diff --git a/_includes/sidebar.html b/_includes/sidebar.html
index 6ec9ed2..4ba306b 100644
--- a/_includes/sidebar.html
+++ b/_includes/sidebar.html
@@ -1,4 +1,4 @@
-<nav>
+<nav aria-label="Site Map" class="site-map"></nav>
     {% for item in site.data.navigation %}
       <a href="{{ item.link }}" {% if page.url == item.link %}class="current"{% endif %}>{{ item.name }}</a>
     {% endfor %}
diff --git a/_sass/main.scss b/_sass/main.scss
index 323ce25..a1b8293 100644
--- a/_sass/main.scss
+++ b/_sass/main.scss
@@ -14,6 +14,7 @@
 
 html {
     width: 100vw;
+    scroll-behavior: smooth;
 }
 
 body {
@@ -50,6 +51,14 @@ a {
     color: #222;
 }
 
+header a {
+    text-decoration: none;
+}
+
+nav a {
+    text-decoration: none;
+}
+
 a:hover {
     text-decoration: underline;
 }
diff --git a/_sass/thesis.scss b/_sass/thesis.scss
index 7629de1..5557bc3 100644
--- a/_sass/thesis.scss
+++ b/_sass/thesis.scss
@@ -17,11 +17,18 @@ figcaption {
 
 // For the table of contents, should probably put this in a container
 
-// remove underline from toc links
-nav a {
-    text-decoration: none;
+//For the animation that plays in the nav as you scroll
+nav.page-table-of-contents a {
+    transition: all 200ms ease-in-out;
+    color: #000;
+    font-weight:normal;
 }
 
+nav.page-table-of-contents li.active > a {
+    color: #000!important;
+    font-weight:bold;
+  }
+
 // modify the spacing of the various levels
 li {
     margin-bottom: 0.2em;
@@ -67,4 +74,14 @@ header li {
         display:block;
         
     }
+}
+
+nav.overall-table-of-contents > ul {
+    padding-inline-start: 0px;
+    
+
+    > li {
+        list-style: none;
+        margin-top: 1em;
+    }
 }
\ No newline at end of file
diff --git a/_thesis/0_Preface/0.1_Aknowledgements.html b/_thesis/0_Preface/0.1_Aknowledgements.html
index 5a14d02..13cca4a 100644
--- a/_thesis/0_Preface/0.1_Aknowledgements.html
+++ b/_thesis/0_Preface/0.1_Aknowledgements.html
@@ -12,167 +12,11 @@ image:
   <meta name="generator" content="pandoc" />
   <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
   <title>Aknowledgements</title>
-  <!-- <style>
-    html {
-      line-height: 1.5;
-      font-family: Georgia, serif;
-      font-size: 20px;
-      color: #1a1a1a;
-      background-color: #fdfdfd;
-    }
-    body {
-      margin: 0 auto;
-      max-width: 36em;
-      padding-left: 50px;
-      padding-right: 50px;
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-      hyphens: auto;
-      overflow-wrap: break-word;
-      text-rendering: optimizeLegibility;
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-      padding-left: 1.7em;
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-    li > ol, li > ul {
-      margin-top: 0;
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-      margin: 1em 0 1em 1.7em;
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-      font-size: 85%;
-      margin: 0;
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-    pre code {
-      padding: 0;
-      overflow: visible;
-      overflow-wrap: normal;
-    }
-    .sourceCode {
-     background-color: transparent;
-     overflow: visible;
-    }
-    hr {
-      background-color: #1a1a1a;
-      border: none;
-      height: 1px;
-      margin: 1em 0;
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-    table {
-      margin: 1em 0;
-      border-collapse: collapse;
-      width: 100%;
-      overflow-x: auto;
-      display: block;
-      font-variant-numeric: lining-nums tabular-nums;
-    }
-    table caption {
-      margin-bottom: 0.75em;
-    }
-    tbody {
-      margin-top: 0.5em;
-      border-top: 1px solid #1a1a1a;
-      border-bottom: 1px solid #1a1a1a;
-    }
-    th {
-      border-top: 1px solid #1a1a1a;
-      padding: 0.25em 0.5em 0.25em 0.5em;
-    }
-    td {
-      padding: 0.125em 0.5em 0.25em 0.5em;
-    }
-    header {
-      margin-bottom: 4em;
-      text-align: center;
-    }
-    #TOC li {
-      list-style: none;
-    }
-    #TOC ul {
-      padding-left: 1.3em;
-    }
-    #TOC > ul {
-      padding-left: 0;
-    }
-    #TOC a:not(:hover) {
-      text-decoration: none;
-    }
-    code{white-space: pre-wrap;}
-    span.smallcaps{font-variant: small-caps;}
-    span.underline{text-decoration: underline;}
-    div.column{display: inline-block; vertical-align: top; width: 50%;}
-    div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
-    ul.task-list{list-style: none;}
-  </style> -->
 
-<!--  -->
-
-<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
-<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
-   -->
 
 <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
+<script src="/assets/js/thesis_scrollspy.js"></script>
 
-
-<!--[if lt IE 9]>
-    <script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
-<![endif]-->
 <link rel="stylesheet" href="/assets/css/styles.css">
 <script src="/assets/js/index.js"></script>
 </head>
@@ -181,15 +25,20 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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 -->
 <!--  -->
+
+<!-- Main Page Body -->
 <p>I would like to thank my supervisor, Professor Johannes Knolle and
 co-supervisor Professor Derek Lee for guidance and support during this
 long process.</p>
@@ -216,6 +65,10 @@ expertise and patience.</p>
 <p>All the I-Stemm team, Katerina, Jeremey, John, ….</p>
 <p>And finally, I’d like the thank the staff of the Camberwell Public
 Library where the majority of this thesis was written.</p>
+<p>We thank Angus MacKinnon for helpful discussions, Sophie Nadel for
+input when preparing the figures.</p>
+
+
 </main>
 </body>
 </html>
diff --git a/_thesis/1_Introduction/1_Intro.html b/_thesis/1_Introduction/1_Intro.html
index 201b807..9da2f0d 100644
--- a/_thesis/1_Introduction/1_Intro.html
+++ b/_thesis/1_Introduction/1_Intro.html
@@ -12,189 +12,11 @@ image:
   <meta name="generator" content="pandoc" />
   <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
   <title>1_Intro</title>
-  <!-- <style>
-    html {
-      line-height: 1.5;
-      font-family: Georgia, serif;
-      font-size: 20px;
-      color: #1a1a1a;
-      background-color: #fdfdfd;
-    }
-    body {
-      margin: 0 auto;
-      max-width: 36em;
-      padding-left: 50px;
-      padding-right: 50px;
-      padding-top: 50px;
-      padding-bottom: 50px;
-      hyphens: auto;
-      overflow-wrap: break-word;
-      text-rendering: optimizeLegibility;
-      font-kerning: normal;
-    }
-    @media (max-width: 600px) {
-      body {
-        font-size: 0.9em;
-        padding: 1em;
-      }
-      h1 {
-        font-size: 1.8em;
-      }
-    }
-    @media print {
-      body {
-        background-color: transparent;
-        color: black;
-        font-size: 12pt;
-      }
-      p, h2, h3 {
-        orphans: 3;
-        widows: 3;
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-    p {
-      margin: 1em 0;
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-      margin-top: 1.4em;
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-      font-size: 85%;
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-    pre code {
-      padding: 0;
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-     background-color: transparent;
-     overflow: visible;
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-      background-color: #1a1a1a;
-      border: none;
-      height: 1px;
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-      margin-top: 0.5em;
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-    th {
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-      padding: 0.25em 0.5em 0.25em 0.5em;
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-    td {
-      padding: 0.125em 0.5em 0.25em 0.5em;
-    }
-    header {
-      margin-bottom: 4em;
-      text-align: center;
-    }
-    #TOC li {
-      list-style: none;
-    }
-    #TOC ul {
-      padding-left: 1.3em;
-    }
-    #TOC > ul {
-      padding-left: 0;
-    }
-    #TOC a:not(:hover) {
-      text-decoration: none;
-    }
-    code{white-space: pre-wrap;}
-    span.smallcaps{font-variant: small-caps;}
-    span.underline{text-decoration: underline;}
-    div.column{display: inline-block; vertical-align: top; width: 50%;}
-    div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
-    ul.task-list{list-style: none;}
-    div.csl-bib-body { }
-    div.csl-entry {
-      clear: both;
-    }
-    .hanging div.csl-entry {
-      margin-left:2em;
-      text-indent:-2em;
-    }
-    div.csl-left-margin {
-      min-width:2em;
-      float:left;
-    }
-    div.csl-right-inline {
-      margin-left:2em;
-      padding-left:1em;
-    }
-    div.csl-indent {
-      margin-left: 2em;
-    }
-  </style> -->
 
-<!--   <script
-       src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js"
-       type="text/javascript"></script>
- -->
-
-<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
-<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
-   -->
 
 <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
+<script src="/assets/js/thesis_scrollspy.js"></script>
 
-
-<!--[if lt IE 9]>
-    <script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
-<![endif]-->
 <link rel="stylesheet" href="/assets/css/styles.css">
 <script src="/assets/js/index.js"></script>
 </head>
@@ -203,7 +25,7 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
 <li><a href="#interacting-quantum-many-body-systems"
 id="toc-interacting-quantum-many-body-systems">Interacting Quantum Many
@@ -215,12 +37,15 @@ id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
 <li><a href="#outline" id="toc-outline">Outline</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 -->
+<!-- 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="#interacting-quantum-many-body-systems"
@@ -235,8 +60,10 @@ id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
 </ul>
 </nav>
  -->
-<h1 id="interacting-quantum-many-body-systems">Interacting Quantum Many
-Body Systems</h1>
+
+<!-- Main Page Body -->
+<section id="interacting-quantum-many-body-systems" class="level1">
+<h1>Interacting Quantum Many Body Systems</h1>
 <p>When you take many objects and let them interact together, it is
 often simpler to describe the behaviour of the group differently from
 the way one would describe the individual objects. Consider a flock of
@@ -321,25 +148,39 @@ antiferromagnetism <span class="citation"
 data-cites="MagnetismCondensedMatter"> [<a
 href="#ref-MagnetismCondensedMatter"
 role="doc-biblioref">12</a>]</span>.</p>
-<p>However, at some point we had to start on the interacting quantum
-many body systems. The properties of some materials cannot be understood
-without a taking into account all three effects and these are
-collectively called strongly correlated materials. The canonical
-examples are superconductivity <span class="citation"
-data-cites="MicroscopicTheorySuperconductivity"> [<a
+<p>The development of Landau-Fermi Liquid theory explained why band
+theory works so well even in cases where an analysis of the relevant
+energies suggests that it should not <span class="citation"
+data-cites="wenQuantumFieldTheory2007"> [<a
+href="#ref-wenQuantumFieldTheory2007"
+role="doc-biblioref">13</a>]</span>. Landau Fermi Liquid theory
+demonstrates that in many cases where electron-electron interactions are
+significant, the system can still be described in terms on generalised
+non-interacting quasiparticles.</p>
+<p>However there are systems where even Landau Fermi Liquid theory
+fails. An effective theoretical description of these systems must
+include electron-electron correlations and they are thus called Strongly
+Correlated Materials <span class="citation"
+data-cites="morosanStronglyCorrelatedMaterials2012"> [<a
+href="#ref-morosanStronglyCorrelatedMaterials2012"
+role="doc-biblioref">14</a>]</span>, Correlated Electron systems or
+Quantum Materials. The canonical examples are superconductivity <span
+class="citation" data-cites="MicroscopicTheorySuperconductivity"> [<a
 href="#ref-MicroscopicTheorySuperconductivity"
-role="doc-biblioref">13</a>]</span>, the fractional quantum hall
+role="doc-biblioref">15</a>]</span>, the fractional quantum hall
 effect <span class="citation"
 data-cites="feldmanFractionalChargeFractional2021"> [<a
 href="#ref-feldmanFractionalChargeFractional2021"
-role="doc-biblioref">14</a>]</span> and the Mott insulators <span
+role="doc-biblioref">16</a>]</span> and the Mott insulators <span
 class="citation"
 data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a
-href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">15</a>,<a
+href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">17</a>,<a
 href="#ref-fisherMottInsulatorsSpin1999"
-role="doc-biblioref">16</a>]</span>. We’ll start by looking at the
+role="doc-biblioref">18</a>]</span>. We’ll start by looking at the
 latter but shall see that there are many links between three topics.</p>
-<h1 id="mott-insulators">Mott Insulators</h1>
+</section>
+<section id="mott-insulators" class="level1">
+<h1>Mott Insulators</h1>
 <p>Mott Insulators are remarkable because their electrical insulator
 properties come from electron-electron interactions. Electrical
 conductivity, the bulk movement of electrons, requires both that there
@@ -375,27 +216,27 @@ many transition metal oxides are erroneously predicted by band theory to
 be conductive <span class="citation"
 data-cites="boerSemiconductorsPartiallyCompletely1937"> [<a
 href="#ref-boerSemiconductorsPartiallyCompletely1937"
-role="doc-biblioref">17</a>]</span> leading to the suggestion that
+role="doc-biblioref">19</a>]</span> leading to the suggestion that
 electron-electron interactions were the cause of this effect <span
 class="citation" data-cites="mottDiscussionPaperBoer1937"> [<a
 href="#ref-mottDiscussionPaperBoer1937"
-role="doc-biblioref">18</a>]</span>. Interest grew with the discovery of
+role="doc-biblioref">20</a>]</span>. Interest grew with the discovery of
 high temperature superconductivity in the cuprates in 1986 <span
 class="citation"
 data-cites="bednorzPossibleHighTcSuperconductivity1986"> [<a
 href="#ref-bednorzPossibleHighTcSuperconductivity1986"
-role="doc-biblioref">19</a>]</span> which is believed to arise as the
+role="doc-biblioref">21</a>]</span> which is believed to arise as the
 result of a doped Mott insulator state <span class="citation"
 data-cites="leeDopingMottInsulator2006"> [<a
 href="#ref-leeDopingMottInsulator2006"
-role="doc-biblioref">20</a>]</span>.</p>
+role="doc-biblioref">22</a>]</span>.</p>
 <p>The canonical toy model of the Mott insulator is the Hubbard
 model <span class="citation"
 data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"> [<a
 href="#ref-gutzwillerEffectCorrelationFerromagnetism1963"
-role="doc-biblioref">21</a>–<a
+role="doc-biblioref">23</a>–<a
 href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
-role="doc-biblioref">23</a>]</span> of <span
+role="doc-biblioref">25</a>]</span> of <span
 class="math inline">\(1/2\)</span> fermions hopping on the lattice with
 hopping parameter <span class="math inline">\(t\)</span> and
 electron-electron repulsion <span class="math inline">\(U\)</span></p>
@@ -425,7 +266,7 @@ class="math inline">\(\mu = \tfrac{U}{2}\)</span> where there is one
 electron per lattice site <span class="citation"
 data-cites="hubbardElectronCorrelationsNarrow1964"> [<a
 href="#ref-hubbardElectronCorrelationsNarrow1964"
-role="doc-biblioref">24</a>]</span>. Here the model can be rewritten in
+role="doc-biblioref">26</a>]</span>. Here the model can be rewritten in
 a symmetric form <span class="math display">\[ H_{\mathrm{H}} = -t
 \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U
 \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} -
@@ -439,12 +280,12 @@ Originally it was proposed that this antiferromagnetic order was the
 cause of the gap opening <span class="citation"
 data-cites="mottMetalInsulatorTransitions1990"> [<a
 href="#ref-mottMetalInsulatorTransitions1990"
-role="doc-biblioref">25</a>]</span>. However, Mott insulators have been
+role="doc-biblioref">27</a>]</span>. However, Mott insulators have been
 found <span class="citation"
 data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
-href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">26</a>,<a
+href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">28</a>,<a
 href="#ref-ribakGaplessExcitationsGround2017"
-role="doc-biblioref">27</a>]</span> without magnetic order. Instead the
+role="doc-biblioref">29</a>]</span> without magnetic order. Instead the
 local moments may form a highly entangled state known as a quantum spin
 liquid, which will be discussed shortly.</p>
 <p>Various theoretical treatments of the Hubbard model have been made,
@@ -452,18 +293,18 @@ including those based on Fermi liquid theory, mean field treatments, the
 local density approximation (LDA) <span class="citation"
 data-cites="slaterMagneticEffectsHartreeFock1951"> [<a
 href="#ref-slaterMagneticEffectsHartreeFock1951"
-role="doc-biblioref">28</a>]</span> and dynamical mean-field
+role="doc-biblioref">30</a>]</span> and dynamical mean-field
 theory <span class="citation"
 data-cites="greinerQuantumPhaseTransition2002"> [<a
 href="#ref-greinerQuantumPhaseTransition2002"
-role="doc-biblioref">29</a>]</span>. None of these approaches are
+role="doc-biblioref">31</a>]</span>. None of these approaches are
 perfect. Strong correlations are poorly described by the Fermi liquid
 theory and the LDA approaches while mean field approximations do poorly
 in low dimensional systems. This theoretical difficulty has made the
 Hubbard model a target for cold atom simulations <span class="citation"
 data-cites="mazurenkoColdatomFermiHubbard2017"> [<a
 href="#ref-mazurenkoColdatomFermiHubbard2017"
-role="doc-biblioref">30</a>]</span>.</p>
+role="doc-biblioref">32</a>]</span>.</p>
 <p>From here the discussion will branch two directions. First, we will
 discuss a limit of the Hubbard model called the Falikov-Kimball Model.
 Second, we will look at quantum spin liquids and the Kitaev honeycomb
@@ -487,9 +328,9 @@ c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
 transition is still poorly understood <span class="citation"
 data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a
 href="#ref-belitzAndersonMottTransition1994"
-role="doc-biblioref">31</a>,<a
+role="doc-biblioref">33</a>,<a
 href="#ref-baskoMetalInsulatorTransition2006"
-role="doc-biblioref">32</a>]</span> the FK model provides a rich test
+role="doc-biblioref">34</a>]</span> the FK model provides a rich test
 bed to explore interaction driven MI transition physics. Despite its
 simplicity, the model has a rich phase diagram in <span
 class="math inline">\(D \geq 2\)</span> dimensions. It shows an Mott
@@ -497,23 +338,23 @@ insulator transition even at high temperature, similar to the
 corresponding Hubbard Model <span class="citation"
 data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
 href="#ref-brandtThermodynamicsCorrelationFunctions1989"
-role="doc-biblioref">33</a>]</span>. In 1D, the ground state
+role="doc-biblioref">35</a>]</span>. In 1D, the ground state
 phenomenology as a function of filling can be rich <span
 class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a
 href="#ref-gruberGroundStatesSpinless1990"
-role="doc-biblioref">34</a>]</span> but the system is disordered for all
+role="doc-biblioref">36</a>]</span> but the system is disordered for all
 <span class="math inline">\(T &gt; 0\)</span> <span class="citation"
 data-cites="kennedyItinerantElectronModel1986"> [<a
 href="#ref-kennedyItinerantElectronModel1986"
-role="doc-biblioref">35</a>]</span>. The model has also been a test-bed
+role="doc-biblioref">37</a>]</span>. The model has also been a test-bed
 for many-body methods, interest took off when an exact dynamical
 mean-field theory solution in the infinite dimensional case was
 found <span class="citation"
 data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a
 href="#ref-antipovCriticalExponentsStrongly2014"
-role="doc-biblioref">36</a>–<a
+role="doc-biblioref">38</a>–<a
 href="#ref-herrmannNonequilibriumDynamicalCluster2016"
-role="doc-biblioref">39</a>]</span>.</p>
+role="doc-biblioref">41</a>]</span>.</p>
 <p>In Chapter 3 I will introduce a generalized FK model in one
 dimension. With the addition of long-range interactions in the
 background field, the model shows a similarly rich phase diagram. I use
@@ -523,16 +364,18 @@ then compare the behaviour of this transitionally invariant model to an
 Anderson model of uncorrelated binary disorder about a background charge
 density wave field which confirms that the fermionic sector only fully
 localizes for very large system sizes.</p>
-<h1 id="quantum-spin-liquids">Quantum Spin Liquids</h1>
+</section>
+<section id="quantum-spin-liquids" class="level1">
+<h1>Quantum Spin Liquids</h1>
 <p>To turn to the other key topic of this thesis, we have discussed the
 question of the magnetic ordering of local moments in the Mott
 insulating state. The local moments may form an AFM ground state.
 Alternatively they may fail to order even at zero temperature <span
 class="citation"
 data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
-href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">26</a>,<a
+href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">28</a>,<a
 href="#ref-ribakGaplessExcitationsGround2017"
-role="doc-biblioref">27</a>]</span>, giving rise to what is known as a
+role="doc-biblioref">29</a>]</span>, giving rise to what is known as a
 quantum spin liquid (QSL) state.</p>
 <p>Landau-Ginzburg-Wilson theory characterises phases of matter as
 inextricably linked to the emergence of long range order via a
@@ -543,29 +386,29 @@ exhibit fractionalised excitations linked to their ground state having
 long range entanglement and non-trivial topological properties <span
 class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a
 href="#ref-broholmQuantumSpinLiquids2020"
-role="doc-biblioref">40</a>]</span>. Quantum spin liquids are the
+role="doc-biblioref">42</a>]</span>. Quantum spin liquids are the
 analogous phase of matter for spin systems. Remarkably the existence of
 QSLs was first suggested by Anderson in 1973 <span class="citation"
 data-cites="andersonResonatingValenceBonds1973"> [<a
 href="#ref-andersonResonatingValenceBonds1973"
-role="doc-biblioref">41</a>]</span>.</p>
+role="doc-biblioref">43</a>]</span>.</p>
 <div id="fig:correlation_spin_orbit_PT" class="fignos">
 <figure>
 <img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png"
 data-short-caption="Phase Diagram" style="width:100.0%"
-alt="Figure 3: From  [42]." />
+alt="Figure 3: From  [44]." />
 <figcaption aria-hidden="true"><span>Figure 3:</span> From <span
 class="citation" data-cites="TrebstPhysRep2022"> [<a
 href="#ref-TrebstPhysRep2022"
-role="doc-biblioref">42</a>]</span>.</figcaption>
+role="doc-biblioref">44</a>]</span>.</figcaption>
 </figure>
 </div>
 <p>The main route to QSLs, though there are others <span
 class="citation"
 data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a
-href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">43</a>–<a
+href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">45</a>–<a
 href="#ref-linExactSymmetryWeaklyinteracting1998"
-role="doc-biblioref">45</a>]</span>, is via frustration of spin models
+role="doc-biblioref">47</a>]</span>, is via frustration of spin models
 that would otherwise order have AFM order. This frustration can come
 geometrically, triangular lattices for instance cannot support AFM
 order. It can also come about as a result of spin-orbit coupling.</p>
@@ -580,16 +423,16 @@ class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with
 strongly anisotropic spin-spin couplings known as Kitaev Materials <span
 class="citation"
 data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a
-href="#ref-TrebstPhysRep2022" role="doc-biblioref">42</a>,<a
-href="#ref-Jackeli2009" role="doc-biblioref">46</a>–<a
-href="#ref-Takagi2019" role="doc-biblioref">49</a>]</span>. Kitaev
+href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>,<a
+href="#ref-Jackeli2009" role="doc-biblioref">48</a>–<a
+href="#ref-Takagi2019" role="doc-biblioref">51</a>]</span>. Kitaev
 materials draw their name from the celebrated Kitaev Honeycomb Model as
 it is believed they will realise the QSL state via the mechanisms of the
 Kitaev Model.</p>
 <p>The Kitaev Honeycomb model <span class="citation"
 data-cites="kitaevAnyonsExactlySolved2006"> [<a
 href="#ref-kitaevAnyonsExactlySolved2006"
-role="doc-biblioref">50</a>]</span> was the first concrete model with a
+role="doc-biblioref">52</a>]</span> was the first concrete model with a
 QSL ground state. It is defined on the honeycomb lattice and provides an
 exactly solvable model whose ground state is a QSL characterized by a
 static <span class="math inline">\(\mathbb Z_2\)</span> gauge field and
@@ -601,37 +444,37 @@ its QSL ground state, it supports a rich phase diagram hosting gapless,
 Abelian and non-Abelian phases <span class="citation"
 data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a
 href="#ref-knolleDynamicsFractionalizationQuantum2015"
-role="doc-biblioref">51</a>]</span> and a finite temperature phase
+role="doc-biblioref">53</a>]</span> and a finite temperature phase
 transition to a thermal metal state <span class="citation"
 data-cites="selfThermallyInducedMetallic2019"> [<a
 href="#ref-selfThermallyInducedMetallic2019"
-role="doc-biblioref">52</a>]</span>. It been proposed that its
+role="doc-biblioref">54</a>]</span>. It been proposed that its
 non-Abelian excitations could be used to support robust topological
 quantum computing [<span class="citation"
 data-cites="kitaev_fault-tolerant_2003"> [<a
 href="#ref-kitaev_fault-tolerant_2003"
-role="doc-biblioref">53</a>]</span>; <span class="citation"
+role="doc-biblioref">55</a>]</span>; <span class="citation"
 data-cites="freedmanTopologicalQuantumComputation2003"> [<a
 href="#ref-freedmanTopologicalQuantumComputation2003"
-role="doc-biblioref">54</a>]</span>;
+role="doc-biblioref">56</a>]</span>;
 nayakNonAbelianAnyonsTopological2008].</p>
 <p>It is by now understood that the Kitaev model on any tri-coordinated
 <span class="math inline">\(z=3\)</span> graph has conserved plaquette
 operators and local symmetries <span class="citation"
 data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007"
-role="doc-biblioref">55</a>,<a href="#ref-Baskaran2008"
-role="doc-biblioref">56</a>]</span> which allow a mapping onto effective
+role="doc-biblioref">57</a>,<a href="#ref-Baskaran2008"
+role="doc-biblioref">58</a>]</span> which allow a mapping onto effective
 free Majorana fermion problems in a background of static <span
 class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation"
 data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a
-href="#ref-Nussinov2009" role="doc-biblioref">57</a>–<a
-href="#ref-hermanns2015weyl" role="doc-biblioref">60</a>]</span>.
+href="#ref-Nussinov2009" role="doc-biblioref">59</a>–<a
+href="#ref-hermanns2015weyl" role="doc-biblioref">62</a>]</span>.
 However, depending on lattice symmetries, finding the ground state flux
 sector and understanding the QSL properties can still be
 challenging <span class="citation"
 data-cites="eschmann2019thermodynamics Peri2020"> [<a
-href="#ref-eschmann2019thermodynamics" role="doc-biblioref">61</a>,<a
-href="#ref-Peri2020" role="doc-biblioref">62</a>]</span>.</p>
+href="#ref-eschmann2019thermodynamics" role="doc-biblioref">63</a>,<a
+href="#ref-Peri2020" role="doc-biblioref">64</a>]</span>.</p>
 <p><strong>paragraph about amorphous lattices</strong></p>
 <p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin
 liquid by extending the Kitaev honeycomb model to random lattices with
@@ -643,14 +486,18 @@ phases with a remarkably simple ground state flux pattern. Furthermore,
 I show that the system undergoes a finite-temperature phase transition
 to a conducting thermal metal state and discuss possible experimental
 realisations.</p>
-<h1 id="outline">Outline</h1>
+</section>
+<section id="outline" class="level1">
+<h1>Outline</h1>
 <p>The next chapter, Chapter 2, will introduce some necessary background
 to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and
 localisation.</p>
 <p>In Chapter 3 I introduce the Long Range Falikov-Kimball Model in
 greater detail. I will present results that. Chapter 4 focusses on the
 Amorphous Kitaev Model.</p>
-<h1 class="unnumbered" id="bibliography">Bibliography</h1>
+</section>
+<section id="bibliography" class="level1 unnumbered">
+<h1 class="unnumbered">Bibliography</h1>
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@@ -244,55 +25,529 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
 <li><a href="#the-falikov-kimball-model"
 id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
 <ul>
-<li><a href="#the-model" id="toc-the-model">The Model</a>
-<ul>
-<li><a href="#particle-hole-symmetry"
-id="toc-particle-hole-symmetry">Particle Hole Symmetry</a></li>
-</ul></li>
-<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
-Diagram</a></li>
-<li><a href="#long-range-ising-models"
-id="toc-long-range-ising-models">Long Range Ising Models</a></li>
+<li><a href="#the-model" id="toc-the-model">The Model</a></li>
+<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
+Diagrams</a></li>
+<li><a href="#long-ranged-ising-model"
+id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
 </ul></li>
+<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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="#the-falikov-kimball-model"
 id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
 <ul>
-<li><a href="#the-model" id="toc-the-model">The Model</a>
-<ul>
-<li><a href="#particle-hole-symmetry"
-id="toc-particle-hole-symmetry">Particle Hole Symmetry</a></li>
-</ul></li>
-<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
-Diagram</a></li>
-<li><a href="#long-range-ising-models"
-id="toc-long-range-ising-models">Long Range Ising Models</a></li>
+<li><a href="#the-model" id="toc-the-model">The Model</a></li>
+<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
+Diagrams</a></li>
+<li><a href="#long-ranged-ising-model"
+id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
 </ul></li>
+<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
 </nav>
  -->
-<h1 id="the-falikov-kimball-model">The Falikov Kimball Model</h1>
-<h2 id="the-model">The Model</h2>
-<p>discuss CDW phase of 2d model as motivation for studying 1d phase
-with long range forces</p>
-<h3 id="particle-hole-symmetry">Particle Hole Symmetry</h3>
-<h2 id="phase-diagram">Phase Diagram</h2>
-<h2 id="long-range-ising-models">Long Range Ising Models</h2>
+
+<!-- Main Page Body -->
+<section id="the-falikov-kimball-model" class="level1">
+<h1>The Falikov Kimball Model</h1>
+<section id="the-model" class="level2">
+<h2>The Model</h2>
+<p>The Falikov-Kimball (FK) model is one of the simplest models of the
+correlated electron problem. It captures the essence of the interaction
+between itinerant and localized electrons. It was originally introduced
+to explain the metal-insulator transition in f-electron systems but in
+its long history it has been interpreted variously as a model of
+electrons and ions, binary alloys or of crystal formation <span
+class="citation"
+data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicovSimpleModelSemiconductorMetal1969 gruberFalicovKimballModelReview1996 gruberFalicovKimballModel2006"> [<a
+href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
+role="doc-biblioref">1</a>–<a href="#ref-gruberFalicovKimballModel2006"
+role="doc-biblioref">4</a>]</span>. In terms of immobile fermions <span
+class="math inline">\(d_i\)</span> and light fermions <span
+class="math inline">\(c_i\)</span> and with chemical potential fixed at
+half-filling, the model reads</p>
+<p><span class="math display">\[\begin{aligned}
+H_{\mathrm{FK}} = &amp; \;U \sum_{i} (d^\dagger_{i}d_{i} -
+\tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle
+i,j\rangle} c^\dagger_{i}c_{j}.\\
+\end{aligned}\]</span></p>
+<p>The connection to the Hubbard model is that we have relabel the up
+and down spin electron states and removed the hopping term for one
+species, the equivalent of taking the limit of infinite mass ratio <span
+class="citation" data-cites="devriesSimplifiedHubbardModel1993"> [<a
+href="#ref-devriesSimplifiedHubbardModel1993"
+role="doc-biblioref">5</a>]</span>.</p>
+<p>Like other exactly solvable models <span class="citation"
+data-cites="smithDisorderFreeLocalization2017"> [<a
+href="#ref-smithDisorderFreeLocalization2017"
+role="doc-biblioref">6</a>]</span> and the Kitaev Model, the FK model
+possesses extensively many conserved degrees of freedom <span
+class="math inline">\([d^\dagger_{i}d_{i}, H] = 0\)</span>. The Hilbert
+space therefore breaks up into a set of sectors in which these operators
+take a definite value. Crucially, this reduces the interaction term
+<span class="math inline">\((d^\dagger_{i}d_{i} -
+\tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\)</span> from being
+quartic in fermion operators to quadratic. This is what makes the FK
+model exactly solvable, in contrast to the Hubbard model.</p>
+<p>Due to Pauli exclusion, maximum filling occurs when each lattice site
+is fully occupied, <span class="math inline">\(\langle n_c + n_d \rangle
+= 2\)</span>. Here we will focus on the half filled case <span
+class="math inline">\(\langle n_c + n_d \rangle = 1\)</span>. Doping the
+model away from the half-filled point leads to rich physics including
+superconductivity <span class="citation"
+data-cites="jedrzejewskiFalicovKimballModels2001"> [<a
+href="#ref-jedrzejewskiFalicovKimballModels2001"
+role="doc-biblioref">7</a>]</span>.</p>
+<p>At half-filling and on bipartite lattices, FK the model is
+particle-hole symmetric. That is, the Hamiltonian anticommutes with the
+particle hole operator <span
+class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a
+consequence the energy spectrum is symmetric about <span
+class="math inline">\(E = 0\)</span> and this is the Fermi energy. The
+particle hole operator corresponds to the substitution <span
+class="math inline">\(c^\dagger_i \rightarrow \epsilon_i c_i,
+d^\dagger_i \rightarrow d_i\)</span> where <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">8</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>.</p>
+<div id="fig:simple_DOS" class="fignos">
+<figure>
+<img src="/assets/thesis/background_chapter/simple_DOS.svg"
+data-short-caption="Cubic Lattice dispersion with disorder"
+style="width:100.0%"
+alt="Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in one dimension. (a) With not external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder." />
+<figcaption aria-hidden="true"><span>Figure 1:</span> The dispersion
+(upper row) and density of states (lower row) obtained from a cubic
+lattice model <span class="math inline">\(H = \sum_{i} V_i
+c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle}
+c^\dagger_{i}c_{j}\)</span> in one dimension. (a) With not external
+potential. (b) With a static charge density wave background <span
+class="math inline">\(V_i = (-1)^i\)</span> (c) A static charge density
+wave background with 2% binary disorder.</figcaption>
+</figure>
+</div>
+<p>We will later add a long range interaction between the localised
+electrons so we will replace the immobile fermions with a classical
+Ising field <span class="math inline">\(S_i = 1 - 2d^\dagger_id_i =
+\pm\tfrac{1}{2}\)</span>.</p>
+<p><span class="math display">\[\begin{aligned}
+H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
+\tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
+\end{aligned}\]</span></p>
+<p>The FK model can be solved exaclty with dynamic mean field theory in
+the infinite dimensional <span class="citation"
+data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a
+href="#ref-antipovCriticalExponentsStrongly2014"
+role="doc-biblioref">9</a>–<a
+href="#ref-herrmannNonequilibriumDynamicalCluster2016"
+role="doc-biblioref">12</a>]</span>.</p>
+<ul>
+<li>displays disorder free localisation</li>
+</ul>
+</section>
+<section id="phase-diagrams" class="level2">
+<h2>Phase Diagrams</h2>
+<div id="fig:fk_phase_diagram" class="fignos">
+<figure>
+<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg"
+data-short-caption="Fermi-Hubbard and Falikov-Kimball Temperatue-Interaction Phase Diagrams"
+style="width:100.0%"
+alt="Figure 2: Schematic Phase diagrams of the Fermi-Hubbard (left) and Falikov-Kimball model (right) showing temperature (T) and repulsive interaction strength (U). Hubbard model diagram adapted from  [13], Falikov-Kimball model from  [14,15]" />
+<figcaption aria-hidden="true"><span>Figure 2:</span> Schematic Phase
+diagrams of the Fermi-Hubbard (left) and Falikov-Kimball model (right)
+showing temperature (T) and repulsive interaction strength (U). Hubbard
+model diagram adapted from <span class="citation"
+data-cites="micnasSuperconductivityNarrowbandSystems1990"> [<a
+href="#ref-micnasSuperconductivityNarrowbandSystems1990"
+role="doc-biblioref">13</a>]</span>, Falikov-Kimball model from <span
+class="citation"
+data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014a"> [<a
+href="#ref-antipovInteractionTunedAndersonMott2016"
+role="doc-biblioref">14</a>,<a
+href="#ref-antipovCriticalExponentsStrongly2014a"
+role="doc-biblioref">15</a>]</span></figcaption>
+</figure>
+</div>
+<ul>
+<li>rich phase diagram in 2d Despite its simplicity, the FK model has a
+rich phase diagram in <span class="math inline">\(D \geq 2\)</span>
+dimensions. For example, it shows an interaction-induced gap opening
+even at high temperatures, similar to the corresponding Hubbard
+Model <span class="citation"
+data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
+href="#ref-brandtThermodynamicsCorrelationFunctions1989"
+role="doc-biblioref">16</a>]</span>.</li>
+</ul>
+<p>At half filling and in dimensions greater than one, the FK model
+exhibits a phase transition at some <span
+class="math inline">\(U\)</span> dependent critical temperature <span
+class="math inline">\(T_c(U)\)</span> to a low temperature charge
+density wave state in which the spins order antiferromagnetically. This
+corresponds to the heavy electrons occupying one of the two sublattices
+A and B <span class="citation"
+data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
+href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
+role="doc-biblioref">17</a>]</span>. In the disordered region above
+<span class="math inline">\(T_c(U)\)</span> there is a transition
+between an Anderson insulator phase at weak interaction and a Mott
+insulator phase in the strongly interacting regime <span
+class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a
+href="#ref-andersonAbsenceDiffusionCertain1958"
+role="doc-biblioref">18</a>]</span>.</p>
+<ul>
+<li>superconductivity when doped</li>
+</ul>
+<p>In 1D, the ground state phenomenology as the model is doped away from
+the half-filled state can be rich <span class="citation"
+data-cites="gruberGroundStatesSpinless1990"> [<a
+href="#ref-gruberGroundStatesSpinless1990"
+role="doc-biblioref">19</a>]</span> but the system is disordered for all
+<span class="math inline">\(T &gt; 0\)</span> <span class="citation"
+data-cites="kennedyItinerantElectronModel1986"> [<a
+href="#ref-kennedyItinerantElectronModel1986"
+role="doc-biblioref">20</a>]</span>.</p>
+<p>In the one dimensional FK model there is no ordered CDW phase <span
+class="citation" data-cites="liebAbsenceMottTransition1968"> [<a
+href="#ref-liebAbsenceMottTransition1968"
+role="doc-biblioref">21</a>]</span>. The supression of phase transition
+is a common phenomena in one dimensional systems. It can be understood
+via Peierls’ argument <span class="citation"
+data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a
+href="#ref-kennedyItinerantElectronModel1986"
+role="doc-biblioref">20</a>,<a
+href="#ref-peierlsIsingModelFerromagnetism1936"
+role="doc-biblioref">22</a>]</span> to be a consequence of the low
+energy penalty for domain walls in one dimensional systems.</p>
+<p>Following Peierls’ argument, consider the difference in free energy
+<span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span>
+between an ordered state and a state with single domain wall in a
+discrete order parameter. Short range interactions produce a constant
+energy penalty for such a domain wall that does not scale with system
+size. In contrast, the number of such single domain wall states scales
+linearly so the entropy is <span class="math inline">\(\propto \ln
+L\)</span>. Thus the entropic contribution dominates (eventually) in the
+thermodynamic limit and no finite temperature order is possible. In two
+dimensions and above, the energy penalty of a domain wall scales like
+<span class="math inline">\(L^{d-1}\)</span> so they can support ordered
+phases.</p>
+</section>
+<section id="long-ranged-ising-model" class="level2">
+<h2>Long Ranged Ising model</h2>
+<p>Our extension to the FK model could now be though of as spinless
+fermions coupled to a long range Ising (LRI) model. The LRI model has
+been extensively studied and its behaviour may be bear relation to the
+behaviour of our modified FK model.</p>
+<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|)
+\tau_i \tau_j = J \sum_{i\neq j} |i - j|^{-\alpha} \tau_i
+\tau_j\]</span></p>
+<p>Renormalisation group analyses show that the LRI model has an ordered
+phase in 1D for $1 &lt; &lt; 2 $ <span class="citation"
+data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
+href="#ref-dysonExistencePhasetransitionOnedimensional1969"
+role="doc-biblioref">23</a>]</span>. Peierls’ argument can be
+extended <span class="citation"
+data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
+href="#ref-thoulessLongRangeOrderOneDimensional1969"
+role="doc-biblioref">24</a>]</span> to long range interactions to
+provide intuition for why this is the case. Again considering the energy
+difference between the ordered state <span
+class="math inline">\(\ket{\ldots\uparrow\uparrow\uparrow\uparrow\ldots}\)</span>
+and a domain wall state <span
+class="math inline">\(\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}\)</span>.
+In the case of the LRI model, careful counting shows that this energy
+penalty is: <span class="math display">\[\Delta E \propto
+\sum_{n=1}^{\infty} n J(n)\]</span></p>
+<p>because each interaction between spins separated across the domain by
+a bond length <span class="math inline">\(n\)</span> can be drawn
+between <span class="math inline">\(n\)</span> equivalent pairs of
+sites. Ruelle proved rigorously for a very general class of 1D systems,
+that if <span class="math inline">\(\Delta E\)</span> or its many-body
+generalisation converges in the thermodynamic limit then the free energy
+is analytic <span class="citation"
+data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a
+href="#ref-ruelleStatisticalMechanicsOnedimensional1968"
+role="doc-biblioref">25</a>]</span>. This rules out a finite order phase
+transition, though not one of the Kosterlitz-Thouless type. Dyson also
+proves this though with a slightly different condition on <span
+class="math inline">\(J(n)\)</span> <span class="citation"
+data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
+href="#ref-dysonExistencePhasetransitionOnedimensional1969"
+role="doc-biblioref">23</a>]</span>.</p>
+<p>With a power law form for <span class="math inline">\(J(n)\)</span>,
+there are three cases to consider:</p>
+<ol type="1">
+<li>$ = 0$ For infinite range interactions the Ising model is exactly
+solveable and mean field theory is exact <span class="citation"
+data-cites="lipkinValidityManybodyApproximation1965"> [<a
+href="#ref-lipkinValidityManybodyApproximation1965"
+role="doc-biblioref">26</a>]</span>.</li>
+<li>$ $ For slowly decaying interactions <span
+class="math inline">\(\sum_n J(n)\)</span> does not converge so the
+Hamiltonian is non-extensive, a case which won’t be further considered
+here.</li>
+<li>$ 1 &lt; &lt; 2 $ A phase transition to an ordered state at a finite
+temperature.</li>
+<li>$ = 2 $ The energy of domain walls diverges logarithmically, and
+this turns out to be a Kostelitz-Thouless transition <span
+class="citation"
+data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
+href="#ref-thoulessLongRangeOrderOneDimensional1969"
+role="doc-biblioref">24</a>]</span>.</li>
+<li>$ 2 &lt; $ For quickly decaying interactions, domain walls have a
+finite energy penalty, hence Peirels’ argument holds and there is no
+phase transition.</li>
+</ol>
+<div id="fig:alpha_diagram" class="fignos">
+<figure>
+<img src="/assets/thesis/background_chapter/alpha_diagram.svg"
+data-short-caption="Long Range Ising Model Behaviour"
+style="width:100.0%" alt="Figure 3: " />
+<figcaption aria-hidden="true"><span>Figure 3:</span> </figcaption>
+</figure>
+</div>
 <div class="sourceCode" id="cb1"><pre
 class="sourceCode python"><code class="sourceCode python"></code></pre></div>
+</section>
+</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-hubbardj.ElectronCorrelationsNarrow1963" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[1] </div><div
+class="csl-right-inline">Hubbard, J., <em><a
+href="https://doi.org/10.1098/rspa.1963.0204">Electron Correlations in
+Narrow Energy Bands</a></em>, Proceedings of the Royal Society of
+London. Series A. Mathematical and Physical Sciences
+<strong>276</strong>, 238 (1963).</div>
+</div>
+<div id="ref-falicovSimpleModelSemiconductorMetal1969" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[2] </div><div class="csl-right-inline">L.
+M. Falicov and J. C. Kimball, <em><a
+href="https://doi.org/10.1103/PhysRevLett.22.997">Simple Model for
+Semiconductor-Metal Transitions: Sm${\mathrm{B}}_{6}$ and
+Transition-Metal Oxides</a></em>, Phys. Rev. Lett. <strong>22</strong>,
+997 (1969).</div>
+</div>
+<div id="ref-gruberFalicovKimballModelReview1996" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[3] </div><div class="csl-right-inline">C.
+Gruber and N. Macris, <em>The Falicov-Kimball Model: A Review of Exact
+Results and Extensions</em>, Helvetica Physica Acta <strong>69</strong>,
+(1996).</div>
+</div>
+<div id="ref-gruberFalicovKimballModel2006" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[4] </div><div class="csl-right-inline">C.
+Gruber and D. Ueltschi, <em>The Falicov-Kimball model</em>, in
+<em>Encyclopedia of mathematical physics. 1, 1,</em> (Elsevier,
+Amsterdam, 2006).</div>
+</div>
+<div id="ref-devriesSimplifiedHubbardModel1993" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[5] </div><div class="csl-right-inline">P.
+de Vries, K. Michielsen, and H. De Raedt, <em><a
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+
 </main>
 </body>
 </html>
diff --git a/_thesis/2_Background/2.2_HKM_Model.html b/_thesis/2_Background/2.2_HKM_Model.html
index a36db86..c7a5ec5 100644
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@@ -266,7 +25,7 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
 <li><a href="#the-kitaev-honeycomb-model"
 id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
@@ -284,12 +43,15 @@ Diagram</a></li>
 </ul></li>
 <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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="#the-kitaev-honeycomb-model"
@@ -310,7 +72,10 @@ Diagram</a></li>
 </ul>
 </nav>
  -->
-<h1 id="the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</h1>
+
+<!-- Main Page Body -->
+<section id="the-kitaev-honeycomb-model" class="level1">
+<h1>The Kitaev Honeycomb Model</h1>
 <p><strong>papers</strong> Jos on dynamics
 https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.115127</p>
 <p><strong>intro</strong> - strong spin orbit coupling leads to
@@ -323,7 +88,8 @@ with long range entanglement (not simple paramagnet)</p>
 <li>experimental probes include inelastic neutron scattering, Raman
 scattering</li>
 </ul>
-<h2 id="the-model">The Model</h2>
+<section id="the-model" class="level2">
+<h2>The Model</h2>
 <div id="fig:intro_figure_by_hand" class="fignos">
 <figure>
 <img
@@ -356,15 +122,24 @@ role="doc-biblioref">1</a>]</span></li>
 <li>has extensively many conserved fluxes</li>
 <li></li>
 </ul>
-<h2 id="a-mapping-to-majorana-fermions">A mapping to Majorana
-Fermions</h2>
-<h2 id="gauge-fields">Gauge Fields</h2>
-<h2 id="anyons-topology-and-the-chern-number">Anyons, Topology and the
-Chern number</h2>
-<h2 id="phase-diagram">Phase Diagram</h2>
+</section>
+<section id="a-mapping-to-majorana-fermions" class="level2">
+<h2>A mapping to Majorana Fermions</h2>
+</section>
+<section id="gauge-fields" class="level2">
+<h2>Gauge Fields</h2>
+</section>
+<section id="anyons-topology-and-the-chern-number" class="level2">
+<h2>Anyons, Topology and the Chern number</h2>
+</section>
+<section id="phase-diagram" class="level2">
+<h2>Phase Diagram</h2>
 <div class="sourceCode" id="cb1"><pre
 class="sourceCode python"><code class="sourceCode python"></code></pre></div>
-<h1 class="unnumbered" id="bibliography">Bibliography</h1>
+</section>
+</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-kugelJahnTellerEffectMagnetism1982" class="csl-entry"
 role="doc-biblioentry">
@@ -375,6 +150,9 @@ Effect and Magnetism: Transition Metal Compounds</a></em>, Sov. Phys.
 Usp. <strong>25</strong>, 231 (1982).</div>
 </div>
 </div>
+</section>
+
+
 </main>
 </body>
 </html>
diff --git a/_thesis/2_Background/2.3_Disorder.html b/_thesis/2_Background/2.3_Disorder.html
index 702a462..244969a 100644
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-   -->
 
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+<script src="/assets/js/thesis_scrollspy.js"></script>
 
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@@ -244,10 +25,10 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
-<li><a href="#disorder-localisation"
-id="toc-disorder-localisation">Disorder &amp; Localisation</a>
+<li><a href="#disorder-and-localisation"
+id="toc-disorder-and-localisation">Disorder and Localisation</a>
 <ul>
 <li><a href="#localisation-anderson-many-body-and-disorder-free"
 id="toc-localisation-anderson-many-body-and-disorder-free">Localisation:
@@ -256,18 +37,23 @@ Anderson, Many Body and Disorder-Free</a></li>
 id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
 <li><a href="#amorphous-magnetism"
 id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
+<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
 </ul></li>
+<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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="#disorder-localisation"
-id="toc-disorder-localisation">Disorder &amp; Localisation</a>
+<li><a href="#disorder-and-localisation"
+id="toc-disorder-and-localisation">Disorder and Localisation</a>
 <ul>
 <li><a href="#localisation-anderson-many-body-and-disorder-free"
 id="toc-localisation-anderson-many-body-and-disorder-free">Localisation:
@@ -276,17 +62,210 @@ Anderson, Many Body and Disorder-Free</a></li>
 id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
 <li><a href="#amorphous-magnetism"
 id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
+<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
 </ul></li>
+<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
 </nav>
  -->
-<h1 id="disorder-localisation">Disorder &amp; Localisation</h1>
-<h2 id="localisation-anderson-many-body-and-disorder-free">Localisation:
-Anderson, Many Body and Disorder-Free</h2>
-<h2 id="disorder-and-spin-liquids">Disorder and Spin liquids</h2>
-<h2 id="amorphous-magnetism">Amorphous Magnetism</h2>
+
+<!-- Main Page Body -->
+<section id="disorder-and-localisation" class="level1">
+<h1>Disorder and Localisation</h1>
+<section id="localisation-anderson-many-body-and-disorder-free"
+class="level2">
+<h2>Localisation: Anderson, Many Body and Disorder-Free</h2>
+</section>
+<section id="disorder-and-spin-liquids" class="level2">
+<h2>Disorder and Spin liquids</h2>
+</section>
+<section id="amorphous-magnetism" class="level2">
+<h2>Amorphous Magnetism</h2>
 <div class="sourceCode" id="cb1"><pre
 class="sourceCode python"><code class="sourceCode python"></code></pre></div>
+</section>
+<section id="localisation" class="level2">
+<h2>Localisation</h2>
+<p>The discovery of localisation in quantum systems surprising at the
+time given the seeming ubiquity of extended Bloch states. Later, when
+thermalisation in quantum systems gained interest, localisation
+phenomena again stood out as counterexamples to the eigenstate
+thermalisation hypothesis <span class="citation"
+data-cites="abaninRecentProgressManybody2017 srednickiChaosQuantumThermalization1994"> [<a
+href="#ref-abaninRecentProgressManybody2017"
+role="doc-biblioref">1</a>,<a
+href="#ref-srednickiChaosQuantumThermalization1994"
+role="doc-biblioref">2</a>]</span>, allowing quantum systems to avoid to
+retain memory of their initial conditions in the face of thermal
+noise.</p>
+<p>The simplest and first discovered kind is Anderson localisation,
+first studied in 1958 <span class="citation"
+data-cites="andersonAbsenceDiffusionCertain1958"> [<a
+href="#ref-andersonAbsenceDiffusionCertain1958"
+role="doc-biblioref">3</a>]</span> in the context of non-interacting
+fermions subject to a static or quenched disorder potential <span
+class="math inline">\(V_j\)</span> drawn uniformly from the interval
+<span class="math inline">\([-W,W]\)</span></p>
+<p><span class="math display">\[
+H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger
+c_j
+\]</span></p>
+<p>this model exhibits exponentially localised eigenfunctions <span
+class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\)</span> which
+cannot contribute to transport processes. Initially it was thought that
+in one dimensional disordered models, all states would be localised,
+however it was later shown that in the presence of correlated disorder,
+bands of extended states can exist <span class="citation"
+data-cites="izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a
+href="#ref-izrailevLocalizationMobilityEdge1999"
+role="doc-biblioref">4</a>–<a
+href="#ref-izrailevAnomalousLocalizationLowDimensional2012"
+role="doc-biblioref">6</a>]</span>.</p>
+<p>Later localisation was found in interacting many-body systems with
+quenched disorder:</p>
+<p><span class="math display">\[
+H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger
+c_j + U\sum_{jk} n_j n_k
+\]</span></p>
+<p>where the number operators <span class="math inline">\(n_j =
+c^\dagger_j c_j\)</span>. Here, in contrast to the Anderson model,
+localisation phenomena can be proven robust to weak perturbations of the
+Hamiltonian. This is called many-body localisation (MBL) <span
+class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016"> [<a
+href="#ref-imbrieManyBodyLocalizationQuantum2016"
+role="doc-biblioref">7</a>]</span>.</p>
+<p>Both MBL and Anderson localisation depend crucially on the presence
+of quenched disorder. This has led to ongoing interest in the
+possibility of disorder-free localisation, in which the disorder
+necessary to generate localisation is generated entirely from the
+dynamics of the model. This contracts with typical models of disordered
+systems in which disorder is explicitly introduced into the Hamilton or
+the initial state.</p>
+<p>The concept of disorder-free localisation was first proposed in the
+context of Helium mixtures <span class="citation"
+data-cites="kagan1984localization"> [<a
+href="#ref-kagan1984localization" role="doc-biblioref">8</a>]</span> and
+then extended to heavy-light mixtures in which multiple species with
+large mass ratios interact. The idea is that the heavier particles act
+as an effective disorder potential for the lighter ones, inducing
+localisation. Two such models <span class="citation"
+data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015"> [<a
+href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
+role="doc-biblioref">9</a>,<a
+href="#ref-schiulazDynamicsManybodyLocalized2015"
+role="doc-biblioref">10</a>]</span> instead find that the models
+thermalise exponentially slowly in system size, which Ref. <span
+class="citation"
+data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"> [<a
+href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
+role="doc-biblioref">9</a>]</span> dubs Quasi-MBL.</p>
+<p>True disorder-free localisation does occur in exactly solvable models
+with extensively many conserved quantities <span class="citation"
+data-cites="smithDisorderFreeLocalization2017"> [<a
+href="#ref-smithDisorderFreeLocalization2017"
+role="doc-biblioref">11</a>]</span>. As conserved quantities have no
+time dynamics this can be thought of as taking the separation of
+timescales to the infinite limit.</p>
+<p>-link to the FK model</p>
+<p>-link to the Kitaev Model</p>
+<p>-link to the physics of amorphous systems</p>
+</section>
+</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-abaninRecentProgressManybody2017" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[1] </div><div class="csl-right-inline">D.
+A. Abanin and Z. Papić, <em><a
+href="https://doi.org/10.1002/andp.201700169">Recent Progress in
+Many-Body Localization</a></em>, ANNALEN DER PHYSIK
+<strong>529</strong>, 1700169 (2017).</div>
+</div>
+<div id="ref-srednickiChaosQuantumThermalization1994" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[2] </div><div class="csl-right-inline">M.
+Srednicki, <em><a href="https://doi.org/10.1103/PhysRevE.50.888">Chaos
+and Quantum Thermalization</a></em>, Phys. Rev. E <strong>50</strong>,
+888 (1994).</div>
+</div>
+<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[3] </div><div class="csl-right-inline">P.
+W. Anderson, <em><a
+href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in
+Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492
+(1958).</div>
+</div>
+<div id="ref-izrailevLocalizationMobilityEdge1999" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[4] </div><div class="csl-right-inline">F.
+M. Izrailev and A. A. Krokhin, <em><a
+href="https://doi.org/10.1103/PhysRevLett.82.4062">Localization and the
+Mobility Edge in One-Dimensional Potentials with Correlated
+Disorder</a></em>, Phys. Rev. Lett. <strong>82</strong>, 4062
+(1999).</div>
+</div>
+<div id="ref-croyAndersonLocalization1D2011" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[5] </div><div class="csl-right-inline">A.
+Croy, P. Cain, and M. Schreiber, <em><a
+href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization
+in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B
+<strong>82</strong>, 107 (2011).</div>
+</div>
+<div id="ref-izrailevAnomalousLocalizationLowDimensional2012"
+class="csl-entry" role="doc-biblioentry">
+<div class="csl-left-margin">[6] </div><div class="csl-right-inline">F.
+M. Izrailev, A. A. Krokhin, and N. M. Makarov, <em><a
+href="https://doi.org/10.1016/j.physrep.2011.11.002">Anomalous
+Localization in Low-Dimensional Systems with Correlated
+Disorder</a></em>, Physics Reports <strong>512</strong>, 125
+(2012).</div>
+</div>
+<div id="ref-imbrieManyBodyLocalizationQuantum2016" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[7] </div><div class="csl-right-inline">J.
+Z. Imbrie, <em><a href="https://doi.org/10.1007/s10955-016-1508-x">On
+Many-Body Localization for Quantum Spin Chains</a></em>, J Stat Phys
+<strong>163</strong>, 998 (2016).</div>
+</div>
+<div id="ref-kagan1984localization" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[8] </div><div class="csl-right-inline">Y.
+Kagan and L. Maksimov, <em>Localization in a System of Interacting
+Particles Diffusing in a Regular Crystal</em>, Zhurnal Eksperimental’noi
+i Teoreticheskoi Fiziki <strong>87</strong>, 348 (1984).</div>
+</div>
+<div id="ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
+class="csl-entry" role="doc-biblioentry">
+<div class="csl-left-margin">[9] </div><div class="csl-right-inline">N.
+ Y. Yao, C.  R. Laumann, J.  I. Cirac, M.  D. Lukin, and J.  E. Moore,
+<em><a
+href="https://doi.org/10.1103/PhysRevLett.117.240601">Quasi-Many-Body
+Localization in Translation-Invariant Systems</a></em>, Phys. Rev. Lett.
+<strong>117</strong>, 240601 (2016).</div>
+</div>
+<div id="ref-schiulazDynamicsManybodyLocalized2015" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[10] </div><div class="csl-right-inline">M.
+Schiulaz, A. Silva, and M. Müller, <em><a
+href="https://doi.org/10.1103/PhysRevB.91.184202">Dynamics in Many-Body
+Localized Quantum Systems Without Disorder</a></em>, Phys. Rev. B
+<strong>91</strong>, 184202 (2015).</div>
+</div>
+<div id="ref-smithDisorderFreeLocalization2017" class="csl-entry"
+role="doc-biblioentry">
+<div class="csl-left-margin">[11] </div><div class="csl-right-inline">A.
+Smith, J. Knolle, D.  L. Kovrizhin, and R. Moessner, <em><a
+href="https://doi.org/10.1103/PhysRevLett.118.266601">Disorder-Free
+Localization</a></em>, Phys. Rev. Lett. <strong>118</strong>, 266601
+(2017).</div>
+</div>
+</div>
+</section>
+
+
 </main>
 </body>
 </html>
diff --git a/_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html b/_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html
index b5438d5..c25a9c8 100644
--- a/_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html
+++ b/_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html
@@ -1,5 +1,5 @@
 ---
-title: The Long Range Falikov-Kimball Model
+title: The Long Range Falikov-Kimball Model - The Model
 excerpt: 
 layout: none
 image: 
@@ -11,253 +11,12 @@ image:
   <meta charset="utf-8" />
   <meta name="generator" content="pandoc" />
   <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
-  <title>The Long Range Falikov-Kimball Model</title>
-  <!-- <style>
-    html {
-      line-height: 1.5;
-      font-family: Georgia, serif;
-      font-size: 20px;
-      color: #1a1a1a;
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+  <title>The Long Range Falikov-Kimball Model - The Model</title>
 
-<!--   <script
-       src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js"
-       type="text/javascript"></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">
 <script src="/assets/js/index.js"></script>
 </head>
@@ -266,122 +25,29 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
-<li><a href="#chapter-summary" id="toc-chapter-summary">Chapter
-Summary</a></li>
-<li><a href="#the-model" id="toc-the-model">The Model</a>
-<ul>
-<li><a href="#localisation" id="toc-localisation">Localisation</a>
-<ul>
-<li><a href="#the-falikov-kimball-model"
-id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a></li>
-</ul></li>
-<li><a href="#falikov-kimball-and-hubbard-models"
-id="toc-falikov-kimball-and-hubbard-models">Falikov Kimball and Hubbard
-models</a>
-<ul>
-<li><a href="#hubbard-model" id="toc-hubbard-model">Hubbard
-model</a></li>
-<li><a href="#falikov-kimball-model"
-id="toc-falikov-kimball-model">Falikov-Kimball model</a></li>
-<li><a href="#thermodynamics-of-the-fk-model"
-id="toc-thermodynamics-of-the-fk-model">Thermodynamics of the FK
-model</a></li>
-</ul></li>
-<li><a href="#long-ranged-ising-model"
-id="toc-long-ranged-ising-model">Long Ranged Ising model</a>
-<ul>
-<li><a href="#thermodynamics"
-id="toc-thermodynamics">Thermodynamics</a></li>
-</ul></li>
-<li><a href="#localisation-1" id="toc-localisation-1">Localisation</a>
-<ul>
-<li><a href="#thermalisation"
-id="toc-thermalisation">Thermalisation</a></li>
-<li><a href="#anderson-localisation"
-id="toc-anderson-localisation">Anderson Localisation</a></li>
-<li><a href="#many-body-localisation"
-id="toc-many-body-localisation">Many Body Localisation</a></li>
-<li><a href="#disorder-free-localisation"
-id="toc-disorder-free-localisation">Disorder Free localisation</a></li>
-</ul></li>
-<li><a href="#introduction" id="toc-introduction">Introduction</a></li>
-<li><a href="#the-long-ranged-falikov-kimball-model"
-id="toc-the-long-ranged-falikov-kimball-model">The Long-Ranged
-Falikov-Kimball Model</a></li>
-<li><a href="#introduction-and-motivation"
-id="toc-introduction-and-motivation">Introduction and
-Motivation</a></li>
-<li><a href="#literature-review" id="toc-literature-review">Literature
-Review</a></li>
-<li><a href="#the-falikov-kimball-model-1"
-id="toc-the-falikov-kimball-model-1">The Falikov-Kimball model</a></li>
-</ul></li>
+<li><a href="#sec:FK-Model" id="toc-sec:FK-Model">The Model</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 -->
+<!-- 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="#chapter-summary" id="toc-chapter-summary">Chapter
-Summary</a></li>
-<li><a href="#the-model" id="toc-the-model">The Model</a>
-<ul>
-<li><a href="#localisation" id="toc-localisation">Localisation</a>
-<ul>
-<li><a href="#the-falikov-kimball-model"
-id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a></li>
-</ul></li>
-<li><a href="#falikov-kimball-and-hubbard-models"
-id="toc-falikov-kimball-and-hubbard-models">Falikov Kimball and Hubbard
-models</a>
-<ul>
-<li><a href="#hubbard-model" id="toc-hubbard-model">Hubbard
-model</a></li>
-<li><a href="#falikov-kimball-model"
-id="toc-falikov-kimball-model">Falikov-Kimball model</a></li>
-<li><a href="#thermodynamics-of-the-fk-model"
-id="toc-thermodynamics-of-the-fk-model">Thermodynamics of the FK
-model</a></li>
-</ul></li>
-<li><a href="#long-ranged-ising-model"
-id="toc-long-ranged-ising-model">Long Ranged Ising model</a>
-<ul>
-<li><a href="#thermodynamics"
-id="toc-thermodynamics">Thermodynamics</a></li>
-</ul></li>
-<li><a href="#localisation-1" id="toc-localisation-1">Localisation</a>
-<ul>
-<li><a href="#thermalisation"
-id="toc-thermalisation">Thermalisation</a></li>
-<li><a href="#anderson-localisation"
-id="toc-anderson-localisation">Anderson Localisation</a></li>
-<li><a href="#many-body-localisation"
-id="toc-many-body-localisation">Many Body Localisation</a></li>
-<li><a href="#disorder-free-localisation"
-id="toc-disorder-free-localisation">Disorder Free localisation</a></li>
-</ul></li>
-<li><a href="#introduction" id="toc-introduction">Introduction</a></li>
-<li><a href="#the-long-ranged-falikov-kimball-model"
-id="toc-the-long-ranged-falikov-kimball-model">The Long-Ranged
-Falikov-Kimball Model</a></li>
-<li><a href="#introduction-and-motivation"
-id="toc-introduction-and-motivation">Introduction and
-Motivation</a></li>
-<li><a href="#literature-review" id="toc-literature-review">Literature
-Review</a></li>
-<li><a href="#the-falikov-kimball-model-1"
-id="toc-the-falikov-kimball-model-1">The Falikov-Kimball model</a></li>
-</ul></li>
+<li><a href="#sec:FK-Model" id="toc-sec:FK-Model">The Model</a></li>
 <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
 </nav>
  -->
+
+<!-- Main Page Body -->
 <p>This chapter expands on work presented in</p>
 <p> <span class="citation"
 data-cites="hodsonOnedimensionalLongrangeFalikovKimball2021"> [<a
@@ -398,468 +64,87 @@ role="doc-biblioref">2</a>]</span>.</p>
 <p><strong>Contributions</strong></p>
 <p>Johannes had the initial idea to use a long range Ising term to
 stablise order in a one dimension Falikov-Kimball model. Josef developed
-a proof of concept during a summer project at Imperial. I wrote the
-simulation code and performed all the analysis presented here.</p>
-<h2 id="chapter-summary">Chapter Summary</h2>
-<p>To evaluate thermodynamic averages we perform a classical Markov
-Chain Monte Carlo random walk over the space of ionic configurations, at
-each step diagonalising the effective electronic Hamiltonian <span
-class="citation"
-data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
-href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
-role="doc-biblioref">3</a>]</span>. Using a binder-cumulant method <span
-class="citation"
-data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a
-href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">4</a>,<a
-href="#ref-musialMonteCarloSimulations2002"
-role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a>]</span>,
-we demonstrate the model has a finite temperature phase transition when
-the interaction is sufficiently long ranged. We then estimate the
-density of states and the inverse participation ratio as a function of
-energy to diagnose localisation properties. We show preliminary results
-that the in-gap states induced at finite temperature are localised while
-the states in the unperturbed bands remain extended, evidence for a
-mobility edge.</p>
-<p>Despite its simplicity, the FK model has a rich phase diagram in
-<span class="math inline">\(D \geq 2\)</span> dimensions. For example,
-it shows an interaction-induced gap opening even at high temperatures,
-similar to the corresponding Hubbard Model <span class="citation"
-data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
-href="#ref-brandtThermodynamicsCorrelationFunctions1989"
-role="doc-biblioref">5</a>]</span>. In 1D, the ground state
-phenomenology as a function of filling can be rich <span
-class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a
-href="#ref-gruberGroundStatesSpinless1990"
-role="doc-biblioref">6</a>]</span> but the system is disordered for all
-<span class="math inline">\(T &gt; 0\)</span> <span class="citation"
-data-cites="kennedyItinerantElectronModel1986"> [<a
-href="#ref-kennedyItinerantElectronModel1986"
-role="doc-biblioref">7</a>]</span>. Moreover, the model has been a
-test-bed for many-body methods, interest took off when an exact DMFT
-solution in the infinite dimensional case was found <span
-class="citation"
-data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a
-href="#ref-antipovCriticalExponentsStrongly2014"
-role="doc-biblioref">8</a>–<a
-href="#ref-herrmannNonequilibriumDynamicalCluster2016"
-role="doc-biblioref">11</a>]</span>.</p>
-<h1 id="the-model">The Model</h1>
-<p>The Falikov-Kimball (FK) model <span
-class="math display">\[\begin{aligned}
-H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
-\tfrac{1}{2}) -\;t \sum_{i} (c^\dagger_{i}c_{i+1} + \textit{h.c.)}\\
-\end{aligned}\]</span> is one of the simplest models of the correlated
-electron problem. It captures the essence of the interaction between
-itinerant <span class="math inline">\(c_i\)</span> and localized
-electrons, equivalent to a model of hopping fermions coupled to a
-classical Ising field <span class="math inline">\(S_i = \pm 1\)</span>.
-It was originally introduced to explain the metal-insulator transition
-in f-electron systems but in its long history it has been interpreted
-variously as a model of electrons and ions, binary alloys or of crystal
-formation <span class="citation"
-data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicovSimpleModelSemiconductorMetal1969 gruberFalicovKimballModelReview1996 gruberFalicovKimballModel2006"> [<a
-href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
-role="doc-biblioref">12</a>–<a href="#ref-gruberFalicovKimballModel2006"
-role="doc-biblioref">15</a>]</span>.</p>
-<p>One can look to the references above to see why one might write the
-FK Model down directly. Here, in order to motivate the connection to
-exactly solvable models, I will explain how the Falikov-Kimball model is
-a particular limit of the Hubbard Model <span
-class="math display">\[\begin{aligned}
-H_{\mathrm{H}} = &amp; \;U \sum_{i} n_{i\downarrow} n_{i\uparrow} -\;t
-\sum_{i\alpha} c^\dagger_{i,\alpha}c_{i+1,\alpha} + \mu \sum_{i\alpha}
-n_{i\alpha} + \textit{h.c.} \\
-\end{aligned}\]</span></p>
-<p>The Hubbard model is one of interacting spin-<span
-class="math inline">\(\tfrac{1}{2}\)</span> electrons hopping on a
-lattice. It has been the subject of intense study for many years but, as
-an interacting many body quantum system, analytic handles on it are few
-and far between. There is still much we don’t know about the Hubbard
-Model.</p>
-<p>Usually the Hubbard model is written with equal hopping for the two
-spins states <span class="math inline">\(t_\uparrow =
-t_\downarrow\)</span> have included a spin dependent hopping term <span
-class="math inline">\(t_\alpha\)</span></p>
-<h2 id="localisation">Localisation</h2>
-<p>The discovery of localisation in quantum systems surprising at the
-time given the seeming ubiquity of extended Bloch states. Later, when
-thermalisation in quantum systems gained interest, localisation
-phenomena again stood out as counterexamples to the eigenstate
-thermalisation hypothesis <span class="citation"
-data-cites="abaninRecentProgressManybody2017 srednickiChaosQuantumThermalization1994"> [<a
-href="#ref-abaninRecentProgressManybody2017"
-role="doc-biblioref">16</a>,<a
-href="#ref-srednickiChaosQuantumThermalization1994"
-role="doc-biblioref">17</a>]</span>, allowing quantum systems to avoid
-to retain memory of their initial conditions in the face of thermal
-noise.</p>
-<p>The simplest and first discovered kind is Anderson localisation,
-first studied in 1958 <span class="citation"
-data-cites="andersonAbsenceDiffusionCertain1958"> [<a
+a proof of concept during a summer project at Imperial along with
+Alexander Belcik. I wrote the simulation code and performed all the
+analysis presented here.</p>
+<p><strong>Chapter Summary</strong></p>
+<p>The paper is organised as follows. First, we introduce the model and
+present its phase diagram. Second, we present the methods used to solve
+it numerically. Last, we investigate the model’s localisation properties
+and conclude.</p>
+<section id="sec:FK-Model" class="level1">
+<h1>The Model</h1>
+<p>Dimensionality is crucial for the physics of both localisation and
+FTPTs. In 1D, disorder generally dominates, even the weakest disorder
+exponentially localises <em>all</em> single particle eigenstates. Only
+longer-range correlations of the disorder potential can potentially
+induce delocalization <span class="citation"
+data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a
+href="#ref-aubryAnalyticityBreakingAnderson1980"
+role="doc-biblioref">3</a>–<a
+href="#ref-dunlapAbsenceLocalizationRandomdimer1990"
+role="doc-biblioref">5</a>]</span>. Thermodynamically, short-range
+interactions cannot overcome thermal defects in 1D which prevents
+ordered phases at nonzero temperature <span class="citation"
+data-cites="andersonAbsenceDiffusionCertain1958 goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a
 href="#ref-andersonAbsenceDiffusionCertain1958"
-role="doc-biblioref">18</a>]</span> in the context of non-interacting
-fermions subject to a static or quenched disorder potential <span
-class="math inline">\(V_j\)</span> drawn uniformly from the interval
-<span class="math inline">\([-W,W]\)</span></p>
-<p><span class="math display">\[
-H = -t\sum_{\langle jk \rangle} c^\daggerger_j c_k + \sum_j V_j
-c_j^\daggerger c_j
-\]</span></p>
-<p>this model exhibits exponentially localised eigenfunctions <span
-class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\)</span> which
-cannot contribute to transport processes. Initially it was thought that
-in one dimensional disordered models, all states would be localised,
-however it was later shown that in the presence of correlated disorder,
-bands of extended states can exist <span class="citation"
-data-cites="izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a
-href="#ref-izrailevLocalizationMobilityEdge1999"
-role="doc-biblioref">19</a>–<a
-href="#ref-izrailevAnomalousLocalizationLowDimensional2012"
-role="doc-biblioref">21</a>]</span>.</p>
-<p>Later localisation was found in interacting many-body systems with
-quenched disorder:</p>
-<p><span class="math display">\[
-H = -t\sum_{\langle jk \rangle} c^\daggerger_j c_k + \sum_j V_j
-c_j^\daggerger c_j + U\sum_{jk} n_j n_k
-\]</span></p>
-<p>where the number operators <span class="math inline">\(n_j =
-c^\dagger_j c_j\)</span>. Here, in contrast to the Anderson model,
-localisation phenomena can be proven robust to weak perturbations of the
-Hamiltonian. This is called many-body localisation (MBL) <span
-class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016"> [<a
-href="#ref-imbrieManyBodyLocalizationQuantum2016"
-role="doc-biblioref">22</a>]</span>.</p>
-<p>Both MBL and Anderson localisation depend crucially on the presence
-of quenched disorder. This has led to ongoing interest in the
-possibility of disorder-free localisation, in which the disorder
-necessary to generate localisation is generated entirely from the
-dynamics of the model. This contracts with typical models of disordered
-systems in which disorder is explicielty introduced into the Hamilton or
-the initial state.</p>
-<p>The concept of disorder-free localisation was first proposed in the
-context of Helium mixtures <span class="citation"
-data-cites="kagan1984localization"> [<a
-href="#ref-kagan1984localization" role="doc-biblioref">23</a>]</span>
-and then extended to heavy-light mixtures in which multiple species with
-large mass ratios interact. The idea is that the heavier particles act
-as an effective disorder potential for the lighter ones, inducing
-localisation. Two such models <span class="citation"
-data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015"> [<a
-href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
-role="doc-biblioref">24</a>,<a
-href="#ref-schiulazDynamicsManybodyLocalized2015"
-role="doc-biblioref">25</a>]</span> instead find that the models
-thermalise exponentially slowly in system size, which Ref. <span
-class="citation"
-data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"> [<a
-href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
-role="doc-biblioref">24</a>]</span> dubs Quasi-MBL.</p>
-<p>True disorder-free localisation does occur in exactly solveable
-models with extensively many conserved quantities <span class="citation"
-data-cites="smithDisorderFreeLocalization2017"> [<a
-href="#ref-smithDisorderFreeLocalization2017"
-role="doc-biblioref">26</a>]</span>. As conserved quantites have no time
-dynamics this can be thought of as taking the separation of timescales
-to the infinite limit.</p>
-<h3 id="the-falikov-kimball-model">The Falikov Kimball Model</h3>
-<p>In the Falikov Kimball (FK) model spinless fermions <span
-class="math inline">\(c_{i\uparrow}\)</span> are coupled via a repulsive
-on-site interaction to a classical degree of freedom <span
-class="math inline">\(n_{i\downarrow}\)</span>.</p>
-<p><span class="math display">\[\begin{aligned}
-H &amp;= -t \sum_{&lt;ij&gt;} c^\daggerger_{i\uparrow}c_{j\uparrow} + U
-\sum_{i} (n_{i \uparrow} - 1/2)( n_{i\downarrow} - 1/2) \\
-       &amp; - \mu \sum_i \left( n_{i \uparrow} + n_{i \downarrow}
-\right) + \sum_{ij} V_{ij} (n_{i\downarrow} - 1/2)(n_{j\downarrow} -
-1/2)
-\end{aligned}\]</span> <strong>replace with hamiltonian from the
-paper</strong></p>
-<p>This notation emphasises that this can also be thought of as an
-asymmetric Hubbard model in which the spin down electrons cannot hop and
-are subject to an additional long range potential. However, to avoid the
-confusion of talking about two distinct species of spinless electrons we
-will use a different interpretation and refer to the classical degrees
-of freedom as the “ionic sector” and the quantum degrees of freedom as
-the “electronic sector”. The final term that induces interactions
-between the classical particles has been added by us to stabilise the
-formation of an ordered phase in 1D. The classical variables commute
-with the Hamiltonian <span class="math inline">\([H, n_{i\downarrow}] =
-0\)</span> so like the lattice gauge model in Ref <span class="citation"
-data-cites="smithDisorderFreeLocalization2017"> [<a
-href="#ref-smithDisorderFreeLocalization2017"
-role="doc-biblioref">26</a>]</span> the FK model has extensively many
-conserved quantities which can act as an effective disorder potential
-for the electronic sector.</p>
-<p>Due to Pauli exclusion, the maximum filling occurs when one of each
-species occupies each lattice site such that <span
-class="math inline">\(\sum_i (n_{i\downarrow} + n_{i\uparrow} )/ N =
-2\)</span>. Here we focus on the half filled case which also displays
-particle-hole symmetry (see later).</p>
-<h2 id="falikov-kimball-and-hubbard-models">Falikov Kimball and Hubbard
-models</h2>
-<p>We will first introduce the standard Hubbard and Falikov-Kimball (FK)
-models then look at some of their properties. We’ll then cover why the
-Falikov-Kimball model represents an interesting system in which to study
-disorder free localisation.</p>
-<h3 id="hubbard-model">Hubbard model</h3>
-<p>The Hubbard model gives a very simple setting in which to study
-interacting, itinerant electrons. It is a tight binding model of spin
-half electrons with finite bandwidth <span
-class="math inline">\(t\)</span> and a repulsive on-site interaction
-<span class="math inline">\(U &gt; 0\)</span>.</p>
-<p><span class="math display">\[
-    H = -\sum_{&lt;ij&gt;,\sigma} t_{\sigma}
-c^\dagger_{i\sigma}c_{j\sigma} + U \sum_{i} (n_{i \uparrow} - 1/2)(
-n_{i\downarrow} - 1/2) - \mu \sum_i \left( n_{i \uparrow} + n_{i
-\downarrow} \right)
-\]</span></p>
-<p>in standard notation. The standard Hubbard model corresponds to the
-case <span class="math inline">\(t_{\uparrow} = t_{\downarrow}\)</span>.
-Here we have used the particle-hole symmetric version of the interaction
-term, which is more often given as <span class="math inline">\(n_{i
-\uparrow} n_{i\downarrow}\)</span>. The difference just amounts to a
-redefinition of the chemical potential.</p>
-<p>Hubbard originally used the model at half filling <span
-class="math inline">\(\mu = 0\)</span> to explain the Mott
-metal-insulator (MI) transition, however it has seen applications to
-high-temperature superconductivity and become target for cold-atom
-optical trap experiments.  <span class="citation"
-data-cites="HubbardModelHalf2013 greiner_quantum_2002 jordens_mott_2008"> [<a
-href="#ref-HubbardModelHalf2013" role="doc-biblioref">27</a>,<a
-href="#ref-greiner_quantum_2002"
-role="doc-biblioref"><strong>greiner_quantum_2002?</strong></a>,<a
-href="#ref-jordens_mott_2008"
-role="doc-biblioref"><strong>jordens_mott_2008?</strong></a>]</span>.
-While simple, only a few analytic results exist, namely the Bethe
-ansatz <span class="citation"
-data-cites="liebAbsenceMottTransition1968"> [<a
-href="#ref-liebAbsenceMottTransition1968"
-role="doc-biblioref">28</a>]</span> which proves the absence of even a
-zero temperature phase transition in the 1D model and Nagaoka’s
-theorem <span class="citation"
-data-cites="nagaokaFerromagnetismNarrowAlmost1966"> [<a
-href="#ref-nagaokaFerromagnetismNarrowAlmost1966"
-role="doc-biblioref">29</a>]</span> which proves that the three
-dimensional model has a ferromagnetic ground state in the vicinity of
-half filling.</p>
-<h3 id="falikov-kimball-model">Falikov-Kimball model</h3>
-<p>The Falikov-Kimball model corresponds to the case <span
-class="math inline">\(t_{\downarrow} = 0\)</span>. It can be interpreted
-as two coupled spinless electron bands with infinite mass ratio. An
-itinerant light species with creation operator <span
-class="math inline">\(c^\dagger_{i\uparrow}\)</span> coupled to an
-infinitely heavy, immobile species with density operator <span
-class="math inline">\(n_{i\downarrow}\)</span>. These are often called c
-and f electrons or electrons and ions. The model was first introduced by
-Hubbard in 1963 as a model of interacting localised and de-localised
-electron bands and gained its name from Falikov and Kimball’s use of it
-to study the MI transition in rare-earth materials <span
-class="citation"
-data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicov_simple_1969"> [<a
-href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
-role="doc-biblioref">12</a>,<a href="#ref-falicov_simple_1969"
-role="doc-biblioref"><strong>falicov_simple_1969?</strong></a>]</span>.</p>
-<p>Here we will use refer to the light spinless species as
-<em>electrons</em> with creation operator <span
-class="math inline">\(c^\dagger_{i}\)</span> and the heavy species as
-<em>ions</em> with density operator <span
-class="math inline">\(n_i\)</span>. When the the density operator of the
-electrons is needed I’ll always use <span
-class="math inline">\(c^\dagger_{i}c_{i}\)</span>. We also set <span
-class="math inline">\(t = 1\)</span>.</p>
-<p><span class="math display">\[
-    H_{\mathrm{FK}} = -\sum_{&lt;ij&gt;} c^\dagger_{i}c_{j} + U \sum_{i}
-(c^\dagger_{i}c_{i} - 1/2)( n_i - 1/2) - \mu \sum_i
-\left(c^\dagger_{i}c_{i} + n_{i}\right)
-\]</span> % ### Particle-Hole Symmetry The Hubbard and FK models on a
-bipartite lattice have particle-hole (PH) symmetry <span
-class="math inline">\(P^\dagger H P = - H\)</span>, accordingly they
-have symmetric energy spectra. The associated symmetry operator <span
-class="math inline">\(P\)</span> exchanges creation and annihilation
-operators along with a sign change between the two sublattices.</p>
-<p><span class="math display">\[ d^\dagger_{i\sigma} = (-1)^i
-c_{i\sigma}\]</span> % The entirely filled state <span
-class="math inline">\(\ket{\Omega} = \sum_{j\rho} c^\dagger_{j\rho}
-\ket{0}\)</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>
-% The number operator <span class="math inline">\(n_{i\sigma} =
-0,1\)</span> counts holes rather than particles: <span
-class="math display">\[ d^\dagger_{i\sigma} d_{i \sigma} = c_{i\sigma}
-c^\dagger_{i\sigma} = 1 - c^\dagger_{i\sigma} c_{i\sigma}\]</span> %
-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:
-<span class="math display">\[ d^\dagger_{i\sigma} d_{j \sigma} =
-(-1)^{i+j} c_{i\sigma} c^\dagger_{j\sigma} = c^\dagger_{i\sigma}
-c_{j\sigma} \]</span> % Since when <span
-class="math inline">\(i\)</span> and <span
-class="math inline">\(j\)</span> label sites on separate sublattices,
-<span class="math inline">\((-1)^{i-j} = -1\)</span> and this is
-absorbed into rearranging the operators via their anti-commutator.</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> % The PH symmetry maps the Hamiltonian to itself with the sign
-of the chemical potential reversed and the density is 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> % 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>
-<h3 id="thermodynamics-of-the-fk-model">Thermodynamics of the FK
-model</h3>
-\begin{figure}
-<p>} \end{figure}</p>
-<p>At half filling and in dimensions greater than one, the FK model
-exhibits a phase transition at some <span
-class="math inline">\(U\)</span> dependent critical temperature <span
-class="math inline">\(T_c(U)\)</span> to a low temperature charge
-density wave state in which the ions occupy one of the two sublattices A
-and B <span class="citation"
-data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
-href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
-role="doc-biblioref">3</a>]</span>. The order parameter is the square of
-the staggered magnetisation: <span class="math display">\[
-M = \sum_{i \in A} n_i - \sum_{i \in B} n_i
-\]</span> % In the disordered phase Ref. <span class="citation"
-data-cites="andersonAbsenceDiffusionCertain1958"> [<a
-href="#ref-andersonAbsenceDiffusionCertain1958"
-role="doc-biblioref">18</a>]</span> identifies an interplay between
-Anderson localisation at weak interaction and a Mott insulator phase in
-the strongly interacting regime.</p>
-<p>In the one dimensional FK model, however, Peierls’ argument <span
-class="citation"
-data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a
-href="#ref-kennedyItinerantElectronModel1986"
-role="doc-biblioref">7</a>,<a
+role="doc-biblioref">6</a>–<a
+href="#ref-kramerLocalizationTheoryExperiment1993"
+role="doc-biblioref">9</a>]</span>. However, the absence of an FTPT in
+the short ranged FK chain is far from obvious because the
+Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the
+fermions <span class="citation"
+data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a
+href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">10</a>–<a
+href="#ref-yosidaMagneticPropertiesCuMn1957"
+role="doc-biblioref">13</a>]</span> decays as <span
+class="math inline">\(r^{-1}\)</span> in 1D <span class="citation"
+data-cites="rusinCalculationRKKYRange2017"> [<a
+href="#ref-rusinCalculationRKKYRange2017"
+role="doc-biblioref">14</a>]</span>. This could in principle induce the
+necessary long-range interactions for the classical Ising
+background <span class="citation"
+data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a
+href="#ref-thoulessLongRangeOrderOneDimensional1969"
+role="doc-biblioref">15</a>,<a
 href="#ref-peierlsIsingModelFerromagnetism1936"
-role="doc-biblioref">30</a>]</span> and the Bethe ansatz <span
-class="citation" data-cites="liebAbsenceMottTransition1968"> [<a
-href="#ref-liebAbsenceMottTransition1968"
-role="doc-biblioref">28</a>]</span> make it clear that there is no
-ordered CDW phase. Peierls’ argument is that one should consider the
-difference in free energy <span class="math inline">\(\Delta F = \Delta
-E - T\Delta S\)</span> between an ordered state and a state with single
-domain wall in the order parameter. In the Ising model this would be
-having the spins pointing up in one part of the model and down in the
-other, for a CDW phase it means having the ions occupy the A sublattice
-in one part and the B sublattice in the other.</p>
-<p>Short range interactions will produce a constant energy penalty for
-such a domain wall that does not scale with system size while in 1D
-there are <span class="math inline">\(L\)</span> such states so the
-domain wall is associated with entropy <span class="math inline">\(S
-\propto \ln L\)</span> which dominates in the thermodynamic limit. The
-slow logarithmic scaling suggests we should be wary of finite size
-scaling effects.</p>
-<p>One dimensional systems are more amenable to numerical and
-experimental study so we add long range staggered interactions to bring
-back the ordered phase:</p>
-<p><span class="math display">\[ H_{\textrm{int}} = 4J \sum_{ij}
-\frac{(-1)^{|i-j|}}{ |i - j|^{\alpha} } (n_i - 1/2) (n_j - 1/2) = J
-\sum_{ij} |i - j|^{-\alpha} \tau_i \tau_j\]</span> % at half-filling the
-modified Hamiltonian is then: <span class="math display">\[
-    H_{\mathrm{FK}}^* &amp;= -\sum_{&lt;ij&gt;} c^\dagger_{i}c_{j} + U
-\sum_{i} (c^\dagger_{i}c_{i} - 1/2)( n_i - 1/2) \\
-    &amp;+ 4J \sum_{ij} \frac{(-1)^{|i-j|}}{ |i - j|^{\alpha} } (n_i -
-1/2) (n_j - 1/2)  \\
-    &amp;= -\sum_{&lt;ij&gt;} c^\dagger_{i}c_{j} + 2U \sum_{i} (-1)^i
-(c^\dagger_{i}c_{i} - 1/2)\tau_i + J \sum_{ij} |i - j|^{-\alpha} \tau_i
-\tau_j  \\
-\]</span> % The form of this interaction comes from interpreting the
-f-electrons as a classical Ising chain using a staggered mapping <span
-class="math inline">\(\tau_i = (-1)^i (2n_i^ f - 1)\)</span> so that
-ferromagnetic order in the <span class="math inline">\(\tau_i\)</span>
-variables corresponds to a CDW state in the <span
-class="math inline">\(n_i^f\)</span> variables. It also preserves the
-particle hole symmetry because for the ions the PH transformation
-corresponds to <span class="math inline">\(n_i \rightarrow 1 -
-n_i\)</span>. When <span class="math inline">\(U = 0\)</span> the model
-decouples into a long ranged Ising model and free fermions.</p>
-<h2 id="long-ranged-ising-model">Long Ranged Ising model</h2>
-<p>Our extension to the FK model could now be though of as spinless
-fermions coupled to a long range Ising (LRI) model. The LRI model has
-been extensively studied and its behaviour may be bear relation to the
-behaviour of our modified FK model.</p>
-<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij}
-J(\abs{i-j}) \tau_i \tau_j = J \sum_{i\neq j} |i - j|^{-\alpha} \tau_i
-\tau_j\]</span> % Rigorous renormalisation group arguments show that the
-LRI model has an ordered phase in 1D for $1 &lt; &lt; 2 $ <span
-class="citation"
-data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
-href="#ref-dysonExistencePhasetransitionOnedimensional1969"
-role="doc-biblioref">31</a>]</span>. Peierls’ argument can be
-extended <span class="citation"
-data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
-href="#ref-thoulessLongRangeOrderOneDimensional1969"
-role="doc-biblioref">32</a>]</span> to provide intuition for why this is
-the case. Again considering the energy difference between the ordered
-state <span
-class="math inline">\(\ket{\ldots\uparrow\uparrow\uparrow\uparrow\ldots}\)</span>
-and a domain wall state <span
-class="math inline">\(\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}\)</span>.
-In the case of the LRI model, careful counting shows that this energy
-penalty is: <span class="math display">\[\Delta E \propto
-\sum_{n=1}^{\infty} n J(n)\]</span> % because each interaction between
-spins separated across the domain by a bond length <span
-class="math inline">\(n\)</span> can be drawn between <span
-class="math inline">\(n\)</span> equivalent pairs of sites. Ruelle
-proved rigorously for a very general class of 1D systems, that if <span
-class="math inline">\(\Delta E\)</span> or its many-body generalisation
-converges in the thermodynamic limit then the free energy is
-analytic <span class="citation"
-data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a
-href="#ref-ruelleStatisticalMechanicsOnedimensional1968"
-role="doc-biblioref">33</a>]</span>. This rules out a finite order phase
-transition, though not one of the Kosterlitz-Thouless type. Dyson also
-proves this though with a slightly different condition on <span
-class="math inline">\(J(n)\)</span> <span class="citation"
-data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
-href="#ref-dysonExistencePhasetransitionOnedimensional1969"
-role="doc-biblioref">31</a>]</span>.</p>
-<p>With a power law form for <span class="math inline">\(J(n)\)</span>,
-there are three cases to consider:</p>
-<ol type="1">
-<li>$ = 0$ For infinite range interactions the Ising model is exactly
-solveable and mean field theory is exact <span class="citation"
-data-cites="lipkinValidityManybodyApproximation1965"> [<a
-href="#ref-lipkinValidityManybodyApproximation1965"
-role="doc-biblioref">34</a>]</span>.</li>
-<li>$ $ For slowly decaying interactions <span
-class="math inline">\(\sum_n J(n)\)</span> does not converge so the
-Hamiltonian is non-extensive, a case which won’t be further considered
-here.</li>
-<li>$ 1 &lt; &lt; 2 $ A phase transition to an ordered state at a finite
-temperature.</li>
-<li>$ = 2 $ The energy of domain walls diverges logarithmically, and
-this turns out to be a Kostelitz-Thouless transition <span
-class="citation"
-data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
-href="#ref-thoulessLongRangeOrderOneDimensional1969"
-role="doc-biblioref">32</a>]</span>.</li>
-<li>$ 2 &lt; $ For quickly decaying interactions, domain walls have a
-finite energy penalty, hence Peirels’ argument holds and there is no
-phase transition.</li>
-</ol>
-<h3 id="thermodynamics">Thermodynamics</h3>
+role="doc-biblioref">16</a>]</span>. However, Kennedy and Lieb
+established rigorously that at half-filling a CDW phase only exists at
+<span class="math inline">\(T = 0\)</span> for the 1D FK model <span
+class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a
+href="#ref-kennedyItinerantElectronModel1986"
+role="doc-biblioref">17</a>]</span>.</p>
+<p>Here, we construct a generalised one-dimensional FK model with
+long-range interactions which induces the otherwise forbidden CDW phase
+at non-zero temperature. We find a rich phase diagram with a CDW FTPT
+and interaction-tuned Anderson versus Mott localized phases similar to
+the 2D FK model <span class="citation"
+data-cites="antipovInteractionTunedAndersonMott2016"> [<a
+href="#ref-antipovInteractionTunedAndersonMott2016"
+role="doc-biblioref">18</a>]</span>. We explore the localization
+properties of the fermionic sector and find that the localisation
+lengths vary dramatically across the phases and for different energies.
+Although moderate system sizes indicate the coexistence of localized and
+delocalized states within the CDW phase, we find quantitatively similar
+behaviour in a model of uncorrelated binary disorder on a CDW
+background. For large system sizes, i.e. for our 1D disorder model we
+can treat linear sizes of several thousand sites, we find that all
+states are eventually localized with a localization length which
+diverges towards zero temperature.</p>
 <p>On bipartite lattices in dimensions 2 and above the FK model exhibits
 a finite temperature phase transition to an ordered charge density wave
 (CDW) phase <span class="citation"
 data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
 href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
-role="doc-biblioref">3</a>]</span>. In this phase, the ions are confined
-to one of the two sublattices, breaking the <span
+role="doc-biblioref">19</a>]</span>. In this phase, the ions are
+confined to one of the two sublattices, breaking the <span
 class="math inline">\(\mathbb{Z}_2\)</span> symmetry.</p>
 <p>In 1D, however, Periel’s argument <span class="citation"
 data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a
-href="#ref-kennedyItinerantElectronModel1986"
-role="doc-biblioref">7</a>,<a
 href="#ref-peierlsIsingModelFerromagnetism1936"
-role="doc-biblioref">30</a>]</span> states that domain walls only
+role="doc-biblioref">16</a>,<a
+href="#ref-kennedyItinerantElectronModel1986"
+role="doc-biblioref">17</a>]</span> states that domain walls only
 introduce a constant energy penalty into the free energy while bringing
 a entropic contribution logarithmic in system size. Hence the 1D model
 does not have a finite temperature phase transition. However 1D systems
@@ -869,253 +154,14 @@ ions in order to stabilise a CDW phase in 1D. This leads to a disordered
 system that is gaped by the CDW background but with correlated
 fluctuations leading to a disorder-free correlation induced mobility
 edge in one dimension.</p>
-<div id="fig:binder" class="fignos">
-<figure>
-<img src="../figure_code/fk_chapter/binder.png"
-data-short-caption="no title" style="width:100.0%"
-alt="Figure 1: Hello I am the figure caption!" />
-<figcaption aria-hidden="true"><span>Figure 1:</span> Hello I am the
-figure caption!</figcaption>
-</figure>
-</div>
-<p>Macro definitions in this cell <span class="math display">\[
-\newcommand{\expval}[1]{\langle #1 \rangle}
-\newcommand{\ket}[1]{|#1\rangle}
-\newcommand{\bra}[1]{\langle#1|}
-\newcommand{\op}[2]{|#1\rangle \langle#2|}
-\]</span></p>
-<p><span class="math display">\[
-\expval{O}, \op{\alpha}{\beta}, \ket{\psi}
-\]</span></p>
-<h2 id="localisation-1">Localisation</h2>
-<h3 id="thermalisation">Thermalisation</h3>
-<p>Isolated classical systems generally thermalise if they are large
-enough. Since classical dynamics is the limit of some underlying quantum
-dynamics, it seems reasonable to suggest that isolated quantum systems
-also thermalise in some related sense. However it is not immediately
-obvious what thermalisation should mean in a quantum setting where the
-presence of unitary time dynamics implies full information about a
-system’s initial state is always preserved.</p>
-<p>A potential solution lies in the eigenstate thermalisation
-hypothesis. It states that if a system thermalises, then we can isolate
-small patches of a larger system, trace everyhing else out, and get a
-thermal density matrix.</p>
-<p>Following Ref. <span class="citation"
-data-cites="abaninRecentProgressManybody2017"> [<a
-href="#ref-abaninRecentProgressManybody2017"
-role="doc-biblioref">16</a>]</span>, consider the time evolution of a
-local operator <span class="math inline">\(\hat{O}\)</span> <span
-class="math display">\[ \expval{\hat{O}}{\psi(t)} = \sum_{\alpha \beta}
-C^*_\alpha C_\beta e^{i(E_\alpha - E_\beta)} O_{\alpha
-\beta}\]</span></p>
-<p>Where <span class="math inline">\(C_\alpha\)</span> are determined by
-the initial state and <span class="math inline">\(O_{\alpha \beta} =
-\expval{\alpha | \hat{O} | \beta}\)</span> are the matrix elements of
-<span class="math inline">\(\hat{O}\)</span> with respect to the energy
-eigenstates. Srednicki <span class="citation"
-data-cites="srednickiChaosQuantumThermalization1994"> [<a
-href="#ref-srednickiChaosQuantumThermalization1994"
-role="doc-biblioref">17</a>]</span> introduced the ansatz that for local
-operators:</p>
-<p><span class="math display">\[O_{\alpha \beta} =
-O(E)\delta_{\alpha\beta} + e^{-S(E)/2} f(E,\omega)
-R_{\alpha\beta}\]</span></p>
-<p>with <span class="math inline">\(E = (E_\alpha + E_\beta)\)</span>,
-<span class="math inline">\(\omega = (E_\alpha - E_\beta)\)</span> and
-<span class="math inline">\(R_{\alpha\beta}\)</span> are sampled from
-some distribution with zero mean and unit variance. The first term
-asserts that the diagonal elements are given by the thermal expectation
-value <span class="math inline">\(O(E) = Tr[e^{-\beta \hat{H}}
-\hat{O}]/\mathcal{Z}\)</span> with <span
-class="math inline">\(\beta\)</span> an effective temperature defined by
-equating the energy to the expectation of the Hamiltonian at that
-temperature <span class="math inline">\(E = Tr[H e^{-\beta
-\hat{H}}/\mathcal{Z}]\)</span>.</p>
-<p>The second term deals with thermodynamic fluctuations scaled by the
-entropy <span class="math inline">\(S(E) = -Tr(\rho \log \rho)\)</span>
-where <span class="math inline">\(\rho = e^{-\beta \hat{H}}\)</span> and
-<span class="math inline">\(\mathcal{Z} = Tr[e^{-\beta
-\hat{H}}]\)</span>.</p>
-<p>With this ansatz the long time average of the observable becomes
-equal to the thermal expectations with fluctuations suppressed by the
-entropic term <span class="math inline">\(e^{-S(E)}\)</span> and the
-rapidly varying phase factors <span class="math inline">\(e^{i(E_\alpha
-- E_\beta)}\)</span>. This statement of the ETH has verified for the
-quantum hard sphere model <span class="citation"
-data-cites="srednickiChaosQuantumThermalization1994"> [<a
-href="#ref-srednickiChaosQuantumThermalization1994"
-role="doc-biblioref">17</a>]</span> and numerically for other
-models <span class="citation"
-data-cites="khatamiFluctuationDissipationTheoremIsolated2013 dalessioQuantumChaosEigenstate2016"> [<a
-href="#ref-khatamiFluctuationDissipationTheoremIsolated2013"
-role="doc-biblioref">35</a>,<a
-href="#ref-dalessioQuantumChaosEigenstate2016"
-role="doc-biblioref">36</a>]</span>.</p>
-<p>An alternate view on ETH is the statement that in thermalising
-systems individual eigenstates look thermal when viewed locally. Take a
-eigenstate <span class="math inline">\(|\alpha\rangle\)</span> with
-energy <span class="math inline">\(E_\alpha\)</span> and as before
-define an effective temperature with <span
-class="math inline">\(E_\alpha = Tr[H e^{-\beta
-\hat{H}}/\mathcal{Z}]\)</span>. This statement of the ETH says that if
-we partition the system into subsystems A and B with a limit taken as B
-becomes very large, B will act as a heat bath for A. Specifically the
-reduced density matrix <span class="math inline">\(\rho_A = Tr_B
-\op{\alpha}{\alpha}\)</span> is equal to the thermal density matrix:</p>
-<p><span class="math display">\[\rho_A = Tr_B |\alpha\rangle \langle
-\alpha| = \mathcal{Z}^{-1} Tr_B [e^{-\beta \hat{H}}] \]</span></p>
-<p>Intuitively, for thermalisation to happen, the degrees of freedom
-must be sufficiently well coupled that energy transport occurs. This
-condition is broken by systems with localised states so a lack of
-thermalisation is often used as a diagnostic tool for localisation.</p>
-<h3 id="anderson-localisation">Anderson Localisation</h3>
-<p>Localisation was first studied by Anderson in 1958 <span
-class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a
-href="#ref-andersonAbsenceDiffusionCertain1958"
-role="doc-biblioref">18</a>]</span> in the context of non-interacting
-fermions subject to a static or quenched disorder potential <span
-class="math inline">\(V_j\)</span> drawn uniformly from the interval
-<span class="math inline">\([-W,W]\)</span>:</p>
-<p><span class="math display">\[
-H = -t\sum_{\expval{jk}} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j
-\]</span></p>
-<p>At sufficiently strong disorder the Anderson model exhibits
-exponentially localised eigenfunctions <span
-class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\)</span> which
-cannot contribute to diffusive transport processes. Except in 1D where
-any disorder strength is sufficient. Intuitively this happens because
-hopping processes between nearby sites become off-resonant, hindering
-the hybridisation that would normally lead to extended Bloch
-states <span class="citation"
-data-cites="kramerLocalizationTheoryExperiment1993"> [<a
-href="#ref-kramerLocalizationTheoryExperiment1993"
-role="doc-biblioref">37</a>]</span>.</p>
-<p>In one and two dimensions, all the states in the Anderson model are
-localised. In three dimensions there are mobility edges. Mobility edges
-are critical energies in the spectrum which separate delocalised states
-in a band from localised states which form a band tail <span
-class="citation" data-cites="abaninRecentProgressManybody2017"> [<a
-href="#ref-abaninRecentProgressManybody2017"
-role="doc-biblioref">16</a>]</span>. An argument due to Lifshitz shows
-that the density of state of the band tail should decay exponentially
-and localised and extended stats cannot co-exist at the same energy as
-they would hybridise into extended states <span class="citation"
-data-cites="kramerLocalizationTheoryExperiment1993"> [<a
-href="#ref-kramerLocalizationTheoryExperiment1993"
-role="doc-biblioref">37</a>]</span>.</p>
-<p>It was thought that mobility edges could not exist in 1D because all
-the states localised in the presence of any amount of disorder. This is
-true for uncorrelated potentials <span class="citation"
-data-cites="goldshteinPurePointSpectrum1977"> [<a
-href="#ref-goldshteinPurePointSpectrum1977"
-role="doc-biblioref">38</a>]</span>. However, it was shown that if the
-disorder potential <span class="math inline">\(V_j\)</span> contains
-spatial correlations mobility edges do exist in 1D <span
-class="citation"
-data-cites="izrailevLocalizationMobilityEdge1999 izrailevAnomalousLocalizationLowDimensional2012"> [<a
-href="#ref-izrailevLocalizationMobilityEdge1999"
-role="doc-biblioref">19</a>,<a
-href="#ref-izrailevAnomalousLocalizationLowDimensional2012"
-role="doc-biblioref">21</a>]</span>. Ref. <span class="citation"
-data-cites="croyAndersonLocalization1D2011"> [<a
-href="#ref-croyAndersonLocalization1D2011"
-role="doc-biblioref">20</a>]</span> extends this work to look at power
-law decay of the correlations: <span class="math display">\[ C(l) =
-\expval{V_i V_{i+l}} \propto l^{-\alpha} \]</span> % Figure
-fig:anderson_dos shows numerical calculations of the Localisation length
-(see later) and density of states for the power law correlated Anderson
-model. At the unperturbed band edges <span class="math inline">\(\abs{E}
-= 2\)</span>, the states transition from extended to localised. The
-behaviour close to the edge takes a universal scaling form with
-exponents dependant on <span class="math inline">\(\alpha\)</span>.</p>
-<h3 id="many-body-localisation">Many Body Localisation</h3>
-<p>A related phenomena known as many body localisation (MBL) was found
-in interacting systems with quenched disorder. A simple example comes
-from adding density-density interactions to the Anderson model:</p>
-<p><span class="math display">\[
-H = -t\sum_{\expval{jk}} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j +
-U\sum_{jk} n_j n_k
-\]</span> % with <span class="math inline">\(n_j = c^\dagger_j
-c_j\)</span> Here, in contrast to the Anderson model, localisation
-phenomena can be proven robust to weak perturbations of the
-Hamiltonian <span class="citation"
-data-cites="imbrieManyBodyLocalizationQuantum2016"> [<a
-href="#ref-imbrieManyBodyLocalizationQuantum2016"
-role="doc-biblioref">22</a>]</span>.</p>
-<p>MBL is defined by the emergence of an extensive number of quasi-local
-operators called local integrals of motions (LIOMs) or l-bits. Following
-Ref. <span class="citation"
-data-cites="abaninRecentProgressManybody2017"> [<a
-href="#ref-abaninRecentProgressManybody2017"
-role="doc-biblioref">16</a>]</span>, using a spin system with variables
-<span class="math inline">\(\sigma^z_i\)</span>, any operator can be
-written in the general form:</p>
-<p><span class="math display">\[ \tau^z_i = \sigma^z_i +
-\sum_{\alpha\beta kl} f_{kl}^{\alpha\beta} \sigma^\alpha_{i+k}
-\sigma_z\beta_{i+k} + ...\]</span> % what defines a MBL system is that
-there exist extensively many <span
-class="math inline">\(\tau^z_i\)</span> for which the coefficients decay
-exponentially with distance <span
-class="math inline">\(f_{kl}^{\alpha\beta} \propto
-e^{-\max(\abs{l},\abs{k}) / \xi}\)</span>. These LIOMs commute with the
-Hamiltonian and each other <span class="math inline">\([\hat{H},
-\tau^z_i] = [\tau^z_i, \tau^z_j] = 0\)</span>. It is this extensive
-number of conserved local charges that leads to the localisation
-properties of MBL. It also has implications for the way entanglement
-grows over time in MBL systems.</p>
-<p>Since the Hamiltonian commutes with all the LIOMs and they are a
-complete operator basis, the Hamiltonian can be written as:</p>
-<p><span class="math display">\[\hat{H} = \sum_{i} h_i \tau^z_i +
-\sum_{ij} J_{ij} \tau^z_i \tau^z_j + \sum_{ijk} J_{ij} \tau^z_i \tau^z_j
-\tau^z_k+ ...\]</span> % Where again the couplings decay exponentially,
-albeit with a different length scale <span
-class="math inline">\(\Bar{\xi}\)</span>. From this form we see that
-distant l-bits can only become entangled on a timescale of:</p>
-<p><span class="math display">\[ t_{\mathrm{ent}}(r) \propto
-\frac{\hbar}{J_0} e^{r/\Bar{\xi}} \]</span> % and hence quantum
-correlations and entanglement propagates logarithmically in MBL
-systems <span class="citation"
-data-cites="imbrieDiagonalizationManyBodyLocalization2016"> [<a
-href="#ref-imbrieDiagonalizationManyBodyLocalization2016"
-role="doc-biblioref">39</a>]</span>.</p>
-<h3 id="disorder-free-localisation">Disorder Free localisation</h3>
-<p>Both Anderson localisation and MBL depend on the presence of quenched
-disorder. Recently the idea of disorder-free localisation has gained
-traction, asking whether the disorder necessary to generate localisation
-can be generated entirely from the dynamics of a model itself.</p>
-<p>The idea was first proposed in the context of Helium mixtures <span
-class="citation" data-cites="kagan1984localization"> [<a
-href="#ref-kagan1984localization" role="doc-biblioref">23</a>]</span>
-and then extended to heavy-light mixtures in which multiple species with
-large mass ratios interact, the idea being that the heavier particles
-act as an effective disorder potential for the lighter ones, inducing
-localisation. Two such models <span class="citation"
-data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015"> [<a
-href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
-role="doc-biblioref">24</a>,<a
-href="#ref-schiulazDynamicsManybodyLocalized2015"
-role="doc-biblioref">25</a>]</span> instead find that the models
-thermalise exponentially slowly in system size, which Ref. <span
-class="citation"
-data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"> [<a
-href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
-role="doc-biblioref">24</a>]</span> dubs Quasi-MBL. A. Smith, J. Knolle
-et al instead looked at models containing an extensive number of
-conserved quantities and demonstrated true disorder free
-localisation <span class="citation"
-data-cites="smithDisorderFreeLocalization2017"> [<a
-href="#ref-smithDisorderFreeLocalization2017"
-role="doc-biblioref">26</a>]</span>.</p>
-<h2 id="introduction">Introduction</h2>
 <p>The presence of the classical field makes the model amenable to an
 exact numerical treatment at finite temperature via a sign problem free
 MCMC algorithm <span class="citation"
 data-cites="devriesGapsDensitiesStates1993 devriesSimplifiedHubbardModel1993 antipovInteractionTunedAndersonMott2016 debskiPossibilityDetectionFinite2016 herrmannSpreadingCorrelationsFalicovKimball2018 maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
-href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
-role="doc-biblioref">3</a>,<a href="#ref-devriesGapsDensitiesStates1993"
-role="doc-biblioref">40</a>–<a
+href="#ref-antipovInteractionTunedAndersonMott2016"
+role="doc-biblioref">18</a>–<a
 href="#ref-herrmannSpreadingCorrelationsFalicovKimball2018"
-role="doc-biblioref">43</a>,<a
+role="doc-biblioref">22</a>,<a
 href="#ref-debskiPossibilityDetectionFinite2016"
 role="doc-biblioref"><strong>debskiPossibilityDetectionFinite2016?</strong></a>]</span>.
 The MCMC treatment motivates a view of the classical background field as
@@ -1127,98 +173,45 @@ model to recent interest in disorder-free localisation <span
 class="citation"
 data-cites="smithDisorderFreeLocalization2017 smithDynamicalLocalizationMathbbZ2018 brenesManyBodyLocalizationDynamics2018"> [<a
 href="#ref-smithDisorderFreeLocalization2017"
-role="doc-biblioref">26</a>,<a
-href="#ref-smithDynamicalLocalizationMathbbZ2018"
-role="doc-biblioref">44</a>,<a
+role="doc-biblioref">23</a>–<a
 href="#ref-brenesManyBodyLocalizationDynamics2018"
-role="doc-biblioref">45</a>]</span>.</p>
-<p>Dimensionality is crucial for the physics of both localisation and
-FTPTs. In 1D, disorder generally dominates, even the weakest disorder
-exponentially localises <em>all</em> single particle eigenstates. Only
-longer-range correlations of the disorder potential can potentially
-induce delocalization <span class="citation"
-data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a
-href="#ref-aubryAnalyticityBreakingAnderson1980"
-role="doc-biblioref">46</a>–<a
-href="#ref-dunlapAbsenceLocalizationRandomdimer1990"
-role="doc-biblioref">48</a>]</span>. Thermodynamically, short-range
-interactions cannot overcome thermal defects in 1D which prevents
-ordered phases at nonzero temperature <span class="citation"
-data-cites="andersonAbsenceDiffusionCertain1958 goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a
-href="#ref-andersonAbsenceDiffusionCertain1958"
-role="doc-biblioref">18</a>,<a
-href="#ref-kramerLocalizationTheoryExperiment1993"
-role="doc-biblioref">37</a>,<a
-href="#ref-goldshteinPurePointSpectrum1977"
-role="doc-biblioref">38</a>,<a
-href="#ref-abrahamsScalingTheoryLocalization1979"
-role="doc-biblioref">49</a>]</span>. However, the absence of an FTPT in
-the short ranged FK chain is far from obvious because the
-Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the
-fermions <span class="citation"
-data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a
-href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">50</a>–<a
-href="#ref-yosidaMagneticPropertiesCuMn1957"
-role="doc-biblioref">53</a>]</span> decays as <span
-class="math inline">\(r^{-1}\)</span> in 1D <span class="citation"
-data-cites="rusinCalculationRKKYRange2017"> [<a
-href="#ref-rusinCalculationRKKYRange2017"
-role="doc-biblioref">54</a>]</span>. This could in principle induce the
-necessary long-range interactions for the classical Ising
-background <span class="citation"
-data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a
-href="#ref-peierlsIsingModelFerromagnetism1936"
-role="doc-biblioref">30</a>,<a
-href="#ref-thoulessLongRangeOrderOneDimensional1969"
-role="doc-biblioref">32</a>]</span>. However, Kennedy and Lieb
-established rigorously that at half-filling a CDW phase only exists at
-<span class="math inline">\(T = 0\)</span> for the 1D FK model <span
-class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a
-href="#ref-kennedyItinerantElectronModel1986"
-role="doc-biblioref">7</a>]</span>.</p>
-<p>Here, we construct a generalised one-dimensional FK model with
-long-range interactions which induces the otherwise forbidden CDW phase
-at non-zero temperature. We find a rich phase diagram with a CDW FTPT
-and interaction-tuned Anderson versus Mott localized phases similar to
-the 2D FK model <span class="citation"
-data-cites="antipovInteractionTunedAndersonMott2016"> [<a
-href="#ref-antipovInteractionTunedAndersonMott2016"
-role="doc-biblioref">42</a>]</span>. We explore the localization
-properties of the fermionic sector and find that the localisation
-lengths vary dramatically across the phases and for different energies.
-Although moderate system sizes indicate the coexistence of localized and
-delocalized states within the CDW phase, we find quantitatively similar
-behaviour in a model of uncorrelated binary disorder on a CDW
-background. For large system sizes, i.e. for our 1D disorder model we
-can treat linear sizes of several thousand sites, we find that all
-states are eventually localized with a localization length which
-diverges towards zero temperature.</p>
-<p>The paper is organised as follows. First, we introduce the model and
-present its phase diagram. Second, we present the methods used to solve
-it numerically. Last, we investigate the model’s localisation properties
-and conclude.</p>
-<h2 id="the-long-ranged-falikov-kimball-model">The Long-Ranged
-Falikov-Kimball Model</h2>
+role="doc-biblioref">25</a>]</span>.</p>
+<p>To evaluate thermodynamic averages we perform a classical Markov
+Chain Monte Carlo random walk over the space of ionic configurations, at
+each step diagonalising the effective electronic Hamiltonian <span
+class="citation"
+data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
+href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
+role="doc-biblioref">19</a>]</span>. Using a binder-cumulant
+method <span class="citation"
+data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a
+href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">26</a>,<a
+href="#ref-musialMonteCarloSimulations2002"
+role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a>]</span>,
+we demonstrate the model has a finite temperature phase transition when
+the interaction is sufficiently long ranged. We then estimate the
+density of states and the inverse participation ratio as a function of
+energy to diagnose localisation properties.</p>
 <p>We interpret the FK model as a model of spinless fermions, <span
-class="math inline">\(c^\dag_{i}\)</span>, hopping on a 1D lattice
+class="math inline">\(c^\dagger_{i}\)</span>, hopping on a 1D lattice
 against a classical Ising spin background, <span
 class="math inline">\(S_i \in {\pm \frac{1}{2}}\)</span>. The fermions
 couple to the spins via an onsite interaction with strength <span
 class="math inline">\(U\)</span> which we supplement by a long-range
-interaction, <span class="math inline">\(J_{ij} = 4\kappa J
-(-1)^{\abs{i-j}} \abs{i-j}^{-\alpha}\)</span>, between the spins. The
-normalisation, <span class="math inline">\(\kappa^{-1} = \sum_{i=1}^{N}
-i^{-\alpha}\)</span>, renders the 0th order mean field critical
-temperature independent of system size. The hopping strength of the
-electrons, <span class="math inline">\(t = 1\)</span>, sets the overall
-energy scale and we concentrate throughout on the particle-hole
-symmetric point at zero chemical potential and half filling <span
-class="citation" data-cites="gruberFalicovKimballModelReview1996"> [<a
+interaction, <span class="math inline">\(J_{ij} = 4\kappa J (-1)^{|i-j|}
+|i-j|^{-\alpha}\)</span>, between the spins. The normalisation, <span
+class="math inline">\(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\)</span>,
+renders the 0th order mean field critical temperature independent of
+system size. The hopping strength of the electrons, <span
+class="math inline">\(t = 1\)</span>, sets the overall energy scale and
+we concentrate throughout on the particle-hole symmetric point at zero
+chemical potential and half filling <span class="citation"
+data-cites="gruberFalicovKimballModelReview1996"> [<a
 href="#ref-gruberFalicovKimballModelReview1996"
-role="doc-biblioref">14</a>]</span>.   <span
+role="doc-biblioref">27</a>]</span>.   <span
 class="math display">\[\begin{aligned}
-H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dag_{i}c_{i} -
-\tfrac{1}{2}) -\;t \sum_{i} (c^\dag_{i}c_{i+1} + \textit{h.c.)}\\
+H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
+\tfrac{1}{2}) -\;t \sum_{i} (c^\dagger_{i}c_{i+1} + \textit{h.c.)}\\
 &amp;  + \sum_{i, j}^{N} J_{ij}  S_i S_j \nonumber
 \label{eq:HFK}\end{aligned}\]</span></p>
 <p>In two or more dimensions, the <span
@@ -1226,17 +219,17 @@ class="math inline">\(J\!=0\!\)</span> FK model has a FTPT to the CDW
 phase with non-zero staggered magnetisation <span
 class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> and
 fermionic order parameter <span class="math inline">\(f = 2
-N^{-1}\abs{\sum_i (-1)^i \; \expval{c^\dag_{i}c_{i}}}\)</span> <span
+N^{-1}|\sum_i (-1)^i \; \expval{c^\dagger_{i}c_{i}}|\)</span> <span
 class="citation"
 data-cites="antipovInteractionTunedAndersonMott2016 maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
-href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
-role="doc-biblioref">3</a>,<a
 href="#ref-antipovInteractionTunedAndersonMott2016"
-role="doc-biblioref">42</a>]</span>. This only exists at zero
+role="doc-biblioref">18</a>,<a
+href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
+role="doc-biblioref">19</a>]</span>. This only exists at zero
 temperature in the short ranged 1D model <span class="citation"
 data-cites="kennedyItinerantElectronModel1986"> [<a
 href="#ref-kennedyItinerantElectronModel1986"
-role="doc-biblioref">7</a>]</span>. To study the CDW phase at finite
+role="doc-biblioref">17</a>]</span>. To study the CDW phase at finite
 temperature in 1D, we add an additional coupling that is both
 long-ranged and staggered by a factor <span
 class="math inline">\((-1)^{|i-j|}\)</span>. The additional coupling
@@ -1256,172 +249,27 @@ universal for <span class="math inline">\(\alpha \leq 3/2\)</span> <span
 class="citation"
 data-cites="ruelleStatisticalMechanicsOnedimensional1968 thoulessLongRangeOrderOneDimensional1969 angeliniRelationsShortrangeLongrange2014"> [<a
 href="#ref-thoulessLongRangeOrderOneDimensional1969"
-role="doc-biblioref">32</a>,<a
+role="doc-biblioref">15</a>,<a
 href="#ref-ruelleStatisticalMechanicsOnedimensional1968"
-role="doc-biblioref">33</a>,<a
+role="doc-biblioref">28</a>,<a
 href="#ref-angeliniRelationsShortrangeLongrange2014"
-role="doc-biblioref">55</a>]</span>. In the following, we choose <span
+role="doc-biblioref">29</a>]</span>. In the following, we choose <span
 class="math inline">\(\alpha = 5/4\)</span> to avoid this additional
 complexity.</p>
 <p>To improve the scaling of finite size effects, we make the
-replacement <span class="math inline">\(\abs{i - j}^{-\alpha}
-\rightarrow \abs{f(i - j)}^{-\alpha}\)</span>, in both <span
+replacement <span class="math inline">\(|i - j|^{-\alpha} \rightarrow
+|f(i - j)|^{-\alpha}\)</span>, in both <span
 class="math inline">\(J_{ij}\)</span> and <span
 class="math inline">\(\kappa\)</span>, where <span
 class="math inline">\(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\)</span>,
 which is smooth across the circular boundary <span class="citation"
 data-cites="fukuiOrderNClusterMonte2009"> [<a
 href="#ref-fukuiOrderNClusterMonte2009"
-role="doc-biblioref">56</a>]</span>. We only consider even system sizes
+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>
-<div class="sourceCode" id="cb1"><pre
-class="sourceCode python"><code class="sourceCode python"></code></pre></div>
-<h2 id="introduction-and-motivation">Introduction and Motivation</h2>
-<p>In this chapter I will introduce an extension of the celebrated
-Falikov-Kimball (FK) model. The FK model is a tight binding hamiltonian
-that models the interaction between two different species of particle.
-It makes the key approximation that one of the species has no quantum
-character at all, which makes it more tractable than a generic
-interacting many-body hamiltonian.</p>
-<p>Though the FK model is possibly one of the simplest lattice models
-you could think up, there is much that is still unknown about it. I will
-review what had already been learned about the model and identify gaps
-in the literature.</p>
-<p>I will then introduce an extension of the model that allows us to
-bring phenomena observed in the 2 and 3D FK model down to one dimension
-where the interplay of dimensionality, localisation and disorder can be
-explored.</p>
-<p>Next I will discuss how the Markov Chain Monte Carlo method
-introduced in can be applied to our Long-Range FK model.</p>
-<p>Finally I will present my results on the Long-Range FK model.</p>
-<h2 id="literature-review">Literature Review</h2>
-<h2 id="the-falikov-kimball-model-1">The Falikov-Kimball model</h2>
-<p>Historically, a lot of progress was made in condensed matter with the
-band theory of solids. This theory models the interaction between a
-single electron and the ionic lattice but it neglects interactions
-between electrons. From band theory we get the concept of a valence
-band, a set of occupied electronic states, and a conduction band just
-above it, containing unoccupied states. Systems are predicted to be
-insulators when there is a large energy gap between the valence and
-conduction band and to be conductors when the gap is less that the
-thermal energy scale <span class="math inline">\(k_bT\)</span>. In many
-systems band theory is enough to make good predictions about electrical,
-mechanical and thermal properties <span class="citation"
-data-cites="ashcroftSolidStatePhysics1976"> [<a
-href="#ref-ashcroftSolidStatePhysics1976"
-role="doc-biblioref">57</a>]</span>.</p>
-<p>Later, the development of Landau Fermi Liquid theory explained why
-band theory works so well even in cases where an analysis of the
-relevant energies suggests that it should not <span class="citation"
-data-cites="wenQuantumFieldTheory2007"> [<a
-href="#ref-wenQuantumFieldTheory2007"
-role="doc-biblioref">58</a>]</span>. Landau Fermi Liquid theory
-demonstrates that in many cases where electron-electron interactions are
-significant, the system can still be described in terms on generalised
-non-interacting quasiparticles.</p>
-<p>However there are systems where even Landau Fermi Liquid theory
-fails. An effective theoretical description of these systems must
-include electron-electron correlations and they are thus called Strongly
-Correlated Materials <span class="citation"
-data-cites="morosanStronglyCorrelatedMaterials2012"> [<a
-href="#ref-morosanStronglyCorrelatedMaterials2012"
-role="doc-biblioref">59</a>]</span>, Correlated Electron systems or
-Quantum Materials. The canonical class of strongly correlated materials
-are the transition metal oxides.</p>
-<p>The transition metal oxides have half filled valence bands and hence
-are predicted to be electrically conductive by band theory.
-Experimentally they are of course found to be insulators, leading Mott
-and others in 1937 to suggest that their insulating character is caused
-by coulomb repulsion between electrons opening an energy gap that
-prevents conduction <span class="citation"
-data-cites="mottDiscussionPaperBoer1937"> [<a
-href="#ref-mottDiscussionPaperBoer1937"
-role="doc-biblioref">60</a>]</span>.</p>
-<p>However a firmer theoretical description of the Mott-transition had
-to wait until 1963 when Martin Gutzwiller <span class="citation"
-data-cites="gutzwillerEffectCorrelationFerromagnetism1963"> [<a
-href="#ref-gutzwillerEffectCorrelationFerromagnetism1963"
-role="doc-biblioref">61</a>]</span>, Junjiro Kanamori <span
-class="citation"
-data-cites="kanamoriElectronCorrelationFerromagnetism1963"> [<a
-href="#ref-kanamoriElectronCorrelationFerromagnetism1963"
-role="doc-biblioref">62</a>]</span> and John Hubbard <span
-class="citation"
-data-cites="hubbardj.ElectronCorrelationsNarrow1963"> [<a
-href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
-role="doc-biblioref">12</a>]</span> independently proposed what would
-become known as the Hubbard Model.</p>
-<p>The Hubbard Model is about the simplest interacting tight-binding
-hamiltonian that could be written down. It is a tight binding model of
-spin half electrons with finite bandwidth <span
-class="math inline">\(t\)</span> and a repulsive on-site interaction
-<span class="math inline">\(U &gt; 0\)</span>. <span
-class="math display">\[H =
--t\sum_{&lt;ij&gt;}c^\dag_{i\sigma}c_{j\sigma} + U \sum_{i} (n_{i
-\uparrow} - 1/2)( n_{i\downarrow} - 1/2) - \mu \sum_i \left( n_{i
-\uparrow} + n_{i \downarrow} \right).\]</span> Where as usual <span
-class="math inline">\(n_{i \sigma} =
-c^\dag_{i\sigma}c_{i\sigma}\)</span> is the number operator. I’m using
-the particle-hole symmetric version of the interaction term which would
-otherwise be written as <span class="math inline">\(n_{i \uparrow}
-n_{i\downarrow}\)</span>. The difference just amounts to a redefinition
-of the chemical potential.</p>
-<p>While it was originally used to explain the Mott metal-insulator
-transition, the Hubbard Model has seen applications to high-temperature
-superconductivity and become the target for cold-atom optical trap
-experiments <span class="citation"
-data-cites="noauthor_hubbard_2013 greiner_quantum_2002 jordens_mott_2008"> [<a
-href="#ref-noauthor_hubbard_2013"
-role="doc-biblioref"><strong>noauthor_hubbard_2013?</strong></a>,<a
-href="#ref-greiner_quantum_2002"
-role="doc-biblioref"><strong>greiner_quantum_2002?</strong></a>,<a
-href="#ref-jordens_mott_2008"
-role="doc-biblioref"><strong>jordens_mott_2008?</strong></a>]</span>.
-Only a few analytic results exist, namely the Bethe ansatz <span
-class="citation" data-cites="lieb_absence_1968"> [<a
-href="#ref-lieb_absence_1968"
-role="doc-biblioref"><strong>lieb_absence_1968?</strong></a>]</span>
-which proves the absence of even a zero temperature phase transition in
-the 1D model and Nagaoka’s theorem <span class="citation"
-data-cites="nagaoka_ferromagnetism_1966"> [<a
-href="#ref-nagaoka_ferromagnetism_1966"
-role="doc-biblioref"><strong>nagaoka_ferromagnetism_1966?</strong></a>]</span>
-which proves that the three dimensional model has a ferromagnetic ground
-state in the vicinity of half filling.</p>
-<p>A full theoretical treatment of the Hubbard Model remains elusive
-because it is an interacting quantum many-body system. Together these
-three properties make for a formidable theoretical challenge. It is for
-this reason that many approaches within condensed matter theory today
-can be interpreted as relaxing one of these three properties. While this
-may limit the applicability of these new models, they can still yield
-useful insights.</p>
-<p>Let’s start with interactions, if we remove them entirely we end up
-back with band theory. If we make the interactions small we’re now able
-to use perturbative methods to take the interactions into account. This
-is what is done in</p>
-<p>If we instead relax the requirement that the system be many-body we
-could say try direct simulations of small numbers of particles.</p>
-<p>Finally we can think about relaxing the requirement that the system
-be quantum. In the extreme case we could make the system entirely
-classical and be left with something like the Ising Model. However,
-there’s another option that still yields something more tractable than
-the Hubbard Model. The trick is that interactions are only problematic
-if they are between quantum particles. Interactions between a quantum
-particle and a classical one are much easier to deal with. Now note that
-in the Hubbard model, electrons with the same spin are prevented from
-occupying the same site by the Pauli Exclusion Principle so the
-interactions are only ever between electrons of opposite spin. Now if we
-imagine splitting the spin half electron into two species and making one
-of them classsical we arrive directly at the Falikov-Kimball Model:</p>
-<p>The Falikov-Kimball (FK) model is one of the simplest models of the
-correlated electron problem. It captures the essence of the interaction
-between itinerant and localized electrons. More plainly, between
-electons that can move and those that can’t.</p>
-<p>Particle hole symmetry</p>
-<div class="sourceCode" id="cb2"><pre
-class="sourceCode python"><code class="sourceCode python"></code></pre></div>
-<p></ij></ij></ij></ij></ij></ij></p>
-<h1 class="unnumbered" id="bibliography">Bibliography</h1>
+</section>
+<section id="bibliography" class="level1 unnumbered">
+<h1 class="unnumbered">Bibliography</h1>
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 </html>
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--- a/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html
+++ b/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html
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@@ -266,11 +25,12 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
+<ul>
+<li><a href="#sec:FK-Methods" id="toc-sec:FK-Methods">Methods</a>
 <ul>
 <li><a href="#markov-chain-monte-carlo"
-id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a>
-<ul>
+id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
 <li><a href="#sampling" id="toc-sampling">Sampling</a></li>
 <li><a href="#markov-chains" id="toc-markov-chains">Markov
 Chains</a></li>
@@ -288,9 +48,6 @@ id="toc-metropolis-hastings">Metropolis-Hastings</a></li>
 <li><a href="#convergence-auto-correlation-and-binning"
 id="toc-convergence-auto-correlation-and-binning">Convergence,
 Auto-correlation and Binning</a></li>
-<li><a href="#applying-mcmc-to-the-fk-model"
-id="toc-applying-mcmc-to-the-fk-model">Applying MCMC to the FK
-model</a></li>
 <li><a href="#proposal-distributions"
 id="toc-proposal-distributions">Proposal Distributions</a></li>
 <li><a href="#perturbation-mcmc" id="toc-perturbation-mcmc">Perturbation
@@ -348,17 +105,21 @@ Trick</a></li>
 </ul></li>
 <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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="#markov-chain-monte-carlo"
-id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a>
+<li><a href="#sec:FK-Methods" id="toc-sec:FK-Methods">Methods</a>
 <ul>
+<li><a href="#markov-chain-monte-carlo"
+id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
 <li><a href="#sampling" id="toc-sampling">Sampling</a></li>
 <li><a href="#markov-chains" id="toc-markov-chains">Markov
 Chains</a></li>
@@ -376,9 +137,6 @@ id="toc-metropolis-hastings">Metropolis-Hastings</a></li>
 <li><a href="#convergence-auto-correlation-and-binning"
 id="toc-convergence-auto-correlation-and-binning">Convergence,
 Auto-correlation and Binning</a></li>
-<li><a href="#applying-mcmc-to-the-fk-model"
-id="toc-applying-mcmc-to-the-fk-model">Applying MCMC to the FK
-model</a></li>
 <li><a href="#proposal-distributions"
 id="toc-proposal-distributions">Proposal Distributions</a></li>
 <li><a href="#perturbation-mcmc" id="toc-perturbation-mcmc">Perturbation
@@ -438,8 +196,15 @@ Trick</a></li>
 </ul>
 </nav>
  -->
-<h1 id="markov-chain-monte-carlo">Markov Chain Monte Carlo</h1>
-<h2 id="sampling">Sampling</h2>
+
+<!-- Main Page Body -->
+<section id="sec:FK-Methods" class="level1">
+<h1>Methods</h1>
+<section id="markov-chain-monte-carlo" class="level2">
+<h2>Markov Chain Monte Carlo</h2>
+</section>
+<section id="sampling" class="level2">
+<h2>Sampling</h2>
 <p>Markov Chain Monte Carlo (MCMC) is a useful method whenever we have a
 probability distribution that we want to sample from but there is not
 direct sampling way to do so.</p>
@@ -477,7 +242,9 @@ of problem happens in many other disciplines too, particularly when
 fitting statistical models using Bayesian inference <span
 class="citation" data-cites="BMCP2021"> [<a href="#ref-BMCP2021"
 role="doc-biblioref">2</a>]</span>.</p>
-<h2 id="markov-chains">Markov Chains</h2>
+</section>
+<section id="markov-chains" class="level2">
+<h2>Markov Chains</h2>
 <p>So what can we do? Well it turns out that if we’re willing to give up
 in the requirement that the samples be uncorrelated then we can use MCMC
 instead.</p>
@@ -501,7 +268,9 @@ are many solutions <span class="citation"
 data-cites="kellyReversibilityStochasticNetworks1981"> [<a
 href="#ref-kellyReversibilityStochasticNetworks1981"
 role="doc-biblioref">6</a>]</span>.</p>
-<h2 id="application-to-the-fk-model">Application to the FK Model</h2>
+</section>
+<section id="application-to-the-fk-model" class="level2">
+<h2>Application to the FK Model</h2>
 <p>We will work in the grand canonical ensemble of fixed temperature,
 chemical potential and volume.</p>
 <p>Since the spin configurations are classical, the Hamiltonian can be
@@ -533,7 +302,8 @@ to the quantum subsystem. <span class="math display">\[\begin{aligned}
 \mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta
 F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}
 \end{aligned}\]</span></p>
-<h3 id="markov-chain-monte-carlo-1">Markov Chain Monte Carlo</h3>
+<section id="markov-chain-monte-carlo-1" class="level3">
+<h3>Markov Chain Monte Carlo</h3>
 <p>Markov Chain Monte Carlo (MCMC) is a technique for evaluating thermal
 expectation values <span class="math inline">\(\expval{O}\)</span> with
 respect to some physical system defined by a set of states <span
@@ -579,8 +349,10 @@ class="math inline">\(\expval{O^2} - \expval{O}^2\)</span> form it would
 have if the estimates were uncorrelated. Methods of estimating the true
 variance of <span class="math inline">\(\expval{O}\)</span> and decided
 how many steps are needed will be considered later.</p>
-<h2 id="the-metropolis-hasting-algorithm">The Metropolis-Hasting
-Algorithm</h2>
+</section>
+</section>
+<section id="the-metropolis-hasting-algorithm" class="level2">
+<h2>The Metropolis-Hasting Algorithm</h2>
 <p>Markov chains are defined by a transition function $(x_{i+1} x_i) $
 giving the probability that a chain in state <span
 class="math inline">\(x_i\)</span> at time <span
@@ -631,7 +403,9 @@ eigenvector equal to <span class="math inline">\(P_i = P(x_i)\)</span>
 with eigenvalue 1 and all other eigenvalues with magnitude less than
 one. The convergence time depends on the magnitude of the second largest
 eigenvalue.</p>
-<h2 id="metropolis-hastings">Metropolis-Hastings</h2>
+</section>
+<section id="metropolis-hastings" class="level2">
+<h2>Metropolis-Hastings</h2>
 <p>In order to actually choose new states according to <span
 class="math inline">\(\mathcal{T}\)</span> one chooses states from a
 proposal distribution <span class="math inline">\(q(x_i \to
@@ -678,83 +452,13 @@ class="math inline">\(f(x,x&#39;) &lt; 1\)</span>.</p>
 class="math inline">\(f(x,x&#39;)\)</span> is as close as possible to
 one, the rate of rejections can be reduced and the algorithm sped
 up.</p>
-<h2 id="convergence-auto-correlation-and-binning">Convergence,
-Auto-correlation and Binning</h2>
+</section>
+<section id="convergence-auto-correlation-and-binning" class="level2">
+<h2>Convergence, Auto-correlation and Binning</h2>
 <p>%Thinning, burn in, multiple runs</p>
-<h2 id="applying-mcmc-to-the-fk-model">Applying MCMC to the FK
-model</h2>
-<p>MCMC can be applied to sample over the classical degrees of freedom
-of the model. We take the full Hamiltonian and split it into a classical
-and a quantum part: <span class="math display">\[\begin{aligned}
-    H_{\mathrm{FK}} &amp;= -\sum_{&lt;ij&gt;} c^\dagger_{i}c_{j} + U
-\sum_{i} (c^\dagger_{i}c_{i} - 1/2)( n_i - 1/2) \\
-    &amp;+ \sum_{ij} J_{ij} (n_i - 1/2) (n_j - 1/2)  - \mu \sum_i
-(c^\dagger_{i}c_{i} + n_i)\\
-    H_q &amp;= -\sum_{&lt;ij&gt;} c^\dagger_{i}c_{j} + \sum_{i}
-\left(U(n_i - 1/2) - \mu\right) c^\dagger_{i}c_{i}\\
-    H_c &amp;= \sum_i \mu n_i - \frac{U}{2}(n_i - 1/2) +
-\sum_{ij}J_{ij}(n_i - 1/2)(n_j - 1/2)
-\end{aligned}
-\]</span> % 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> % 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">\[\begin{aligned}
-Z^k    &amp;= \Tr e^{-\beta F^k} = \sum_j e^{-\beta \sum_i m^j_i
-\epsilon^k_i}\\
-       &amp;= \sum_j \prod_i e^{- \beta m^j_i \epsilon^k_i}= \prod_i
-\sum_j e^{- \beta m^j_i \epsilon^k_i}\\
-       &amp;= \prod_i (1 + e^{- \beta \epsilon^k_i})\\
-F^k    &amp;= -1/\beta \sum_k \ln{(1 + e^{- \beta \epsilon^k_i})}
-\end{aligned}\]</span> % Observables can then be calculated from the
-partition function, for examples the occupation numbers:</p>
-<p><span class="math display">\[\begin{aligned}
-\tex{N} &amp;= \frac{1}{\beta} \frac{1}{Z} \frac{\partial Z}{\partial
-\mu} = - \frac{\partial F}{\partial \mu}\\
-    &amp;= \frac{1}{\beta} \frac{1}{Z} \frac{\partial}{\partial \mu}
-\sum_k e^{-\beta (H^k + F^k)}\\
-    &amp;= 1/Z \sum_k (N^k_{\mathrm{ion}} + N^k_{\mathrm{electron}})
-e^{-\beta (H^k + F^k)}\\
-\end{aligned}\]</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> % The MCMC algorithm consists of performing a
-random walk over the states <span class="math inline">\(\{ n^k_i
-\}\)</span>. In the simplest case the proposal distribution corresponds
-to flipping a random site from occupied to unoccupied or vice versa,
-since this proposal is symmetric the acceptance function becomes: <span
-class="math display">\[\begin{aligned}
-P(k) &amp;= \mathcal{Z}^{-1} e^{-\beta(H^k + F^k)} \\
-\mathcal{A}(k \to k&#39;) &amp;= \min\left(1,
-\frac{P(k&#39;)}{P(k)}\right) = \min\left(1, e^{\beta(H^{k&#39;} +
-F^{k&#39;})-\beta(H^k + F^k)}\right)
-\end{aligned}\]</span> % At each step <span
-class="math inline">\(F^k\)</span> is calculated by diagonalising the
-tri-diagonal matrix representation of <span
-class="math inline">\(H_q\)</span> with open boundary conditions.
-Observables are simply averages over the their value at each step of the
-random walk. The full spectrum and eigenbasis is too large to save to
-disk so usually running averages of key observables are taken as the
-walk progresses.</p>
-<h2 id="proposal-distributions">Proposal Distributions</h2>
+</section>
+<section id="proposal-distributions" class="level2">
+<h2>Proposal Distributions</h2>
 <p>In a MCMC method a key property is the proportion of the time that
 proposals are accepted, the acceptance rate. If this rate is too low the
 random walk is trying to take overly large steps in energy space which
@@ -773,7 +477,9 @@ at a time.</p>
 occasionally proposing a flip of the entire state. This works because
 near half-filling, flipping the occupations of all the sites will
 produce a state at or near the energy of the current one.</p>
-<h2 id="perturbation-mcmc">Perturbation MCMC</h2>
+</section>
+<section id="perturbation-mcmc" class="level2">
+<h2>Perturbation MCMC</h2>
 <p>The matrix diagonalisation is the most computationally expensive step
 of the process, a speed up can be obtained by modifying the proposal
 distribution to depend on the classical part of the energy, a trick
@@ -796,7 +502,9 @@ data-cites="huangAcceleratedMonteCarlo2017"> [<a
 href="#ref-huangAcceleratedMonteCarlo2017"
 role="doc-biblioref">10</a>]</span> does this with restricted Boltzmann
 machines whose form is very similar to a classical spin model.</p>
-<h2 id="scaling">Scaling</h2>
+</section>
+<section id="scaling" class="level2">
+<h2>Scaling</h2>
 <p>In order to reduce the effects of the boundary conditions and the
 finite size of the system we redefine and normalise the coupling matrix
 to have 0 derivative at its furthest extent rather than cutting off
@@ -808,7 +516,9 @@ J(x) &amp;= \frac{J_0 J&#39;(x)}{\sum_y J&#39;(y)}
 \end{aligned}\]</span> % The scaling ensures that, in the ordered phase,
 the overall potential felt by each site due to the rest of the system is
 independent of system size.</p>
-<h2 id="binder-cumulants">Binder Cumulants</h2>
+</section>
+<section id="binder-cumulants" class="level2">
+<h2>Binder Cumulants</h2>
 <p>The Binder cumulant is defined as: <span class="math display">\[U_B =
 1 - \frac{\tex{\mu_4}}{3\tex{\mu_2}^2}\]</span> % where <span
 class="math display">\[\mu_n = \tex{(m - \tex{m})^n}\]</span> % are the
@@ -822,9 +532,11 @@ data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a
 href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">11</a>,<a
 href="#ref-musialMonteCarloSimulations2002"
 role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a>]</span>.</p>
-<h2 id="markov-chain-monte-carlo-in-practice">Markov Chain Monte-Carlo
-in Practice</h2>
-<h3 id="quick-intro-to-mcmc">Quick Intro to MCMC</h3>
+</section>
+<section id="markov-chain-monte-carlo-in-practice" class="level2">
+<h2>Markov Chain Monte-Carlo in Practice</h2>
+<section id="quick-intro-to-mcmc" class="level3">
+<h3>Quick Intro to MCMC</h3>
 <p>The main paper relies on extensively to evaluate thermal expectation
 values within the model by walking over states of the classical spin
 system <span class="math inline">\(S_i\)</span>. For a classical system,
@@ -898,7 +610,9 @@ time) the probability <span class="math inline">\(p_t(\s;\s_0)\)</span>
 approaches the thermal distribution <span class="math inline">\(P(\s;
 \beta) = \mathcal{Z}^{-1} e^{-\beta F(\s)}\)</span>. This turns out to
 be quite easy to achieve using the Metropolis-Hasting algorithm.</p>
-<h3 id="convergence-time">Convergence Time</h3>
+</section>
+<section id="convergence-time" class="level3">
+<h3>Convergence Time</h3>
 <p>Considering <span class="math inline">\(p(\s)\)</span> as a vector
 <span class="math inline">\(\vec{p}\)</span> whose jth entry is the
 probability of the jth state <span class="math inline">\(p_j =
@@ -928,7 +642,9 @@ class="math inline">\(\lambda_1\)</span>. In practice this means that
 one throws away the data from the beginning of the random walk in order
 reduce the dependence on the initial conditions and be close enough to
 the target distribution.</p>
-<h3 id="auto-correlation-time">Auto-correlation Time</h3>
+</section>
+<section id="auto-correlation-time" class="level3">
+<h3>Auto-correlation Time</h3>
 <div id="fig:m_autocorr" class="fignos">
 <figure>
 <img src="../figure_code/fk_chapter/lsr/figs/m_autocorr.png"
@@ -1002,8 +718,9 @@ convergence time and the auto-correlation time as much as possible. In
 order to explain how, we need to introduce the Metropolis-Hasting (MH)
 algorithm and how it gives an explicit form for the transition
 function.</p>
-<h3 id="the-metropolis-hastings-algorithm">The Metropolis-Hastings
-Algorithm</h3>
+</section>
+<section id="the-metropolis-hastings-algorithm" class="level3">
+<h3>The Metropolis-Hastings Algorithm</h3>
 <p>MH breaks up the transition function into a proposal distribution
 <span class="math inline">\(q(\s \to \s&#39;)\)</span> and an acceptance
 function <span class="math inline">\(\mathcal{A}(\s \to
@@ -1047,7 +764,10 @@ class="sourceCode python"><code class="sourceCode python"><span id="cb2-1"><a hr
 <p>This has the effect of always accepting proposed states that are
 lower in energy and sometimes accepting those that are higher in energy
 than the current state.</p>
-<h2 id="two-step-trick">Two Step Trick</h2>
+</section>
+</section>
+<section id="two-step-trick" class="level2">
+<h2>Two Step Trick</h2>
 <p>Here, we incorporate a modification to the standard
 Metropolis-Hastings algorithm <span class="citation"
 data-cites="hastingsMonteCarloSampling1970"> [<a
@@ -1106,8 +826,9 @@ sampled from while offering sufficient flexibility that they can be
 adjusted to match the target distribution. Our proposed method is
 considerably simpler and does not require training while still reaping
 some of the benefits of reduced computation.</p>
-<h2 id="detailed-balance-for-the-two-step-method">Detailed Balance for
-the two step method</h2>
+</section>
+<section id="detailed-balance-for-the-two-step-method" class="level2">
+<h2>Detailed Balance for the two step method</h2>
 <p>Given a MCMC algorithm with target distribution <span
 class="math inline">\(\pi(a)\)</span> and transition function <span
 class="math inline">\(\mathcal{T}\)</span> the detailed balance
@@ -1158,7 +879,8 @@ r_c\right) \min\left(1, r_q\right)}{ \min\left(1, 1/r_c\right)
 <p>which simplifies to <span class="math inline">\(r_c r_q\)</span> as
 <span class="math inline">\(\min(1,r)/\min(1,1/r) = r\)</span> for <span
 class="math inline">\(r &gt; 0\)</span>.</p>
-<h3 id="two-step-trick-1">Two Step Trick</h3>
+<section id="two-step-trick-1" class="level3">
+<h3>Two Step Trick</h3>
 <p>Our method already relies heavily on the split between the classical
 and quantum sector to derive a sign problem free MCMC algorithm but it
 turns out that there is a further trick we can play with it. The free
@@ -1182,8 +904,9 @@ class="sourceCode python"><code class="sourceCode python"><span id="cb4-1"><a hr
 <span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a>    </span>
 <span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a>        states[i] <span class="op">=</span> current_state</span>
 <span id="cb4-15"><a href="#cb4-15" aria-hidden="true" tabindex="-1"></a>    </span></code></pre></div>
-<h3 id="tuning-the-proposal-distribution">Tuning the proposal
-distribution</h3>
+</section>
+<section id="tuning-the-proposal-distribution" class="level3">
+<h3>Tuning the proposal distribution</h3>
 <div id="fig:autocorr_multiple_proposals" class="fignos">
 <figure>
 <img
@@ -1227,8 +950,12 @@ the best choice. However for some simulations at very high temperature
 flipping more spins is warranted. Tuning the proposal distribution
 automatically seems like something that would not yield enough benefit
 for the additional complexity it would require.</p>
-<h2 id="diagnostics-of-localisation">Diagnostics of Localisation</h2>
-<h3 id="inverse-participation-ratio">Inverse Participation Ratio</h3>
+</section>
+</section>
+<section id="diagnostics-of-localisation" class="level2">
+<h2>Diagnostics of Localisation</h2>
+<section id="inverse-participation-ratio" class="level3">
+<h3>Inverse Participation Ratio</h3>
 <p>The inverse participation ratio is defined for a normalised wave
 function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i
 \abs{\psi_i}^2 = 1\)</span> as its fourth moment <span class="citation"
@@ -1287,7 +1014,10 @@ with <span class="math inline">\(M\)</span> dimensions each taking <span
 class="math inline">\(N\)</span> distinct values.</p>
 <p>Detailed and Global balance equation Mixing times Cluster updates and
 Critical slowing down Effective Sample Size</p>
-<h2 id="markov-chain-monte-carlo-2">Markov Chain Monte-Carlo</h2>
+</section>
+</section>
+<section id="markov-chain-monte-carlo-2" class="level2">
+<h2>Markov Chain Monte-Carlo</h2>
 <p>Dimensionality can be both a blessing and a curse. In I’ll discuss
 the fact that statistical physics can be somewhat boring in one
 dimension where most simple models have no phase transitions. This
@@ -1398,7 +1128,9 @@ probability <span class="math inline">\(p_t(\s;\s_0)\)</span> approaches
 the thermal distribution <span class="math inline">\(P(\s; \beta) =
 \Z^{-1} e^{-\beta F(\s)}\)</span>. This turns out to be quite easy to
 achieve using the Metropolis-Hasting algorithm.</p>
-<h2 id="convergence-time-1">Convergence Time</h2>
+</section>
+<section id="convergence-time-1" class="level2">
+<h2>Convergence Time</h2>
 <p>Considering <span class="math inline">\(p(\s)\)</span> as a vector
 <span class="math inline">\(\vec{p}\)</span> whose jth entry is the
 probability of the jth state <span class="math inline">\(p_j =
@@ -1427,7 +1159,9 @@ class="math inline">\(\lambda_1\)</span>. In practice this means that
 one throws away the data from the beginning of the random walk in order
 reduce the dependence on the initial conditions and be close enough to
 the target distribution.</p>
-<h2 id="auto-correlation-time-1">Auto-correlation Time</h2>
+</section>
+<section id="auto-correlation-time-1" class="level2">
+<h2>Auto-correlation Time</h2>
 <div id="fig:m_autocorr" class="fignos">
 <figure>
 <img src="figs/lsr/m_autocorr.png"
@@ -1497,8 +1231,9 @@ convergence time and the auto-correlation time as much as possible. In
 order to explain how, we need to introduce the Metropolis-Hasting (MH)
 algorithm and how it gives an explicit form for the transition
 function.</p>
-<h2 id="the-metropolis-hastings-algorithm-1">The Metropolis-Hastings
-Algorithm</h2>
+</section>
+<section id="the-metropolis-hastings-algorithm-1" class="level2">
+<h2>The Metropolis-Hastings Algorithm</h2>
 <p>MH breaks up the transition function into a proposal distribution
 <span class="math inline">\(q(\s \to \s&#39;)\)</span> and an acceptance
 function <span class="math inline">\(\A(\s \to \s&#39;)\)</span>. <span
@@ -1542,8 +1277,9 @@ class="sourceCode python"><code class="sourceCode python"><span id="cb5-1"><a hr
 <p>This has the effect of always accepting proposed states that are
 lower in energy and sometimes accepting those that are higher in energy
 than the current state.</p>
-<h2 id="choosing-the-proposal-distribution">Choosing the proposal
-distribution</h2>
+</section>
+<section id="choosing-the-proposal-distribution" class="level2">
+<h2>Choosing the proposal distribution</h2>
 <p><img src="figs/lsr/autocorr_multiple_proposals.png" title="fig:"
 id="fig:comparison"
 alt="t = 1, \alpha = 1.25, J = U = 5 [fig:comparison]" /> Simulations
@@ -1581,7 +1317,9 @@ simulations at very high temperature flipping more spins is warranted.
 Tuning the proposal distribution automatically seems like something that
 would not yield enough benefit for the additional complexity it would
 require.</p>
-<h2 id="two-step-trick-2">Two Step Trick</h2>
+</section>
+<section id="two-step-trick-2" class="level2">
+<h2>Two Step Trick</h2>
 <p>Our method already relies heavily on the split between the classical
 and quantum sector to derive a sign problem free MCMC algorithm but it
 turns out that there is a further trick we can play with it. The free
@@ -1604,10 +1342,10 @@ class="sourceCode python"><code class="sourceCode python"><span id="cb6-1"><a hr
 <span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a>    </span>
 <span id="cb6-13"><a href="#cb6-13" aria-hidden="true" tabindex="-1"></a>        states[i] <span class="op">=</span> current_state</span>
 <span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a>    </span></code></pre></div>
-<div class="sourceCode" id="cb7"><pre
-class="sourceCode python"><code class="sourceCode python"></code></pre></div>
-<p></ij></ij></p>
-<h1 class="unnumbered" id="bibliography">Bibliography</h1>
+</section>
+</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-devroyeRandomSampling1986" class="csl-entry"
 role="doc-biblioentry">
@@ -1745,6 +1483,7 @@ Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492
 (1958).</div>
 </div>
 </div>
+</section>
 <section class="footnotes footnotes-end-of-document"
 role="doc-endnotes">
 <hr />
@@ -1779,6 +1518,8 @@ involving a sum over the auto-correlation function.<a href="#fnref4"
 class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section>
+
+
 </main>
 </body>
 </html>
diff --git a/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html b/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html
index 787945f..e3c6203 100644
--- a/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html
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 <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
+<script src="/assets/js/thesis_scrollspy.js"></script>
 
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 <script src="/assets/js/index.js"></script>
 </head>
@@ -266,44 +25,49 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
-<li><a href="#the-phase-diagram" id="toc-the-phase-diagram">The Phase
+<li><a href="#sec:FK-results" id="toc-sec:FK-results">Results</a>
+<ul>
+<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
 Diagram</a></li>
 <li><a href="#localisation-properties"
 id="toc-localisation-properties">Localisation Properties</a></li>
-<li><a href="#discussion-conclusion"
-id="toc-discussion-conclusion">Discussion &amp; Conclusion</a></li>
-<li><a href="#acknowledgments"
-id="toc-acknowledgments">Acknowledgments</a></li>
-<li><a href="#uncorrelated-disorder-model"
-id="toc-uncorrelated-disorder-model"><span id="app:disorder_model"
-label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li>
+</ul></li>
+<li><a href="#discussion-and-conclusion-secamk-conclusion"
+id="toc-discussion-and-conclusion-secamk-conclusion">Discussion and
+Conclusion {sec:AMK-Conclusion}</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 -->
+<!-- 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="#the-phase-diagram" id="toc-the-phase-diagram">The Phase
+<li><a href="#sec:FK-results" id="toc-sec:FK-results">Results</a>
+<ul>
+<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
 Diagram</a></li>
 <li><a href="#localisation-properties"
 id="toc-localisation-properties">Localisation Properties</a></li>
-<li><a href="#discussion-conclusion"
-id="toc-discussion-conclusion">Discussion &amp; Conclusion</a></li>
-<li><a href="#acknowledgments"
-id="toc-acknowledgments">Acknowledgments</a></li>
-<li><a href="#uncorrelated-disorder-model"
-id="toc-uncorrelated-disorder-model"><span id="app:disorder_model"
-label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li>
+</ul></li>
+<li><a href="#discussion-and-conclusion-secamk-conclusion"
+id="toc-discussion-and-conclusion-secamk-conclusion">Discussion and
+Conclusion {sec:AMK-Conclusion}</a></li>
 <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
 </nav>
  -->
+
+<!-- Main Page Body -->
+<section id="sec:FK-results" class="level1">
+<h1>Results</h1>
 <div id="fig:phase_diagram" class="fignos">
 <figure>
 <img src="pdf_figs/phase_diagram.svg"
@@ -342,7 +106,17 @@ parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha =
 1.25\)</span> except where explicitly varied.</figcaption>
 </figure>
 </div>
-<h1 id="the-phase-diagram">The Phase Diagram</h1>
+<div id="fig:binder" class="fignos">
+<figure>
+<img src="/assets/thesis/fk_chapter/binder.png"
+data-short-caption="no title" style="width:100.0%"
+alt="Figure 2: Hello I am the figure caption!" />
+<figcaption aria-hidden="true"><span>Figure 2:</span> Hello I am the
+figure caption!</figcaption>
+</figure>
+</div>
+<section id="phase-diagram" class="level2">
+<h2>Phase Diagram</h2>
 <p>Figs. [<a href="#fig:phase_diagram" data-reference-type="ref"
 data-reference="fig:phase_diagram">1</a>a] and [<a
 href="#fig:phase_diagram" data-reference-type="ref"
@@ -385,7 +159,9 @@ induced by the translational symmetry breaking in the CDW state below
 href="#fig:band_opening" data-reference-type="ref"
 data-reference="fig:band_opening">3</a>a]. The Anderson phase is gapless
 but, as we explain below, shows localised fermionic eigenstates.</p>
-<h1 id="localisation-properties">Localisation Properties</h1>
+</section>
+<section id="localisation-properties" class="level2">
+<h2>Localisation Properties</h2>
 <p>The MCMC formulation suggests viewing the spin configurations as a
 form of annealed binary disorder whose probability distribution is given
 by the Boltzmann weight <span class="math inline">\(e^{-\beta
@@ -453,8 +229,8 @@ temperatures?</p>
 <div id="fig:indiv_IPR" class="fignos">
 <figure>
 <img src="pdf_figs/indiv_IPR.svg"
-alt="Figure 2: Energy resolved DOS(\omega) and \tau (the scaling exponent of IPR(\omega) against system size N). The left column shows the Anderson phase U = 2 at high T = 2.5 and the CDW phase at low T = 1.5 temperature. IPRs are evaluated for one of the in-gap states \omega_0/U = 0.057 and the center of the band \omega_1 U = 0.81. The right column shows instead the Mott and CDW phases at U = 5 with \omega_0/U = 0.24 and \omega_1/U = 0.571. For all the plots J = 5,\;\alpha = 1.25 and the fits for \tau use system sizes greater than 60. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalisation of the states." />
-<figcaption aria-hidden="true"><span>Figure 2:</span> Energy resolved
+alt="Figure 3: Energy resolved DOS(\omega) and \tau (the scaling exponent of IPR(\omega) against system size N). The left column shows the Anderson phase U = 2 at high T = 2.5 and the CDW phase at low T = 1.5 temperature. IPRs are evaluated for one of the in-gap states \omega_0/U = 0.057 and the center of the band \omega_1 U = 0.81. The right column shows instead the Mott and CDW phases at U = 5 with \omega_0/U = 0.24 and \omega_1/U = 0.571. For all the plots J = 5,\;\alpha = 1.25 and the fits for \tau use system sizes greater than 60. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalisation of the states." />
+<figcaption aria-hidden="true"><span>Figure 3:</span> Energy resolved
 DOS(<span class="math inline">\(\omega\)</span>) and <span
 class="math inline">\(\tau\)</span> (the scaling exponent of IPR(<span
 class="math inline">\(\omega\)</span>) against system size <span
@@ -485,8 +261,8 @@ states.</figcaption>
 <div id="fig:band_opening" class="fignos">
 <figure>
 <img src="pdf_figs/gap_openingboth.svg"
-alt="Figure 3: The DOS (a and c) and scaling exponent \tau (b and d) as a function of energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at U=2 while (c) and (d) show the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown a white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. U = 5,\;J = 5,\;\alpha = 1.25" />
-<figcaption aria-hidden="true"><span>Figure 3:</span> The DOS (a and c)
+alt="Figure 4: The DOS (a and c) and scaling exponent \tau (b and d) as a function of energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at U=2 while (c) and (d) show the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown a white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. U = 5,\;J = 5,\;\alpha = 1.25" />
+<figcaption aria-hidden="true"><span>Figure 4:</span> The DOS (a and c)
 and scaling exponent <span class="math inline">\(\tau\)</span> (b and d)
 as a function of energy and temperature. (a) and (b) show the system
 transitioning from the CDW phase to the gapless Anderson insulating one
@@ -511,8 +287,8 @@ alt="The DOS (a) and scaling exponent \tau (b) as a function of energy for the C
 <div id="fig:indiv_IPR_disorder" class="fignos">
 <figure>
 <img src="pdf_figs/indiv_IPR_disorder.svg"
-alt="Figure 4: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 &lt; \rho &lt; 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation  [2], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N &gt; 400" />
-<figcaption aria-hidden="true"><span>Figure 4:</span> A comparison of
+alt="Figure 5: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 &lt; \rho &lt; 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation  [2], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N &gt; 400" />
+<figcaption aria-hidden="true"><span>Figure 5:</span> A comparison of
 the full FK model to a simple binary disorder model (DM) with a CDW wave
 background perturbed by uncorrelated defects at density <span
 class="math inline">\(0 &lt; \rho &lt; 1\)</span> matched to the largest
@@ -624,7 +400,11 @@ interactions, here manifest as a peculiar binary potential, and
 localization can be very intricate and the added advantage of our 1D
 model is that we can explore very large system sizes for a complete
 understanding.</p>
-<h1 id="discussion-conclusion">Discussion &amp; Conclusion</h1>
+</section>
+</section>
+<section id="discussion-and-conclusion-secamk-conclusion"
+class="level1">
+<h1>Discussion and Conclusion {sec:AMK-Conclusion}</h1>
 <p>The FK model is one of the simplest non-trivial models of interacting
 fermions. We studied its thermodynamic and localisation properties
 brought down in dimensionality to 1D by adding a novel long-ranged
@@ -666,17 +446,7 @@ thermal domain wall defects. Finally, the rich physics of our model
 should be realizable in systems with long-range interactions, such as
 trapped ion quantum simulators, where one can also explore the fully
 interacting regime with a dynamical background field.</p>
-<h1 id="acknowledgments">Acknowledgments</h1>
-<p>We wish to acknowledge the support of Alexander Belcik who was
-involved with the initial stages of the project. We thank Angus
-MacKinnon for helpful discussions, Sophie Nadel for input when preparing
-the figures and acknowledge support from the Imperial-TUM flagship
-partnership. This work was supported in part by the Engineering and
-Physical Sciences Research Council (EPSRC) <a
-href="https://gtr.ukri.org/project/145404DD-ABAD-4CFB-A2D8-152A6AFCCEB7#/tabOverview">Project
-No. 2120140</a>.</p>
-<h1 id="uncorrelated-disorder-model"><span id="app:disorder_model"
-label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</h1>
+<p><strong>UNCORRELATED DISORDER MODEL</strong></p>
 <p>The disorder model referred to in the main text is defined by
 replacing the spin degree of freedom in the FK model <span
 class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder
@@ -698,7 +468,9 @@ H_{\mathrm{DM}} = &amp; \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} -
 \nonumber\end{aligned}\]</span></p>
 <div class="sourceCode" id="cb1"><pre
 class="sourceCode python"><code class="sourceCode python"></code></pre></div>
-<h1 class="unnumbered" id="bibliography">Bibliography</h1>
+</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-binderFiniteSizeScaling1981" class="csl-entry"
 role="doc-biblioentry">
@@ -790,6 +562,9 @@ Model</a></em>, J. Phys. A: Math. Gen. <strong>30</strong>, L711
 (1997).</div>
 </div>
 </div>
+</section>
+
+
 </main>
 </body>
 </html>
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 <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
+<script src="/assets/js/thesis_scrollspy.js"></script>
 
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 </head>
@@ -204,7 +26,7 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
 <li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a>
 <ul>
@@ -253,12 +75,15 @@ Statistics</a></li>
 </ul></li>
 <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a>
@@ -310,7 +135,10 @@ Statistics</a></li>
 </ul>
 </nav>
  -->
-<h2 id="gauge-fields">Gauge Fields</h2>
+
+<!-- Main Page Body -->
+<section id="gauge-fields" class="level2">
+<h2>Gauge Fields</h2>
 <p>The bond operators <span class="math inline">\(u_{ij}\)</span> are
 useful because they label a bond sector <span
 class="math inline">\(\mathcal{\tilde{L}}_u\)</span> in which we can
@@ -337,7 +165,8 @@ to its neighbour in each plaquette operator. This is consistent with the
 earlier observation that each <span class="math inline">\(W_p\)</span>
 takes values <span class="math inline">\(\pm 1\)</span> for even paths
 and <span class="math inline">\(\pm i\)</span> for odd paths.</p>
-<h3 id="vortices-and-their-movements">Vortices and their movements</h3>
+<section id="vortices-and-their-movements" class="level3">
+<h3>Vortices and their movements</h3>
 <div id="fig:types_of_dual_loops_animated" class="fignos">
 <figure>
 <img
@@ -381,8 +210,9 @@ a closed loop on the dual lattice. Applying such a bond flip leaves the
 vortex sector unchanged. We can also do the same thing but move the
 vortex around one the non-contractible loops of the lattice (fig. <a
 href="#fig:types_of_dual_loops_animated">1</a> (d)).</p>
-<h3 id="dual-loops-and-gauge-symmetries">Dual Loops and gauge
-symmetries</h3>
+</section>
+<section id="dual-loops-and-gauge-symmetries" class="level3">
+<h3>Dual Loops and gauge symmetries</h3>
 <div id="fig:gauge_symmetries" class="fignos">
 <figure>
 <img
@@ -428,7 +258,9 @@ class="math inline">\(D_j\)</span>s, the non-contractible loops.</p>
 <p><strong>The plaquette operators and topological fluxes are the gauge
 invariant quantities which determine the physics of the
 model</strong></p>
-<h3 id="composition-of-wilson-loops">Composition of Wilson loops</h3>
+</section>
+<section id="composition-of-wilson-loops" class="level3">
+<h3>Composition of Wilson loops</h3>
 <div id="fig:plaquette_addition_by_hand" class="fignos">
 <figure>
 <img
@@ -475,8 +307,9 @@ discrete version of Stoke’s theorem.</figcaption>
 </div>
 <p>Takeaway: Wilson loops can always be decomposed into products of
 plaquettes operators unless they are non-contractable.</p>
-<h3 id="gauge-degeneracy-and-the-euler-equation">Gauge Degeneracy and
-the Euler Equation</h3>
+</section>
+<section id="gauge-degeneracy-and-the-euler-equation" class="level3">
+<h3>Gauge Degeneracy and the Euler Equation</h3>
 <div id="fig:state_decomposition_animated" class="fignos">
 <figure>
 <img
@@ -626,8 +459,9 @@ chosen from a tree since loops can be removed by a gauge
 transformation.</figcaption>
 </figure>
 </div>
-<h3 id="counting-edges-plaquettes-and-vertices">Counting edges,
-plaquettes and vertices</h3>
+</section>
+<section id="counting-edges-plaquettes-and-vertices" class="level3">
+<h3>Counting edges, plaquettes and vertices</h3>
 <p>It is useful to know how the trivalent structure of the lattice
 constrains the number of bonds <span class="math inline">\(B\)</span>,
 plaquettes <span class="math inline">\(P\)</span> and vertices <span
@@ -667,7 +501,10 @@ fig. <a href="#fig:flood_fill">7</a> but for the amorphous
 lattice.</figcaption>
 </figure>
 </div>
-<h2 id="the-projector">The Projector</h2>
+</section>
+</section>
+<section id="the-projector" class="level2">
+<h2>The Projector</h2>
 <p>The projection from the extended space to the physical space will not
 be particularly important for the results presented here. However, the
 theory remains useful to explain why this is.</p>
@@ -782,7 +619,8 @@ determined the single particle eigenstates of a bond sector, the true
 many body ground state has the same energy as either the empty state
 with <span class="math inline">\(n_i = 0\)</span> or a state with a
 single fermion in the lowest level.</p>
-<h3 id="ground-state-degeneracy">Ground State Degeneracy</h3>
+<section id="ground-state-degeneracy" class="level3">
+<h3>Ground State Degeneracy</h3>
 <div id="fig:loops_and_dual_loops" class="fignos">
 <figure>
 <img
@@ -848,7 +686,9 @@ degeneracy in the Abelian phase and a threefold degeneracy in the
 non-Abelian phase.</figcaption>
 </figure>
 </div>
-<h3 id="quick-breather">Quick Breather</h3>
+</section>
+<section id="quick-breather" class="level3">
+<h3>Quick Breather</h3>
 <p>Let’s consider where are with the model now. We can map the spin
 Hamiltonian to a Majorana Hamiltonian in an extended Hilbert space.
 Along with that mapping comes a gauge field <span
@@ -877,7 +717,10 @@ fermions in the system grows.</p>
 basis, we would need to include the full symmetrisation over the gauge
 fields. However, this was not necessary for any of the results that will
 be presented here.</p>
-<h2 id="the-ground-state">The Ground State</h2>
+</section>
+</section>
+<section id="the-ground-state" class="level2">
+<h2>The Ground State</h2>
 <p>We have shown that the Hamiltonian is gauge invariant. As a result,
 only the flux sector and the two topological fluxes affect the spectrum
 of the Hamiltonian. Thus, we can label the many body ground state by a
@@ -981,7 +824,8 @@ lattices <span class="citation" data-cites="lieb_flux_1994"> [<a
 href="#ref-lieb_flux_1994" role="doc-biblioref">5</a>]</span> and is
 supported by numerical evidence. As noted before, any flux that differs
 from the ground state is an excitation which we call a vortex.</p>
-<h3 id="finite-size-effects">Finite size effects</h3>
+<section id="finite-size-effects" class="level3">
+<h3>Finite size effects</h3>
 <p>This guess only works for larger lattices. To rigorously test it, we
 would want to directly enumerate the <span
 class="math inline">\(2^N\)</span> vortex sectors for a smaller lattice
@@ -1026,7 +870,9 @@ class="math inline">\(\phi_0\)</span> correctly predicts the ground
 state for hundreds of thousands of lattices with up to twenty
 plaquettes. For larger lattices, we verified that random perturbations
 around the predicted ground state never yield a lower energy state.</p>
-<h3 id="chiral-symmetry">Chiral Symmetry</h3>
+</section>
+<section id="chiral-symmetry" class="level3">
+<h3>Chiral Symmetry</h3>
 <p>The discussion above shows that the ground state has a twofold
 <strong>chiral</strong> degeneracy which arises because the global sign
 of the odd plaquettes does not matter.</p>
@@ -1043,15 +889,19 @@ class="math inline">\(\phi\)</span> fluxes <span class="citation"
 data-cites="yaoExactChiralSpin2007"> [<a
 href="#ref-yaoExactChiralSpin2007"
 role="doc-biblioref">7</a>]</span>.</p>
-<h2 id="phases-of-the-kitaev-model">Phases of the Kitaev Model</h2>
+</section>
+</section>
+<section id="phases-of-the-kitaev-model" class="level2">
+<h2>Phases of the Kitaev Model</h2>
 <p>discuss the Abelian A phase / toric code phase / anisotropic
 phase</p>
 <p>the isotropic gapless phase of the standard model</p>
 <p>The isotropic gapped phase with the addition of a magnetic field</p>
-<h2 id="what-is-so-great-about-two-dimensions">What is so great about
-two dimensions?</h2>
-<h3 id="topology-chirality-and-edge-modes">Topology, chirality and edge
-modes</h3>
+</section>
+<section id="what-is-so-great-about-two-dimensions" class="level2">
+<h2>What is so great about two dimensions?</h2>
+<section id="topology-chirality-and-edge-modes" class="level3">
+<h3>Topology, chirality and edge modes</h3>
 <p>Most thermodynamic and quantum phases studied can be characterised by
 a local order parameter. That is, a function or operator that only
 requires knowledge about some fixed sized patch of the system that does
@@ -1067,7 +917,9 @@ distinguish it from standard symmetry breaking.</p>
 defined on a graph that is embedded either into the plane or onto the
 torus. The extension to surfaces like the torus but with more than one
 handle is relatively easy.</p>
-<h3 id="anyonic-statistics">Anyonic Statistics</h3>
+</section>
+<section id="anyonic-statistics" class="level3">
+<h3>Anyonic Statistics</h3>
 <p><strong>NB: I’m thinking about moving this section to the overall
 intro, but it’s nice to be able to refer to specifics of the Kitaev
 model also so I’m not sure. It currently repeats a discussion of the
@@ -1213,7 +1065,10 @@ href="#ref-hastingsDynamicallyGeneratedLogical2021"
 role="doc-biblioref">17</a>,<a
 href="#ref-kitaevFaulttolerantQuantumComputation2003"
 role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p>
-<h1 class="unnumbered" id="bibliography">Bibliography</h1>
+</section>
+</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-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry"
 role="doc-biblioentry">
@@ -1343,6 +1198,9 @@ href="https://doi.org/10.22331/q-2021-10-19-564">Dynamically Generated
 Logical Qubits</a></em>, Quantum <strong>5</strong>, 564 (2021).</div>
 </div>
 </div>
+</section>
+
+
 </main>
 </body>
 </html>
diff --git a/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html b/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html
index 50223aa..68770dc 100644
--- a/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html
+++ b/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html
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-   -->
 
 <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
+<script src="/assets/js/thesis_scrollspy.js"></script>
 
-
-<!--[if lt IE 9]>
-    <script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
-<![endif]-->
 <link rel="stylesheet" href="/assets/css/styles.css">
 <script src="/assets/js/index.js"></script>
 </head>
@@ -204,11 +26,9 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
-<li><a href="#contributions"
-id="toc-contributions">Contributions</a></li>
-<li><a href="#introduction" id="toc-introduction">Introduction</a>
+<li><a href="#sec:AMK-Model" id="toc-sec:AMK-Model">The Model</a>
 <ul>
 <li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
 Systems</a></li>
@@ -239,17 +59,18 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
 </ul></li>
 <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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="#contributions"
-id="toc-contributions">Contributions</a></li>
-<li><a href="#introduction" id="toc-introduction">Introduction</a>
+<li><a href="#sec:AMK-Model" id="toc-sec:AMK-Model">The Model</a>
 <ul>
 <li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
 Systems</a></li>
@@ -282,7 +103,9 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
 </ul>
 </nav>
  -->
-<h1 id="contributions">Contributions</h1>
+
+<!-- Main Page Body -->
+<p><strong>Contributions</strong></p>
 <p>The material in this chapter expands on work presented in</p>
 <p><strong>Insert citation of amorphous Kitaev paper here</strong></p>
 <p>which was a joint project of the first three authors with advice and
@@ -301,7 +124,8 @@ rest of the programming for Koala while pair programming and
 ’whiteboard’ing, this included the phase diagram, edge mode and finite
 temperature analyses as well as the derivation of the projector in the
 amorphous case.</p>
-<h1 id="introduction">Introduction</h1>
+<section id="sec:AMK-Model" class="level1">
+<h1>The Model</h1>
 <div id="fig:intro_figure_by_hand" class="fignos">
 <figure>
 <img
@@ -361,7 +185,8 @@ about because the model has extensively many conserved degrees of
 freedom. These conserved quantities can be factored out as classical
 degrees of freedom, leaving behind a non-interacting quantum model that
 is easy to solve.</p>
-<h2 id="amorphous-systems">Amorphous Systems</h2>
+<section id="amorphous-systems" class="level2">
+<h2>Amorphous Systems</h2>
 <p><strong>Insert discussion of why a generalisation to the amorphous
 case is interesting</strong></p>
 <p>This chapter details the physics of the Kitaev model on amorphous
@@ -391,7 +216,9 @@ that there is a phase transition to a thermal metal state.</p>
 and the motivations for doing so. It also discusses how a well known
 quantum error correcting code defined on the Kitaev Honeycomb model
 could be generalised to the amorphous case.</p>
-<h2 id="glossary">Glossary</h2>
+</section>
+<section id="glossary" class="level2">
+<h2>Glossary</h2>
 <ul>
 <li><p>Lattice: The underlying graph on which the models are defined.
 Composed of sites (vertices), bonds (edges) and plaquettes
@@ -465,12 +292,16 @@ class="math inline">\(A_\alpha\)</span> means <span
 class="math inline">\(J_\alpha &gt;&gt; J_\beta, J_\gamma\)</span>.</li>
 <li>The B phase: The roughly isotropic region of the phase diagram.</li>
 </ul>
-<h2 id="the-kitaev-model">The Kitaev Model</h2>
-<h3 id="commutation-relations">Commutation relations</h3>
+</section>
+<section id="the-kitaev-model" class="level2">
+<h2>The Kitaev Model</h2>
+<section id="commutation-relations" class="level3">
+<h3>Commutation relations</h3>
 <p>Before diving into the Hamiltonian of the Kitaev model, the following
 describes the key commutation relations of spins, fermions and
 Majoranas.</p>
-<h4 id="spins">Spins</h4>
+<section id="spins" class="level4">
+<h4>Spins</h4>
 <p>Skip this is you are familiar with the algebra of the Pauli matrices.
 Scalars like <span class="math inline">\(\delta_{ij}\)</span> should be
 understood to be multiplied by an implicit identity <span
@@ -502,7 +333,9 @@ be computed relatively easily by applying the above relations yielding:
 i \epsilon^{\alpha\beta\gamma}\]</span> and <span
 class="math display">\[[\sigma^\alpha \sigma^\beta, \sigma^\gamma] =
 0\]</span></p>
-<h4 id="fermions-and-majoranas">Fermions and Majoranas</h4>
+</section>
+<section id="fermions-and-majoranas" class="level4">
+<h4>Fermions and Majoranas</h4>
 <p>The fermionic creation and anhilation operators are defined by the
 canonical anticommutation relations <span
 class="math display">\[\begin{aligned}
@@ -540,7 +373,10 @@ alt="Figure 2: A visual introduction to the Kitaev Model." />
 introduction to the Kitaev Model.</figcaption>
 </figure>
 </div>
-<h3 id="the-hamiltonian">The Hamiltonian</h3>
+</section>
+</section>
+<section id="the-hamiltonian" class="level3">
+<h3>The Hamiltonian</h3>
 <p>To start from the fundamentals, the Kitaev Honeycomb model is a model
 of interacting spin<span class="math inline">\(-1/2\)</span>s on the
 vertices of a honeycomb lattice. Each bond in the lattice is assigned a
@@ -649,9 +485,11 @@ of a plaquette operator away from the ground state as
 Hilbert space into a set of ‘vortex sectors’ labelled by that particular
 flux configuration <span class="math inline">\(\phi_i = \pm 1,\pm
 i\)</span>.</p>
-<h3 id="from-spins-to-majorana-operators">From Spins to Majorana
-operators</h3>
-<h4 id="for-a-single-spin">For a single spin</h4>
+</section>
+<section id="from-spins-to-majorana-operators" class="level3">
+<h3>From Spins to Majorana operators</h3>
+<section id="for-a-single-spin" class="level4">
+<h4>For a single spin</h4>
 <p>Let us start by considering only one site and its <span
 class="math inline">\(\sigma^x, \sigma^y\)</span> and <span
 class="math inline">\(\sigma^z\)</span> operators which live in a two
@@ -704,7 +542,9 @@ instead get <span class="math display">\[
 \tilde{\sigma}^x\tilde{\sigma}^y\tilde{\sigma}^z = iD \]</span></p>
 <p>This makes sense if we promise to confine ourselves to the physical
 subspace <span class="math inline">\(D = 1\)</span>.</p>
-<h4 id="for-multiple-spins">For multiple spins</h4>
+</section>
+<section id="for-multiple-spins" class="level4">
+<h4>For multiple spins</h4>
 <p>This construction easily generalises to the case of multiple spins.
 We get a set of 4 Majoranas <span class="math inline">\(b^x_j,\;
 b^y_j,\;b^z_j,\; c_j\)</span> and a <span class="math inline">\(D_j =
@@ -756,8 +596,11 @@ degree of degeneracy.</p>
 <p>In summary, Majorana bond operators <span
 class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span
 class="math inline">\(\mathbb{Z_2}\)</span> gauge field!</p>
-<h3 id="partitioning-the-hilbert-space-into-bond-sectors">Partitioning
-the Hilbert Space into Bond sectors</h3>
+</section>
+</section>
+<section id="partitioning-the-hilbert-space-into-bond-sectors"
+class="level3">
+<h3>Partitioning the Hilbert Space into Bond sectors</h3>
 <p>Similarly to the story with the plaquette operators from the spin
 language, we can divide the Hilbert space <span
 class="math inline">\(\mathcal{L}\)</span> into sectors labelled by a
@@ -781,7 +624,10 @@ confined entirely within the physical subspace <span
 class="math inline">\(\mathcal{L}_p\)</span> and, indeed, we will see
 that they are not. However, it will be helpful to first develop the
 theory of the Majorana Hamiltonian further.</p>
-<h2 id="the-majorana-hamiltonian">The Majorana Hamiltonian</h2>
+</section>
+</section>
+<section id="the-majorana-hamiltonian" class="level2">
+<h2>The Majorana Hamiltonian</h2>
 <p>We now have a quadratic Hamiltonian <span class="math display">\[
 \tilde{H} =  \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha}
 u_{ij} c_i c_j\]</span> in which most of the Majorana degrees of freedom
@@ -833,8 +679,9 @@ can take half the absolute value of the whole set to recover <span
 class="math inline">\(\sum_m \epsilon_m\)</span> easily.</p>
 <p>Takeaway: the Majorana Hamiltonian is quadratic within a Bond
 Sector.</p>
-<h3 id="mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
-back from Bond Sectors to the Physical Subspace</h3>
+<section id="mapping-back-from-bond-sectors-to-the-physical-subspace"
+class="level3">
+<h3>Mapping back from Bond Sectors to the Physical Subspace</h3>
 <p>At this point, given a particular bond configuration <span
 class="math inline">\(u_{ij} = \pm 1\)</span>, we can construct a
 quadratic Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span>
@@ -891,7 +738,9 @@ namely how to construct a set of gauge invariant quantities out of the
 be the plaquette operators.</p>
 <p>Takeaway: The Bond Sectors overlap with the physical subspace but are
 not contained within it.</p>
-<h3 id="open-boundary-conditions">Open boundary conditions</h3>
+</section>
+<section id="open-boundary-conditions" class="level3">
+<h3>Open boundary conditions</h3>
 <p>Care must be taken when defining open boundary conditions. Simply
 removing bonds from the lattice leaves behind unpaired <span
 class="math inline">\(b^\alpha\)</span> operators that must be paired in
@@ -907,7 +756,11 @@ which we set to 1 when calculating the projector.</p>
 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>
-<h1 class="unnumbered" id="bibliography">Bibliography</h1>
+</section>
+</section>
+</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-marsalTopologicalWeaireThorpe2020" class="csl-entry"
 role="doc-biblioentry">
@@ -953,6 +806,9 @@ class="csl-right-inline">J.-P. Blaizot and G. Ripka, <em>Quantum Theory
 of Finite Systems</em> (The MIT Press, 1986).</div>
 </div>
 </div>
+</section>
+
+
 </main>
 </body>
 </html>
diff --git a/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html b/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html
index 94c1bde..147aec9 100644
--- a/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html
+++ b/_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html
@@ -13,252 +13,11 @@ image:
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   <meta name="description" content="The methods used to study the Amorphous Kitaev Model." />
   <title>The Amorphous Kitaev Model - Introduction</title>
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+<script src="/assets/js/thesis_scrollspy.js"></script>
 
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@@ -267,9 +26,9 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
-<li><a href="#methods" id="toc-methods">Methods</a>
+<li><a href="#sec:AMK-Methods" id="toc-sec:AMK-Methods">Methods</a>
 <ul>
 <li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li>
 <li><a href="#graph-representation" id="toc-graph-representation">Graph
@@ -295,15 +54,18 @@ Markers</a></li>
 </ul></li>
 <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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="#methods" id="toc-methods">Methods</a>
+<li><a href="#sec:AMK-Methods" id="toc-sec:AMK-Methods">Methods</a>
 <ul>
 <li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li>
 <li><a href="#graph-representation" id="toc-graph-representation">Graph
@@ -331,9 +93,12 @@ Markers</a></li>
 </ul>
 </nav>
  -->
+
+<!-- Main Page Body -->
 <div class="sourceCode" id="cb1"><pre
 class="sourceCode python"><code class="sourceCode python"></code></pre></div>
-<h1 id="methods">Methods</h1>
+<section id="sec:AMK-Methods" class="level1">
+<h1>Methods</h1>
 <p>The practical implementation of what is described in this section is
 available as a Python package called Koala (Kitaev On Amorphous
 LAttices) <span class="citation"
@@ -341,7 +106,8 @@ data-cites="tomImperialCMTHKoalaFirst2022"> [<a
 href="#ref-tomImperialCMTHKoalaFirst2022"
 role="doc-biblioref"><strong>tomImperialCMTHKoalaFirst2022?</strong></a>]</span>.
 All results and figures were generated with Koala.</p>
-<h2 id="voronisation">Voronisation</h2>
+<section id="voronisation" class="level2">
+<h2>Voronisation</h2>
 <p>To study the properties of the amorphous Kitaev model, we need to
 sample from the space of possible trivalent graphs.</p>
 <p>A simple method is to use a Voronoi partition of the torus <span
@@ -396,7 +162,9 @@ is shown here to help the reader identify corresponding
 edges.</figcaption>
 </figure>
 </div>
-<h2 id="graph-representation">Graph Representation</h2>
+</section>
+<section id="graph-representation" class="level2">
+<h2>Graph Representation</h2>
 <p>Three keys pieces of information allow us to represent amorphous
 lattices.</p>
 <p>Most of the graph connectivity is encoded by an ordered list of edges
@@ -451,7 +219,9 @@ similar idea, we unwrap the torus to one unit cell and keep track of
 which bonds cross the cell boundaries.</figcaption>
 </figure>
 </div>
-<h2 id="colouring-the-bonds">Colouring the Bonds</h2>
+</section>
+<section id="colouring-the-bonds" class="level2">
+<h2>Colouring the Bonds</h2>
 <p>The Kitaev Model requires that each edge in the lattice be assigned a
 label <span class="math inline">\(x\)</span>, <span
 class="math inline">\(y\)</span> or <span
@@ -522,8 +292,8 @@ valid 3-edge-colourings of amorphous lattices. Colors that differ from
 the leftmost panel are highlighted.</figcaption>
 </figure>
 </div>
-<h3 id="four-colourings-and-three-colourings">Four-colourings and
-three-colourings</h3>
+<section id="four-colourings-and-three-colourings" class="level3">
+<h3>Four-colourings and three-colourings</h3>
 <p><strong>add diagram of this</strong></p>
 <p>A four-face-colouring can be converted into a three-edge-colouring
 quite easily: 1. Assume the faces of G can be four-coloured with labels
@@ -558,8 +328,9 @@ suggests that Voronoi lattices may have additional structures that make
 them three-edge-colourable. Intuitively, it seems that the kinds of
 toroidal graphs that cannot be three-edge-coloured could never be
 generated by a Voronoi partition with more than a few seed points.</p>
-<h3 id="finding-lattice-colourings-with-minisat">Finding Lattice
-colourings with miniSAT</h3>
+</section>
+<section id="finding-lattice-colourings-with-minisat" class="level3">
+<h3>Finding Lattice colourings with miniSAT</h3>
 <p>Some issues are harder in theory than in practice.
 Three-edge-colouring cubic toroidal graphs appears to be one of those
 things.</p>
@@ -673,8 +444,9 @@ generating a lattice. For larger systems, the time taken to perform the
 diagonalisation dominates.</figcaption>
 </figure>
 </div>
-<h3 id="does-it-matter-which-colouring-we-choose">Does it matter which
-colouring we choose?</h3>
+</section>
+<section id="does-it-matter-which-colouring-we-choose" class="level3">
+<h3>Does it matter which colouring we choose?</h3>
 <p>In the isotropic case <span class="math inline">\(J^\alpha =
 1\)</span>, it is easy to show that choosing a particular valid
 colouring cannot make a difference. As the choice of how we define the
@@ -687,8 +459,11 @@ one colouring into that generated by another.</p>
 question whether particular physical properties could arise by
 engineering the colouring in this phase though we expect them to exhibit
 a self averaging behaviour.</p>
-<h2 id="mapping-between-flux-sectors-and-bond-sectors">Mapping between
-flux sectors and bond sectors</h2>
+</section>
+</section>
+<section id="mapping-between-flux-sectors-and-bond-sectors"
+class="level2">
+<h2>Mapping between flux sectors and bond sectors</h2>
 <p>Constructing the Majorana representation of the model requires the
 particular bond configuration <span class="math inline">\(u_{jk} = \pm
 1\)</span>. However, the large number of gauge symmetries of the bond
@@ -743,7 +518,9 @@ flux will remain because the starting and target flux sectors differed
 by an odd number of fluxes.</figcaption>
 </figure>
 </div>
-<h2 id="chern-markers">Chern Markers</h2>
+</section>
+<section id="chern-markers" class="level2">
+<h2>Chern Markers</h2>
 <p>We know that the standard Kitaev model supports both Abelian and
 non-Abelian phases. Therefore, how can we assess whether this is also
 the case for the amorphous Kitaev model?</p>
@@ -759,7 +536,10 @@ system.</p>
 <p><strong>Expand on definition here</strong></p>
 <p><strong>Discuss link between Chern number and Anyonic
 Statistics</strong></p>
-<h1 class="unnumbered" id="bibliography">Bibliography</h1>
+</section>
+</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-mitchellAmorphousTopologicalInsulators2018"
 class="csl-entry" role="doc-biblioentry">
@@ -903,6 +683,9 @@ Clustering and Graph Coloring Algorithms</a></em>, J. ACM
 <strong>30</strong>, 417 (1983).</div>
 </div>
 </div>
+</section>
+
+
 </main>
 </body>
 </html>
diff --git a/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html b/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html
index 4327d4f..f1a0107 100644
--- a/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html
+++ b/_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html
@@ -13,189 +13,11 @@ image:
   <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
   <meta name="description" content="The Amorphous Kitaev model is a chiral spin liquid!" />
   <title>The Amorphous Kitaev Model - Results</title>
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-<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
-   -->
 
 <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
+<script src="/assets/js/thesis_scrollspy.js"></script>
 
-
-<!--[if lt IE 9]>
-    <script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
-<![endif]-->
 <link rel="stylesheet" href="/assets/css/styles.css">
 <script src="/assets/js/index.js"></script>
 </head>
@@ -204,9 +26,9 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
-<li><a href="#results" id="toc-results">Results</a>
+<li><a href="#sec:AMK-Results" id="toc-sec:AMK-Results">Results</a>
 <ul>
 <li><a href="#the-ground-state-flux-sector"
 id="toc-the-ground-state-flux-sector">The Ground State Flux
@@ -226,32 +48,27 @@ non-Abelian?</a></li>
 id="toc-anderson-transition-to-a-thermal-metal">Anderson Transition to a
 Thermal Metal</a></li>
 </ul></li>
+<li><a href="#sec:AMK-Conclusion" id="toc-sec:AMK-Conclusion">Discussion
+and Conclusion</a>
+<ul>
 <li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li>
-<li><a href="#discussion" id="toc-discussion">Discussion</a>
-<ul>
-<li><a href="#limits-of-the-ground-state-conjecture"
-id="toc-limits-of-the-ground-state-conjecture">Limits of the ground
-state conjecture</a></li>
-</ul></li>
-<li><a href="#outlook" id="toc-outlook">Outlook</a>
-<ul>
-<li><a href="#experimental-realisations-and-signatures"
-id="toc-experimental-realisations-and-signatures">Experimental
-Realisations and Signatures</a></li>
-<li><a href="#generalisations"
-id="toc-generalisations">Generalisations</a></li>
+<li><a href="#discussion" id="toc-discussion">Discussion</a></li>
+<li><a href="#outlook" id="toc-outlook">Outlook</a></li>
 </ul></li>
 <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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="#results" id="toc-results">Results</a>
+<li><a href="#sec:AMK-Results" id="toc-sec:AMK-Results">Results</a>
 <ul>
 <li><a href="#the-ground-state-flux-sector"
 id="toc-the-ground-state-flux-sector">The Ground State Flux
@@ -271,27 +88,23 @@ non-Abelian?</a></li>
 id="toc-anderson-transition-to-a-thermal-metal">Anderson Transition to a
 Thermal Metal</a></li>
 </ul></li>
+<li><a href="#sec:AMK-Conclusion" id="toc-sec:AMK-Conclusion">Discussion
+and Conclusion</a>
+<ul>
 <li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li>
-<li><a href="#discussion" id="toc-discussion">Discussion</a>
-<ul>
-<li><a href="#limits-of-the-ground-state-conjecture"
-id="toc-limits-of-the-ground-state-conjecture">Limits of the ground
-state conjecture</a></li>
-</ul></li>
-<li><a href="#outlook" id="toc-outlook">Outlook</a>
-<ul>
-<li><a href="#experimental-realisations-and-signatures"
-id="toc-experimental-realisations-and-signatures">Experimental
-Realisations and Signatures</a></li>
-<li><a href="#generalisations"
-id="toc-generalisations">Generalisations</a></li>
+<li><a href="#discussion" id="toc-discussion">Discussion</a></li>
+<li><a href="#outlook" id="toc-outlook">Outlook</a></li>
 </ul></li>
 <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
 </ul>
 </nav>
  -->
-<h1 id="results">Results</h1>
-<h2 id="the-ground-state-flux-sector">The Ground State Flux Sector</h2>
+
+<!-- Main Page Body -->
+<section id="sec:AMK-Results" class="level1">
+<h1>Results</h1>
+<section id="the-ground-state-flux-sector" class="level2">
+<h2>The Ground State Flux Sector</h2>
 <p>Here I will discuss the numerical evidence that our guess for the
 ground state flux sector is correct. We will do this by enumerating all
 the flux sectors of many separate system realisations. However there are
@@ -345,8 +158,9 @@ conjecture. In these cases, the energy difference between the true
 ground state and our prediction was on the order of <span
 class="math inline">\(10^{-6} J\)</span>. It is unclear whether this is
 a finite size effect or something else.</p>
-<h2 id="spontaneous-chiral-symmetry-breaking">Spontaneous Chiral
-Symmetry Breaking</h2>
+</section>
+<section id="spontaneous-chiral-symmetry-breaking" class="level2">
+<h2>Spontaneous Chiral Symmetry Breaking</h2>
 <p>The spin Kitaev Hamiltonian is real and therefore has time reversal
 symmetry (TRS). However, the flux <span
 class="math inline">\(\phi_p\)</span> through any plaquette with an odd
@@ -368,7 +182,9 @@ href="#ref-Peri2020" role="doc-biblioref">4</a>]</span>. This
 spontaneously broken symmetry avoids the need to explicitly break TRS
 with a magnetic field term as is done in the original honeycomb
 model.</p>
-<h2 id="ground-state-phase-diagram">Ground State Phase Diagram</h2>
+</section>
+<section id="ground-state-phase-diagram" class="level2">
+<h2>Ground State Phase Diagram</h2>
 <p>As previously discussed, the standard Honeycomb model has a Abelian,
 gapped phase in the anisotropic region (the A phase) and is gapless in
 the isotropic region. The introduction of a magnetic field breaks the
@@ -430,7 +246,8 @@ class="math inline">\(0\)</span> to <span class="math inline">\(\pm
 <strong>citation</strong>.</figcaption>
 </figure>
 </div>
-<h3 id="is-it-abelian-or-non-abelian">Is it Abelian or non-Abelian?</h3>
+<section id="is-it-abelian-or-non-abelian" class="level3">
+<h3>Is it Abelian or non-Abelian?</h3>
 <p>The two phases of the amorphous model are clearly gapped, though
 later I’ll double check this with finite size scaling.</p>
 <p>The next question is: do these phases support excitations with
@@ -527,7 +344,9 @@ with fixed <span class="math inline">\(r = 0.3\)</span> nicely confirms
 that the isotropic phase is non-Abelian.</figcaption>
 </figure>
 </div>
-<h3 id="edge-modes">Edge Modes</h3>
+</section>
+<section id="edge-modes" class="level3">
+<h3>Edge Modes</h3>
 <p>Chiral Spin Liquids support topological protected edge modes on open
 boundary conditions <span class="citation"
 data-cites="qi_general_2006"> [<a href="#ref-qi_general_2006"
@@ -562,8 +381,10 @@ each energy window. Cutting the boundary fills the gap with localised
 states.</figcaption>
 </figure>
 </div>
-<h2 id="anderson-transition-to-a-thermal-metal">Anderson Transition to a
-Thermal Metal</h2>
+</section>
+</section>
+<section id="anderson-transition-to-a-thermal-metal" class="level2">
+<h2>Anderson Transition to a Thermal Metal</h2>
 <p>Previous work on the honeycomb model at finite temperature has shown
 that the B phase undergoes a thermal transition from a quantum spin
 liquid phase a to a <strong>thermal metal</strong> phase <span
@@ -716,7 +537,12 @@ plots the density of vortices is <span class="math inline">\(\rho =
 \infty\)</span> limit.</figcaption>
 </figure>
 </div>
-<h1 id="conclusion">Conclusion</h1>
+</section>
+</section>
+<section id="sec:AMK-Conclusion" class="level1">
+<h1>Discussion and Conclusion</h1>
+<section id="conclusion" class="level2">
+<h2>Conclusion</h2>
 <p>In this chapter we have looked at an extension of the Kitaev
 honeycomb model to amorphous lattices with coordination number three. We
 discussed a method to construct arbitrary trivalent lattices using
@@ -735,9 +561,10 @@ spin liquid phase.</p>
 <p>Finally we showed evidence that the amorphous system undergoes an
 Anderson transition to a thermal metal phase, driven by the
 proliferation of vortices with increasing temperature.</p>
-<h1 id="discussion">Discussion</h1>
-<h2 id="limits-of-the-ground-state-conjecture">Limits of the ground
-state conjecture</h2>
+</section>
+<section id="discussion" class="level2">
+<h2>Discussion</h2>
+<p><strong>Limits of the ground state conjecture</strong></p>
 <p>We found a small number of lattices for which the ground state
 conjecture did not correctly predict the true ground state flux sector.
 I see two possibilities for what could cause this.</p>
@@ -753,12 +580,13 @@ colouring for a lattice affects the physical properties in the toric
 code A phase. It is possible that some property of the particular
 colouring chosen is what leads to failure of the ground state conjecture
 here.</p>
-<h1 id="outlook">Outlook</h1>
+</section>
+<section id="outlook" class="level2">
+<h2>Outlook</h2>
 <p>This exactly solvable chiral QSL provides a first example of a
 topological quantum many-body phase in amorphous magnets, which raises a
 number of questions for future research.</p>
-<h2 id="experimental-realisations-and-signatures">Experimental
-Realisations and Signatures</h2>
+<p><strong>Experimental Realisations and Signatures</strong></p>
 <p>The obvious question is whether amorphous Kitaev materials could be
 physically realised.</p>
 <p>Most crystals can as exists in a metastable amorphous state if they
@@ -793,7 +621,7 @@ behavior <span class="citation"
 data-cites="misumiQuantumSpinLiquid2020"> [<a
 href="#ref-misumiQuantumSpinLiquid2020"
 role="doc-biblioref">29</a>]</span>.</p>
-<h2 id="generalisations">Generalisations</h2>
+<p><strong>Generalisations</strong></p>
 <p>The model presented here could be generalized in several ways.</p>
 <p>First, it would be interesting to study the stability of the chiral
 amorphous Kitaev QSL with respect to perturbations\ <span
@@ -831,7 +659,10 @@ 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>
-<h1 class="unnumbered" id="bibliography">Bibliography</h1>
+</section>
+</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-kitaevAnyonsExactlySolved2006" class="csl-entry"
 role="doc-biblioentry">
@@ -1188,6 +1019,9 @@ Wu, D. Arovas, and H.-H. Hung, <em>Γ-Matrix Generalization of the Kitaev
 Model</em>, Physical Review B <strong>79</strong>, 134427 (2009).</div>
 </div>
 </div>
+</section>
+
+
 </main>
 </body>
 </html>
diff --git a/_thesis/5_Conclusion/5_Conclusion.html b/_thesis/5_Conclusion/5_Conclusion.html
index 8a4f405..7c468cb 100644
--- a/_thesis/5_Conclusion/5_Conclusion.html
+++ b/_thesis/5_Conclusion/5_Conclusion.html
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-      margin-bottom: 4em;
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-    span.smallcaps{font-variant: small-caps;}
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-    div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
-    ul.task-list{list-style: none;}
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-<!--  -->
-
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-<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
-   -->
 
 <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
+<script src="/assets/js/thesis_scrollspy.js"></script>
 
-
-<!--[if lt IE 9]>
-    <script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
-<![endif]-->
 <link rel="stylesheet" href="/assets/css/styles.css">
 <script src="/assets/js/index.js"></script>
 </head>
@@ -181,17 +25,20 @@ image:
 <!--Capture the table of contents from pandoc as a jekyll variable  -->
 {% capture tableOfContents %}
 <br>
-Contents:
+<nav aria-label="Table of Contents" class="page-table-of-contents">
 <ul>
 <li><a href="#discussion" id="toc-discussion">Discussion</a></li>
 <li><a href="#outlook" id="toc-outlook">Outlook</a></li>
 </ul>
+</nav>
 {% endcapture %}
 
-<!-- Give the table of contents to header as a variable -->
+<!-- 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="#discussion" id="toc-discussion">Discussion</a></li>
@@ -199,8 +46,16 @@ Contents:
 </ul>
 </nav>
  -->
-<h2 id="discussion">Discussion</h2>
-<h2 id="outlook">Outlook</h2>
+
+<!-- Main Page Body -->
+<section id="discussion" class="level2">
+<h2>Discussion</h2>
+</section>
+<section id="outlook" class="level2">
+<h2>Outlook</h2>
+</section>
+
+
 </main>
 </body>
 </html>
diff --git a/_thesis/6_Appendices/A.1_Particle_Hole_Symmetry.html b/_thesis/6_Appendices/A.1_Particle_Hole_Symmetry.html
new file mode 100644
index 0000000..f993732
--- /dev/null
+++ b/_thesis/6_Appendices/A.1_Particle_Hole_Symmetry.html
@@ -0,0 +1,128 @@
+---
+title: A.1_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>A.1_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 -->
+<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>
+</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>
diff --git a/_thesis/6_Appendices/A.2_Markov_Chain_Monte_Carlo.html b/_thesis/6_Appendices/A.2_Markov_Chain_Monte_Carlo.html
new file mode 100644
index 0000000..94d110e
--- /dev/null
+++ b/_thesis/6_Appendices/A.2_Markov_Chain_Monte_Carlo.html
@@ -0,0 +1,146 @@
+---
+title: A.2_Markov_Chain_Monte_Carlo
+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>A.2_Markov_Chain_Monte_Carlo</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="#markov-chain-monte-carlo"
+id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a>
+<ul>
+<li><a href="#applying-mcmc-to-the-fk-model"
+id="toc-applying-mcmc-to-the-fk-model">Applying MCMC to the FK
+model</a></li>
+</ul></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="#markov-chain-monte-carlo"
+id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a>
+<ul>
+<li><a href="#applying-mcmc-to-the-fk-model"
+id="toc-applying-mcmc-to-the-fk-model">Applying MCMC to the FK
+model</a></li>
+</ul></li>
+</ul>
+</nav>
+ -->
+
+<!-- Main Page Body -->
+<section id="markov-chain-monte-carlo" class="level1">
+<h1>Markov Chain Monte Carlo</h1>
+<section id="applying-mcmc-to-the-fk-model" class="level2">
+<h2>Applying MCMC to the FK model</h2>
+<p>MCMC can be applied to sample over the classical degrees of freedom
+of the model. We take the full Hamiltonian and split it into a classical
+and a quantum part: <span class="math display">\[\begin{aligned}
+    H_{\mathrm{FK}} &amp;= -\sum_{&lt;ij&gt;} c^\dagger_{i}c_{j} + U
+\sum_{i} (c^\dagger_{i}c_{i} - 1/2)( n_i - 1/2) \\
+    &amp;+ \sum_{ij} J_{ij} (n_i - 1/2) (n_j - 1/2)  - \mu \sum_i
+(c^\dagger_{i}c_{i} + n_i)\\
+    H_q &amp;= -\sum_{&lt;ij&gt;} c^\dagger_{i}c_{j} + \sum_{i}
+\left(U(n_i - 1/2) - \mu\right) c^\dagger_{i}c_{i}\\
+    H_c &amp;= \sum_i \mu n_i - \frac{U}{2}(n_i - 1/2) +
+\sum_{ij}J_{ij}(n_i - 1/2)(n_j - 1/2)
+\end{aligned}
+\]</span></p>
+<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> % 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">\[\begin{aligned}
+Z^k    &amp;= \Tr e^{-\beta F^k} = \sum_j e^{-\beta \sum_i m^j_i
+\epsilon^k_i}\\
+       &amp;= \sum_j \prod_i e^{- \beta m^j_i \epsilon^k_i}= \prod_i
+\sum_j e^{- \beta m^j_i \epsilon^k_i}\\
+       &amp;= \prod_i (1 + e^{- \beta \epsilon^k_i})\\
+F^k    &amp;= -1/\beta \sum_k \ln{(1 + e^{- \beta \epsilon^k_i})}
+\end{aligned}\]</span> % Observables can then be calculated from the
+partition function, for examples the occupation numbers:</p>
+<p><span class="math display">\[\begin{aligned}
+\tex{N} &amp;= \frac{1}{\beta} \frac{1}{Z} \frac{\partial Z}{\partial
+\mu} = - \frac{\partial F}{\partial \mu}\\
+    &amp;= \frac{1}{\beta} \frac{1}{Z} \frac{\partial}{\partial \mu}
+\sum_k e^{-\beta (H^k + F^k)}\\
+    &amp;= 1/Z \sum_k (N^k_{\mathrm{ion}} + N^k_{\mathrm{electron}})
+e^{-\beta (H^k + F^k)}\\
+\end{aligned}\]</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> % The MCMC algorithm consists of performing a
+random walk over the states <span class="math inline">\(\{ n^k_i
+\}\)</span>. In the simplest case the proposal distribution corresponds
+to flipping a random site from occupied to unoccupied or vice versa,
+since this proposal is symmetric the acceptance function becomes: <span
+class="math display">\[\begin{aligned}
+P(k) &amp;= \mathcal{Z}^{-1} e^{-\beta(H^k + F^k)} \\
+\mathcal{A}(k \to k&#39;) &amp;= \min\left(1,
+\frac{P(k&#39;)}{P(k)}\right) = \min\left(1, e^{\beta(H^{k&#39;} +
+F^{k&#39;})-\beta(H^k + F^k)}\right)
+\end{aligned}\]</span> % At each step <span
+class="math inline">\(F^k\)</span> is calculated by diagonalising the
+tri-diagonal matrix representation of <span
+class="math inline">\(H_q\)</span> with open boundary conditions.
+Observables are simply averages over the their value at each step of the
+random walk. The full spectrum and eigenbasis is too large to save to
+disk so usually running averages of key observables are taken as the
+walk progresses.</p>
+<div class="sourceCode" id="cb1"><pre
+class="sourceCode python"><code class="sourceCode python"></code></pre></div>
+<p></ij></ij></p>
+</section>
+</section>
+
+
+</main>
+</body>
+</html>
diff --git a/_thesis/6_Appendices/A.3_Lattice_Generation.html b/_thesis/6_Appendices/A.3_Lattice_Generation.html
new file mode 100644
index 0000000..a05380c
--- /dev/null
+++ b/_thesis/6_Appendices/A.3_Lattice_Generation.html
@@ -0,0 +1,60 @@
+---
+title: A.3_Lattice_Generation
+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>A.3_Lattice_Generation</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="#lattice-generation" id="toc-lattice-generation">Lattice
+Generation</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="#lattice-generation" id="toc-lattice-generation">Lattice
+Generation</a></li>
+</ul>
+</nav>
+ -->
+
+<!-- Main Page Body -->
+<section id="lattice-generation" class="level1">
+<h1>Lattice Generation</h1>
+<div class="sourceCode" id="cb1"><pre
+class="sourceCode python"><code class="sourceCode python"></code></pre></div>
+</section>
+
+
+</main>
+</body>
+</html>
diff --git a/_thesis/6_Appendices/A.4_Lattice_Colouring.html b/_thesis/6_Appendices/A.4_Lattice_Colouring.html
new file mode 100644
index 0000000..f33f121
--- /dev/null
+++ b/_thesis/6_Appendices/A.4_Lattice_Colouring.html
@@ -0,0 +1,60 @@
+---
+title: A.4_Lattice_Colouring
+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>A.4_Lattice_Colouring</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="#lattice-colouring" id="toc-lattice-colouring">Lattice
+Colouring</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="#lattice-colouring" id="toc-lattice-colouring">Lattice
+Colouring</a></li>
+</ul>
+</nav>
+ -->
+
+<!-- Main Page Body -->
+<section id="lattice-colouring" class="level1">
+<h1>Lattice Colouring</h1>
+<div class="sourceCode" id="cb1"><pre
+class="sourceCode python"><code class="sourceCode python"></code></pre></div>
+</section>
+
+
+</main>
+</body>
+</html>
diff --git a/_thesis/6_Appendices/A.5_The_Projector.html b/_thesis/6_Appendices/A.5_The_Projector.html
new file mode 100644
index 0000000..4f04f76
--- /dev/null
+++ b/_thesis/6_Appendices/A.5_The_Projector.html
@@ -0,0 +1,58 @@
+---
+title: A.5_The_Projector
+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>A.5_The_Projector</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="#the-projector" id="toc-the-projector">The
+Projector</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="#the-projector" id="toc-the-projector">The
+Projector</a></li>
+</ul>
+</nav>
+ -->
+
+<!-- Main Page Body -->
+<section id="the-projector" class="level1">
+<h1>The Projector</h1>
+</section>
+
+
+</main>
+</body>
+</html>
diff --git a/_thesis/toc.html b/_thesis/toc.html
index 27212eb..228ebb4 100644
--- a/_thesis/toc.html
+++ b/_thesis/toc.html
@@ -10,32 +10,31 @@
     <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/2.2_HKM_Model.html#the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a></li>
-    <li><a href="./2_Background/2.3_Disorder.html#disorder-&-localisation">Disorder & Localisation</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#chapter-summary">Chapter 3: The Long Range Falikov-Kimball Model</a></li>
+  <li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#the-model">Chapter 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/3.2_LRFK_Methods.html#markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
-    <li><a href="./3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html#the-phase-diagram">The Phase Diagram</a></li>
-    <li><a href="./3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html#localisation-properties">Localisation Properties</a></li>
-    <li><a href="./3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html#discussion-&-conclusion">Discussion & Conclusion</a></li>
-    <li><a href="./3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html#acknowledgments">Acknowledgments</a></li>
-    <li><a href="./3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html#[]{#app:disorder_model-label="app:disorder_model"}-uncorrelated-disorder-model">[]{#app:disorder_model label="app:disorder_model"} UNCORRELATED DISORDER 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">Chapter 4: The Amorphous Kitaev Model</a></li>
     <ul>
-    <li><a href="./4_Amorphous_Kitaev_Model/4.1_AMK_Model.html#contributions">Contributions</a></li>
-    <li><a href="./4_Amorphous_Kitaev_Model/4.1_AMK_Model.html#introduction">Introduction</a></li>
+    <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#conclusion">Conclusion</a></li>
-    <li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#discussion">Discussion</a></li>
-    <li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#outlook">Outlook</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">Conclusion</a></li>
   <li><a href="./6_Appendices/A.1_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Appendices</a></li>
     <ul>
     <li><a href="./6_Appendices/A.1_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Markov Chain Monte Carlo</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_Lattice_Generation.html#lattice-generation">Lattice Generation</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.3_Lattice_Colouring.html#lattice-colouring">Lattice Colouring</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.4_The_Projector.html#the-projector">The Projector</a></li>
+    <li><a href="./6_Appendices/A.5_The_Projector.html#the-projector">The Projector</a></li>
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+       style="fill:none;stroke:#000000;stroke-width:0.8;stroke-linecap:square;stroke-linejoin:miter;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
+       id="path3588" /><path
+       d="M 308.01067,127.144 H 437.976"
+       style="fill:none;stroke:#000000;stroke-width:0.8;stroke-linecap:square;stroke-linejoin:miter;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
+       id="path3590" /></g></svg>
diff --git a/thesis.html b/thesis.html
index e7336f6..6485640 100644
--- a/thesis.html
+++ b/thesis.html
@@ -7,6 +7,6 @@ permalink: /thesis/
 This is my work-in-progress thesis. It will be available as a traditional PDF too but I wanted to make it available as nicely rendered website too!
 
 <h2>Contents</h2>
-<nav>
+<nav aria-label="Thesis Table of Contents" class="overall-table-of-contents">
 {% include_relative _thesis/toc.html %}
 </nav>
\ No newline at end of file