@ anyway, an arbitrary pairing of the unpaired <span
 class="math inline">\(b^\alpha\)</span> operators could be performed.
 &lt;/i,j&gt;&lt;/i,j&gt;</p>
 <p>Next Section: <a
-href="../4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html#amk-methods">Methods</a></p>
+href="../4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html">Methods</a></p>
 </section>
 </section>
 </section>

_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html

@@ -541,7 +541,7 @@ system.</p>
 <p><strong>Discuss link between Chern number and Anyonic
 Statistics</strong></p>
 <p>Next Section: <a
-href="../4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#amk-results">Results</a></p>
+href="../4_Amorphous_Kitaev_Model/4.3_AMK_Results.html">Results</a></p>
 </section>
 </section>
 <section id="bibliography" class="level1 unnumbered">

_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html

@@ -665,8 +665,8 @@ href="#ref-Wu2009" role="doc-biblioref">47</a>]</span></p>
 quantum many body phases albeit material candidates aplenty. We expect
 our exact chiral amorphous spin liquid to find many generalisation to
 realistic amorphous quantum magnets and beyond.</p>
-<p>Next Chapter: <a
+<p>Next Chapter: <a href="../5_Conclusion/5_Conclusion.html">5
-href="../5_Conclusion/5_Conclusion.html#discussion">5 Conclusion</a></p>
+Conclusion</a></p>
 </section>
 </section>
 <section id="bibliography" class="level1 unnumbered">

_thesis/5_Conclusion/5_Conclusion.html

@@ -27,6 +27,14 @@ image:
 <br>
 <nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
+<li><a href="#material-realisations"
+id="toc-material-realisations">Material Realisations</a>
+<ul>
+<li><a href="#amorphous-materials"
+id="toc-amorphous-materials">Amorphous Materials</a></li>
+<li><a href="#metal-organic-frameworks"
+id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
+</ul></li>
 <li><a href="#discussion" id="toc-discussion">Discussion</a></li>
 <li><a href="#outlook" id="toc-outlook">Outlook</a></li>
 </ul>
@@ -41,6 +49,14 @@ image:
 <!-- Table of Contents -->
 <!-- <nav id="TOC" role="doc-toc">
 <ul>
+<li><a href="#material-realisations"
+id="toc-material-realisations">Material Realisations</a>
+<ul>
+<li><a href="#amorphous-materials"
+id="toc-amorphous-materials">Amorphous Materials</a></li>
+<li><a href="#metal-organic-frameworks"
+id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
+</ul></li>
 <li><a href="#discussion" id="toc-discussion">Discussion</a></li>
 <li><a href="#outlook" id="toc-outlook">Outlook</a></li>
 </ul>
@@ -52,13 +68,22 @@ image:
 <p>5 Conclusion</p>
 <hr />
 </div>
-<section id="discussion" class="level2">
+<section id="material-realisations" class="level1">
-<h2>Discussion</h2>
+<h1>Material Realisations</h1>
+<section id="amorphous-materials" class="level2">
+<h2>Amorphous Materials</h2>
 </section>
-<section id="outlook" class="level2">
+<section id="metal-organic-frameworks" class="level2">
-<h2>Outlook</h2>
+<h2>Metal Organic Frameworks</h2>
+</section>
+</section>
+<section id="discussion" class="level1">
+<h1>Discussion</h1>
+</section>
+<section id="outlook" class="level1">
+<h1>Outlook</h1>
 <p>Next Chapter: <a
-href="../6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">Appendices</a></p>
+href="../6_Appendices/A.1.2_Fermion_Free_Energy.html">Appendices</a></p>
 </section>
 
 

_thesis/6_Appendices/A.1.2_Fermion_Free_Energy.html

@@ -0,0 +1,110 @@
+---
+title: Particle-Hole Symmetry
+excerpt: 
+layout: none
+image: 
+
+---
+<!DOCTYPE html>
+<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
+<head>
+  <meta charset="utf-8" />
+  <meta name="generator" content="pandoc" />
+  <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
+  <title>Particle-Hole Symmetry</title>
+
+
+<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
+<script src="/assets/js/thesis_scrollspy.js"></script>
+
+<link rel="stylesheet" href="/assets/css/styles.css">
+<script src="/assets/js/index.js"></script>
+</head>
+<body>
+
+<!--Capture the table of contents from pandoc as a jekyll variable  -->
+{% capture tableOfContents %}
+<br>
+<nav aria-label="Table of Contents" class="page-table-of-contents">
+<ul>
+<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>
+</ul>
+</nav>
+{% endcapture %}
+
+<!-- Give the table of contents to header as a variable so it can be put into the sidebar-->
+{% include header.html extra=tableOfContents %}
+
+<main>
+
+<!-- Table of Contents -->
+<!-- <nav id="TOC" role="doc-toc">
+<ul>
+<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>
+</ul>
+</nav>
+ -->
+
+<!-- Main Page Body -->
+<div id="page-header">
+<p>Appendices</p>
+<hr />
+</div>
+<section id="evaluation-of-the-fermion-free-energy" class="level1">
+<h1>Evaluation of the Fermion Free Energy</h1>
+<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">\[\begin{aligned}
+F^k &amp;= -1/\beta \ln{\mathcal{Z}^k} \\
+\end{aligned}\]</span> % Such that the overall partition function is:
+<span class="math display">\[\begin{aligned}
+\mathcal{Z} &amp;= \sum_k e^{- \beta H^k} Z^k \\
+&amp;= \sum_k e^{-\beta (H^k + F^k)} \\
+\end{aligned}\]</span></p>
+<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">\[\begin{aligned}
+Z^k    &amp;= \mathrm{Tr} e^{-\beta F^k} = \sum_j e^{-\beta \sum_i m^j_i
+\epsilon^k_i}\\
+       &amp;= \sum_j \prod_i e^{- \beta m^j_i \epsilon^k_i}= \prod_i
+\sum_j e^{- \beta m^j_i \epsilon^k_i}\\
+       &amp;= \prod_i (1 + e^{- \beta \epsilon^k_i})\\
+F^k    &amp;= -1/\beta \sum_k \ln{(1 + e^{- \beta \epsilon^k_i})}
+\end{aligned}\]</span> % Observables can then be calculated from the
+partition function, for examples the occupation numbers:</p>
+<p><span class="math display">\[\begin{aligned}
+\langle N \rangle &amp;= \frac{1}{\beta} \frac{1}{Z} \frac{\partial
+Z}{\partial \mu} = - \frac{\partial F}{\partial \mu}\\
+    &amp;= \frac{1}{\beta} \frac{1}{Z} \frac{\partial}{\partial \mu}
+\sum_k e^{-\beta (H^k + F^k)}\\
+    &amp;= 1/Z \sum_k (N^k_{\mathrm{ion}} + N^k_{\mathrm{electron}})
+e^{-\beta (H^k + F^k)}\\
+\end{aligned}\]</span> % with the definitions:</p>
+<p><span class="math display">\[\begin{aligned}
+N^k_{\mathrm{ion}} &amp;= - \frac{\partial H^k}{\partial \mu} = \sum_i
+n^k_i\\
+N^k_{\mathrm{electron}} &amp;= - \frac{\partial F^k}{\partial \mu} =
+\sum_i \left(1 + e^{\beta \epsilon^k_i}\right)^{-1}\\
+\end{aligned}\]</span></p>
+<p>Next Section: <a
+href="../6_Appendices/A.1_Particle_Hole_Symmetry-Copy1.html">Particle-Hole
+Symmetry</a></p>
+</section>
+
+
+</main>
+</body>
+</html>

_thesis/6_Appendices/A.1_Particle_Hole_Symmetry-Copy1.html

@@ -0,0 +1,135 @@
+---
+title: Particle-Hole Symmetry
+excerpt: 
+layout: none
+image: 
+
+---
+<!DOCTYPE html>
+<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
+<head>
+  <meta charset="utf-8" />
+  <meta name="generator" content="pandoc" />
+  <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
+  <title>Particle-Hole Symmetry</title>
+
+
+<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
+<script src="/assets/js/thesis_scrollspy.js"></script>
+
+<link rel="stylesheet" href="/assets/css/styles.css">
+<script src="/assets/js/index.js"></script>
+</head>
+<body>
+
+<!--Capture the table of contents from pandoc as a jekyll variable  -->
+{% capture tableOfContents %}
+<br>
+<nav aria-label="Table of Contents" class="page-table-of-contents">
+<ul>
+<li><a href="#particle-hole-symmetry"
+id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
+<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
+</ul>
+</nav>
+{% endcapture %}
+
+<!-- Give the table of contents to header as a variable so it can be put into the sidebar-->
+{% include header.html extra=tableOfContents %}
+
+<main>
+
+<!-- Table of Contents -->
+<!-- <nav id="TOC" role="doc-toc">
+<ul>
+<li><a href="#particle-hole-symmetry"
+id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
+<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
+</ul>
+</nav>
+ -->
+
+<!-- Main Page Body -->
+<div id="page-header">
+<p>Appendices</p>
+<hr />
+</div>
+<section id="particle-hole-symmetry" class="level1">
+<h1>Particle-Hole Symmetry</h1>
+<p>The Hubbard and FK models on a bipartite lattice have particle-hole
+(PH) symmetry <span class="math inline">\(\mathcal{P}^\dagger H
+\mathcal{P} = - H\)</span>, accordingly they have symmetric energy
+spectra. The associated symmetry operator <span
+class="math inline">\(\mathcal{P}\)</span> exchanges creation and
+annihilation operators along with a sign change between the two
+sublattices. In the language of the Hubbard model of electrons <span
+class="math inline">\(c_{\alpha,i}\)</span> with spin <span
+class="math inline">\(\alpha\)</span> at site <span
+class="math inline">\(i\)</span> the particle hole operator corresponds
+to the substitution of new fermion operators <span
+class="math inline">\(d^\dagger_{\alpha,i}\)</span> and number operators
+<span class="math inline">\(m_{\alpha,i}\)</span> where</p>
+<p><span class="math display">\[d^\dagger_{\alpha,i} = \epsilon_i
+c_{\alpha,i}\]</span> <span class="math display">\[m_{\alpha,i} =
+d^\dagger_{\alpha,i}d_{\alpha,i}\]</span></p>
+<p>the lattices must be bipartite because to make this work we set <span
+class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and
+<span class="math inline">\(-1\)</span> for the even sublattice <span
+class="citation" data-cites="gruberFalicovKimballModel2005"> [<a
+href="#ref-gruberFalicovKimballModel2005"
+role="doc-biblioref">1</a>]</span>.</p>
+<p>The entirely filled state <span class="math inline">\(\ket{\Omega} =
+\sum_{\alpha,i} c^\dagger_{\alpha,i} \ket{0}\)</span> becomes the new
+vacuum state <span class="math display">\[d_{i\sigma} \ket{\Omega} =
+(-1)^i c^\dagger_{i\sigma} \sum_{j\rho} c^\dagger_{j\rho} \ket{0} =
+0.\]</span></p>
+<p>The number operator <span class="math inline">\(m_{\alpha,i} =
+0,1\)</span> counts holes rather than electrons <span
+class="math display">\[ m_{\alpha,i} = c_{\alpha,i} c^\dagger_{\alpha,i}
+= 1 - c^\dagger_{\alpha,i} c_{\alpha,i}.\]</span></p>
+<p>With the last equality following from the fermionic commutation
+relations. In the case of nearest neighbour hopping on a bipartite
+lattice this transformation also leaves the hopping term unchanged
+because <span class="math inline">\(\epsilon_i \epsilon_j = -1\)</span>
+when <span class="math inline">\(i\)</span> and <span
+class="math inline">\(j\)</span> are on different sublattices: <span
+class="math display">\[ d^\dagger_{\alpha,i} d_{\alpha,j} = \epsilon_i
+\epsilon_j c_{\alpha,i} c^\dagger_{\alpha,j} = c^\dagger_{\alpha,i}
+c_{\alpha,j} \]</span></p>
+<p>Defining the particle density <span
+class="math inline">\(\rho\)</span> as the number of fermions per site:
+<span class="math display">\[
+    \rho = \frac{1}{N} \sum_i \left( n_{i \uparrow} + n_{i \downarrow}
+\right)
+\]</span></p>
+<p>The PH symmetry maps the Hamiltonian to itself with the sign of the
+chemical potential reversed and the density inverted about half filling:
+<span class="math display">\[ \text{PH} : H(t, U, \mu) \rightarrow H(t,
+U, -\mu) \]</span> <span class="math display">\[ \rho \rightarrow 2 -
+\rho \]</span></p>
+<p>The Hamiltonian is symmetric under PH at <span
+class="math inline">\(\mu = 0\)</span> and so must all the observables,
+hence half filling <span class="math inline">\(\rho = 1\)</span> occurs
+here. This symmetry and known observable acts as a useful test for the
+numerical calculations.</p>
+<p>Next Section: <a
+href="../6_Appendices/A.2_Markov_Chain_Monte_Carlo.html">Markov Chain
+Monte Carlo</a></p>
+</section>
+<section id="bibliography" class="level1 unnumbered">
+<h1 class="unnumbered">Bibliography</h1>
+<div id="refs" class="references csl-bib-body" role="doc-bibliography">
+<div id="ref-gruberFalicovKimballModel2005" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[1] </div><div class="csl-right-inline">C.
+Gruber and D. Ueltschi, <em><a
+href="http://arxiv.org/abs/math-ph/0502041">The Falicov-Kimball
+Model</a></em>, arXiv:math-Ph/0502041 (2005).</div>
+</div>
+</div>
+</section>
+
+
+</main>
+</body>
+</html>

_thesis/6_Appendices/A.1_Particle_Hole_Symmetry.html

@@ -113,8 +113,8 @@ hence half filling <span class="math inline">\(\rho = 1\)</span> occurs
 here. This symmetry and known observable acts as a useful test for the
 numerical calculations.</p>
 <p>Next Section: <a
-href="../6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#applying-mcmc-to-the-fk-model">Applying
+href="../6_Appendices/A.2_Markov_Chain_Monte_Carlo.html">Applying MCMC
-MCMC to the FK model</a></p>
+to the FK model</a></p>
 </section>
 <section id="bibliography" class="level1 unnumbered">
 <h1 class="unnumbered">Bibliography</h1>

_thesis/6_Appendices/A.3_Lattice_Generation.html

@@ -57,7 +57,7 @@ Generation</a></li>
 <div class="sourceCode" id="cb1"><pre
 class="sourceCode python"><code class="sourceCode python"></code></pre></div>
 <p>Next Section: <a
-href="../6_Appendices/A.4_Lattice_Colouring.html#lattice-colouring">Lattice
+href="../6_Appendices/A.4_Lattice_Colouring.html">Lattice
 Colouring</a></p>
 </section>
 

_thesis/toc.html

@@ -1,35 +1,43 @@
   <ul>
-  <li><a href="./1_Introduction/1_Intro.html#interacting-quantum-many-body-systems">1 Introduction</a></li>
+  <li><a href="./1_Introduction/1_Intro.html">1 Introduction</a></li>
     <ul>
-    <li><a href="./1_Introduction/1_Intro.html#interacting-quantum-many-body-systems">Interacting Quantum Many Body Systems</a></li>
+    <li><a href="./1_Introduction/1_Intro.html">Interacting Quantum Many Body Systems</a></li>
     <li><a href="./1_Introduction/1_Intro.html#mott-insulators">Mott Insulators</a></li>
     <li><a href="./1_Introduction/1_Intro.html#quantum-spin-liquids">Quantum Spin Liquids</a></li>
   </ul>
-  <li><a href="./2_Background/2.1_FK_Model.html#the-falikov-kimball-model">2 Background</a></li>
+  <li><a href="./2_Background/2.1_FK_Model.html">2 Background</a></li>
     <ul>
-    <li><a href="./2_Background/2.1_FK_Model.html#the-falikov-kimball-model">The Falikov Kimball Model</a></li>
+    <li><a href="./2_Background/2.1_FK_Model.html">The Falikov Kimball Model</a></li>
     <li><a href="./2_Background/2.2_HKM_Model.html#the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a></li>
     <li><a href="./2_Background/2.3_Disorder.html#disorder-and-localisation">Disorder and Localisation</a></li>
   </ul>
-  <li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#the-model">3 The Long Range Falikov-Kimball Model</a></li>
+  <li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falikov-Kimball Model</a></li>
     <ul>
-    <li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#the-model">The Model</a></li>
+    <li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">The Model</a></li>
     <li><a href="./3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html#methods">Methods</a></li>
     <li><a href="./3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html#results">Results</a></li>
     <li><a href="./3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html#discussion-and-conclusion">Discussion and Conclusion</a></li>
   </ul>
-  <li><a href="./4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html#gauge-fields">4 The Amorphous Kitaev Model</a></li>
+  <li><a href="./4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">4 The Amorphous Kitaev Model</a></li>
     <ul>
     <li><a href="./4_Amorphous_Kitaev_Model/4.1_AMK_Model.html#the-model">The Model</a></li>
     <li><a href="./4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html#methods">Methods</a></li>
     <li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#results">Results</a></li>
     <li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#discussion-and-conclusion">Discussion and Conclusion</a></li>
   </ul>
-  <li><a href="./5_Conclusion/5_Conclusion.html#discussion">5 Conclusion</a></li>
+  <li><a href="./5_Conclusion/5_Conclusion.html">5 Conclusion</a></li>
-  <li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">Appendices</a></li>
     <ul>
+    <li><a href="./5_Conclusion/5_Conclusion.html">Material Realisations</a></li>
+    <li><a href="./5_Conclusion/5_Conclusion.html#discussion">Discussion</a></li>
+    <li><a href="./5_Conclusion/5_Conclusion.html#outlook">Outlook</a></li>
+  </ul>
+  <li><a href="./6_Appendices/A.1.2_Fermion_Free_Energy.html">Appendices</a></li>
+    <ul>
+    <li><a href="./6_Appendices/A.1.2_Fermion_Free_Energy.html">Evaluation of the Fermion Free Energy</a></li>
+    <li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry-Copy1.html#particle-hole-symmetry">Particle-Hole Symmetry</a></li>
     <li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">Particle-Hole Symmetry</a></li>
     <li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
+    <li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#[\[app:balance\]]">[\[app:balance\]]</a></li>
     <li><a href="./6_Appendices/A.3_Lattice_Generation.html#lattice-generation">Lattice Generation</a></li>
     <li><a href="./6_Appendices/A.4_Lattice_Colouring.html#lattice-colouring">Lattice Colouring</a></li>
     <li><a href="./6_Appendices/A.5_The_Projector.html#the-projector">The Projector</a></li>