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_includes/header.html
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<a href="/feed.xml"><img class="icon" src="/assets/icons/rss.svg">RSS</a>
</p>
{% include sidebar.html%}
{{ include.extra }}
<hr>
</header>

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div.csl-entry div {
display: inline;
}
header li {
list-style: none;
a {
text-decoration: none;
margin-bottom: 0.5em;
display:block;
}
}

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_thesis/0_Preface/0.1_Aknowledgements.html
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@ -177,9 +177,19 @@ image:
<script src="/assets/js/index.js"></script>
</head>
<body>
{% include header.html %}
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<!-- -->
<p>I would like to thank my supervisor, Professor Johannes Knolle and
co-supervisor Professor Derek Lee for guidance and support during this
long process.</p>

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<script src="/assets/js/index.js"></script>
</head>
<body>
{% include header.html %}
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#interacting-quantum-many-body-systems"
id="toc-interacting-quantum-many-body-systems">Interacting Quantum Many
Body Systems</a></li>
<li><a href="#mott-insulators" id="toc-mott-insulators">Mott
Insulators</a></li>
<li><a href="#quantum-spin-liquids"
id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
<li><a href="#outline" id="toc-outline">Outline</a></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#interacting-quantum-many-body-systems"
id="toc-interacting-quantum-many-body-systems">Interacting Quantum Many
Body Systems</a></li>
<li><a href="#mott-insulators" id="toc-mott-insulators">Mott
Insulators</a></li>
<li><a href="#quantum-spin-liquids"
id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
<li><a href="#outline" id="toc-outline">Outline</a></li>
</ul>
</nav>
<p><strong>Interacting Quantum Many Body Systems</strong></p>
-->
<h1 id="interacting-quantum-many-body-systems">Interacting Quantum Many
Body Systems</h1>
<p>When you take many objects and let them interact together, it is
often simpler to describe the behaviour of the group differently from
the way one would describe the individual objects. Consider a flock of
@ -219,13 +246,13 @@ natural description of this phenomena is couched in terms of the flock
rather than of the individual birds.</p>
<p>The behaviours of the flock are an <em>emergent phenomena</em>. The
starlings are only interacting with their immediate six or seven
neighbours <span class="citation"
neighbours <span class="citation"
data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"> [<a
href="#ref-king2012murmurations" role="doc-biblioref">1</a>,<a
href="#ref-balleriniInteractionRulingAnimal2008"
role="doc-biblioref">2</a>]</span>, what a physicist would call a
<em>local interaction</em>. There is much philosophical debate about how
exactly to define emergence <span class="citation"
exactly to define emergence <span class="citation"
data-cites="andersonMoreDifferent1972 kivelsonDefiningEmergencePhysics2016"> [<a
href="#ref-andersonMoreDifferent1972" role="doc-biblioref">3</a>,<a
href="#ref-kivelsonDefiningEmergencePhysics2016"
@ -247,12 +274,12 @@ href="creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA
</div>
<p>To give an example closer to the topic at hand, our understanding of
thermodynamics began with bulk properties like heat, energy, pressure
and temperature <span class="citation"
and temperature <span class="citation"
data-cites="saslowHistoryThermodynamicsMissing2020"> [<a
href="#ref-saslowHistoryThermodynamicsMissing2020"
role="doc-biblioref">5</a>]</span>. It was only later that we gained an
understanding of how these properties emerge from microscopic
interactions between very large numbers of particles <span
interactions between very large numbers of particles <span
class="citation" data-cites="flammHistoryOutlookStatistical1998"> [<a
href="#ref-flammHistoryOutlookStatistical1998"
role="doc-biblioref">6</a>]</span>.</p>
@ -265,50 +292,51 @@ these three ingredients nature builds all manner of weird and wonderful
materials.</p>
<p>Historically, we made initial headway in the study of many-body
systems, ignoring interactions and quantum properties. The ideal gas law
and the Drude classical electron gas <span class="citation"
and the Drude classical electron gas <span class="citation"
data-cites="ashcroftSolidStatePhysics1976"> [<a
href="#ref-ashcroftSolidStatePhysics1976"
role="doc-biblioref">7</a>]</span> are good examples. Including
interactions into many-body physics leads to the Ising model <span
interactions into many-body physics leads to the Ising model <span
class="citation" data-cites="isingBeitragZurTheorie1925"> [<a
href="#ref-isingBeitragZurTheorie1925"
role="doc-biblioref">8</a>]</span>, Landau theory <span class="citation"
role="doc-biblioref">8</a>]</span>, Landau theory <span class="citation"
data-cites="landau2013fluid"> [<a href="#ref-landau2013fluid"
role="doc-biblioref">9</a>]</span> and the classical theory of phase
transitions <span class="citation"
transitions <span class="citation"
data-cites="jaegerEhrenfestClassificationPhase1998"> [<a
href="#ref-jaegerEhrenfestClassificationPhase1998"
role="doc-biblioref">10</a>]</span>. In contrast, condensed matter
theory got it state in quantum many-body theory. Bloch’s theorem <span
theory got it state in quantum many-body theory. Bloch’s theorem <span
class="citation"
data-cites="blochÜberQuantenmechanikElektronen1929"> [<a
href="#ref-blochÜberQuantenmechanikElektronen1929"
role="doc-biblioref">11</a>]</span> predicted the properties of
non-interacting electrons in crystal lattices, leading to band theory.
In the same vein, advances were made in understanding the quantum
origins of magnetism, including ferromagnetism and antiferromagnetism
<span class="citation" data-cites="MagnetismCondensedMatter"> [<a
origins of magnetism, including ferromagnetism and
antiferromagnetism <span class="citation"
data-cites="MagnetismCondensedMatter"> [<a
href="#ref-MagnetismCondensedMatter"
role="doc-biblioref">12</a>]</span>.</p>
<p>However, at some point we had to start on the interacting quantum
many body systems. The properties of some materials cannot be understood
without a taking into account all three effects and these are
collectively called strongly correlated materials. The canonical
examples are superconductivity <span class="citation"
examples are superconductivity <span class="citation"
data-cites="MicroscopicTheorySuperconductivity"> [<a
href="#ref-MicroscopicTheorySuperconductivity"
role="doc-biblioref">13</a>]</span>, the fractional quantum hall effect
<span class="citation"
role="doc-biblioref">13</a>]</span>, the fractional quantum hall
effect <span class="citation"
data-cites="feldmanFractionalChargeFractional2021"> [<a
href="#ref-feldmanFractionalChargeFractional2021"
role="doc-biblioref">14</a>]</span> and the Mott insulators <span
role="doc-biblioref">14</a>]</span> and the Mott insulators <span
class="citation"
data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a
href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">15</a>,<a
href="#ref-fisherMottInsulatorsSpin1999"
role="doc-biblioref">16</a>]</span>. We’ll start by looking at the
latter but shall see that there are many links between three topics.</p>
<p><strong>Mott Insulators</strong></p>
<h1 id="mott-insulators">Mott Insulators</h1>
<p>Mott Insulators are remarkable because their electrical insulator
properties come from electron-electron interactions. Electrical
conductivity, the bulk movement of electrons, requires both that there
@ -328,7 +356,7 @@ methods.</p>
<figure>
<img src="/assets/thesis/intro_chapter/venn_diagram.svg"
data-short-caption="Interacting Quantum Many Body Systems Venn Diagram"
style="width:100.0%"
style="width:57.0%"
alt="Figure 2: Three key adjectives. Many Body, the fact of describing systems in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials." />
<figcaption aria-hidden="true"><span>Figure 2:</span> Three key
adjectives. Many Body, the fact of describing systems in the limit of
@ -341,25 +369,25 @@ to what are called strongly correlated materials.</figcaption>
</div>
<p>The theory of Mott insulators developed out of the observation that
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
href="#ref-boerSemiconductorsPartiallyCompletely1937"
role="doc-biblioref">17</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
href="#ref-mottDiscussionPaperBoer1937"
role="doc-biblioref">18</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"
data-cites="bednorzPossibleHighTcSuperconductivity1986"> [<a
href="#ref-bednorzPossibleHighTcSuperconductivity1986"
role="doc-biblioref">19</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
href="#ref-leeDopingMottInsulator2006"
role="doc-biblioref">20</a>]</span>.</p>
<p>The canonical toy model of the Mott insulator is the Hubbard model
<span class="citation"
<p>The canonical toy model of the Mott insulator is the Hubbard
model <span class="citation"
data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"> [<a
href="#ref-gutzwillerEffectCorrelationFerromagnetism1963"
role="doc-biblioref">21</a>–<a
@ -391,7 +419,7 @@ class="math inline">\(|0\rangle, |\uparrow\rangle, |\downarrow\rangle,
|\uparrow\downarrow\rangle\)</span> depending on the filing.</p>
<p>The Mott insulating phase occurs at half filling <span
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
href="#ref-hubbardElectronCorrelationsNarrow1964"
role="doc-biblioref">24</a>]</span>. Here the model can be rewritten in
@ -405,11 +433,11 @@ cost energy <span class="math inline">\(U\)</span>, hence the system has
a finite bandgap and is an interaction driven Mott insulator. Depending
on the lattice, the local moments may then order antiferromagnetically.
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
href="#ref-mottMetalInsulatorTransitions1990"
role="doc-biblioref">25</a>]</span>. However, Mott insulators have been
found <span class="citation"
found <span class="citation"
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">26</a>,<a
href="#ref-ribakGaplessExcitationsGround2017"
@ -418,18 +446,18 @@ local moments may form a highly entangled state known as a quantum spin
liquid, which will be discussed shortly.</p>
<p>Various theoretical treatments of the Hubbard model have been made,
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
href="#ref-slaterMagneticEffectsHartreeFock1951"
role="doc-biblioref">28</a>]</span> and dynamical mean-field theory
<span class="citation"
role="doc-biblioref">28</a>]</span> and dynamical mean-field
theory <span class="citation"
data-cites="greinerQuantumPhaseTransition2002"> [<a
href="#ref-greinerQuantumPhaseTransition2002"
role="doc-biblioref">29</a>]</span>. None of these approaches are
perfect. Strong correlations are poorly described by the Fermi liquid
theory and the LDA approaches while mean field approximations do poorly
in low dimensional systems. This theoretical difficulty has made the
Hubbard model a target for cold atom simulations <span class="citation"
Hubbard model a target for cold atom simulations <span class="citation"
data-cites="mazurenkoColdatomFermiHubbard2017"> [<a
href="#ref-mazurenkoColdatomFermiHubbard2017"
role="doc-biblioref">30</a>]</span>.</p>
@ -453,7 +481,7 @@ c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
\tfrac{1}{2}). \\
\end{aligned}\]</span></p>
<p>Given that the physics of states near the metal-insulator (MI)
transition is still poorly understood <span class="citation"
transition is still poorly understood <span class="citation"
data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a
href="#ref-belitzAndersonMottTransition1994"
role="doc-biblioref">31</a>,<a
@ -492,27 +520,79 @@ then compare the behaviour of this transitionally invariant model to an
Anderson model of uncorrelated binary disorder about a background charge
density wave field which confirms that the fermionic sector only fully
localizes for very large system sizes.</p>
<p><strong>An exactly solvable Quantum Spin Liquid</strong></p>
<h1 id="quantum-spin-liquids">Quantum Spin Liquids</h1>
<p>To turn to the other key topic of this thesis, we have discussed the
question of the magnetic ordering of local moments in the Mott
insulating state. The local moments may form an AFM ground state.
Alternatively they may fail to order even at zero temperature <span
Alternatively they may fail to order even at zero temperature <span
class="citation"
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">26</a>,<a
href="#ref-ribakGaplessExcitationsGround2017"
role="doc-biblioref">27</a>]</span>, giving rise to what is known as a
quantum spin liquid (QSL) state.</p>
<p>Landau theory characterises phases of matter as inextricably linked
to the emergence of long range order via a spontaneously broken
symmetry. The fractional quantum Hall (FQH) state, discovered in the
1980s is an explicit example of an electronic system that falls outside
of the Landau paradigm. FQH systems exhibit fractionalised excitations
linked to their ground state having long range entanglement and
non-trivial topological properties <span class="citation"
data-cites="broholmQuantumSpinLiquids2020"> [<a
href="#ref-broholmQuantumSpinLiquids2020"
role="doc-biblioref">40</a>]</span>. Quantum spin liquids are the
analogous phase of matter for spin systems. Remarkably the existence of
QSLs was first suggested by Anderson in 1973 <span class="citation"
data-cites="andersonResonatingValenceBonds1973"> [<a
href="#ref-andersonResonatingValenceBonds1973"
role="doc-biblioref">41</a>]</span>.</p>
<div id="fig:correlation_spin_orbit_PT" class="fignos">
<figure>
<img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png"
data-short-caption="Phase Diagram" style="width:100.0%"
alt="Figure 3: From  [42]." />
<figcaption aria-hidden="true"><span>Figure 3:</span> From <span
class="citation" data-cites="TrebstPhysRep2022"> [<a
href="#ref-TrebstPhysRep2022"
role="doc-biblioref">42</a>]</span>.</figcaption>
</figure>
</div>
<p>The main route to QSLs, though there are others <span
class="citation"
data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a
href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">43</a>–<a
href="#ref-linExactSymmetryWeaklyinteracting1998"
role="doc-biblioref">45</a>]</span>, is via frustration of spin models
that would otherwise order have AFM order. This frustration can come
geometrically, triangular lattices for instance cannot support AFM
order. It can also come about as a result of spin-orbit coupling.</p>
<p>Electron spin naturally couples to magnetic fields. Spin-orbit
coupling is a relativistic effect, that very roughly corresponds to the
fact that in the frame of reference of a moving electron, the electric
field of nearby nuclei look like magnetic field to which the electron
spin couples. In certain transition metal based compounds, such as those
based on Iridium and Rutheniun, crystal field effects, strong spin-orbit
coupling and narrow bandwidths lead to effective spin-<span
class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with
strongly anisotropic spin-spin couplings <span class="citation"
data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022"
role="doc-biblioref">42</a>]</span>.</p>
<p>The celebrated Kitaev model <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">46</a>]</span></p>
<p>QSLs are a long range entangled ground state of a highly
frustated</p>
<ul>
<li><p>QSLs introduced by anderson 1973 <span class="citation"
data-cites="andersonResonatingValenceBonds1973"> [<a
href="#ref-andersonResonatingValenceBonds1973"
role="doc-biblioref">40</a>]</span></p></li>
<li><p>Geometric frustration that prevents magnetic ordering is an
important part of getting a QSL, suggests exploring the lattice and
avenue of interest.</p></li>
<li><p>QSLs introduced by anderson 1973</p></li>
<li><p>Frustration can be geometric, such as AFM couplings on a
triangular lattice. It can also come from anisotropic couplings induced
via spin-orbit coupling.</p></li>
</ul>
<p>Geometric frustration or spin-orbit coupling can prevent magnetic
ordering is an important part of getting a QSL, suggests exploring the
lattice and avenue of interest.</p>
<ul>
<li><p>Spin orbit effect is a relativistic effect that couples electron
spin to orbital angular moment. Very roughly, an electron sees the
electric field of the nucleus as a magnetic field due to its movement
@ -526,17 +606,6 @@ elements</p></li>
surface</p></li>
<li><p>the chern number</p></li>
</ul>
<div id="fig:correlation_spin_orbit_PT" class="fignos">
<figure>
<img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png"
data-short-caption="Phase Diagram" style="width:100.0%"
alt="Figure 3: From  [41]." />
<figcaption aria-hidden="true"><span>Figure 3:</span> From <span
class="citation" data-cites="TrebstPhysRep2022"> [<a
href="#ref-TrebstPhysRep2022"
role="doc-biblioref">41</a>]</span>.</figcaption>
</figure>
</div>
<p>kinds of mott insulators: Mott-Heisenberg (AFM order below Néel
temperature) Mott-Hubbard (no long-range order of local magnetic
moments) Mott-Anderson (disorder + correlations) Wigner Crystal</p>
@ -867,9 +936,17 @@ href="https://doi.org/10.1103/PhysRevB.94.245114">Nonequilibrium
Dynamical Cluster Approximation Study of the Falicov-Kimball
Model</a></em>, Phys. Rev. B <strong>94</strong>, 245114 (2016).</div>
</div>
<div id="ref-broholmQuantumSpinLiquids2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[40] </div><div class="csl-right-inline">C.
Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T.
Senthil, <em><a href="https://doi.org/10.1126/science.aay0668">Quantum
Spin Liquids</a></em>, Science <strong>367</strong>, eaay0668
(2020).</div>
</div>
<div id="ref-andersonResonatingValenceBonds1973" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[40] </div><div class="csl-right-inline">P.
<div class="csl-left-margin">[41] </div><div class="csl-right-inline">P.
W. Anderson, <em><a
href="https://doi.org/10.1016/0025-5408(73)90167-0">Resonating Valence
Bonds: A New Kind of Insulator?</a></em>, Materials Research Bulletin
@ -877,12 +954,43 @@ Bonds: A New Kind of Insulator?</a></em>, Materials Research Bulletin
</div>
<div id="ref-TrebstPhysRep2022" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[41] </div><div class="csl-right-inline">S.
<div class="csl-left-margin">[42] </div><div class="csl-right-inline">S.
Trebst and C. Hickey, <em><a
href="https://doi.org/10.1016/j.physrep.2021.11.003">Kitaev
Materials</a></em>, Physics Reports <strong>950</strong>, 1
(2022).</div>
</div>
<div id="ref-balentsNodalLiquidTheory1998" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[43] </div><div class="csl-right-inline">L.
Balents, M. P. A. Fisher, and C. Nayak, <em><a
href="https://doi.org/10.1142/S0217979298000570">Nodal Liquid Theory of
the Pseudo-Gap Phase of High-Tc Superconductors</a></em>, Int. J. Mod.
Phys. B <strong>12</strong>, 1033 (1998).</div>
</div>
<div id="ref-balentsDualOrderParameter1999" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[44] </div><div class="csl-right-inline">L.
Balents, M. P. A. Fisher, and C. Nayak, <em><a
href="https://doi.org/10.1103/PhysRevB.60.1654">Dual Order Parameter for
the Nodal Liquid</a></em>, Phys. Rev. B <strong>60</strong>, 1654
(1999).</div>
</div>
<div id="ref-linExactSymmetryWeaklyinteracting1998" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[45] </div><div
class="csl-right-inline">H.-H. Lin, L. Balents, and M. P. A. Fisher,
<em><a href="https://doi.org/10.1103/PhysRevB.58.1794">Exact SO(8)
Symmetry in the Weakly-Interacting Two-Leg Ladder</a></em>, Phys. Rev. B
<strong>58</strong>, 1794 (1998).</div>
</div>
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[46] </div><div class="csl-right-inline">A.
Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons
in an Exactly Solved Model and Beyond</a></em>, Annals of Physics
<strong>321</strong>, 2 (2006).</div>
</div>
</div>
</main>
</body>

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_thesis/2_Background/2.1_FK_Model.html
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@ -240,10 +240,33 @@ image:
<script src="/assets/js/index.js"></script>
</head>
<body>
{% include header.html %}
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#the-falikov-kimball-model"
id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
<ul>
<li><a href="#the-model" id="toc-the-model">The Model</a>
<ul>
<li><a href="#particle-hole-symmetry"
id="toc-particle-hole-symmetry">Particle Hole Symmetry</a></li>
</ul></li>
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
Diagram</a></li>
<li><a href="#long-range-ising-models"
id="toc-long-range-ising-models">Long Range Ising Models</a></li>
</ul></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#the-falikov-kimball-model"
id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
@ -260,6 +283,7 @@ id="toc-long-range-ising-models">Long Range Ising Models</a></li>
</ul></li>
</ul>
</nav>
-->
<h1 id="the-falikov-kimball-model">The Falikov Kimball Model</h1>
<h2 id="the-model">The Model</h2>
<p>discuss CDW phase of 2d model as motivation for studying 1d phase

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</head>
<body>
{% include header.html %}
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{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#the-kitaev-honeycomb-model"
id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
<ul>
<li><a href="#the-model" id="toc-the-model">The Model</a></li>
<li><a href="#a-mapping-to-majorana-fermions"
id="toc-a-mapping-to-majorana-fermions">A mapping to Majorana
Fermions</a></li>
<li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a></li>
<li><a href="#anyons-topology-and-the-chern-number"
id="toc-anyons-topology-and-the-chern-number">Anyons, Topology and the
Chern number</a></li>
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
Diagram</a></li>
</ul></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#the-kitaev-honeycomb-model"
id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
@ -283,6 +307,7 @@ Diagram</a></li>
</ul></li>
</ul>
</nav>
-->
<h1 id="the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</h1>
<p><strong>intro</strong> - strong spin orbit coupling leads to
anisotropic spin exchange (as opposed to isotropic exchange like the
@ -319,7 +344,7 @@ Majorana <span class="math inline">\(c_i\)</span> per site.</figcaption>
</div>
<ul>
<li>strong spin orbit coupling yields spatial anisotropic spin exchange
leading to compass models <span class="citation"
leading to compass models <span class="citation"
data-cites="kugelJahnTellerEffectMagnetism1982"> [<a
href="#ref-kugelJahnTellerEffectMagnetism1982"
role="doc-biblioref">1</a>]</span></li>

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</head>
<body>
{% include header.html %}
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{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#disorder-localisation"
id="toc-disorder-localisation">Disorder &amp; Localisation</a>
<ul>
<li><a href="#localisation-anderson-many-body-and-disorder-free"
id="toc-localisation-anderson-many-body-and-disorder-free">Localisation:
Anderson, Many Body and Disorder-Free</a></li>
<li><a href="#disorder-and-spin-liquids"
id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
<li><a href="#amorphous-magnetism"
id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
</ul></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#disorder-localisation"
id="toc-disorder-localisation">Disorder &amp; Localisation</a>
@ -258,6 +279,7 @@ id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
</ul></li>
</ul>
</nav>
-->
<h1 id="disorder-localisation">Disorder &amp; Localisation</h1>
<h2 id="localisation-anderson-many-body-and-disorder-free">Localisation:
Anderson, Many Body and Disorder-Free</h2>

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</head>
<body>
{% include header.html %}
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#markov-chain-monte-carlo"
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a>
<ul>
<li><a href="#sampling" id="toc-sampling">Sampling</a></li>
<li><a href="#markov-chains" id="toc-markov-chains">Markov
Chains</a></li>
<li><a href="#application-to-the-fk-model"
id="toc-application-to-the-fk-model">Application to the FK Model</a>
<ul>
<li><a href="#markov-chain-monte-carlo-1"
id="toc-markov-chain-monte-carlo-1">Markov Chain Monte Carlo</a></li>
</ul></li>
<li><a href="#the-metropolis-hasting-algorithm"
id="toc-the-metropolis-hasting-algorithm">The Metropolis-Hasting
Algorithm</a></li>
<li><a href="#metropolis-hastings"
id="toc-metropolis-hastings">Metropolis-Hastings</a></li>
<li><a href="#convergence-auto-correlation-and-binning"
id="toc-convergence-auto-correlation-and-binning">Convergence,
Auto-correlation and Binning</a></li>
<li><a href="#applying-mcmc-to-the-fk-model"
id="toc-applying-mcmc-to-the-fk-model">Applying MCMC to the FK
model</a></li>
<li><a href="#proposal-distributions"
id="toc-proposal-distributions">Proposal Distributions</a></li>
<li><a href="#perturbation-mcmc" id="toc-perturbation-mcmc">Perturbation
MCMC</a></li>
<li><a href="#scaling" id="toc-scaling">Scaling</a></li>
<li><a href="#binder-cumulants" id="toc-binder-cumulants">Binder
Cumulants</a></li>
<li><a href="#markov-chain-monte-carlo-in-practice"
id="toc-markov-chain-monte-carlo-in-practice">Markov Chain Monte-Carlo
in Practice</a>
<ul>
<li><a href="#quick-intro-to-mcmc" id="toc-quick-intro-to-mcmc">Quick
Intro to MCMC</a></li>
<li><a href="#convergence-time" id="toc-convergence-time">Convergence
Time</a></li>
<li><a href="#auto-correlation-time"
id="toc-auto-correlation-time">Auto-correlation Time</a></li>
<li><a href="#the-metropolis-hastings-algorithm"
id="toc-the-metropolis-hastings-algorithm">The Metropolis-Hastings
Algorithm</a></li>
</ul></li>
<li><a href="#two-step-trick" id="toc-two-step-trick">Two Step
Trick</a></li>
<li><a href="#detailed-balance-for-the-two-step-method"
id="toc-detailed-balance-for-the-two-step-method">Detailed Balance for
the two step method</a>
<ul>
<li><a href="#two-step-trick-1" id="toc-two-step-trick-1">Two Step
Trick</a></li>
<li><a href="#tuning-the-proposal-distribution"
id="toc-tuning-the-proposal-distribution">Tuning the proposal
distribution</a></li>
</ul></li>
<li><a href="#diagnostics-of-localisation"
id="toc-diagnostics-of-localisation">Diagnostics of Localisation</a>
<ul>
<li><a href="#inverse-participation-ratio"
id="toc-inverse-participation-ratio">Inverse Participation
Ratio</a></li>
</ul></li>
<li><a href="#markov-chain-monte-carlo-2"
id="toc-markov-chain-monte-carlo-2">Markov Chain Monte-Carlo</a></li>
<li><a href="#convergence-time-1"
id="toc-convergence-time-1">Convergence Time</a></li>
<li><a href="#auto-correlation-time-1"
id="toc-auto-correlation-time-1">Auto-correlation Time</a></li>
<li><a href="#the-metropolis-hastings-algorithm-1"
id="toc-the-metropolis-hastings-algorithm-1">The Metropolis-Hastings
Algorithm</a></li>
<li><a href="#choosing-the-proposal-distribution"
id="toc-choosing-the-proposal-distribution">Choosing the proposal
distribution</a></li>
<li><a href="#two-step-trick-2" id="toc-two-step-trick-2">Two Step
Trick</a></li>
</ul></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#markov-chain-monte-carlo"
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a>
@ -347,6 +435,7 @@ Trick</a></li>
</ul></li>
</ul>
</nav>
-->
<h1 id="markov-chain-monte-carlo">Markov Chain Monte Carlo</h1>
<h2 id="sampling">Sampling</h2>
<p>Markov Chain Monte Carlo (MCMC) is a useful method whenever we have a
@ -360,7 +449,7 @@ The fact they’re uncorrelated is key as we’ll see later. Examples of
direct sampling methods range from the trivial: take n random bits to
generate integers uniformly between 0 and <span
class="math inline">\(2^n\)</span> to more complex methods such as
inverse transform sampling and rejection sampling <span class="citation"
inverse transform sampling and rejection sampling <span class="citation"
data-cites="devroyeRandomSampling1986"> [<a
href="#ref-devroyeRandomSampling1986"
role="doc-biblioref">1</a>]</span>.</p>
@ -383,7 +472,7 @@ with system size. Even if we could calculate <span
class="math inline">\(\mathcal{Z}\)</span>, sampling from an
exponentially large number of options quickly become tricky. This kind
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"
role="doc-biblioref">2</a>]</span>.</p>
<h2 id="markov-chains">Markov Chains</h2>
@ -393,7 +482,7 @@ instead.</p>
<p>MCMC defines a weighted random walk over the states <span
class="math inline">\((S_0, S_1, S_2, ...)\)</span>, such that in the
long time limit, states are visited according to their probability <span
class="math inline">\(p(S)\)</span>. <span class="citation"
class="math inline">\(p(S)\)</span>. <span class="citation"
data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"> [<a
href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">3</a>–<a
href="#ref-wolffMonteCarloErrors2004"
@ -447,7 +536,7 @@ F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}
expectation values <span class="math inline">\(\expval{O}\)</span> with
respect to some physical system defined by a set of states <span
class="math inline">\(\{x: x \in S\}\)</span> and a free energy <span
class="math inline">\(F(x)\)</span> <span class="citation"
class="math inline">\(F(x)\)</span> <span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a>]</span>. The thermal expectation value is
@ -526,7 +615,7 @@ P(x) \mathcal{T}(x \rightarrow x&#39;) = P(x&#39;) \mathcal{T}(x&#39;
\rightarrow x)
\]</span> % In practice most algorithms are constructed to satisfy
detailed balance though there are arguments that relaxing the condition
can lead to faster algorithms <span class="citation"
can lead to faster algorithms <span class="citation"
data-cites="kapferSamplingPolytopeHarddisk2013"> [<a
href="#ref-kapferSamplingPolytopeHarddisk2013"
role="doc-biblioref">8</a>]</span>.</p>
@ -558,7 +647,7 @@ x_{i}\)</span>. Now <span class="math inline">\(\mathcal{T}(x\to x&#39;)
<p>The Metropolis-Hasting algorithm is a slight extension of the
original Metropolis algorithm that allows for non-symmetric proposal
distributions $q(xx’) q(x’x) $. It can be derived starting from detailed
balance <span class="citation"
balance <span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a>]</span>: <span
@ -671,7 +760,7 @@ problematic because it means very few new samples will be generated. If
it is too high it implies the steps are too small, a problem because
then the walk will take longer to explore the state space and the
samples will be highly correlated. Ideal values for the acceptance rate
can be calculated under certain assumptions <span class="citation"
can be calculated under certain assumptions <span class="citation"
data-cites="robertsWeakConvergenceOptimal1997"> [<a
href="#ref-robertsWeakConvergenceOptimal1997"
role="doc-biblioref">9</a>]</span>. Here we monitor the acceptance rate
@ -686,7 +775,7 @@ produce a state at or near the energy of the current one.</p>
<p>The matrix diagonalisation is the most computationally expensive step
of the process, a speed up can be obtained by modifying the proposal
distribution to depend on the classical part of the energy, a trick
gleaned from Ref. <span class="citation"
gleaned from Ref. <span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a>]</span>: <span class="math display">\[
@ -700,7 +789,7 @@ without performing the diagonalisation at no cost to the accuracy of the
MCMC method.</p>
<p>An extension of this idea is to try to define a classical model with
a similar free energy dependence on the classical state as the full
quantum, Ref. <span class="citation"
quantum, Ref. <span class="citation"
data-cites="huangAcceleratedMonteCarlo2017"> [<a
href="#ref-huangAcceleratedMonteCarlo2017"
role="doc-biblioref">10</a>]</span> does this with restricted Boltzmann
@ -725,8 +814,8 @@ central moments of the order parameter m: <span class="math display">\[m
= \sum_i (-1)^i (2n_i - 1) / N\]</span> % The Binder cumulant evaluated
against temperature can be used as a diagnostic for the existence of a
phase transition. If multiple such curves are plotted for different
system sizes, a crossing indicates the location of a critical point
<span class="citation"
system sizes, a crossing indicates the location of a critical
point <span class="citation"
data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a
href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">11</a>,<a
href="#ref-musialMonteCarloSimulations2002"
@ -757,7 +846,7 @@ very expensive operation!~\footnote{The effort involved in exact
diagonalisation scales like <span class="math inline">\(N^2\)</span> for
systems with a tri-diagonal matrix representation (open boundary
conditions and nearest neighbour hopping) and like <span
class="math inline">\(N^3\)</span> for a generic matrix <span
class="math inline">\(N^3\)</span> for a generic matrix <span
class="citation"
data-cites="bolchQueueingNetworksMarkov2006 usmaniInversionTridiagonalJacobi1994"> [<a
href="#ref-bolchQueueingNetworksMarkov2006"
@ -877,7 +966,7 @@ auto-correlation time <span class="math inline">\(\tau(O)\)</span>
informally as the number of MCMC samples of some observable O that are
statistically equal to one independent sample or equivalently as the
number of MCMC steps after which the samples are correlated below some
cutoff, see <span class="citation"
cutoff, see <span class="citation"
data-cites="krauthIntroductionMonteCarlo1996"> [<a
href="#ref-krauthIntroductionMonteCarlo1996"
role="doc-biblioref">14</a>]</span> for a more rigorous definition
@ -1020,7 +1109,7 @@ the two step method</h2>
<p>Given a MCMC algorithm with target distribution <span
class="math inline">\(\pi(a)\)</span> and transition function <span
class="math inline">\(\mathcal{T}\)</span> the detailed balance
condition is sufficient (along with some technical constraints <span
condition is sufficient (along with some technical constraints <span
class="citation" data-cites="wolffMonteCarloErrors2004"> [<a
href="#ref-wolffMonteCarloErrors2004"
role="doc-biblioref">5</a>]</span>) to guarantee that in the long time
@ -1140,7 +1229,7 @@ for the additional complexity it would require.</p>
<h3 id="inverse-participation-ratio">Inverse Participation Ratio</h3>
<p>The inverse participation ratio is defined for a normalised wave
function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i
\abs{\psi_i}^2 = 1\)</span> as its fourth moment <span class="citation"
\abs{\psi_i}^2 = 1\)</span> as its fourth moment <span class="citation"
data-cites="kramerLocalizationTheoryExperiment1993"> [<a
href="#ref-kramerLocalizationTheoryExperiment1993"
role="doc-biblioref">17</a>]</span>: <span class="math display">\[
@ -1154,7 +1243,7 @@ fractal dimensionality <span class="math inline">\(d &gt; d* &gt;
P(L) \goeslike L^{d*}
\]</span> % For extended states <span class="math inline">\(d* =
0\)</span> while for localised ones <span class="math inline">\(d* =
0\)</span>. In this work we take use an energy resolved IPR <span
0\)</span>. In this work we take use an energy resolved IPR <span
class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a
href="#ref-andersonAbsenceDiffusionCertain1958"
role="doc-biblioref">18</a>]</span>: <span class="math display">\[

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<script src="/assets/js/index.js"></script>
</head>
<body>
{% include header.html %}
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#the-phase-diagram" id="toc-the-phase-diagram">The Phase
Diagram</a></li>
<li><a href="#localisation-properties"
id="toc-localisation-properties">Localisation Properties</a></li>
<li><a href="#discussion-conclusion"
id="toc-discussion-conclusion">Discussion &amp; Conclusion</a></li>
<li><a href="#acknowledgments"
id="toc-acknowledgments">Acknowledgments</a></li>
<li><a href="#uncorrelated-disorder-model"
id="toc-uncorrelated-disorder-model"><span id="app:disorder_model"
label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#the-phase-diagram" id="toc-the-phase-diagram">The Phase
Diagram</a></li>
@ -280,6 +301,7 @@ id="toc-uncorrelated-disorder-model"><span id="app:disorder_model"
label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li>
</ul>
</nav>
-->
<div id="fig:phase_diagram" class="fignos">
<figure>
<img src="pdf_figs/phase_diagram.svg"

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@ -200,10 +200,65 @@ image:
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</head>
<body>
{% include header.html %}
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a>
<ul>
<li><a href="#vortices-and-their-movements"
id="toc-vortices-and-their-movements">Vortices and their
movements</a></li>
<li><a href="#dual-loops-and-gauge-symmetries"
id="toc-dual-loops-and-gauge-symmetries">Dual Loops and gauge
symmetries</a></li>
<li><a href="#composition-of-wilson-loops"
id="toc-composition-of-wilson-loops">Composition of Wilson
loops</a></li>
<li><a href="#gauge-degeneracy-and-the-euler-equation"
id="toc-gauge-degeneracy-and-the-euler-equation">Gauge Degeneracy and
the Euler Equation</a></li>
<li><a href="#counting-edges-plaquettes-and-vertices"
id="toc-counting-edges-plaquettes-and-vertices">Counting edges,
plaquettes and vertices</a></li>
</ul></li>
<li><a href="#the-projector" id="toc-the-projector">The Projector</a>
<ul>
<li><a href="#ground-state-degeneracy"
id="toc-ground-state-degeneracy">Ground State Degeneracy</a></li>
<li><a href="#quick-breather" id="toc-quick-breather">Quick
Breather</a></li>
</ul></li>
<li><a href="#the-ground-state" id="toc-the-ground-state">The Ground
State</a>
<ul>
<li><a href="#finite-size-effects" id="toc-finite-size-effects">Finite
size effects</a></li>
<li><a href="#chiral-symmetry" id="toc-chiral-symmetry">Chiral
Symmetry</a></li>
</ul></li>
<li><a href="#phases-of-the-kitaev-model"
id="toc-phases-of-the-kitaev-model">Phases of the Kitaev Model</a></li>
<li><a href="#what-is-so-great-about-two-dimensions"
id="toc-what-is-so-great-about-two-dimensions">What is so great about
two dimensions?</a>
<ul>
<li><a href="#topology-chirality-and-edge-modes"
id="toc-topology-chirality-and-edge-modes">Topology, chirality and edge
modes</a></li>
<li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic
Statistics</a></li>
</ul></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a>
<ul>
@ -252,6 +307,7 @@ Statistics</a></li>
</ul></li>
</ul>
</nav>
-->
<h2 id="gauge-fields">Gauge Fields</h2>
<p>The bond operators <span class="math inline">\(u_{ij}\)</span> are
useful because they label a bond sector <span
@ -539,7 +595,7 @@ symmetries</strong> and <strong><span class="math inline">\(2^2 =
4\)</span> topological sectors</strong>.</p>
<p>The topological sector forms the basis of proposals to construct
topologically protected qubits since the four sectors can only be mixed
by a highly non-local perturbations <span class="citation"
by a highly non-local perturbations <span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003"> [<a
href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref">1</a>]</span>.</p>
@ -675,8 +731,8 @@ any information about the underlying lattice.</p>
<p><span class="math display">\[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i
\prod_i^{2N} b^y_i \prod_i^{2N} b^z_i \prod_i^{2N} c_i\]</span></p>
<p>The product over <span class="math inline">\(c_i\)</span> operators
reduces to a determinant of the Q matrix and the fermion parity, see
<span class="citation"
reduces to a determinant of the Q matrix and the fermion parity,
see <span class="citation"
data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
role="doc-biblioref">2</a>]</span>. The only difference from the
@ -702,7 +758,7 @@ depend only on the lattice structure.</p>
<p><span class="math inline">\(\hat{\pi} = \prod{i}^{N} (1 -
2\hat{n}_i)\)</span> is the parity of the particular many body state
determined by fermionic occupation numbers <span
class="math inline">\(n_i\)</span>. As discussed in <span
class="math inline">\(n_i\)</span>. As discussed in <span
class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
role="doc-biblioref">2</a>]</span>, <span
@ -711,7 +767,7 @@ that <span class="math inline">\([\hat{\pi}, D_i] = 0\)</span>.</p>
<p>This implies that <span class="math inline">\(det(Q^u) \prod -i
u_{ij}\)</span> is also a gauge invariant quantity. In translation
invariant models this quantity which can be related to the parity of the
number of vortex pairs in the system <span class="citation"
number of vortex pairs in the system <span class="citation"
data-cites="yaoAlgebraicSpinLiquid2009"> [<a
href="#ref-yaoAlgebraicSpinLiquid2009"
role="doc-biblioref">3</a>]</span>.</p>
@ -743,7 +799,7 @@ vortex pair, transporting one of them around the major or minor
diameters of the torus and, then, annihilating them again.</figcaption>
</figure>
</div>
<p>More general arguments <span class="citation"
<p>More general arguments <span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">4</a>,<a
@ -837,7 +893,7 @@ definition, the vortex free sector.</p>
<p>On the Honeycomb, Lieb’s theorem implies that the ground state
corresponds to the state where all <span class="math inline">\(u_{jk} =
1\)</span>. This implies that the flux free sector is the ground state
sector <span class="citation" data-cites="lieb_flux_1994"> [<a
sector <span class="citation" data-cites="lieb_flux_1994"> [<a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span>.</p>
<p>Lieb’s theorem does not generalise easily to the amorphous case.
However, we can get some intuition by examining the problem that will
@ -918,8 +974,8 @@ i)^{n_{\mathrm{sides}}},
class="math inline">\(n_{\mathrm{sides}}\)</span> is the number of edges
that form each plaquette and the choice of sign gives a twofold chiral
ground state degeneracy.</p>
<p>This conjecture is consistent with Lieb’s theorem on regular lattices
<span class="citation" data-cites="lieb_flux_1994"> [<a
<p>This conjecture is consistent with Lieb’s theorem on regular
lattices <span class="citation" data-cites="lieb_flux_1994"> [<a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span> and is
supported by numerical evidence. As noted before, any flux that differs
from the ground state is an excitation which we call a vortex.</p>
@ -973,7 +1029,7 @@ around the predicted ground state never yield a lower energy state.</p>
<strong>chiral</strong> degeneracy which arises because the global sign
of the odd plaquettes does not matter.</p>
<p>This happens because we have broken the time reversal symmetry of the
original model by adding odd plaquettes <span class="citation"
original model by adding odd plaquettes <span class="citation"
data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"> [<a
href="#ref-Chua2011" role="doc-biblioref">7</a>–<a
href="#ref-WangHaoranPRB2021" role="doc-biblioref">14</a>]</span>.</p>
@ -981,7 +1037,7 @@ href="#ref-WangHaoranPRB2021" role="doc-biblioref">14</a>]</span>.</p>
to a magnetic field, we get two degenerate ground states of different
handedness. Practically speaking, one ground state is related to the
other by inverting the imaginary <span
class="math inline">\(\phi\)</span> fluxes <span class="citation"
class="math inline">\(\phi\)</span> fluxes <span class="citation"
data-cites="yaoExactChiralSpin2007"> [<a
href="#ref-yaoExactChiralSpin2007"
role="doc-biblioref">8</a>]</span>.</p>
@ -1111,18 +1167,18 @@ and construct the set <span class="math inline">\((+1, +1), (+1, -1),
<figure>
<img src="/assets/thesis/amk_chapter/topological_fluxes.png"
data-short-caption="Topological Fluxes" style="width:57.0%"
alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make  [15]." />
alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make  [15]." />
<figcaption aria-hidden="true"><span>Figure 14:</span> Wilson loops that
wind the major or minor diameters of the torus measure flux winding
through the hole of the doughnut/torus or through the filling. If they
made doughnuts that both had a jam filling and a hole, this analogy
would be a lot easier to make <span class="citation"
would be a lot easier to make <span class="citation"
data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis"
role="doc-biblioref">15</a>]</span>.</figcaption>
</figure>
</div>
<p>However, in the non-Abelian phase we have to wrangle with monodromy
<span class="citation"
<p>However, in the non-Abelian phase we have to wrangle with
monodromy <span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">4</a>,<a
@ -1134,7 +1190,7 @@ them around the torus in such a way that, rather than annihilating to
the vacuum, we annihilate them to create an excited state instead of a
ground state. This means that we end up with only three degenerate
ground states in the non-Abelian phase <span class="math inline">\((+1,
+1), (+1, -1), (-1, +1)\)</span> <span class="citation"
+1), (+1, -1), (-1, +1)\)</span> <span class="citation"
data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"> [<a
href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">3</a>,<a
href="#ref-chungTopologicalQuantumPhase2010"
@ -1146,8 +1202,8 @@ the state physical. Hence, the process does not give a fourth ground
state.</p>
<p>Recently, the topology has notably gained interest because of
proposals to use this ground state degeneracy to implement both
passively fault tolerant and actively stabilised quantum computations
<span class="citation"
passively fault tolerant and actively stabilised quantum
computations <span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a
href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref">1</a>,<a

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@ -200,10 +200,51 @@ image:
<script src="/assets/js/index.js"></script>
</head>
<body>
{% include header.html %}
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#contributions"
id="toc-contributions">Contributions</a></li>
<li><a href="#introduction" id="toc-introduction">Introduction</a>
<ul>
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
Systems</a></li>
<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
<li><a href="#the-kitaev-model" id="toc-the-kitaev-model">The Kitaev
Model</a>
<ul>
<li><a href="#commutation-relations"
id="toc-commutation-relations">Commutation relations</a></li>
<li><a href="#the-hamiltonian" id="toc-the-hamiltonian">The
Hamiltonian</a></li>
<li><a href="#from-spins-to-majorana-operators"
id="toc-from-spins-to-majorana-operators">From Spins to Majorana
operators</a></li>
<li><a href="#partitioning-the-hilbert-space-into-bond-sectors"
id="toc-partitioning-the-hilbert-space-into-bond-sectors">Partitioning
the Hilbert Space into Bond sectors</a></li>
</ul></li>
<li><a href="#the-majorana-hamiltonian"
id="toc-the-majorana-hamiltonian">The Majorana Hamiltonian</a>
<ul>
<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace"
id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
back from Bond Sectors to the Physical Subspace</a></li>
<li><a href="#open-boundary-conditions"
id="toc-open-boundary-conditions">Open boundary conditions</a></li>
</ul></li>
</ul></li>
</ul>
{% endcapture %}
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{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#contributions"
id="toc-contributions">Contributions</a></li>
@ -238,6 +279,7 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
</ul></li>
</ul>
</nav>
-->
<h1 id="contributions">Contributions</h1>
<p>The material in this chapter expands on work presented in</p>
<p><strong>Insert citation of amorphous Kitaev paper here</strong></p>
@ -245,7 +287,7 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
guidance from Willian and Johannes. The project grew out of an interest
Gino, Peru and I had in studying amorphous systems, coupled with
Johannes’ expertise on the Kitaev model. The idea to use voronoi
partitions came from <span class="citation"
partitions came from <span class="citation"
data-cites="marsalTopologicalWeaireThorpe2020"> [<a
href="#ref-marsalTopologicalWeaireThorpe2020"
role="doc-biblioref">1</a>]</span> and Gino did the implementation of
@ -289,7 +331,7 @@ material. Candidate materials, such as <span
class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have
sufficiently strong spin-orbit coupling and the correct lattice
structure to behave according to the Kitaev Honeycomb model with small
corrections <span class="citation"
corrections <span class="citation"
data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"> [<a
href="#ref-banerjeeProximateKitaevQuantum2016"
role="doc-biblioref">2</a>,<a href="#ref-trebstKitaevMaterials2022"
@ -301,14 +343,14 @@ after quantum spin liquid state. Its excitations are anyons, particles
that can only exist in two dimensions that break the normal
fermion/boson dichotomy. Anyons have been the subject of much attention
because, among other reasons, they can be braided through spacetime to
achieve noise tolerant quantum computations <span class="citation"
achieve noise tolerant quantum computations <span class="citation"
data-cites="freedmanTopologicalQuantumComputation2003"> [<a
href="#ref-freedmanTopologicalQuantumComputation2003"
role="doc-biblioref">3</a>]</span>.</p>
<p>Third, and perhaps most importantly, this model is a rare many body
interacting quantum system that can be treated analytically. It is
exactly solvable. We can explicitly write down its many body ground
states in terms of single particle states <span class="citation"
states in terms of single particle states <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a>]</span>. The solubility of the Kitaev
@ -326,7 +368,7 @@ lattices.</p>
look at the gauge symmetries of the model as well as its solution via a
transformation to a Majorana hamiltonian. This discussion shows that,
for the the model to be solvable, it needs only be defined on a
trivalent, tri-edge-colourable lattice <span class="citation"
trivalent, tri-edge-colourable lattice <span class="citation"
data-cites="Nussinov2009"> [<a href="#ref-Nussinov2009"
role="doc-biblioref">5</a>]</span>.</p>
<p>The methods section discusses how to generate such lattices and
@ -512,7 +554,7 @@ on site <span class="math inline">\(j\)</span> and <span
class="math inline">\(\langle j,k\rangle_\alpha\)</span> is a pair of
nearest-neighbour indices connected by an <span
class="math inline">\(\alpha\)</span>-bond with exchange coupling <span
class="math inline">\(J^\alpha\)</span> <span class="citation"
class="math inline">\(J^\alpha\)</span> <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a>]</span>. For notational brevity, it is useful
@ -743,7 +785,7 @@ theory of the Majorana Hamiltonian further.</p>
u_{ij} c_i c_j\]</span> in which most of the Majorana degrees of freedom
have paired along bonds to become a classical gauge field <span
class="math inline">\(u_{ij}\)</span>. What follows is relatively
standard theory for quadratic Majorana Hamiltonians <span
standard theory for quadratic Majorana Hamiltonians <span
class="citation" data-cites="BlaizotRipka1986"> [<a
href="#ref-BlaizotRipka1986" role="doc-biblioref">6</a>]</span>.</p>
<p>Because of the antisymmetry of the matrix with entries <span

73
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@ -200,10 +200,44 @@ image:
<script src="/assets/js/index.js"></script>
</head>
<body>
{% include header.html %}
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{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#methods" id="toc-methods">Methods</a>
<ul>
<li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li>
<li><a href="#graph-representation" id="toc-graph-representation">Graph
Representation</a></li>
<li><a href="#colouring-the-bonds"
id="toc-colouring-the-bonds">Colouring the Bonds</a>
<ul>
<li><a href="#four-colourings-and-three-colourings"
id="toc-four-colourings-and-three-colourings">Four-colourings and
three-colourings</a></li>
<li><a href="#finding-lattice-colourings-with-minisat"
id="toc-finding-lattice-colourings-with-minisat">Finding Lattice
colourings with miniSAT</a></li>
<li><a href="#does-it-matter-which-colouring-we-choose"
id="toc-does-it-matter-which-colouring-we-choose">Does it matter which
colouring we choose?</a></li>
</ul></li>
<li><a href="#mapping-between-flux-sectors-and-bond-sectors"
id="toc-mapping-between-flux-sectors-and-bond-sectors">Mapping between
flux sectors and bond sectors</a></li>
<li><a href="#chern-markers" id="toc-chern-markers">Chern
Markers</a></li>
</ul></li>
</ul>
{% endcapture %}
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{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#methods" id="toc-methods">Methods</a>
<ul>
@ -231,10 +265,11 @@ Markers</a></li>
</ul></li>
</ul>
</nav>
-->
<h1 id="methods">Methods</h1>
<p>The practical implementation of what is described in this section is
available as a Python package called Koala (Kitaev On Amorphous
LAttices) <span class="citation"
LAttices) <span class="citation"
data-cites="tomImperialCMTHKoalaFirst2022"> [<a
href="#ref-tomImperialCMTHKoalaFirst2022"
role="doc-biblioref"><strong>tomImperialCMTHKoalaFirst2022?</strong></a>]</span>.
@ -242,7 +277,7 @@ All results and figures were generated with Koala.</p>
<h2 id="voronisation">Voronisation</h2>
<p>To study the properties of the amorphous Kitaev model, we need to
sample from the space of possible trivalent graphs.</p>
<p>A simple method is to use a Voronoi partition of the torus <span
<p>A simple method is to use a Voronoi partition of the torus <span
class="citation"
data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020 florescu_designer_2009"> [<a
href="#ref-mitchellAmorphousTopologicalInsulators2018"
@ -259,16 +294,16 @@ the graph is embedded into the plane. It is also trivalent in that every
vertex is connected to exactly three edges <strong>cite</strong>.</p>
<p>Ideally, we would sample uniformly from the space of possible
trivalent graphs. Indeed, there has been some work on how to do this
using a Markov Chain Monte Carlo approach <span class="citation"
using a Markov Chain Monte Carlo approach <span class="citation"
data-cites="alyamiUniformSamplingDirected2016"> [<a
href="#ref-alyamiUniformSamplingDirected2016"
role="doc-biblioref">4</a>]</span>. However, it does not guarantee that
the resulting graph is planar, which we must ensure so that the edges
can be 3-coloured.</p>
<p>In practice, we use a standard algorithm <span class="citation"
<p>In practice, we use a standard algorithm <span class="citation"
data-cites="barberQuickhullAlgorithmConvex1996"> [<a
href="#ref-barberQuickhullAlgorithmConvex1996"
role="doc-biblioref">5</a>]</span> from Scipy <span class="citation"
role="doc-biblioref">5</a>]</span> from Scipy <span class="citation"
data-cites="virtanenSciPyFundamentalAlgorithms2020"> [<a
href="#ref-virtanenSciPyFundamentalAlgorithms2020"
role="doc-biblioref">6</a>]</span> which computes the Voronoi partition
@ -368,7 +403,7 @@ onto the plane without any edges crossing. Bridgeless graphs do not
contain any edges that, when removed, would partition the graph into
disconnected components.</p>
<p>This problem must be distinguished from that considered by the famous
four-colour theorem <span class="citation"
four-colour theorem <span class="citation"
data-cites="appelEveryPlanarMap1989"> [<a
href="#ref-appelEveryPlanarMap1989" role="doc-biblioref">7</a>]</span>.
The 4-colour theorem is concerned with assigning colours to the
@ -379,7 +414,7 @@ colouring.</p>
embedded onto the plane without any edges crossing. Here we are
concerned with Toroidal graphs, which can be embedded onto the torus
without any edges crossing. In fact, toroidal graphs require up to seven
colours <span class="citation"
colours <span class="citation"
data-cites="heawoodMapColouringTheorems"> [<a
href="#ref-heawoodMapColouringTheorems"
role="doc-biblioref">8</a>]</span>. The complete graph <span
@ -389,22 +424,22 @@ that requires seven colours.</p>
edge-colour any graph. An <span
class="math inline">\(\mathcal{O}(mn)\)</span> algorithm exists to do it
for a graph with <span class="math inline">\(m\)</span> edges and <span
class="math inline">\(n\)</span> vertices <span class="citation"
class="math inline">\(n\)</span> vertices <span class="citation"
data-cites="gEstimateChromaticClass1964"> [<a
href="#ref-gEstimateChromaticClass1964"
role="doc-biblioref">9</a>]</span>. Restricting ourselves to graphs with
<span class="math inline">\(\Delta = 3\)</span> like ours, those can be
four-edge-coloured in linear time <span class="citation"
four-edge-coloured in linear time <span class="citation"
data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"> [<a
href="#ref-skulrattanakulchai4edgecoloringGraphsMaximum2002"
role="doc-biblioref">10</a>]</span>.</p>
<p>However, three-edge-colouring them is more difficult. Cubic, planar,
bridgeless graphs can be three-edge-coloured if and only if they can be
four-face-coloured <span class="citation"
four-face-coloured <span class="citation"
data-cites="tait1880remarks"> [<a href="#ref-tait1880remarks"
role="doc-biblioref">11</a>]</span>. An <span
class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists here
<span class="citation" data-cites="robertson1996efficiently"> [<a
class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists
here <span class="citation" data-cites="robertson1996efficiently"> [<a
href="#ref-robertson1996efficiently"
role="doc-biblioref">12</a>]</span>. However, it is not clear whether
this extends to cubic, <strong>toroidal</strong> bridgeless graphs.</p>
@ -466,17 +501,17 @@ solver. A SAT problem is a set of statements about some number of
boolean variables , such as “<span class="math inline">\(x_1\)</span> or
not <span class="math inline">\(x_3\)</span> is true”, and looks for an
assignment <span class="math inline">\(x_i \in {0,1}\)</span> that
satisfies all the statements <span class="citation"
satisfies all the statements <span class="citation"
data-cites="Karp1972"> [<a href="#ref-Karp1972"
role="doc-biblioref">13</a>]</span>.</p>
<p>General purpose, high performance programs for solving SAT problems
have been an area of active research for decades <span class="citation"
have been an area of active research for decades <span class="citation"
data-cites="alounehComprehensiveStudyAnalysis2019"> [<a
href="#ref-alounehComprehensiveStudyAnalysis2019"
role="doc-biblioref">14</a>]</span>. Such programs are useful because,
by the Cook-Levin theorem, any NP problem can be encoded in polynomial
time as an instance of a SAT problem . This property is what makes SAT
one of the subset of NP problems called NP-Complete <span
one of the subset of NP problems called NP-Complete <span
class="citation"
data-cites="cookComplexityTheoremprovingProcedures1971 levin1973universal"> [<a
href="#ref-cookComplexityTheoremprovingProcedures1971"
@ -494,7 +529,7 @@ could be used to speed up its solution, using a SAT solver appears to be
a reasonable first method to try. As will be discussed later, this
turned out to work well enough and looking for a better solution was not
necessary.</p>
<p>We use a solver called <code>MiniSAT</code> <span class="citation"
<p>We use a solver called <code>MiniSAT</code> <span class="citation"
data-cites="imms-sat18"> [<a href="#ref-imms-sat18"
role="doc-biblioref">17</a>]</span>. Like most modern SAT solvers,
<code>MiniSAT</code> requires the input problem to be specified in
@ -554,7 +589,7 @@ a graph and assigns them a colour that is not already disallowed. This
does not work for our purposes because it is not designed to look for a
particular n-colouring. However, it does include the option of using a
heuristic function that determine the order in which vertices will be
coloured <span class="citation"
coloured <span class="citation"
data-cites="kosowski2004classical matulaSmallestlastOrderingClustering1983"> [<a
href="#ref-kosowski2004classical" role="doc-biblioref">18</a>,<a
href="#ref-matulaSmallestlastOrderingClustering1983"

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@ -200,10 +200,55 @@ image:
<script src="/assets/js/index.js"></script>
</head>
<body>
{% include header.html %}
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{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#results" id="toc-results">Results</a>
<ul>
<li><a href="#the-ground-state-flux-sector"
id="toc-the-ground-state-flux-sector">The Ground State Flux
Sector</a></li>
<li><a href="#spontaneous-chiral-symmetry-breaking"
id="toc-spontaneous-chiral-symmetry-breaking">Spontaneous Chiral
Symmetry Breaking</a></li>
<li><a href="#ground-state-phase-diagram"
id="toc-ground-state-phase-diagram">Ground State Phase Diagram</a>
<ul>
<li><a href="#is-it-abelian-or-non-abelian"
id="toc-is-it-abelian-or-non-abelian">Is it Abelian or
non-Abelian?</a></li>
<li><a href="#edge-modes" id="toc-edge-modes">Edge Modes</a></li>
</ul></li>
<li><a href="#anderson-transition-to-a-thermal-metal"
id="toc-anderson-transition-to-a-thermal-metal">Anderson Transition to a
Thermal Metal</a></li>
</ul></li>
<li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li>
<li><a href="#discussion" id="toc-discussion">Discussion</a>
<ul>
<li><a href="#limits-of-the-ground-state-conjecture"
id="toc-limits-of-the-ground-state-conjecture">Limits of the ground
state conjecture</a></li>
</ul></li>
<li><a href="#outlook" id="toc-outlook">Outlook</a>
<ul>
<li><a href="#experimental-realisations-and-signatures"
id="toc-experimental-realisations-and-signatures">Experimental
Realisations and Signatures</a></li>
<li><a href="#generalisations"
id="toc-generalisations">Generalisations</a></li>
</ul></li>
</ul>
{% endcapture %}
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{% include header.html extra=tableOfContents %}
<main>
<nav id="TOC" role="doc-toc">
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#results" id="toc-results">Results</a>
<ul>
@ -242,6 +287,7 @@ id="toc-generalisations">Generalisations</a></li>
</ul></li>
</ul>
</nav>
-->
<h1 id="results">Results</h1>
<h2 id="the-ground-state-flux-sector">The Ground State Flux Sector</h2>
<p>Here I will discuss the numerical evidence that our guess for the
@ -249,7 +295,7 @@ ground state flux sector is correct. We will do this by enumerating all
the flux sectors of many separate system realisations. However there are
some issues we will need to address to make this argument work.</p>
<p>We have two seemingly irreconcilable problems. Finite size effects
have a large energetic contribution for small systems <span
have a large energetic contribution for small systems <span
class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">1</a>]</span> so we would like to perform our
@ -308,7 +354,7 @@ relatively regular pattern for the imaginary fluxes with only a global
two-fold chiral degeneracy.</p>
<p>Thus, states with a fixed flux sector spontaneously break time
reversal symmetry. This was first described by Yao and Kivelson for a
translation invariant Kitaev model with odd sided plaquettes <span
translation invariant Kitaev model with odd sided plaquettes <span
class="citation" data-cites="Yao2011"> [<a href="#ref-Yao2011"
role="doc-biblioref">2</a>]</span>.</p>
<p>So we have flux sectors that come in degenerate pairs, where time
@ -348,9 +394,9 @@ straight lines <span class="math inline">\(|J^x| = |J^y| +
class="math inline">\(x,y,z\)</span>, shown as dotted line on ~<a
href="#fig:phase_diagram">1</a> (Right). We find that on the amorphous
lattice these boundaries exhibit an inward curvature, similar to
honeycomb Kitaev models with flux <span class="citation"
honeycomb Kitaev models with flux <span class="citation"
data-cites="Nasu_Thermal_2015"> [<a href="#ref-Nasu_Thermal_2015"
role="doc-biblioref">5</a>]</span> or bond <span class="citation"
role="doc-biblioref">5</a>]</span> or bond <span class="citation"
data-cites="knolle_dynamics_2016"> [<a href="#ref-knolle_dynamics_2016"
role="doc-biblioref">6</a>]</span> disorder.</p>
<div id="fig:phase_diagram" class="fignos">
@ -387,7 +433,7 @@ class="math inline">\(0\)</span> to <span class="math inline">\(\pm
later I’ll double check this with finite size scaling.</p>
<p>The next question is: do these phases support excitations with
Abelian or non-Abelian statistics? To answer that we turn to Chern
numbers <span class="citation"
numbers <span class="citation"
data-cites="berryQuantalPhaseFactors1984 simonHolonomyQuantumAdiabatic1983 thoulessQuantizedHallConductance1982"> [<a
href="#ref-berryQuantalPhaseFactors1984" role="doc-biblioref">7</a>–<a
href="#ref-thoulessQuantizedHallConductance1982"
@ -399,17 +445,17 @@ to its Chern number <strong>[citation]</strong>. However the Chern
number is only defined for the translation invariant case because it
relies on integrals defined in k-space.</p>
<p>A family of real space generalisations of the Chern number that work
for amorphous systems exist called local topological markers <span
for amorphous systems exist called local topological markers <span
class="citation"
data-cites="bianco_mapping_2011 Hastings_Almost_2010 mitchellAmorphousTopologicalInsulators2018"> [<a
href="#ref-bianco_mapping_2011" role="doc-biblioref">10</a>–<a
href="#ref-mitchellAmorphousTopologicalInsulators2018"
role="doc-biblioref">12</a>]</span> and indeed Kitaev defines one in his
original paper on the model <span class="citation"
original paper on the model <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">1</a>]</span>.</p>
<p>Here we use the crosshair marker of <span class="citation"
<p>Here we use the crosshair marker of <span class="citation"
data-cites="peru_preprint"> [<a href="#ref-peru_preprint"
role="doc-biblioref">13</a>]</span> because it works well on smaller
systems. We calculate the projector <span class="math inline">\(P =
@ -438,13 +484,14 @@ character of the phases.</p>
<p>In the A phase of the amorphous model we find that <span
class="math inline">\(\nu=0\)</span> and hence the excitations have
Abelian character, similar to the honeycomb model. This phase is thus
the amorphous analogue of the Abelian toric-code quantum spin liquid
<span class="citation" data-cites="kitaev_fault-tolerant_2003"> [<a
the amorphous analogue of the Abelian toric-code quantum spin
liquid <span class="citation"
data-cites="kitaev_fault-tolerant_2003"> [<a
href="#ref-kitaev_fault-tolerant_2003"
role="doc-biblioref">14</a>]</span>.</p>
<p>The B phase has <span class="math inline">\(\nu=\pm1\)</span> so is a
non-Abelian <em>chiral spin liquid</em> (CSL) similar to that of the
Yao-Kivelson model <span class="citation"
Yao-Kivelson model <span class="citation"
data-cites="yaoExactChiralSpin2007"> [<a
href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>]</span>.
The CSL state is the the magnetic analogue of the fractional quantum
@ -455,9 +502,9 @@ this phase.</p>
<img
src="/assets/thesis/amk_chapter/results/phase_diagram_chern/phase_diagram_chern.svg"
data-short-caption="Local Chern Markers" style="width:100.0%"
alt="Figure 2: (Center) The crosshair marker  [13], a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = \pm 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian." />
alt="Figure 2: (Center) The crosshair marker  [13], a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = \pm 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian." />
<figcaption aria-hidden="true"><span>Figure 2:</span> (Center) The
crosshair marker <span class="citation" data-cites="peru_preprint"> [<a
crosshair marker <span class="citation" data-cites="peru_preprint"> [<a
href="#ref-peru_preprint" role="doc-biblioref">13</a>]</span>, a local
topological marker, evaluated on the Amorphous Kitaev Model. The marker
is defined around a point, denoted by the dotted crosshair. Information
@ -480,7 +527,7 @@ that the isotropic phase is non-Abelian.</figcaption>
</div>
<h3 id="edge-modes">Edge Modes</h3>
<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"
role="doc-biblioref">15</a>]</span>. fig. <a
href="#fig:edge_modes">3</a> shows the probability density of one such
@ -517,31 +564,31 @@ states.</figcaption>
Thermal Metal</h2>
<p>Previous work on the honeycomb model at finite temperature has shown
that the B phase undergoes a thermal transition from a quantum spin
liquid phase a to a <strong>thermal metal</strong> phase <span
liquid phase a to a <strong>thermal metal</strong> phase <span
class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a>]</span>.</p>
<p>This happens because at finite temperature, thermal fluctuations lead
to spontaneous vortex-pair formation. As discussed previously these
fluxes are dressed by Majorana bounds states and the composite object is
an Ising-type non-Abelian anyon <span class="citation"
an Ising-type non-Abelian anyon <span class="citation"
data-cites="Beenakker2013"> [<a href="#ref-Beenakker2013"
role="doc-biblioref">17</a>]</span>. The interactions between these
anyons are oscillatory similar to the RKKY exchange and decay
exponentially with separation <span class="citation"
exponentially with separation <span class="citation"
data-cites="Laumann2012 Lahtinen_2011 lahtinenTopologicalLiquidNucleation2012"> [<a
href="#ref-Laumann2012" role="doc-biblioref">18</a>–<a
href="#ref-lahtinenTopologicalLiquidNucleation2012"
role="doc-biblioref">20</a>]</span>. At sufficient density, the anyons
hybridise to a macroscopically degenerate state known as <em>thermal
metal</em> <span class="citation" data-cites="Laumann2012"> [<a
metal</em> <span class="citation" data-cites="Laumann2012"> [<a
href="#ref-Laumann2012" role="doc-biblioref">18</a>]</span>. At close
range the oscillatory behaviour of the interactions can be modelled by a
random sign which forms the basis for a random matrix theory description
of the thermal metal state.</p>
<p>The amorphous chiral spin liquid undergoes the same form of Anderson
transition to a thermal metal state. Markov Chain Monte Carlo would be
necessary to simulate this in full detail <span class="citation"
necessary to simulate this in full detail <span class="citation"
data-cites="selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a>]</span> but in order to avoid that
@ -635,7 +682,7 @@ model onto a Majorana model with interactions that take random signs
which can itself be mapped onto a coarser lattice with lower energy
excitations and so on. This can be repeating indefinitely, showing the
model must have excitations at arbitrarily low energies in the
thermodynamic limit <span class="citation"
thermodynamic limit <span class="citation"
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000"
@ -650,10 +697,10 @@ field.</p>
src="/assets/thesis/amk_chapter/results/DOS_oscillations/DOS_oscillations.svg"
data-short-caption="Distinctive Oscillations in the Density of States"
style="width:100.0%"
alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state  [16,21]. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is \rho = 0.5 corresponding to the T = \infty limit." />
alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state  [16,21]. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is \rho = 0.5 corresponding to the T = \infty limit." />
<figcaption aria-hidden="true"><span>Figure 6:</span> Density of states
at high temperature showing the logarithmic divergence at zero energy
and oscillations characteristic of the thermal metal state <span
and oscillations characteristic of the thermal metal state <span
class="citation"
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019"
@ -713,20 +760,20 @@ Realisations and Signatures</h2>
<p>The obvious question is whether amorphous Kitaev materials could be
physically realised.</p>
<p>Most crystals can as exists in a metastable amorphous state if they
are cooled rapidly, freezing them into a disordered configuration <span
are cooled rapidly, freezing them into a disordered configuration <span
class="citation"
data-cites="Weaire1976 Petrakovski1981 Kaneyoshi2018"> [<a
href="#ref-Weaire1976" role="doc-biblioref">22</a>–<a
href="#ref-Kaneyoshi2018" role="doc-biblioref">24</a>]</span>. Indeed
quenching has been used by humans to control the hardness of steel or
iron for thousands of years. It would therefore be interesting to study
amorphous version of candidate Kitaev materials <span class="citation"
amorphous version of candidate Kitaev materials <span class="citation"
data-cites="trebstKitaevMaterials2022"> [<a
href="#ref-trebstKitaevMaterials2022"
role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a>]</span>
such as <span class="math inline">\(\alpha-\textrm{RuCl}_3\)</span> to
see whether they maintain even approximate fixed coordination number
locally as is the case with amorphous Silicon and Germanium <span
locally as is the case with amorphous Silicon and Germanium <span
class="citation" data-cites="Weaire1971 betteridge1973possible"> [<a
href="#ref-Weaire1971" role="doc-biblioref">25</a>,<a
href="#ref-betteridge1973possible"
@ -747,7 +794,7 @@ role="doc-biblioref">29</a>]</span>.</p>
<h2 id="generalisations">Generalisations</h2>
<p>The model presented here could be generalized in several ways.</p>
<p>First, it would be interesting to study the stability of the chiral
amorphous Kitaev QSL with respect to perturbations <span
amorphous Kitaev QSL with respect to perturbations\ <span
class="citation"
data-cites="Rau2014 Chaloupka2010 Chaloupka2013 Chaloupka2015 Winter2016"> [<a
href="#ref-Rau2014" role="doc-biblioref">30</a>–<a
@ -760,7 +807,7 @@ j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} +
\sigma_j\sigma_k\]</span> With a view to more realistic prospects of
observation, it would be interesting to see if the properties of the
Kitaev-Heisenberg model generalise from the honeycomb to the amorphous
case[<span class="citation" data-cites="Chaloupka2010"> [<a
case [<span class="citation" data-cites="Chaloupka2010"> [<a
href="#ref-Chaloupka2010" role="doc-biblioref">31</a>]</span>; <span
class="citation" data-cites="Chaloupka2015"> [<a
href="#ref-Chaloupka2015" role="doc-biblioref">33</a>]</span>; <span
@ -773,7 +820,7 @@ href="#ref-manousakisSpinTextonehalfHeisenberg1991"
role="doc-biblioref">37</a>]</span>;].</p>
<p>Finally it might be possible to look at generalizations to
higher-spin models or those on random networks with different
coordination numbers <span class="citation"
coordination numbers <span class="citation"
data-cites="Baskaran2008 Yao2009 Nussinov2009 Yao2011 Chua2011 Natori2020 Chulliparambil2020 Chulliparambil2021 Seifert2020 WangHaoranPRB2021 Wu2009"> [<a
href="#ref-Yao2011" role="doc-biblioref">2</a>,<a
href="#ref-Baskaran2008" role="doc-biblioref">38</a>–<a

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