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_includes/sidebar.html
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<nav> <nav aria-label="Site Map" class="site-map"></nav>
{% for item in site.data.navigation %} {% for item in site.data.navigation %}
<a href="{{ item.link }}" {% if page.url == item.link %}class="current"{% endif %}>{{ item.name }}</a> <a href="{{ item.link }}" {% if page.url == item.link %}class="current"{% endif %}>{{ item.name }}</a>
{% endfor %} {% endfor %}

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html { html {
width: 100vw; width: 100vw;
scroll-behavior: smooth;
} }
body { body {
@ -50,6 +51,14 @@ a {
color: #222; color: #222;
} }
header a {
text-decoration: none;
}
nav a {
text-decoration: none;
}
a:hover { a:hover {
text-decoration: underline; text-decoration: underline;
} }

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_sass/thesis.scss
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@ -17,11 +17,18 @@ figcaption {
// For the table of contents, should probably put this in a container // For the table of contents, should probably put this in a container
// remove underline from toc links //For the animation that plays in the nav as you scroll
nav a { nav.page-table-of-contents a {
text-decoration: none; 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 // modify the spacing of the various levels
li { li {
margin-bottom: 0.2em; margin-bottom: 0.2em;
@ -68,3 +75,13 @@ header li {
} }
} }
nav.overall-table-of-contents > ul {
padding-inline-start: 0px;
> li {
list-style: none;
margin-top: 1em;
}
}

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_thesis/0_Preface/0.1_Aknowledgements.html
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<script src="/assets/js/thesis_scrollspy.js"></script>
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@ -181,15 +25,20 @@ image:
<!--Capture the table of contents from pandoc as a jekyll variable --> <!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %} {% capture tableOfContents %}
<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
</nav>
{% endcapture %} {% 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 %} {% include header.html extra=tableOfContents %}
<main> <main>
<!-- Table of Contents -->
<!-- --> <!-- -->
<!-- Main Page Body -->
<p>I would like to thank my supervisor, Professor Johannes Knolle and <p>I would like to thank my supervisor, Professor Johannes Knolle and
co-supervisor Professor Derek Lee for guidance and support during this co-supervisor Professor Derek Lee for guidance and support during this
long process.</p> long process.</p>
@ -216,6 +65,10 @@ expertise and patience.</p>
<p>All the I-Stemm team, Katerina, Jeremey, John, ….</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 <p>And finally, I’d like the thank the staff of the Camberwell Public
Library where the majority of this thesis was written.</p> 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> </main>
</body> </body>
</html> </html>

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_thesis/1_Introduction/1_Intro.html
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@ -12,189 +12,11 @@ image:
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@ -203,7 +25,7 @@ image:
<!--Capture the table of contents from pandoc as a jekyll variable --> <!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %} {% capture tableOfContents %}
<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#interacting-quantum-many-body-systems" <li><a href="#interacting-quantum-many-body-systems"
id="toc-interacting-quantum-many-body-systems">Interacting Quantum Many 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="#outline" id="toc-outline">Outline</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav>
{% endcapture %} {% 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 %} {% include header.html extra=tableOfContents %}
<main> <main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#interacting-quantum-many-body-systems" <li><a href="#interacting-quantum-many-body-systems"
@ -235,8 +60,10 @@ id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
</ul> </ul>
</nav> </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 <p>When you take many objects and let them interact together, it is
often simpler to describe the behaviour of the group differently from often simpler to describe the behaviour of the group differently from
the way one would describe the individual objects. Consider a flock of the way one would describe the individual objects. Consider a flock of
@ -321,25 +148,39 @@ antiferromagnetism <span class="citation"
data-cites="MagnetismCondensedMatter"> [<a data-cites="MagnetismCondensedMatter"> [<a
href="#ref-MagnetismCondensedMatter" href="#ref-MagnetismCondensedMatter"
role="doc-biblioref">12</a>]</span>.</p> role="doc-biblioref">12</a>]</span>.</p>
<p>However, at some point we had to start on the interacting quantum <p>The development of Landau-Fermi Liquid theory explained why band
many body systems. The properties of some materials cannot be understood theory works so well even in cases where an analysis of the relevant
without a taking into account all three effects and these are energies suggests that it should not <span class="citation"
collectively called strongly correlated materials. The canonical data-cites="wenQuantumFieldTheory2007"> [<a
examples are superconductivity <span class="citation" href="#ref-wenQuantumFieldTheory2007"
data-cites="MicroscopicTheorySuperconductivity"> [<a 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" 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" effect <span class="citation"
data-cites="feldmanFractionalChargeFractional2021"> [<a data-cites="feldmanFractionalChargeFractional2021"> [<a
href="#ref-feldmanFractionalChargeFractional2021" 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" class="citation"
data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a 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" 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> 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 <p>Mott Insulators are remarkable because their electrical insulator
properties come from electron-electron interactions. Electrical properties come from electron-electron interactions. Electrical
conductivity, the bulk movement of electrons, requires both that there 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" be conductive <span class="citation"
data-cites="boerSemiconductorsPartiallyCompletely1937"> [<a data-cites="boerSemiconductorsPartiallyCompletely1937"> [<a
href="#ref-boerSemiconductorsPartiallyCompletely1937" 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 electron-electron interactions were the cause of this effect <span
class="citation" data-cites="mottDiscussionPaperBoer1937"> [<a class="citation" data-cites="mottDiscussionPaperBoer1937"> [<a
href="#ref-mottDiscussionPaperBoer1937" 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 high temperature superconductivity in the cuprates in 1986 <span
class="citation" class="citation"
data-cites="bednorzPossibleHighTcSuperconductivity1986"> [<a data-cites="bednorzPossibleHighTcSuperconductivity1986"> [<a
href="#ref-bednorzPossibleHighTcSuperconductivity1986" 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" result of a doped Mott insulator state <span class="citation"
data-cites="leeDopingMottInsulator2006"> [<a data-cites="leeDopingMottInsulator2006"> [<a
href="#ref-leeDopingMottInsulator2006" 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 <p>The canonical toy model of the Mott insulator is the Hubbard
model <span class="citation" model <span class="citation"
data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"> [<a data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"> [<a
href="#ref-gutzwillerEffectCorrelationFerromagnetism1963" href="#ref-gutzwillerEffectCorrelationFerromagnetism1963"
role="doc-biblioref">21</a>–<a role="doc-biblioref">23</a>–<a
href="#ref-hubbardj.ElectronCorrelationsNarrow1963" 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 class="math inline">\(1/2\)</span> fermions hopping on the lattice with
hopping parameter <span class="math inline">\(t\)</span> and hopping parameter <span class="math inline">\(t\)</span> and
electron-electron repulsion <span class="math inline">\(U\)</span></p> 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" electron per lattice site <span class="citation"
data-cites="hubbardElectronCorrelationsNarrow1964"> [<a data-cites="hubbardElectronCorrelationsNarrow1964"> [<a
href="#ref-hubbardElectronCorrelationsNarrow1964" 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 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_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U
\sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \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" cause of the gap opening <span class="citation"
data-cites="mottMetalInsulatorTransitions1990"> [<a data-cites="mottMetalInsulatorTransitions1990"> [<a
href="#ref-mottMetalInsulatorTransitions1990" 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" found <span class="citation"
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a 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" 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 local moments may form a highly entangled state known as a quantum spin
liquid, which will be discussed shortly.</p> liquid, which will be discussed shortly.</p>
<p>Various theoretical treatments of the Hubbard model have been made, <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" local density approximation (LDA) <span class="citation"
data-cites="slaterMagneticEffectsHartreeFock1951"> [<a data-cites="slaterMagneticEffectsHartreeFock1951"> [<a
href="#ref-slaterMagneticEffectsHartreeFock1951" 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" theory <span class="citation"
data-cites="greinerQuantumPhaseTransition2002"> [<a data-cites="greinerQuantumPhaseTransition2002"> [<a
href="#ref-greinerQuantumPhaseTransition2002" 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 perfect. Strong correlations are poorly described by the Fermi liquid
theory and the LDA approaches while mean field approximations do poorly theory and the LDA approaches while mean field approximations do poorly
in low dimensional systems. This theoretical difficulty has made the in low dimensional systems. This theoretical difficulty has made the
Hubbard model a target for cold atom simulations <span class="citation" Hubbard model a target for cold atom simulations <span class="citation"
data-cites="mazurenkoColdatomFermiHubbard2017"> [<a data-cites="mazurenkoColdatomFermiHubbard2017"> [<a
href="#ref-mazurenkoColdatomFermiHubbard2017" 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 <p>From here the discussion will branch two directions. First, we will
discuss a limit of the Hubbard model called the Falikov-Kimball Model. discuss a limit of the Hubbard model called the Falikov-Kimball Model.
Second, we will look at quantum spin liquids and the Kitaev honeycomb 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" transition is still poorly understood <span class="citation"
data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a
href="#ref-belitzAndersonMottTransition1994" href="#ref-belitzAndersonMottTransition1994"
role="doc-biblioref">31</a>,<a role="doc-biblioref">33</a>,<a
href="#ref-baskoMetalInsulatorTransition2006" 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 bed to explore interaction driven MI transition physics. Despite its
simplicity, the model has a rich phase diagram in <span simplicity, the model has a rich phase diagram in <span
class="math inline">\(D \geq 2\)</span> dimensions. It shows an Mott 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" corresponding Hubbard Model <span class="citation"
data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
href="#ref-brandtThermodynamicsCorrelationFunctions1989" 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 phenomenology as a function of filling can be rich <span
class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a
href="#ref-gruberGroundStatesSpinless1990" 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" <span class="math inline">\(T &gt; 0\)</span> <span class="citation"
data-cites="kennedyItinerantElectronModel1986"> [<a data-cites="kennedyItinerantElectronModel1986"> [<a
href="#ref-kennedyItinerantElectronModel1986" 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 for many-body methods, interest took off when an exact dynamical
mean-field theory solution in the infinite dimensional case was mean-field theory solution in the infinite dimensional case was
found <span class="citation" found <span class="citation"
data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a
href="#ref-antipovCriticalExponentsStrongly2014" href="#ref-antipovCriticalExponentsStrongly2014"
role="doc-biblioref">36</a>–<a role="doc-biblioref">38</a>–<a
href="#ref-herrmannNonequilibriumDynamicalCluster2016" 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 <p>In Chapter 3 I will introduce a generalized FK model in one
dimension. With the addition of long-range interactions in the dimension. With the addition of long-range interactions in the
background field, the model shows a similarly rich phase diagram. I use 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 Anderson model of uncorrelated binary disorder about a background charge
density wave field which confirms that the fermionic sector only fully density wave field which confirms that the fermionic sector only fully
localizes for very large system sizes.</p> 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 <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 question of the magnetic ordering of local moments in the Mott
insulating state. The local moments may form an AFM ground state. insulating state. The local moments may form an AFM ground state.
Alternatively they may fail to order even at zero temperature <span Alternatively they may fail to order even at zero temperature <span
class="citation" class="citation"
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a 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" 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> quantum spin liquid (QSL) state.</p>
<p>Landau-Ginzburg-Wilson theory characterises phases of matter as <p>Landau-Ginzburg-Wilson theory characterises phases of matter as
inextricably linked to the emergence of long range order via a 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 long range entanglement and non-trivial topological properties <span
class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a
href="#ref-broholmQuantumSpinLiquids2020" 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 analogous phase of matter for spin systems. Remarkably the existence of
QSLs was first suggested by Anderson in 1973 <span class="citation" QSLs was first suggested by Anderson in 1973 <span class="citation"
data-cites="andersonResonatingValenceBonds1973"> [<a data-cites="andersonResonatingValenceBonds1973"> [<a
href="#ref-andersonResonatingValenceBonds1973" 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"> <div id="fig:correlation_spin_orbit_PT" class="fignos">
<figure> <figure>
<img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png" <img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png"
data-short-caption="Phase Diagram" style="width:100.0%" 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 <figcaption aria-hidden="true"><span>Figure 3:</span> From <span
class="citation" data-cites="TrebstPhysRep2022"> [<a class="citation" data-cites="TrebstPhysRep2022"> [<a
href="#ref-TrebstPhysRep2022" href="#ref-TrebstPhysRep2022"
role="doc-biblioref">42</a>]</span>.</figcaption> role="doc-biblioref">44</a>]</span>.</figcaption>
</figure> </figure>
</div> </div>
<p>The main route to QSLs, though there are others <span <p>The main route to QSLs, though there are others <span
class="citation" class="citation"
data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a 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" 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 that would otherwise order have AFM order. This frustration can come
geometrically, triangular lattices for instance cannot support AFM geometrically, triangular lattices for instance cannot support AFM
order. It can also come about as a result of spin-orbit coupling.</p> 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 strongly anisotropic spin-spin couplings known as Kitaev Materials <span
class="citation" class="citation"
data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a
href="#ref-TrebstPhysRep2022" role="doc-biblioref">42</a>,<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>,<a
href="#ref-Jackeli2009" role="doc-biblioref">46</a>–<a href="#ref-Jackeli2009" role="doc-biblioref">48</a>–<a
href="#ref-Takagi2019" role="doc-biblioref">49</a>]</span>. Kitaev href="#ref-Takagi2019" role="doc-biblioref">51</a>]</span>. Kitaev
materials draw their name from the celebrated Kitaev Honeycomb Model as 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 it is believed they will realise the QSL state via the mechanisms of the
Kitaev Model.</p> Kitaev Model.</p>
<p>The Kitaev Honeycomb model <span class="citation" <p>The Kitaev Honeycomb model <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006" 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 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 exactly solvable model whose ground state is a QSL characterized by a
static <span class="math inline">\(\mathbb Z_2\)</span> gauge field and 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" Abelian and non-Abelian phases <span class="citation"
data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a
href="#ref-knolleDynamicsFractionalizationQuantum2015" 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" transition to a thermal metal state <span class="citation"
data-cites="selfThermallyInducedMetallic2019"> [<a data-cites="selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019" 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 non-Abelian excitations could be used to support robust topological
quantum computing [<span class="citation" quantum computing [<span class="citation"
data-cites="kitaev_fault-tolerant_2003"> [<a data-cites="kitaev_fault-tolerant_2003"> [<a
href="#ref-kitaev_fault-tolerant_2003" 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 data-cites="freedmanTopologicalQuantumComputation2003"> [<a
href="#ref-freedmanTopologicalQuantumComputation2003" href="#ref-freedmanTopologicalQuantumComputation2003"
role="doc-biblioref">54</a>]</span>; role="doc-biblioref">56</a>]</span>;
nayakNonAbelianAnyonsTopological2008].</p> nayakNonAbelianAnyonsTopological2008].</p>
<p>It is by now understood that the Kitaev model on any tri-coordinated <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 <span class="math inline">\(z=3\)</span> graph has conserved plaquette
operators and local symmetries <span class="citation" operators and local symmetries <span class="citation"
data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007" data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007"
role="doc-biblioref">55</a>,<a href="#ref-Baskaran2008" role="doc-biblioref">57</a>,<a href="#ref-Baskaran2008"
role="doc-biblioref">56</a>]</span> which allow a mapping onto effective role="doc-biblioref">58</a>]</span> which allow a mapping onto effective
free Majorana fermion problems in a background of static <span free Majorana fermion problems in a background of static <span
class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation" class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation"
data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a
href="#ref-Nussinov2009" role="doc-biblioref">57</a>–<a href="#ref-Nussinov2009" role="doc-biblioref">59</a>–<a
href="#ref-hermanns2015weyl" role="doc-biblioref">60</a>]</span>. href="#ref-hermanns2015weyl" role="doc-biblioref">62</a>]</span>.
However, depending on lattice symmetries, finding the ground state flux However, depending on lattice symmetries, finding the ground state flux
sector and understanding the QSL properties can still be sector and understanding the QSL properties can still be
challenging <span class="citation" challenging <span class="citation"
data-cites="eschmann2019thermodynamics Peri2020"> [<a data-cites="eschmann2019thermodynamics Peri2020"> [<a
href="#ref-eschmann2019thermodynamics" role="doc-biblioref">61</a>,<a href="#ref-eschmann2019thermodynamics" role="doc-biblioref">63</a>,<a
href="#ref-Peri2020" role="doc-biblioref">62</a>]</span>.</p> href="#ref-Peri2020" role="doc-biblioref">64</a>]</span>.</p>
<p><strong>paragraph about amorphous lattices</strong></p> <p><strong>paragraph about amorphous lattices</strong></p>
<p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin <p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin
liquid by extending the Kitaev honeycomb model to random lattices with 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 I show that the system undergoes a finite-temperature phase transition
to a conducting thermal metal state and discuss possible experimental to a conducting thermal metal state and discuss possible experimental
realisations.</p> 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 <p>The next chapter, Chapter 2, will introduce some necessary background
to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and
localisation.</p> localisation.</p>
<p>In Chapter 3 I introduce the Long Range Falikov-Kimball Model in <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 greater detail. I will present results that. Chapter 4 focusses on the
Amorphous Kitaev Model.</p> Amorphous Kitaev Model.</p>
<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="refs" class="references csl-bib-body" role="doc-bibliography">
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@ -735,9 +582,25 @@ role="doc-biblioentry">
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<div class="csl-left-margin">[13] </div><div class="csl-right-inline">J. <div class="csl-left-margin">[15] </div><div class="csl-right-inline">J.
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<div class="csl-left-margin">[15] </div><div class="csl-right-inline">N. <div class="csl-left-margin">[17] </div><div class="csl-right-inline">N.
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Transition Metals</a></em>, Proc. Phys. Soc. A <strong>62</strong>, 416 Transition Metals</a></em>, Proc. Phys. Soc. A <strong>62</strong>, 416
@ -761,7 +624,7 @@ Transition Metals</a></em>, Proc. Phys. Soc. A <strong>62</strong>, 416
</div> </div>
<div id="ref-fisherMottInsulatorsSpin1999" class="csl-entry" <div id="ref-fisherMottInsulatorsSpin1999" class="csl-entry"
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<div class="csl-left-margin">[16] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[18] </div><div class="csl-right-inline">M.
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@ -772,16 +635,15 @@ edited by A. Comtet, T. Jolicœur, S. Ouvry, and F. David, Vol. 69
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</main> </main>
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<ul> <ul>
<li><a href="#the-falikov-kimball-model" <li><a href="#the-falikov-kimball-model"
id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a> id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
<ul> <ul>
<li><a href="#the-model" id="toc-the-model">The Model</a> <li><a href="#the-model" id="toc-the-model">The Model</a></li>
<ul> <li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
<li><a href="#particle-hole-symmetry" Diagrams</a></li>
id="toc-particle-hole-symmetry">Particle Hole Symmetry</a></li> <li><a href="#long-ranged-ising-model"
</ul></li> id="toc-long-ranged-ising-model">Long Ranged Ising model</a></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>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav>
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<main> <main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#the-falikov-kimball-model" <li><a href="#the-falikov-kimball-model"
id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a> id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
<ul> <ul>
<li><a href="#the-model" id="toc-the-model">The Model</a> <li><a href="#the-model" id="toc-the-model">The Model</a></li>
<ul> <li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
<li><a href="#particle-hole-symmetry" Diagrams</a></li>
id="toc-particle-hole-symmetry">Particle Hole Symmetry</a></li> <li><a href="#long-ranged-ising-model"
</ul></li> id="toc-long-ranged-ising-model">Long Ranged Ising model</a></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>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
<h1 id="the-falikov-kimball-model">The Falikov Kimball Model</h1>
<h2 id="the-model">The Model</h2> <!-- Main Page Body -->
<p>discuss CDW phase of 2d model as motivation for studying 1d phase <section id="the-falikov-kimball-model" class="level1">
with long range forces</p> <h1>The Falikov Kimball Model</h1>
<h3 id="particle-hole-symmetry">Particle Hole Symmetry</h3> <section id="the-model" class="level2">
<h2 id="phase-diagram">Phase Diagram</h2> <h2>The Model</h2>
<h2 id="long-range-ising-models">Long Range Ising Models</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>
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class="sourceCode python"><code class="sourceCode python"></code></pre></div> class="sourceCode python"><code class="sourceCode python"></code></pre></div>
</section>
</section>
<section id="bibliography" class="level1 unnumbered">
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@ -266,7 +25,7 @@ image:
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<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#the-kitaev-honeycomb-model" <li><a href="#the-kitaev-honeycomb-model"
id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a> id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
@ -284,12 +43,15 @@ Diagram</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
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<main> <main>
<!-- Table of Contents -->
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<ul> <ul>
<li><a href="#the-kitaev-honeycomb-model" <li><a href="#the-kitaev-honeycomb-model"
@ -310,7 +72,10 @@ Diagram</a></li>
</ul> </ul>
</nav> </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 <p><strong>papers</strong> Jos on dynamics
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.115127</p> https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.115127</p>
<p><strong>intro</strong> - strong spin orbit coupling leads to <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 <li>experimental probes include inelastic neutron scattering, Raman
scattering</li> scattering</li>
</ul> </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"> <div id="fig:intro_figure_by_hand" class="fignos">
<figure> <figure>
<img <img
@ -356,15 +122,24 @@ role="doc-biblioref">1</a>]</span></li>
<li>has extensively many conserved fluxes</li> <li>has extensively many conserved fluxes</li>
<li></li> <li></li>
</ul> </ul>
<h2 id="a-mapping-to-majorana-fermions">A mapping to Majorana </section>
Fermions</h2> <section id="a-mapping-to-majorana-fermions" class="level2">
<h2 id="gauge-fields">Gauge Fields</h2> <h2>A mapping to Majorana Fermions</h2>
<h2 id="anyons-topology-and-the-chern-number">Anyons, Topology and the </section>
Chern number</h2> <section id="gauge-fields" class="level2">
<h2 id="phase-diagram">Phase Diagram</h2> <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 <div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div> 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="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry" <div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
@ -375,6 +150,9 @@ Effect and Magnetism: Transition Metal Compounds</a></em>, Sov. Phys.
Usp. <strong>25</strong>, 231 (1982).</div> Usp. <strong>25</strong>, 231 (1982).</div>
</div> </div>
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</section>
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<!--Capture the table of contents from pandoc as a jekyll variable --> <!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %} {% capture tableOfContents %}
<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#disorder-localisation" <li><a href="#disorder-and-localisation"
id="toc-disorder-localisation">Disorder &amp; Localisation</a> id="toc-disorder-and-localisation">Disorder and Localisation</a>
<ul> <ul>
<li><a href="#localisation-anderson-many-body-and-disorder-free" <li><a href="#localisation-anderson-many-body-and-disorder-free"
id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: 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> id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
<li><a href="#amorphous-magnetism" <li><a href="#amorphous-magnetism"
id="toc-amorphous-magnetism">Amorphous Magnetism</a></li> id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav>
{% endcapture %} {% endcapture %}
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<li><a href="#disorder-localisation" <li><a href="#disorder-and-localisation"
id="toc-disorder-localisation">Disorder &amp; Localisation</a> id="toc-disorder-and-localisation">Disorder and Localisation</a>
<ul> <ul>
<li><a href="#localisation-anderson-many-body-and-disorder-free" <li><a href="#localisation-anderson-many-body-and-disorder-free"
id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: 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> id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
<li><a href="#amorphous-magnetism" <li><a href="#amorphous-magnetism"
id="toc-amorphous-magnetism">Amorphous Magnetism</a></li> id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
<h1 id="disorder-localisation">Disorder &amp; Localisation</h1>
<h2 id="localisation-anderson-many-body-and-disorder-free">Localisation: <!-- Main Page Body -->
Anderson, Many Body and Disorder-Free</h2> <section id="disorder-and-localisation" class="level1">
<h2 id="disorder-and-spin-liquids">Disorder and Spin liquids</h2> <h1>Disorder and Localisation</h1>
<h2 id="amorphous-magnetism">Amorphous Magnetism</h2> <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 <div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div> 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>
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@ -266,11 +25,12 @@ image:
<!--Capture the table of contents from pandoc as a jekyll variable --> <!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %} {% capture tableOfContents %}
<br> <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> <ul>
<li><a href="#markov-chain-monte-carlo" <li><a href="#markov-chain-monte-carlo"
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a> id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
<ul>
<li><a href="#sampling" id="toc-sampling">Sampling</a></li> <li><a href="#sampling" id="toc-sampling">Sampling</a></li>
<li><a href="#markov-chains" id="toc-markov-chains">Markov <li><a href="#markov-chains" id="toc-markov-chains">Markov
Chains</a></li> Chains</a></li>
@ -288,9 +48,6 @@ id="toc-metropolis-hastings">Metropolis-Hastings</a></li>
<li><a href="#convergence-auto-correlation-and-binning" <li><a href="#convergence-auto-correlation-and-binning"
id="toc-convergence-auto-correlation-and-binning">Convergence, id="toc-convergence-auto-correlation-and-binning">Convergence,
Auto-correlation and Binning</a></li> 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" <li><a href="#proposal-distributions"
id="toc-proposal-distributions">Proposal Distributions</a></li> id="toc-proposal-distributions">Proposal Distributions</a></li>
<li><a href="#perturbation-mcmc" id="toc-perturbation-mcmc">Perturbation <li><a href="#perturbation-mcmc" id="toc-perturbation-mcmc">Perturbation
@ -348,17 +105,21 @@ Trick</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav>
{% endcapture %} {% 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 %} {% include header.html extra=tableOfContents %}
<main> <main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#markov-chain-monte-carlo" <li><a href="#sec:FK-Methods" id="toc-sec:FK-Methods">Methods</a>
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a>
<ul> <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="#sampling" id="toc-sampling">Sampling</a></li>
<li><a href="#markov-chains" id="toc-markov-chains">Markov <li><a href="#markov-chains" id="toc-markov-chains">Markov
Chains</a></li> Chains</a></li>
@ -376,9 +137,6 @@ id="toc-metropolis-hastings">Metropolis-Hastings</a></li>
<li><a href="#convergence-auto-correlation-and-binning" <li><a href="#convergence-auto-correlation-and-binning"
id="toc-convergence-auto-correlation-and-binning">Convergence, id="toc-convergence-auto-correlation-and-binning">Convergence,
Auto-correlation and Binning</a></li> 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" <li><a href="#proposal-distributions"
id="toc-proposal-distributions">Proposal Distributions</a></li> id="toc-proposal-distributions">Proposal Distributions</a></li>
<li><a href="#perturbation-mcmc" id="toc-perturbation-mcmc">Perturbation <li><a href="#perturbation-mcmc" id="toc-perturbation-mcmc">Perturbation
@ -438,8 +196,15 @@ Trick</a></li>
</ul> </ul>
</nav> </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 <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 probability distribution that we want to sample from but there is not
direct sampling way to do so.</p> 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 fitting statistical models using Bayesian inference <span
class="citation" data-cites="BMCP2021"> [<a href="#ref-BMCP2021" class="citation" data-cites="BMCP2021"> [<a href="#ref-BMCP2021"
role="doc-biblioref">2</a>]</span>.</p> 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 <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 in the requirement that the samples be uncorrelated then we can use MCMC
instead.</p> instead.</p>
@ -501,7 +268,9 @@ are many solutions <span class="citation"
data-cites="kellyReversibilityStochasticNetworks1981"> [<a data-cites="kellyReversibilityStochasticNetworks1981"> [<a
href="#ref-kellyReversibilityStochasticNetworks1981" href="#ref-kellyReversibilityStochasticNetworks1981"
role="doc-biblioref">6</a>]</span>.</p> 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, <p>We will work in the grand canonical ensemble of fixed temperature,
chemical potential and volume.</p> chemical potential and volume.</p>
<p>Since the spin configurations are classical, the Hamiltonian can be <p>Since the spin configurations are classical, the Hamiltonian can be
@ -533,7 +302,8 @@ to the quantum subsystem. <span class="math display">\[\begin{aligned}
\mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta \mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta
F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]} F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}
\end{aligned}\]</span></p> \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 <p>Markov Chain Monte Carlo (MCMC) is a technique for evaluating thermal
expectation values <span class="math inline">\(\expval{O}\)</span> with expectation values <span class="math inline">\(\expval{O}\)</span> with
respect to some physical system defined by a set of states <span 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 have if the estimates were uncorrelated. Methods of estimating the true
variance of <span class="math inline">\(\expval{O}\)</span> and decided variance of <span class="math inline">\(\expval{O}\)</span> and decided
how many steps are needed will be considered later.</p> how many steps are needed will be considered later.</p>
<h2 id="the-metropolis-hasting-algorithm">The Metropolis-Hasting </section>
Algorithm</h2> </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) $ <p>Markov chains are defined by a transition function $(x_{i+1} x_i) $
giving the probability that a chain in state <span giving the probability that a chain in state <span
class="math inline">\(x_i\)</span> at time <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 with eigenvalue 1 and all other eigenvalues with magnitude less than
one. The convergence time depends on the magnitude of the second largest one. The convergence time depends on the magnitude of the second largest
eigenvalue.</p> 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 <p>In order to actually choose new states according to <span
class="math inline">\(\mathcal{T}\)</span> one chooses states from a class="math inline">\(\mathcal{T}\)</span> one chooses states from a
proposal distribution <span class="math inline">\(q(x_i \to 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 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 one, the rate of rejections can be reduced and the algorithm sped
up.</p> up.</p>
<h2 id="convergence-auto-correlation-and-binning">Convergence, </section>
Auto-correlation and Binning</h2> <section id="convergence-auto-correlation-and-binning" class="level2">
<h2>Convergence, Auto-correlation and Binning</h2>
<p>%Thinning, burn in, multiple runs</p> <p>%Thinning, burn in, multiple runs</p>
<h2 id="applying-mcmc-to-the-fk-model">Applying MCMC to the FK </section>
model</h2> <section id="proposal-distributions" class="level2">
<p>MCMC can be applied to sample over the classical degrees of freedom <h2>Proposal Distributions</h2>
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>
<p>In a MCMC method a key property is the proportion of the time that <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 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 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 occasionally proposing a flip of the entire state. This works because
near half-filling, flipping the occupations of all the sites will near half-filling, flipping the occupations of all the sites will
produce a state at or near the energy of the current one.</p> 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 <p>The matrix diagonalisation is the most computationally expensive step
of the process, a speed up can be obtained by modifying the proposal 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 distribution to depend on the classical part of the energy, a trick
@ -796,7 +502,9 @@ data-cites="huangAcceleratedMonteCarlo2017"> [<a
href="#ref-huangAcceleratedMonteCarlo2017" href="#ref-huangAcceleratedMonteCarlo2017"
role="doc-biblioref">10</a>]</span> does this with restricted Boltzmann role="doc-biblioref">10</a>]</span> does this with restricted Boltzmann
machines whose form is very similar to a classical spin model.</p> 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 <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 finite size of the system we redefine and normalise the coupling matrix
to have 0 derivative at its furthest extent rather than cutting off 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, \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 the overall potential felt by each site due to the rest of the system is
independent of system size.</p> 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 = <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 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 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-binderFiniteSizeScaling1981" role="doc-biblioref">11</a>,<a
href="#ref-musialMonteCarloSimulations2002" href="#ref-musialMonteCarloSimulations2002"
role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a>]</span>.</p> role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a>]</span>.</p>
<h2 id="markov-chain-monte-carlo-in-practice">Markov Chain Monte-Carlo </section>
in Practice</h2> <section id="markov-chain-monte-carlo-in-practice" class="level2">
<h3 id="quick-intro-to-mcmc">Quick Intro to MCMC</h3> <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 <p>The main paper relies on extensively to evaluate thermal expectation
values within the model by walking over states of the classical spin values within the model by walking over states of the classical spin
system <span class="math inline">\(S_i\)</span>. For a classical system, 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; approaches the thermal distribution <span class="math inline">\(P(\s;
\beta) = \mathcal{Z}^{-1} e^{-\beta F(\s)}\)</span>. This turns out to \beta) = \mathcal{Z}^{-1} e^{-\beta F(\s)}\)</span>. This turns out to
be quite easy to achieve using the Metropolis-Hasting algorithm.</p> 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 <p>Considering <span class="math inline">\(p(\s)\)</span> as a vector
<span class="math inline">\(\vec{p}\)</span> whose jth entry is the <span class="math inline">\(\vec{p}\)</span> whose jth entry is the
probability of the jth state <span class="math inline">\(p_j = 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 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 reduce the dependence on the initial conditions and be close enough to
the target distribution.</p> 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"> <div id="fig:m_autocorr" class="fignos">
<figure> <figure>
<img src="../figure_code/fk_chapter/lsr/figs/m_autocorr.png" <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) order to explain how, we need to introduce the Metropolis-Hasting (MH)
algorithm and how it gives an explicit form for the transition algorithm and how it gives an explicit form for the transition
function.</p> function.</p>
<h3 id="the-metropolis-hastings-algorithm">The Metropolis-Hastings </section>
Algorithm</h3> <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 <p>MH breaks up the transition function into a proposal distribution
<span class="math inline">\(q(\s \to \s&#39;)\)</span> and an acceptance <span class="math inline">\(q(\s \to \s&#39;)\)</span> and an acceptance
function <span class="math inline">\(\mathcal{A}(\s \to 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 <p>This has the effect of always accepting proposed states that are
lower in energy and sometimes accepting those that are higher in energy lower in energy and sometimes accepting those that are higher in energy
than the current state.</p> 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 <p>Here, we incorporate a modification to the standard
Metropolis-Hastings algorithm <span class="citation" Metropolis-Hastings algorithm <span class="citation"
data-cites="hastingsMonteCarloSampling1970"> [<a 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 adjusted to match the target distribution. Our proposed method is
considerably simpler and does not require training while still reaping considerably simpler and does not require training while still reaping
some of the benefits of reduced computation.</p> some of the benefits of reduced computation.</p>
<h2 id="detailed-balance-for-the-two-step-method">Detailed Balance for </section>
the two step method</h2> <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 <p>Given a MCMC algorithm with target distribution <span
class="math inline">\(\pi(a)\)</span> and transition function <span class="math inline">\(\pi(a)\)</span> and transition function <span
class="math inline">\(\mathcal{T}\)</span> the detailed balance 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 <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 <span class="math inline">\(\min(1,r)/\min(1,1/r) = r\)</span> for <span
class="math inline">\(r &gt; 0\)</span>.</p> 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 <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 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 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="cb4-13"><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="cb4-14"><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> <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 </section>
distribution</h3> <section id="tuning-the-proposal-distribution" class="level3">
<h3>Tuning the proposal distribution</h3>
<div id="fig:autocorr_multiple_proposals" class="fignos"> <div id="fig:autocorr_multiple_proposals" class="fignos">
<figure> <figure>
<img <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 flipping more spins is warranted. Tuning the proposal distribution
automatically seems like something that would not yield enough benefit automatically seems like something that would not yield enough benefit
for the additional complexity it would require.</p> for the additional complexity it would require.</p>
<h2 id="diagnostics-of-localisation">Diagnostics of Localisation</h2> </section>
<h3 id="inverse-participation-ratio">Inverse Participation Ratio</h3> </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 <p>The inverse participation ratio is defined for a normalised wave
function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i 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" \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> class="math inline">\(N\)</span> distinct values.</p>
<p>Detailed and Global balance equation Mixing times Cluster updates and <p>Detailed and Global balance equation Mixing times Cluster updates and
Critical slowing down Effective Sample Size</p> 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 <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 the fact that statistical physics can be somewhat boring in one
dimension where most simple models have no phase transitions. This 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) = 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 \Z^{-1} e^{-\beta F(\s)}\)</span>. This turns out to be quite easy to
achieve using the Metropolis-Hasting algorithm.</p> 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 <p>Considering <span class="math inline">\(p(\s)\)</span> as a vector
<span class="math inline">\(\vec{p}\)</span> whose jth entry is the <span class="math inline">\(\vec{p}\)</span> whose jth entry is the
probability of the jth state <span class="math inline">\(p_j = 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 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 reduce the dependence on the initial conditions and be close enough to
the target distribution.</p> 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"> <div id="fig:m_autocorr" class="fignos">
<figure> <figure>
<img src="figs/lsr/m_autocorr.png" <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) order to explain how, we need to introduce the Metropolis-Hasting (MH)
algorithm and how it gives an explicit form for the transition algorithm and how it gives an explicit form for the transition
function.</p> function.</p>
<h2 id="the-metropolis-hastings-algorithm-1">The Metropolis-Hastings </section>
Algorithm</h2> <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 <p>MH breaks up the transition function into a proposal distribution
<span class="math inline">\(q(\s \to \s&#39;)\)</span> and an acceptance <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 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 <p>This has the effect of always accepting proposed states that are
lower in energy and sometimes accepting those that are higher in energy lower in energy and sometimes accepting those that are higher in energy
than the current state.</p> than the current state.</p>
<h2 id="choosing-the-proposal-distribution">Choosing the proposal </section>
distribution</h2> <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:" <p><img src="figs/lsr/autocorr_multiple_proposals.png" title="fig:"
id="fig:comparison" id="fig:comparison"
alt="t = 1, \alpha = 1.25, J = U = 5 [fig:comparison]" /> Simulations 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 Tuning the proposal distribution automatically seems like something that
would not yield enough benefit for the additional complexity it would would not yield enough benefit for the additional complexity it would
require.</p> 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 <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 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 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="cb6-12"><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="cb6-13"><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> <span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a> </span></code></pre></div>
<div class="sourceCode" id="cb7"><pre </section>
class="sourceCode python"><code class="sourceCode python"></code></pre></div> </section>
<p></ij></ij></p> <section id="bibliography" class="level1 unnumbered">
<h1 class="unnumbered" id="bibliography">Bibliography</h1> <h1 class="unnumbered">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-devroyeRandomSampling1986" class="csl-entry" <div id="ref-devroyeRandomSampling1986" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
@ -1745,6 +1483,7 @@ Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492
(1958).</div> (1958).</div>
</div> </div>
</div> </div>
</section>
<section class="footnotes footnotes-end-of-document" <section class="footnotes footnotes-end-of-document"
role="doc-endnotes"> role="doc-endnotes">
<hr /> <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> class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol> </ol>
</section> </section>
</main> </main>
</body> </body>
</html> </html>

337
_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html
View File

@ -12,252 +12,11 @@ image:
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@ -266,44 +25,49 @@ image:
<!--Capture the table of contents from pandoc as a jekyll variable --> <!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %} {% capture tableOfContents %}
<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <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> Diagram</a></li>
<li><a href="#localisation-properties" <li><a href="#localisation-properties"
id="toc-localisation-properties">Localisation Properties</a></li> id="toc-localisation-properties">Localisation Properties</a></li>
<li><a href="#discussion-conclusion" </ul></li>
id="toc-discussion-conclusion">Discussion &amp; Conclusion</a></li> <li><a href="#discussion-and-conclusion-secamk-conclusion"
<li><a href="#acknowledgments" id="toc-discussion-and-conclusion-secamk-conclusion">Discussion and
id="toc-acknowledgments">Acknowledgments</a></li> Conclusion {sec:AMK-Conclusion}</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>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav>
{% endcapture %} {% 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 %} {% include header.html extra=tableOfContents %}
<main> <main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <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> Diagram</a></li>
<li><a href="#localisation-properties" <li><a href="#localisation-properties"
id="toc-localisation-properties">Localisation Properties</a></li> id="toc-localisation-properties">Localisation Properties</a></li>
<li><a href="#discussion-conclusion" </ul></li>
id="toc-discussion-conclusion">Discussion &amp; Conclusion</a></li> <li><a href="#discussion-and-conclusion-secamk-conclusion"
<li><a href="#acknowledgments" id="toc-discussion-and-conclusion-secamk-conclusion">Discussion and
id="toc-acknowledgments">Acknowledgments</a></li> Conclusion {sec:AMK-Conclusion}</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>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
<!-- Main Page Body -->
<section id="sec:FK-results" class="level1">
<h1>Results</h1>
<div id="fig:phase_diagram" class="fignos"> <div id="fig:phase_diagram" class="fignos">
<figure> <figure>
<img src="pdf_figs/phase_diagram.svg" <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> 1.25\)</span> except where explicitly varied.</figcaption>
</figure> </figure>
</div> </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" <p>Figs. [<a href="#fig:phase_diagram" data-reference-type="ref"
data-reference="fig:phase_diagram">1</a>a] and [<a data-reference="fig:phase_diagram">1</a>a] and [<a
href="#fig:phase_diagram" data-reference-type="ref" 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" href="#fig:band_opening" data-reference-type="ref"
data-reference="fig:band_opening">3</a>a]. The Anderson phase is gapless data-reference="fig:band_opening">3</a>a]. The Anderson phase is gapless
but, as we explain below, shows localised fermionic eigenstates.</p> 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 <p>The MCMC formulation suggests viewing the spin configurations as a
form of annealed binary disorder whose probability distribution is given form of annealed binary disorder whose probability distribution is given
by the Boltzmann weight <span class="math inline">\(e^{-\beta by the Boltzmann weight <span class="math inline">\(e^{-\beta
@ -453,8 +229,8 @@ temperatures?</p>
<div id="fig:indiv_IPR" class="fignos"> <div id="fig:indiv_IPR" class="fignos">
<figure> <figure>
<img src="pdf_figs/indiv_IPR.svg" <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." /> 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 2:</span> Energy resolved <figcaption aria-hidden="true"><span>Figure 3:</span> Energy resolved
DOS(<span class="math inline">\(\omega\)</span>) and <span DOS(<span class="math inline">\(\omega\)</span>) and <span
class="math inline">\(\tau\)</span> (the scaling exponent of IPR(<span class="math inline">\(\tau\)</span> (the scaling exponent of IPR(<span
class="math inline">\(\omega\)</span>) against system size <span class="math inline">\(\omega\)</span>) against system size <span
@ -485,8 +261,8 @@ states.</figcaption>
<div id="fig:band_opening" class="fignos"> <div id="fig:band_opening" class="fignos">
<figure> <figure>
<img src="pdf_figs/gap_openingboth.svg" <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" /> 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 3:</span> The DOS (a and c) <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) 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 as a function of energy and temperature. (a) and (b) show the system
transitioning from the CDW phase to the gapless Anderson insulating one 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"> <div id="fig:indiv_IPR_disorder" class="fignos">
<figure> <figure>
<img src="pdf_figs/indiv_IPR_disorder.svg" <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" /> 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 4:</span> A comparison of <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 the full FK model to a simple binary disorder model (DM) with a CDW wave
background perturbed by uncorrelated defects at density <span background perturbed by uncorrelated defects at density <span
class="math inline">\(0 &lt; \rho &lt; 1\)</span> matched to the largest 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 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 model is that we can explore very large system sizes for a complete
understanding.</p> 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 <p>The FK model is one of the simplest non-trivial models of interacting
fermions. We studied its thermodynamic and localisation properties fermions. We studied its thermodynamic and localisation properties
brought down in dimensionality to 1D by adding a novel long-ranged 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 should be realizable in systems with long-range interactions, such as
trapped ion quantum simulators, where one can also explore the fully trapped ion quantum simulators, where one can also explore the fully
interacting regime with a dynamical background field.</p> interacting regime with a dynamical background field.</p>
<h1 id="acknowledgments">Acknowledgments</h1> <p><strong>UNCORRELATED DISORDER MODEL</strong></p>
<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>The disorder model referred to in the main text is defined by <p>The disorder model referred to in the main text is defined by
replacing the spin degree of freedom in the FK model <span replacing the spin degree of freedom in the FK model <span
class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder 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> \nonumber\end{aligned}\]</span></p>
<div class="sourceCode" id="cb1"><pre <div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div> 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="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" <div id="ref-binderFiniteSizeScaling1981" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
@ -790,6 +562,9 @@ Model</a></em>, J. Phys. A: Math. Gen. <strong>30</strong>, L711
(1997).</div> (1997).</div>
</div> </div>
</div> </div>
</section>
</main> </main>
</body> </body>
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@ -204,7 +26,7 @@ image:
<!--Capture the table of contents from pandoc as a jekyll variable --> <!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %} {% capture tableOfContents %}
<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a> <li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a>
<ul> <ul>
@ -253,12 +75,15 @@ Statistics</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav>
{% endcapture %} {% endcapture %}
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<main> <main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a> <li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a>
@ -310,7 +135,10 @@ Statistics</a></li>
</ul> </ul>
</nav> </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 <p>The bond operators <span class="math inline">\(u_{ij}\)</span> are
useful because they label a bond sector <span useful because they label a bond sector <span
class="math inline">\(\mathcal{\tilde{L}}_u\)</span> in which we can 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> earlier observation that each <span class="math inline">\(W_p\)</span>
takes values <span class="math inline">\(\pm 1\)</span> for even paths takes values <span class="math inline">\(\pm 1\)</span> for even paths
and <span class="math inline">\(\pm i\)</span> for odd paths.</p> 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"> <div id="fig:types_of_dual_loops_animated" class="fignos">
<figure> <figure>
<img <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 sector unchanged. We can also do the same thing but move the
vortex around one the non-contractible loops of the lattice (fig. <a vortex around one the non-contractible loops of the lattice (fig. <a
href="#fig:types_of_dual_loops_animated">1</a> (d)).</p> href="#fig:types_of_dual_loops_animated">1</a> (d)).</p>
<h3 id="dual-loops-and-gauge-symmetries">Dual Loops and gauge </section>
symmetries</h3> <section id="dual-loops-and-gauge-symmetries" class="level3">
<h3>Dual Loops and gauge symmetries</h3>
<div id="fig:gauge_symmetries" class="fignos"> <div id="fig:gauge_symmetries" class="fignos">
<figure> <figure>
<img <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 <p><strong>The plaquette operators and topological fluxes are the gauge
invariant quantities which determine the physics of the invariant quantities which determine the physics of the
model</strong></p> 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"> <div id="fig:plaquette_addition_by_hand" class="fignos">
<figure> <figure>
<img <img
@ -475,8 +307,9 @@ discrete version of Stoke’s theorem.</figcaption>
</div> </div>
<p>Takeaway: Wilson loops can always be decomposed into products of <p>Takeaway: Wilson loops can always be decomposed into products of
plaquettes operators unless they are non-contractable.</p> plaquettes operators unless they are non-contractable.</p>
<h3 id="gauge-degeneracy-and-the-euler-equation">Gauge Degeneracy and </section>
the Euler Equation</h3> <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"> <div id="fig:state_decomposition_animated" class="fignos">
<figure> <figure>
<img <img
@ -626,8 +459,9 @@ chosen from a tree since loops can be removed by a gauge
transformation.</figcaption> transformation.</figcaption>
</figure> </figure>
</div> </div>
<h3 id="counting-edges-plaquettes-and-vertices">Counting edges, </section>
plaquettes and vertices</h3> <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 <p>It is useful to know how the trivalent structure of the lattice
constrains the number of bonds <span class="math inline">\(B\)</span>, constrains the number of bonds <span class="math inline">\(B\)</span>,
plaquettes <span class="math inline">\(P\)</span> and vertices <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> lattice.</figcaption>
</figure> </figure>
</div> </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 <p>The projection from the extended space to the physical space will not
be particularly important for the results presented here. However, the be particularly important for the results presented here. However, the
theory remains useful to explain why this is.</p> 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 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 with <span class="math inline">\(n_i = 0\)</span> or a state with a
single fermion in the lowest level.</p> 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"> <div id="fig:loops_and_dual_loops" class="fignos">
<figure> <figure>
<img <img
@ -848,7 +686,9 @@ degeneracy in the Abelian phase and a threefold degeneracy in the
non-Abelian phase.</figcaption> non-Abelian phase.</figcaption>
</figure> </figure>
</div> </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 <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. Hamiltonian to a Majorana Hamiltonian in an extended Hilbert space.
Along with that mapping comes a gauge field <span 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 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 fields. However, this was not necessary for any of the results that will
be presented here.</p> 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, <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 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 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 href="#ref-lieb_flux_1994" role="doc-biblioref">5</a>]</span> and is
supported by numerical evidence. As noted before, any flux that differs supported by numerical evidence. As noted before, any flux that differs
from the ground state is an excitation which we call a vortex.</p> 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 <p>This guess only works for larger lattices. To rigorously test it, we
would want to directly enumerate the <span would want to directly enumerate the <span
class="math inline">\(2^N\)</span> vortex sectors for a smaller lattice 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 state for hundreds of thousands of lattices with up to twenty
plaquettes. For larger lattices, we verified that random perturbations plaquettes. For larger lattices, we verified that random perturbations
around the predicted ground state never yield a lower energy state.</p> 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 <p>The discussion above shows that the ground state has a twofold
<strong>chiral</strong> degeneracy which arises because the global sign <strong>chiral</strong> degeneracy which arises because the global sign
of the odd plaquettes does not matter.</p> 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 data-cites="yaoExactChiralSpin2007"> [<a
href="#ref-yaoExactChiralSpin2007" href="#ref-yaoExactChiralSpin2007"
role="doc-biblioref">7</a>]</span>.</p> 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 <p>discuss the Abelian A phase / toric code phase / anisotropic
phase</p> phase</p>
<p>the isotropic gapless phase of the standard model</p> <p>the isotropic gapless phase of the standard model</p>
<p>The isotropic gapped phase with the addition of a magnetic field</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 </section>
two dimensions?</h2> <section id="what-is-so-great-about-two-dimensions" class="level2">
<h3 id="topology-chirality-and-edge-modes">Topology, chirality and edge <h2>What is so great about two dimensions?</h2>
modes</h3> <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 <p>Most thermodynamic and quantum phases studied can be characterised by
a local order parameter. That is, a function or operator that only a local order parameter. That is, a function or operator that only
requires knowledge about some fixed sized patch of the system that does 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 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 torus. The extension to surfaces like the torus but with more than one
handle is relatively easy.</p> 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 <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 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 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 role="doc-biblioref">17</a>,<a
href="#ref-kitaevFaulttolerantQuantumComputation2003" href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p> 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="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry" <div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry"
role="doc-biblioentry"> 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> Logical Qubits</a></em>, Quantum <strong>5</strong>, 564 (2021).</div>
</div> </div>
</div> </div>
</section>
</main> </main>
</body> </body>
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<title>The Amorphous Kitaev Model - Introduction</title> <title>The Amorphous Kitaev Model - Introduction</title>
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@ -204,11 +26,9 @@ image:
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<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#contributions" <li><a href="#sec:AMK-Model" id="toc-sec:AMK-Model">The Model</a>
id="toc-contributions">Contributions</a></li>
<li><a href="#introduction" id="toc-introduction">Introduction</a>
<ul> <ul>
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous <li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
Systems</a></li> Systems</a></li>
@ -239,17 +59,18 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
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<main> <main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#contributions" <li><a href="#sec:AMK-Model" id="toc-sec:AMK-Model">The Model</a>
id="toc-contributions">Contributions</a></li>
<li><a href="#introduction" id="toc-introduction">Introduction</a>
<ul> <ul>
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous <li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
Systems</a></li> Systems</a></li>
@ -282,7 +103,9 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
</ul> </ul>
</nav> </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>The material in this chapter expands on work presented in</p>
<p><strong>Insert citation of amorphous Kitaev paper here</strong></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 <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 ’whiteboard’ing, this included the phase diagram, edge mode and finite
temperature analyses as well as the derivation of the projector in the temperature analyses as well as the derivation of the projector in the
amorphous case.</p> 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"> <div id="fig:intro_figure_by_hand" class="fignos">
<figure> <figure>
<img <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 freedom. These conserved quantities can be factored out as classical
degrees of freedom, leaving behind a non-interacting quantum model that degrees of freedom, leaving behind a non-interacting quantum model that
is easy to solve.</p> 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 <p><strong>Insert discussion of why a generalisation to the amorphous
case is interesting</strong></p> case is interesting</strong></p>
<p>This chapter details the physics of the Kitaev model on amorphous <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 and the motivations for doing so. It also discusses how a well known
quantum error correcting code defined on the Kitaev Honeycomb model quantum error correcting code defined on the Kitaev Honeycomb model
could be generalised to the amorphous case.</p> could be generalised to the amorphous case.</p>
<h2 id="glossary">Glossary</h2> </section>
<section id="glossary" class="level2">
<h2>Glossary</h2>
<ul> <ul>
<li><p>Lattice: The underlying graph on which the models are defined. <li><p>Lattice: The underlying graph on which the models are defined.
Composed of sites (vertices), bonds (edges) and plaquettes 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> 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> <li>The B phase: The roughly isotropic region of the phase diagram.</li>
</ul> </ul>
<h2 id="the-kitaev-model">The Kitaev Model</h2> </section>
<h3 id="commutation-relations">Commutation relations</h3> <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 <p>Before diving into the Hamiltonian of the Kitaev model, the following
describes the key commutation relations of spins, fermions and describes the key commutation relations of spins, fermions and
Majoranas.</p> 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. <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 Scalars like <span class="math inline">\(\delta_{ij}\)</span> should be
understood to be multiplied by an implicit identity <span 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 i \epsilon^{\alpha\beta\gamma}\]</span> and <span
class="math display">\[[\sigma^\alpha \sigma^\beta, \sigma^\gamma] = class="math display">\[[\sigma^\alpha \sigma^\beta, \sigma^\gamma] =
0\]</span></p> 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 <p>The fermionic creation and anhilation operators are defined by the
canonical anticommutation relations <span canonical anticommutation relations <span
class="math display">\[\begin{aligned} 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> introduction to the Kitaev Model.</figcaption>
</figure> </figure>
</div> </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 <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 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 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 Hilbert space into a set of ‘vortex sectors’ labelled by that particular
flux configuration <span class="math inline">\(\phi_i = \pm 1,\pm flux configuration <span class="math inline">\(\phi_i = \pm 1,\pm
i\)</span>.</p> i\)</span>.</p>
<h3 id="from-spins-to-majorana-operators">From Spins to Majorana </section>
operators</h3> <section id="from-spins-to-majorana-operators" class="level3">
<h4 id="for-a-single-spin">For a single spin</h4> <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 <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">\(\sigma^x, \sigma^y\)</span> and <span
class="math inline">\(\sigma^z\)</span> operators which live in a two 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> \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 <p>This makes sense if we promise to confine ourselves to the physical
subspace <span class="math inline">\(D = 1\)</span>.</p> 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. <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,\; 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">\(D_j = b^y_j,\;b^z_j,\; c_j\)</span> and a <span class="math inline">\(D_j =
@ -756,8 +596,11 @@ degree of degeneracy.</p>
<p>In summary, Majorana bond operators <span <p>In summary, Majorana bond operators <span
class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span
class="math inline">\(\mathbb{Z_2}\)</span> gauge field!</p> class="math inline">\(\mathbb{Z_2}\)</span> gauge field!</p>
<h3 id="partitioning-the-hilbert-space-into-bond-sectors">Partitioning </section>
the Hilbert Space into Bond sectors</h3> </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 <p>Similarly to the story with the plaquette operators from the spin
language, we can divide the Hilbert space <span language, we can divide the Hilbert space <span
class="math inline">\(\mathcal{L}\)</span> into sectors labelled by a 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 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 that they are not. However, it will be helpful to first develop the
theory of the Majorana Hamiltonian further.</p> 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">\[ <p>We now have a quadratic Hamiltonian <span class="math display">\[
\tilde{H} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} \tilde{H} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha}
u_{ij} c_i c_j\]</span> in which most of the Majorana degrees of freedom 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> class="math inline">\(\sum_m \epsilon_m\)</span> easily.</p>
<p>Takeaway: the Majorana Hamiltonian is quadratic within a Bond <p>Takeaway: the Majorana Hamiltonian is quadratic within a Bond
Sector.</p> Sector.</p>
<h3 id="mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping <section id="mapping-back-from-bond-sectors-to-the-physical-subspace"
back from Bond Sectors to the Physical Subspace</h3> class="level3">
<h3>Mapping back from Bond Sectors to the Physical Subspace</h3>
<p>At this point, given a particular bond configuration <span <p>At this point, given a particular bond configuration <span
class="math inline">\(u_{ij} = \pm 1\)</span>, we can construct a class="math inline">\(u_{ij} = \pm 1\)</span>, we can construct a
quadratic Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> 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> be the plaquette operators.</p>
<p>Takeaway: The Bond Sectors overlap with the physical subspace but are <p>Takeaway: The Bond Sectors overlap with the physical subspace but are
not contained within it.</p> 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 <p>Care must be taken when defining open boundary conditions. Simply
removing bonds from the lattice leaves behind unpaired <span removing bonds from the lattice leaves behind unpaired <span
class="math inline">\(b^\alpha\)</span> operators that must be paired in 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 anyway, an arbitrary pairing of the unpaired <span
class="math inline">\(b^\alpha\)</span> operators could be performed. class="math inline">\(b^\alpha\)</span> operators could be performed.
&lt;/i,j&gt;&lt;/i,j&gt;</p> &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="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry" <div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry"
role="doc-biblioentry"> 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> of Finite Systems</em> (The MIT Press, 1986).</div>
</div> </div>
</div> </div>
</section>
</main> </main>
</body> </body>
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@ -267,9 +26,9 @@ image:
<!--Capture the table of contents from pandoc as a jekyll variable --> <!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %} {% capture tableOfContents %}
<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#methods" id="toc-methods">Methods</a> <li><a href="#sec:AMK-Methods" id="toc-sec:AMK-Methods">Methods</a>
<ul> <ul>
<li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li> <li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li>
<li><a href="#graph-representation" id="toc-graph-representation">Graph <li><a href="#graph-representation" id="toc-graph-representation">Graph
@ -295,15 +54,18 @@ Markers</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
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{% include header.html extra=tableOfContents %} {% include header.html extra=tableOfContents %}
<main> <main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#methods" id="toc-methods">Methods</a> <li><a href="#sec:AMK-Methods" id="toc-sec:AMK-Methods">Methods</a>
<ul> <ul>
<li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li> <li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li>
<li><a href="#graph-representation" id="toc-graph-representation">Graph <li><a href="#graph-representation" id="toc-graph-representation">Graph
@ -331,9 +93,12 @@ Markers</a></li>
</ul> </ul>
</nav> </nav>
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<div class="sourceCode" id="cb1"><pre <div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div> 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 <p>The practical implementation of what is described in this section is
available as a Python package called Koala (Kitaev On Amorphous available as a Python package called Koala (Kitaev On Amorphous
LAttices) <span class="citation" LAttices) <span class="citation"
@ -341,7 +106,8 @@ data-cites="tomImperialCMTHKoalaFirst2022"> [<a
href="#ref-tomImperialCMTHKoalaFirst2022" href="#ref-tomImperialCMTHKoalaFirst2022"
role="doc-biblioref"><strong>tomImperialCMTHKoalaFirst2022?</strong></a>]</span>. role="doc-biblioref"><strong>tomImperialCMTHKoalaFirst2022?</strong></a>]</span>.
All results and figures were generated with Koala.</p> 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 <p>To study the properties of the amorphous Kitaev model, we need to
sample from the space of possible trivalent graphs.</p> sample from the space of possible trivalent graphs.</p>
<p>A simple method is to use a Voronoi partition of the torus <span <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> edges.</figcaption>
</figure> </figure>
</div> </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 <p>Three keys pieces of information allow us to represent amorphous
lattices.</p> lattices.</p>
<p>Most of the graph connectivity is encoded by an ordered list of edges <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> which bonds cross the cell boundaries.</figcaption>
</figure> </figure>
</div> </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 <p>The Kitaev Model requires that each edge in the lattice be assigned a
label <span class="math inline">\(x\)</span>, <span label <span class="math inline">\(x\)</span>, <span
class="math inline">\(y\)</span> or <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> the leftmost panel are highlighted.</figcaption>
</figure> </figure>
</div> </div>
<h3 id="four-colourings-and-three-colourings">Four-colourings and <section id="four-colourings-and-three-colourings" class="level3">
three-colourings</h3> <h3>Four-colourings and three-colourings</h3>
<p><strong>add diagram of this</strong></p> <p><strong>add diagram of this</strong></p>
<p>A four-face-colouring can be converted into a three-edge-colouring <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 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 them three-edge-colourable. Intuitively, it seems that the kinds of
toroidal graphs that cannot be three-edge-coloured could never be toroidal graphs that cannot be three-edge-coloured could never be
generated by a Voronoi partition with more than a few seed points.</p> generated by a Voronoi partition with more than a few seed points.</p>
<h3 id="finding-lattice-colourings-with-minisat">Finding Lattice </section>
colourings with miniSAT</h3> <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. <p>Some issues are harder in theory than in practice.
Three-edge-colouring cubic toroidal graphs appears to be one of those Three-edge-colouring cubic toroidal graphs appears to be one of those
things.</p> things.</p>
@ -673,8 +444,9 @@ generating a lattice. For larger systems, the time taken to perform the
diagonalisation dominates.</figcaption> diagonalisation dominates.</figcaption>
</figure> </figure>
</div> </div>
<h3 id="does-it-matter-which-colouring-we-choose">Does it matter which </section>
colouring we choose?</h3> <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 = <p>In the isotropic case <span class="math inline">\(J^\alpha =
1\)</span>, it is easy to show that choosing a particular valid 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 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 question whether particular physical properties could arise by
engineering the colouring in this phase though we expect them to exhibit engineering the colouring in this phase though we expect them to exhibit
a self averaging behaviour.</p> a self averaging behaviour.</p>
<h2 id="mapping-between-flux-sectors-and-bond-sectors">Mapping between </section>
flux sectors and bond sectors</h2> </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 <p>Constructing the Majorana representation of the model requires the
particular bond configuration <span class="math inline">\(u_{jk} = \pm particular bond configuration <span class="math inline">\(u_{jk} = \pm
1\)</span>. However, the large number of gauge symmetries of the bond 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> by an odd number of fluxes.</figcaption>
</figure> </figure>
</div> </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 <p>We know that the standard Kitaev model supports both Abelian and
non-Abelian phases. Therefore, how can we assess whether this is also non-Abelian phases. Therefore, how can we assess whether this is also
the case for the amorphous Kitaev model?</p> the case for the amorphous Kitaev model?</p>
@ -759,7 +536,10 @@ system.</p>
<p><strong>Expand on definition here</strong></p> <p><strong>Expand on definition here</strong></p>
<p><strong>Discuss link between Chern number and Anyonic <p><strong>Discuss link between Chern number and Anyonic
Statistics</strong></p> 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="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-mitchellAmorphousTopologicalInsulators2018" <div id="ref-mitchellAmorphousTopologicalInsulators2018"
class="csl-entry" role="doc-biblioentry"> 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> <strong>30</strong>, 417 (1983).</div>
</div> </div>
</div> </div>
</section>
</main> </main>
</body> </body>
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<title>The Amorphous Kitaev Model - Results</title> <title>The Amorphous Kitaev Model - Results</title>
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@ -204,9 +26,9 @@ image:
<!--Capture the table of contents from pandoc as a jekyll variable --> <!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %} {% capture tableOfContents %}
<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#results" id="toc-results">Results</a> <li><a href="#sec:AMK-Results" id="toc-sec:AMK-Results">Results</a>
<ul> <ul>
<li><a href="#the-ground-state-flux-sector" <li><a href="#the-ground-state-flux-sector"
id="toc-the-ground-state-flux-sector">The Ground State Flux 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 id="toc-anderson-transition-to-a-thermal-metal">Anderson Transition to a
Thermal Metal</a></li> Thermal Metal</a></li>
</ul></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="#conclusion" id="toc-conclusion">Conclusion</a></li>
<li><a href="#discussion" id="toc-discussion">Discussion</a> <li><a href="#discussion" id="toc-discussion">Discussion</a></li>
<ul> <li><a href="#outlook" id="toc-outlook">Outlook</a></li>
<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>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav>
{% endcapture %} {% 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 %} {% include header.html extra=tableOfContents %}
<main> <main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#results" id="toc-results">Results</a> <li><a href="#sec:AMK-Results" id="toc-sec:AMK-Results">Results</a>
<ul> <ul>
<li><a href="#the-ground-state-flux-sector" <li><a href="#the-ground-state-flux-sector"
id="toc-the-ground-state-flux-sector">The Ground State Flux 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 id="toc-anderson-transition-to-a-thermal-metal">Anderson Transition to a
Thermal Metal</a></li> Thermal Metal</a></li>
</ul></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="#conclusion" id="toc-conclusion">Conclusion</a></li>
<li><a href="#discussion" id="toc-discussion">Discussion</a> <li><a href="#discussion" id="toc-discussion">Discussion</a></li>
<ul> <li><a href="#outlook" id="toc-outlook">Outlook</a></li>
<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>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </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 <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 ground state flux sector is correct. We will do this by enumerating all
the flux sectors of many separate system realisations. However there are 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 ground state and our prediction was on the order of <span
class="math inline">\(10^{-6} J\)</span>. It is unclear whether this is class="math inline">\(10^{-6} J\)</span>. It is unclear whether this is
a finite size effect or something else.</p> a finite size effect or something else.</p>
<h2 id="spontaneous-chiral-symmetry-breaking">Spontaneous Chiral </section>
Symmetry Breaking</h2> <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 <p>The spin Kitaev Hamiltonian is real and therefore has time reversal
symmetry (TRS). However, the flux <span symmetry (TRS). However, the flux <span
class="math inline">\(\phi_p\)</span> through any plaquette with an odd 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 spontaneously broken symmetry avoids the need to explicitly break TRS
with a magnetic field term as is done in the original honeycomb with a magnetic field term as is done in the original honeycomb
model.</p> 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, <p>As previously discussed, the standard Honeycomb model has a Abelian,
gapped phase in the anisotropic region (the A phase) and is gapless in gapped phase in the anisotropic region (the A phase) and is gapless in
the isotropic region. The introduction of a magnetic field breaks the 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> <strong>citation</strong>.</figcaption>
</figure> </figure>
</div> </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 <p>The two phases of the amorphous model are clearly gapped, though
later I’ll double check this with finite size scaling.</p> later I’ll double check this with finite size scaling.</p>
<p>The next question is: do these phases support excitations with <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> that the isotropic phase is non-Abelian.</figcaption>
</figure> </figure>
</div> </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 <p>Chiral Spin Liquids support topological protected edge modes on open
boundary conditions <span class="citation" boundary conditions <span class="citation"
data-cites="qi_general_2006"> [<a href="#ref-qi_general_2006" 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> states.</figcaption>
</figure> </figure>
</div> </div>
<h2 id="anderson-transition-to-a-thermal-metal">Anderson Transition to a </section>
Thermal Metal</h2> </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 <p>Previous work on the honeycomb model at finite temperature has shown
that the B phase undergoes a thermal transition from a quantum spin that the B phase undergoes a thermal transition from a quantum spin
liquid phase a to a <strong>thermal metal</strong> phase <span 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> \infty\)</span> limit.</figcaption>
</figure> </figure>
</div> </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 <p>In this chapter we have looked at an extension of the Kitaev
honeycomb model to amorphous lattices with coordination number three. We honeycomb model to amorphous lattices with coordination number three. We
discussed a method to construct arbitrary trivalent lattices using 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 <p>Finally we showed evidence that the amorphous system undergoes an
Anderson transition to a thermal metal phase, driven by the Anderson transition to a thermal metal phase, driven by the
proliferation of vortices with increasing temperature.</p> proliferation of vortices with increasing temperature.</p>
<h1 id="discussion">Discussion</h1> </section>
<h2 id="limits-of-the-ground-state-conjecture">Limits of the ground <section id="discussion" class="level2">
state conjecture</h2> <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 <p>We found a small number of lattices for which the ground state
conjecture did not correctly predict the true ground state flux sector. conjecture did not correctly predict the true ground state flux sector.
I see two possibilities for what could cause this.</p> 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 code A phase. It is possible that some property of the particular
colouring chosen is what leads to failure of the ground state conjecture colouring chosen is what leads to failure of the ground state conjecture
here.</p> 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 <p>This exactly solvable chiral QSL provides a first example of a
topological quantum many-body phase in amorphous magnets, which raises a topological quantum many-body phase in amorphous magnets, which raises a
number of questions for future research.</p> number of questions for future research.</p>
<h2 id="experimental-realisations-and-signatures">Experimental <p><strong>Experimental Realisations and Signatures</strong></p>
Realisations and Signatures</h2>
<p>The obvious question is whether amorphous Kitaev materials could be <p>The obvious question is whether amorphous Kitaev materials could be
physically realised.</p> physically realised.</p>
<p>Most crystals can as exists in a metastable amorphous state if they <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 data-cites="misumiQuantumSpinLiquid2020"> [<a
href="#ref-misumiQuantumSpinLiquid2020" href="#ref-misumiQuantumSpinLiquid2020"
role="doc-biblioref">29</a>]</span>.</p> 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>The model presented here could be generalized in several ways.</p>
<p>First, it would be interesting to study the stability of the chiral <p>First, it would be interesting to study the stability of the chiral
amorphous Kitaev QSL with respect to perturbations\ <span 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 quantum many body phases albeit material candidates aplenty. We expect
our exact chiral amorphous spin liquid to find many generalisation to our exact chiral amorphous spin liquid to find many generalisation to
realistic amorphous quantum magnets and beyond.</p> 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="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry" <div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry"> 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> Model</em>, Physical Review B <strong>79</strong>, 134427 (2009).</div>
</div> </div>
</div> </div>
</section>
</main> </main>
</body> </body>
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@ -181,17 +25,20 @@ image:
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{% capture tableOfContents %} {% capture tableOfContents %}
<br> <br>
Contents: <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#discussion" id="toc-discussion">Discussion</a></li> <li><a href="#discussion" id="toc-discussion">Discussion</a></li>
<li><a href="#outlook" id="toc-outlook">Outlook</a></li> <li><a href="#outlook" id="toc-outlook">Outlook</a></li>
</ul> </ul>
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<main> <main>
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<ul> <ul>
<li><a href="#discussion" id="toc-discussion">Discussion</a></li> <li><a href="#discussion" id="toc-discussion">Discussion</a></li>
@ -199,8 +46,16 @@ Contents:
</ul> </ul>
</nav> </nav>
--> -->
<h2 id="discussion">Discussion</h2>
<h2 id="outlook">Outlook</h2> <!-- Main Page Body -->
<section id="discussion" class="level2">
<h2>Discussion</h2>
</section>
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@ -0,0 +1,128 @@
---
title: A.1_Particle_Hole_Symmetry
excerpt:
layout: none
image:
---
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<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>
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</ul>
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<!-- 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>

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<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"
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<li><a href="#applying-mcmc-to-the-fk-model"
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<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>
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<ul> <ul>
<li><a href="./2_Background/2.1_FK_Model.html#the-falikov-kimball-model">The Falikov Kimball Model</a></li> <li><a href="./2_Background/2.1_FK_Model.html#the-falikov-kimball-model">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/2.2_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> </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> <ul>
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#the-model">The Model</a></li> <li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#the-model">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/3.2_LRFK_Methods.html#methods">Methods</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#results">Results</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-and-conclusion">Discussion and Conclusion</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>
</ul> </ul>
<li><a href="./4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html#gauge-fields">Chapter 4: The Amorphous Kitaev Model</a></li> <li><a href="./4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html#gauge-fields">Chapter 4: The Amorphous Kitaev Model</a></li>
<ul> <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#the-model">The Model</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/4.2_AMK_Methods.html#methods">Methods</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#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-and-conclusion">Discussion and 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>
</ul> </ul>
<li><a href="./5_Conclusion/5_Conclusion.html#discussion">Conclusion</a></li> <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> <li><a href="./6_Appendices/A.1_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Appendices</a></li>
<ul> <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_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_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_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.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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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! 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> <h2>Contents</h2>
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