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@ -204,43 +204,35 @@ image:
<main> <main>
<nav id="TOC" role="doc-toc"> <nav id="TOC" role="doc-toc">
<ul> <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-and-the-hubbard-model"
id="toc-mott-insulators-and-the-hubbard-model">Mott Insulators and The
Hubbard Model</a></li>
<li><a href="#outline" id="toc-outline">Outline</a></li> <li><a href="#outline" id="toc-outline">Outline</a></li>
</ul> </ul>
</nav> </nav>
<h1 id="interacting-quantum-many-body-systems">Interacting Quantum Many <p><strong>Interacting Quantum Many Body Systems</strong></p>
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 than often simpler to describe the behaviour of the group differently from
one would describe the individual objects. Consider a flock (technically the way one would describe the individual objects. Consider a flock of
called a <em>murmuration</em>) of starlings like fig. <a starlings like that of fig. <a href="#fig:Studland_Starlings">1</a>.
href="#fig:Studland_Starlings">1</a>. Watching the flock you’ll see that Watching the flock you’ll see that it has a distinct outline, that waves
it has a distinct outline, that waves of density will sometimes of density will sometimes propagate through the closely packed birds and
propagate through the closely packed birds and that the flock seems to that the flock seems to respond to predators as a distinct object. The
respond to predators as a distinct object. The natural description of natural description of this phenomena is couched in terms of the flock
this phenomena is couched in terms of the flock rather than the rather than of the individual birds.</p>
individual birds.</p> <p>The behaviours of the flock are an <em>emergent phenomena</em>. The
<p>The behaviours of the flock are an emergent phenomena. The starlings starlings are only interacting with their immediate six or seven
are only interacting with their immediate six or seven neighbours<span neighbours <span class="citation"
class="citation" data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"> [<a
data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"><sup><a
href="#ref-king2012murmurations" role="doc-biblioref">1</a>,<a href="#ref-king2012murmurations" role="doc-biblioref">1</a>,<a
href="#ref-balleriniInteractionRulingAnimal2008" href="#ref-balleriniInteractionRulingAnimal2008"
role="doc-biblioref">2</a></sup></span>. This is what a physicist would role="doc-biblioref">2</a>]</span>, what a physicist would call a
call a <em>local interaction</em>. There is much philosophical debate <em>local interaction</em>. There is much philosophical debate about how
about how exactly to define emergence<span class="citation" exactly to define emergence <span class="citation"
data-cites="andersonMoreDifferent1972 kivelsonDefiningEmergencePhysics2016"><sup><a data-cites="andersonMoreDifferent1972 kivelsonDefiningEmergencePhysics2016"> [<a
href="#ref-andersonMoreDifferent1972" role="doc-biblioref">3</a>,<a href="#ref-andersonMoreDifferent1972" role="doc-biblioref">3</a>,<a
href="#ref-kivelsonDefiningEmergencePhysics2016" href="#ref-kivelsonDefiningEmergencePhysics2016"
role="doc-biblioref">4</a></sup></span> but for our purposes it enough role="doc-biblioref">4</a>]</span> but for our purposes it enough to say
to say that emergence is the fact that the aggregate behaviour of many that emergence is the fact that the aggregate behaviour of many
interacting objects may be very different from the individual behaviour interacting objects may necessitate a description very different from
of those objects.</p> that of the individual objects.</p>
<div id="fig:Studland_Starlings" class="fignos"> <div id="fig:Studland_Starlings" class="fignos">
<figure> <figure>
<img src="/assets/thesis/intro_chapter/Studland_Starlings.jpeg" <img src="/assets/thesis/intro_chapter/Studland_Starlings.jpeg"
@ -253,17 +245,17 @@ href="creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA
3.0</a></figcaption> 3.0</a></figcaption>
</figure> </figure>
</div> </div>
<p>To give another example, our understanding of thermodynamics began <p>To give an example closer to the topic at hand, our understanding of
with bulk properties like heat, energy, pressure and temperature<span thermodynamics began with bulk properties like heat, energy, pressure
class="citation" and temperature <span class="citation"
data-cites="saslowHistoryThermodynamicsMissing2020"><sup><a data-cites="saslowHistoryThermodynamicsMissing2020"> [<a
href="#ref-saslowHistoryThermodynamicsMissing2020" href="#ref-saslowHistoryThermodynamicsMissing2020"
role="doc-biblioref">5</a></sup></span>. It was only later that we role="doc-biblioref">5</a>]</span>. It was only later that we gained an
gained an understanding of how these properties emerge from microscopic 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"><sup><a class="citation" data-cites="flammHistoryOutlookStatistical1998"> [<a
href="#ref-flammHistoryOutlookStatistical1998" href="#ref-flammHistoryOutlookStatistical1998"
role="doc-biblioref">6</a></sup></span>.</p> role="doc-biblioref">6</a>]</span>.</p>
<p>Condensed Matter is, at its heart, the study of what behaviours <p>Condensed Matter is, at its heart, the study of what behaviours
emerge from large numbers of interacting quantum objects at low energy. emerge from large numbers of interacting quantum objects at low energy.
When these three properties are present together: a large number of When these three properties are present together: a large number of
@ -273,60 +265,65 @@ these three ingredients nature builds all manner of weird and wonderful
materials.</p> materials.</p>
<p>Historically, we made initial headway in the study of many-body <p>Historically, we made initial headway in the study of many-body
systems, ignoring interactions and quantum properties. The ideal gas law 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"><sup><a data-cites="ashcroftSolidStatePhysics1976"> [<a
href="#ref-ashcroftSolidStatePhysics1976" href="#ref-ashcroftSolidStatePhysics1976"
role="doc-biblioref">7</a></sup></span> are good examples. Including 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"><sup><a class="citation" data-cites="isingBeitragZurTheorie1925"> [<a
href="#ref-isingBeitragZurTheorie1925" href="#ref-isingBeitragZurTheorie1925"
role="doc-biblioref">8</a></sup></span>, Landau theory<span role="doc-biblioref">8</a>]</span>, Landau theory <span class="citation"
class="citation" data-cites="landau2013fluid"><sup><a data-cites="landau2013fluid"> [<a href="#ref-landau2013fluid"
href="#ref-landau2013fluid" role="doc-biblioref">9</a></sup></span> and role="doc-biblioref">9</a>]</span> and the classical theory of phase
the classical theory of phase transitions<span class="citation" transitions <span class="citation"
data-cites="jaegerEhrenfestClassificationPhase1998"><sup><a data-cites="jaegerEhrenfestClassificationPhase1998"> [<a
href="#ref-jaegerEhrenfestClassificationPhase1998" href="#ref-jaegerEhrenfestClassificationPhase1998"
role="doc-biblioref">10</a></sup></span>. In contrast, condensed matter 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" class="citation"
data-cites="blochÜberQuantenmechanikElektronen1929"><sup><a data-cites="blochÜberQuantenmechanikElektronen1929"> [<a
href="#ref-blochÜberQuantenmechanikElektronen1929" href="#ref-blochÜberQuantenmechanikElektronen1929"
role="doc-biblioref">11</a></sup></span> predicted the properties of role="doc-biblioref">11</a>]</span> predicted the properties of
non-interacting electrons in crystal lattices, leading to band theory. non-interacting electrons in crystal lattices, leading to band theory.
In the same vein, advances were made in understanding the quantum In the same vein, advances were made in understanding the quantum
origins of magnetism, including ferromagnetism and origins of magnetism, including ferromagnetism and antiferromagnetism
antiferromagnetism<span class="citation" <span class="citation" data-cites="MagnetismCondensedMatter"> [<a
data-cites="MagnetismCondensedMatter"><sup><a
href="#ref-MagnetismCondensedMatter" href="#ref-MagnetismCondensedMatter"
role="doc-biblioref">12</a></sup></span>.</p> role="doc-biblioref">12</a>]</span>.</p>
<p>However, at some point we had to start on the interacting quantum <p>However, at some point we had to start on the interacting quantum
many body systems. Some phenomena cannot be understood without a taking many body systems. The properties of some materials cannot be understood
into account all three effects. The canonical examples are without a taking into account all three effects and these are
superconductivity<span class="citation" collectively called strongly correlated materials. The canonical
data-cites="MicroscopicTheorySuperconductivity"><sup><a examples are superconductivity <span class="citation"
data-cites="MicroscopicTheorySuperconductivity"> [<a
href="#ref-MicroscopicTheorySuperconductivity" href="#ref-MicroscopicTheorySuperconductivity"
role="doc-biblioref">13</a></sup></span>, the fractional quantum hall role="doc-biblioref">13</a>]</span>, the fractional quantum hall effect
effect<span class="citation" <span class="citation"
data-cites="feldmanFractionalChargeFractional2021"><sup><a data-cites="feldmanFractionalChargeFractional2021"> [<a
href="#ref-feldmanFractionalChargeFractional2021" href="#ref-feldmanFractionalChargeFractional2021"
role="doc-biblioref">14</a></sup></span> and the Mott insulators<span role="doc-biblioref">14</a>]</span> and the Mott insulators <span
class="citation" class="citation"
data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"><sup><a data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a
href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">15</a>,<a href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">15</a>,<a
href="#ref-fisherMottInsulatorsSpin1999" href="#ref-fisherMottInsulatorsSpin1999"
role="doc-biblioref">16</a></sup></span>. We will discuss the latter in role="doc-biblioref">16</a>]</span>. We’ll start by looking at the
more detail.</p> latter but shall see that there are many links between three topics.</p>
<p>Electrical conductivity, the bulk movement of electrons, requires <p><strong>Mott Insulators</strong></p>
both that there are electronic states very close in energy to the ground <p>Mott Insulators are remarkable because their electrical insulator
state and that those states are delocalised so that they can contribute properties come from electron-electron interactions. Electrical
to macroscopic transport. Band insulators are systems whose Fermi level conductivity, the bulk movement of electrons, requires both that there
falls within a gap in the density of states and thus fail the first are electronic states very close in energy to the ground state and that
criteria. Anderson Insulators have only localised electronic states near those states are delocalised so that they can contribute to macroscopic
the fermi level and therefore fail the second criteria. We will discuss transport. Band insulators are systems whose Fermi level falls within a
Anderson insulators and disorder in a later section.</p> gap in the density of states and thus fail the first criteria. Band
insulators derive their character from the characteristics of the
underlying lattice. Anderson Insulators have only localised electronic
states near the fermi level and therefore fail the second criteria. We
will discuss Anderson insulators and disorder in a later section.</p>
<p>Both band and Anderson insulators occur without electron-electron <p>Both band and Anderson insulators occur without electron-electron
interactions. Mott insulators, by contrast, are by these interactions interactions. Mott insulators, by contrast, require a many body picture
and hence elude band theory and single-particle methods.</p> to understand and thus elude band theory and single-particle
methods.</p>
<div id="fig:venn_diagram" class="fignos"> <div id="fig:venn_diagram" class="fignos">
<figure> <figure>
<img src="/assets/thesis/intro_chapter/venn_diagram.svg" <img src="/assets/thesis/intro_chapter/venn_diagram.svg"
@ -342,146 +339,202 @@ or indirectly. When taken together, these three properties can give rise
to what are called strongly correlated materials.</figcaption> to what are called strongly correlated materials.</figcaption>
</figure> </figure>
</div> </div>
<h1 id="mott-insulators-and-the-hubbard-model">Mott Insulators and The
Hubbard Model</h1>
<p>The theory of Mott insulators developed out of the observation that <p>The theory of Mott insulators developed out of the observation that
many transition metal oxides are erroneously predicted by band theory to many transition metal oxides are erroneously predicted by band theory to
be conductive<span class="citation" be conductive <span class="citation"
data-cites="boerSemiconductorsPartiallyCompletely1937"><sup><a data-cites="boerSemiconductorsPartiallyCompletely1937"> [<a
href="#ref-boerSemiconductorsPartiallyCompletely1937" href="#ref-boerSemiconductorsPartiallyCompletely1937"
role="doc-biblioref">17</a></sup></span> leading to the suggestion that 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"><sup><a class="citation" data-cites="mottDiscussionPaperBoer1937"> [<a
href="#ref-mottDiscussionPaperBoer1937" href="#ref-mottDiscussionPaperBoer1937"
role="doc-biblioref">18</a></sup></span>. Interest grew with the role="doc-biblioref">18</a>]</span>. Interest grew with the discovery of
discovery of high temperature superconductivity in the cuprates in high temperature superconductivity in the cuprates in 1986 <span
1986<span class="citation" class="citation"
data-cites="bednorzPossibleHighTcSuperconductivity1986"><sup><a data-cites="bednorzPossibleHighTcSuperconductivity1986"> [<a
href="#ref-bednorzPossibleHighTcSuperconductivity1986" href="#ref-bednorzPossibleHighTcSuperconductivity1986"
role="doc-biblioref">19</a></sup></span> which is believed to arise as role="doc-biblioref">19</a>]</span> which is believed to arise as the
the result of doping a Mott insulator state<span class="citation" result of a doped Mott insulator state <span class="citation"
data-cites="leeDopingMottInsulator2006"><sup><a data-cites="leeDopingMottInsulator2006"> [<a
href="#ref-leeDopingMottInsulator2006" href="#ref-leeDopingMottInsulator2006"
role="doc-biblioref">20</a></sup></span>.</p> role="doc-biblioref">20</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
model<span class="citation" <span class="citation"
data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"><sup><a data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"> [<a
href="#ref-gutzwillerEffectCorrelationFerromagnetism1963" href="#ref-gutzwillerEffectCorrelationFerromagnetism1963"
role="doc-biblioref">21</a>–<a role="doc-biblioref">21</a>–<a
href="#ref-hubbardj.ElectronCorrelationsNarrow1963" href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
role="doc-biblioref">23</a></sup></span> of <span role="doc-biblioref">23</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>
<p><span class="math display">\[ H = -t \sum_{\langle i,j \rangle <p><span class="math display">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j
\alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow} \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow}
n_{i\downarrow} - \mu \sum_{i,\alpha} n_{i\alpha}\]</span></p> n_{i\downarrow} - \mu \sum_{i,\alpha} n_{i\alpha}\]</span></p>
<p>where <span class="math inline">\(c^\dagger_{i\alpha}\)</span> <p>where <span class="math inline">\(c^\dagger_{i\alpha}\)</span>
creates a spin <span class="math inline">\(\alpha\)</span> electron at creates a spin <span class="math inline">\(\alpha\)</span> electron at
site <span class="math inline">\(i\)</span> and the number operator site <span class="math inline">\(i\)</span> and the number operator
<span class="math inline">\(n_{i\alpha}\)</span> measures the number of <span class="math inline">\(n_{i\alpha}\)</span> measures the number of
electrons with spin <span class="math inline">\(\alpha\)</span> at site electrons with spin <span class="math inline">\(\alpha\)</span> at site
<span class="math inline">\(i\)</span>. In the non-interacting limit <span class="math inline">\(i\)</span>. The sum runs over lattice
<span class="math inline">\(U &lt;&lt; t\)</span>, the model reduces to neighbours <span class="math inline">\(\langle i,j \rangle\)</span>
free fermions and the many-body ground state is a separable product of including both <span class="math inline">\(\langle i,j \rangle\)</span>
Bloch waves filled up to the Fermi level. In the interacting limit <span and <span class="math inline">\(\langle j,i \rangle\)</span> so that the
class="math inline">\(U &gt;&gt; t\)</span> on the other hand, the model is Hermition.</p>
system breaks up into a product of local moments, each in one the four <p>In the non-interacting limit <span class="math inline">\(U &lt;&lt;
states <span class="math inline">\(|0\rangle, |\uparrow\rangle, t\)</span>, the model reduces to free fermions and the many-body ground
|\downarrow\rangle, |\uparrow\downarrow\rangle\)</span> depending on the state is a separable product of Bloch waves filled up to the Fermi
filing.</p> level. In the interacting limit <span class="math inline">\(U &gt;&gt;
t\)</span> on the other hand, the system breaks up into a product of
local moments, each in one the four states <span
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 <p>The Mott insulating phase occurs at half filling <span
class="math inline">\(\mu = \tfrac{U}{2}\)</span> where there is one 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"><sup><a data-cites="hubbardElectronCorrelationsNarrow1964"> [<a
href="#ref-hubbardElectronCorrelationsNarrow1964" href="#ref-hubbardElectronCorrelationsNarrow1964"
role="doc-biblioref">24</a></sup></span>. Here the model can be role="doc-biblioref">24</a>]</span>. Here the model can be rewritten in
rewritten in a symmetric form <span class="math display">\[ H = -t a symmetric form <span class="math display">\[ H_{\mathrm{H}} = -t
\sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U
\sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} -
\tfrac{1}{2})\]</span></p> \tfrac{1}{2})\]</span></p>
<p>The basic reason that the half filled state is insulating seems is <p>The basic reason that the half filled state is insulating seems is
trivial. Any excitation must include states of double occupancy that trivial. Any excitation must include states of double occupancy that
cost energy <span class="math inline">\(U\)</span>, hence the system has cost energy <span class="math inline">\(U\)</span>, hence the system has
a finite bandgap and is an interaction driven Mott insulator. Originally a finite bandgap and is an interaction driven Mott insulator. Depending
it was proposed that antiferromagnetic order was a necessary condition on the lattice, the local moments may then order antiferromagnetically.
for the Mott insulator transition<span class="citation" Originally it was proposed that this antiferromagnetic order was the
data-cites="mottMetalInsulatorTransitions1990"><sup><a cause of the gap opening <span class="citation"
data-cites="mottMetalInsulatorTransitions1990"> [<a
href="#ref-mottMetalInsulatorTransitions1990" href="#ref-mottMetalInsulatorTransitions1990"
role="doc-biblioref">25</a></sup></span> but later examples were found role="doc-biblioref">25</a>]</span>. However, Mott insulators have been
without magnetic order <strong>cite</strong>.</p> found <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> without magnetic order. Instead the
local moments may form a highly entangled state known as a quantum spin
liquid, which will be discussed shortly.</p>
<p>Various theoretical treatments of the Hubbard model have been made, <p>Various theoretical treatments of the Hubbard model have been made,
including those based on Fermi liquid theory, mean field treatments, the 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"><sup><a data-cites="slaterMagneticEffectsHartreeFock1951"> [<a
href="#ref-slaterMagneticEffectsHartreeFock1951" href="#ref-slaterMagneticEffectsHartreeFock1951"
role="doc-biblioref">26</a></sup></span> and dynamical mean-field role="doc-biblioref">28</a>]</span> and dynamical mean-field theory
theory<span class="citation" <span class="citation"
data-cites="greinerQuantumPhaseTransition2002"><sup><a data-cites="greinerQuantumPhaseTransition2002"> [<a
href="#ref-greinerQuantumPhaseTransition2002" href="#ref-greinerQuantumPhaseTransition2002"
role="doc-biblioref">27</a></sup></span>. None of these approaches is role="doc-biblioref">29</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"><sup><a data-cites="mazurenkoColdatomFermiHubbard2017"> [<a
href="#ref-mazurenkoColdatomFermiHubbard2017" href="#ref-mazurenkoColdatomFermiHubbard2017"
role="doc-biblioref">28</a></sup></span>.</p> role="doc-biblioref">30</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 go down the rabbit hole of strongly correlated systems Second, we will look at quantum spin liquids and the Kitaev honeycomb
without magnetic order. This will lead us to Quantum spin liquids and model.</p>
the Kitaev honeycomb model.</p> <p><strong>The Falikov-Kimball Model</strong></p>
<p><strong>An exactly solvable model of the Mott Insulator</strong> - <p>Though not the original reason for its introduction, the
demonstrate mott insulator in hubbard model, briefly tease the falikov Falikov-Kimball (FK) model is the limit of the Hubbard model as the mass
kimball model in order to lay the ground work to talk about the falikov ratio of the spin up and spin down electron is taken to infinity. This
kimball model later</p> gives a model with two fermion species, one itinerant and one entirely
<ul> immobile. The number operators for the immobile fermions are therefore
<li>FK model has extensively many conserved charges which makes it conserved quantities and can be be treated like classical degrees of
tractable</li> freedom. For our purposes it will be useful to replace the immobile
<li>Disorder free localisation</li> fermions with a classical Ising background field <span
</ul> class="math inline">\(S_i = \pm1\)</span>.</p>
<p><strong>An exactly solvable Quantum Spin Liquid</strong> - <p><span class="math display">\[\begin{aligned}
relationship between mott insulators and spin liquids: the electrons in H_{\mathrm{FK}} = &amp; -\;t \sum_{\langle i,j \rangle}
a mott insulator form local moments that normally form an AFM ground c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
state but if they don’t, due to frustration or other reason, the local \tfrac{1}{2}). \\
moments may form a QSL at T=0 instead.<span class="citation" \end{aligned}\]</span></p>
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"><sup><a <p>Given that the physics of states near the metal-insulator (MI)
href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">29</a>,<a transition is still poorly understood <span class="citation"
data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a
href="#ref-belitzAndersonMottTransition1994"
role="doc-biblioref">31</a>,<a
href="#ref-baskoMetalInsulatorTransition2006"
role="doc-biblioref">32</a>]</span> the FK model provides a rich test
bed to explore interaction driven MI transition physics. Despite its
simplicity, the model has a rich phase diagram in <span
class="math inline">\(D \geq 2\)</span> dimensions. It shows an Mott
insulator transition even at high temperature, similar to the
corresponding Hubbard Model <span class="citation"
data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
href="#ref-brandtThermodynamicsCorrelationFunctions1989"
role="doc-biblioref">33</a>]</span>. In 1D, the ground state
phenomenology as a function of filling can be rich <span
class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a
href="#ref-gruberGroundStatesSpinless1990"
role="doc-biblioref">34</a>]</span> but the system is disordered for all
<span class="math inline">\(T &gt; 0\)</span> <span class="citation"
data-cites="kennedyItinerantElectronModel1986"> [<a
href="#ref-kennedyItinerantElectronModel1986"
role="doc-biblioref">35</a>]</span>. The model has also been a test-bed
for many-body methods, interest took off when an exact dynamical
mean-field theory solution in the infinite dimensional case was
found <span class="citation"
data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a
href="#ref-antipovCriticalExponentsStrongly2014"
role="doc-biblioref">36</a>–<a
href="#ref-herrmannNonequilibriumDynamicalCluster2016"
role="doc-biblioref">39</a>]</span>.</p>
<p>In Chapter 3 I will introduce a generalized FK model in one
dimension. With the addition of long-range interactions in the
background field, the model shows a similarly rich phase diagram. I use
an exact Markov chain Monte Carlo method to map the phase diagram and
compute the energy-resolved localization properties of the fermions. I
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>
<p>To turn to the other key topic of this thesis, we have discussed the
question of the magnetic ordering of local moments in the Mott
insulating state. The local moments may form an AFM ground state.
Alternatively they may fail to order even at zero temperature <span
class="citation"
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">26</a>,<a
href="#ref-ribakGaplessExcitationsGround2017" href="#ref-ribakGaplessExcitationsGround2017"
role="doc-biblioref">30</a></sup></span></p> role="doc-biblioref">27</a>]</span>, giving rise to what is known as a
quantum spin liquid (QSL) state.</p>
<p>QSLs are a long range entangled ground state of a highly
frustated</p>
<ul> <ul>
<li><p>QSLs introduced by anderson 1973<span class="citation" <li><p>QSLs introduced by anderson 1973 <span class="citation"
data-cites="andersonResonatingValenceBonds1973"><sup><a data-cites="andersonResonatingValenceBonds1973"> [<a
href="#ref-andersonResonatingValenceBonds1973" href="#ref-andersonResonatingValenceBonds1973"
role="doc-biblioref">31</a></sup></span></p></li> 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>Spin orbit effect is a relativistic effect that couples electron <li><p>Spin orbit effect is a relativistic effect that couples electron
spin to orbital angular moment. Very roughly, an electron sees the spin to orbital angular moment. Very roughly, an electron sees the
electric field of the nucleus as a magnetic field due to its movement electric field of the nucleus as a magnetic field due to its movement
and the electron spin couples to this.</p></li> and the electron spin couples to this. Can be strong in heavy
<li><p>can be string in heavy elements</p></li> elements</p></li>
<li><p>The Kitaev Model</p></li> <li><p>The Kitaev Model as a canonical QSL</p></li>
<li><p>Kitaev model has extensively many conserved charges too</p></li> <li><p>Kitaev model has extensively many conserved charges too</p></li>
<li><p>Frustration</p></li>
<li><p>anyons</p></li> <li><p>anyons</p></li>
<li><p>fractionalisation</p></li> <li><p>fractionalisation</p></li>
<li><p>Topology -&gt; GS degeneracy depends on the genus of the <li><p>Topology -&gt; GS degeneracy depends on the genus of the
surface</p></li> surface</p></li>
<li><p>the chern number</p></li> <li><p>the chern number</p></li>
<li><p>quasiparticles</p></li>
<li><p>topological order</p></li>
<li><p>protected edge states</p></li>
<li><p>Abelian and non-Abelian anyons</p></li>
</ul> </ul>
<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: From32." /> alt="Figure 3: From  [41]." />
<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"><sup><a class="citation" data-cites="TrebstPhysRep2022"> [<a
href="#ref-TrebstPhysRep2022" href="#ref-TrebstPhysRep2022"
role="doc-biblioref">32</a></sup></span>.</figcaption> role="doc-biblioref">41</a>]</span>.</figcaption>
</figure> </figure>
</div> </div>
<p>kinds of mott insulators: Mott-Heisenberg (AFM order below Néel <p>kinds of mott insulators: Mott-Heisenberg (AFM order below Néel
@ -507,263 +560,327 @@ designed to fill this gap and present the results.</p>
<p>Finally in chapter 4 I will summarise the results and discuss what <p>Finally in chapter 4 I will summarise the results and discuss what
implications they have for our understanding interacting many-body implications they have for our understanding interacting many-body
quantum systems.</p> quantum systems.</p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2" <div id="refs" class="references csl-bib-body" role="doc-bibliography">
role="doc-bibliography">
<div id="ref-king2012murmurations" class="csl-entry" <div id="ref-king2012murmurations" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div <div class="csl-left-margin">[1] </div><div class="csl-right-inline">A.
class="csl-right-inline">King, A. J. &amp; Sumpter, D. J. Murmurations. J. King and D. J. Sumpter, <em>Murmurations</em>, Current Biology
<em>Current Biology</em> <strong>22</strong>, R112–R114 (2012).</div> <strong>22</strong>, R112 (2012).</div>
</div> </div>
<div id="ref-balleriniInteractionRulingAnimal2008" class="csl-entry" <div id="ref-balleriniInteractionRulingAnimal2008" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div <div class="csl-left-margin">[2] </div><div class="csl-right-inline">M.
class="csl-right-inline">Ballerini, M. <em>et al.</em> <a Ballerini et al., <em><a
href="https://doi.org/10.1073/pnas.0711437105">Interaction ruling animal href="https://doi.org/10.1073/pnas.0711437105">Interaction Ruling Animal
collective behavior depends on topological rather than metric distance: Collective Behavior Depends on Topological Rather Than Metric Distance:
Evidence from a field study</a>. <em>Proceedings of the National Academy Evidence from a Field Study</a></em>, Proceedings of the National
of Sciences</em> <strong>105</strong>, 1232–1237 (2008).</div> Academy of Sciences <strong>105</strong>, 1232 (2008).</div>
</div> </div>
<div id="ref-andersonMoreDifferent1972" class="csl-entry" <div id="ref-andersonMoreDifferent1972" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div <div class="csl-left-margin">[3] </div><div class="csl-right-inline">P.
class="csl-right-inline">Anderson, P. W. <a W. Anderson, <em><a
href="https://doi.org/10.1126/science.177.4047.393">More Is href="https://doi.org/10.1126/science.177.4047.393">More Is
Different</a>. <em>Science</em> <strong>177</strong>, 393–396 Different</a></em>, Science <strong>177</strong>, 393 (1972).</div>
(1972).</div>
</div> </div>
<div id="ref-kivelsonDefiningEmergencePhysics2016" class="csl-entry" <div id="ref-kivelsonDefiningEmergencePhysics2016" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div <div class="csl-left-margin">[4] </div><div class="csl-right-inline">S.
class="csl-right-inline">Kivelson, S. &amp; Kivelson, S. A. <a Kivelson and S. A. Kivelson, <em><a
href="https://doi.org/10.1038/npjquantmats.2016.24">Defining emergence href="https://doi.org/10.1038/npjquantmats.2016.24">Defining Emergence
in physics</a>. <em>npj Quant Mater</em> <strong>1</strong>, 1–2 in Physics</a></em>, Npj Quant Mater <strong>1</strong>, 1 (2016).</div>
(2016).</div>
</div> </div>
<div id="ref-saslowHistoryThermodynamicsMissing2020" class="csl-entry" <div id="ref-saslowHistoryThermodynamicsMissing2020" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div <div class="csl-left-margin">[5] </div><div class="csl-right-inline">W.
class="csl-right-inline">Saslow, W. M. <a M. Saslow, <em><a href="https://doi.org/10.3390/e22010077">A History of
href="https://doi.org/10.3390/e22010077">A History of Thermodynamics: Thermodynamics: The Missing Manual</a></em>, Entropy (Basel)
The Missing Manual</a>. <em>Entropy (Basel)</em> <strong>22</strong>, 77 <strong>22</strong>, 77 (2020).</div>
(2020).</div>
</div> </div>
<div id="ref-flammHistoryOutlookStatistical1998" class="csl-entry" <div id="ref-flammHistoryOutlookStatistical1998" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div <div class="csl-left-margin">[6] </div><div class="csl-right-inline">D.
class="csl-right-inline">Flamm, D. History and outlook of statistical Flamm, <em><a
physics. Preprint at <a href="https://doi.org/10.48550/arXiv.physics/9803005">History and
href="https://doi.org/10.48550/arXiv.physics/9803005">https://doi.org/10.48550/arXiv.physics/9803005</a> Outlook of Statistical Physics</a></em>, arXiv:physics/9803005.</div>
(1998).</div>
</div> </div>
<div id="ref-ashcroftSolidStatePhysics1976" class="csl-entry" <div id="ref-ashcroftSolidStatePhysics1976" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div <div class="csl-left-margin">[7] </div><div class="csl-right-inline">N.
class="csl-right-inline">Ashcroft, N. W. &amp; Mermin, N. D. <em>Solid W. Ashcroft and N. D. Mermin, <em>Solid State Physics</em> (Holt,
State Physics</em>. (Holt, Rinehart and Winston, 1976).</div> Rinehart and Winston, 1976).</div>
</div> </div>
<div id="ref-isingBeitragZurTheorie1925" class="csl-entry" <div id="ref-isingBeitragZurTheorie1925" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div <div class="csl-left-margin">[8] </div><div class="csl-right-inline">E.
class="csl-right-inline">Ising, E. <a Ising, <em><a href="https://doi.org/10.1007/BF02980577">Beitrag zur
href="https://doi.org/10.1007/BF02980577">Beitrag zur Theorie des Theorie des Ferromagnetismus</a></em>, Z. Physik <strong>31</strong>,
Ferromagnetismus</a>. <em>Z. Physik</em> <strong>31</strong>, 253–258 253 (1925).</div>
(1925).</div>
</div> </div>
<div id="ref-landau2013fluid" class="csl-entry" role="doc-biblioentry"> <div id="ref-landau2013fluid" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div <div class="csl-left-margin">[9] </div><div class="csl-right-inline">L.
class="csl-right-inline">Landau, L. D. &amp; Lifshitz, E. M. <em>Fluid D. Landau and E. M. Lifshitz, <em>Fluid Mechanics: Landau and Lifshitz:
mechanics: Landau and lifshitz: Course of theoretical physics, volume Course of Theoretical Physics, Volume 6</em>, Vol. 6 (Elsevier,
6</em>. vol. 6 (Elsevier, 2013).</div> 2013).</div>
</div> </div>
<div id="ref-jaegerEhrenfestClassificationPhase1998" class="csl-entry" <div id="ref-jaegerEhrenfestClassificationPhase1998" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div <div class="csl-left-margin">[10] </div><div class="csl-right-inline">G.
class="csl-right-inline">Jaeger, G. <a Jaeger, <em><a href="https://doi.org/10.1007/s004070050021">The
href="https://doi.org/10.1007/s004070050021">The Ehrenfest Ehrenfest Classification of Phase Transitions: Introduction and
Classification of Phase Transitions: Introduction and Evolution</a>. Evolution</a></em>, Arch Hist Exact Sc. <strong>53</strong>, 51
<em>Arch Hist Exact Sc.</em> <strong>53</strong>, 51–81 (1998).</div> (1998).</div>
</div> </div>
<div id="ref-blochÜberQuantenmechanikElektronen1929" class="csl-entry" <div id="ref-blochÜberQuantenmechanikElektronen1929" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div <div class="csl-left-margin">[11] </div><div class="csl-right-inline">F.
class="csl-right-inline">Bloch, F. <a Bloch, <em><a href="https://doi.org/10.1007/BF01339455">Über die
href="https://doi.org/10.1007/BF01339455">Über die Quantenmechanik der Quantenmechanik der Elektronen in Kristallgittern</a></em>, Z. Physik
Elektronen in Kristallgittern</a>. <em>Z. Physik</em> <strong>52</strong>, 555 (1929).</div>
<strong>52</strong>, 555–600 (1929).</div>
</div> </div>
<div id="ref-MagnetismCondensedMatter" class="csl-entry" <div id="ref-MagnetismCondensedMatter" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div <div class="csl-left-margin">[12] </div><div class="csl-right-inline">S.
class="csl-right-inline">Blundell, S. <em>Magnetism in Condensed Blundell, <em>Magnetism in Condensed Matter</em> (OUP Oxford,
Matter</em>. (OUP Oxford, 2001).</div> 2001).</div>
</div> </div>
<div id="ref-MicroscopicTheorySuperconductivity" class="csl-entry" <div id="ref-MicroscopicTheorySuperconductivity" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div <div class="csl-left-margin">[13] </div><div class="csl-right-inline">J.
class="csl-right-inline">Bardeen, J., Cooper, L. N. &amp; Schrieffer, J. Bardeen, L. N. Cooper, and J. R. Schrieffer, <em><a
R. <a href="https://doi.org/10.1103/PhysRev.106.162">Microscopic Theory href="https://doi.org/10.1103/PhysRev.106.162">Microscopic Theory of
of Superconductivity</a>. <em>Phys. Rev.</em> <strong>106</strong>, Superconductivity</a></em>, Phys. Rev. <strong>106</strong>, 162
162–164 (1957).</div> (1957).</div>
</div> </div>
<div id="ref-feldmanFractionalChargeFractional2021" class="csl-entry" <div id="ref-feldmanFractionalChargeFractional2021" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div <div class="csl-left-margin">[14] </div><div class="csl-right-inline">D.
class="csl-right-inline">Feldman, D. E. &amp; Halperin, B. I. <a E. Feldman and B. I. Halperin, <em><a
href="https://doi.org/10.1088/1361-6633/ac03aa">Fractional charge and href="https://doi.org/10.1088/1361-6633/ac03aa">Fractional Charge and
fractional statistics in the quantum Hall effects</a>. <em>Rep. Prog. Fractional Statistics in the Quantum Hall Effects</a></em>, Rep. Prog.
Phys.</em> <strong>84</strong>, 076501 (2021).</div> Phys. <strong>84</strong>, 076501 (2021).</div>
</div> </div>
<div id="ref-mottBasisElectronTheory1949" class="csl-entry" <div id="ref-mottBasisElectronTheory1949" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div <div class="csl-left-margin">[15] </div><div class="csl-right-inline">N.
class="csl-right-inline">Mott, N. F. <a F. Mott, <em><a href="https://doi.org/10.1088/0370-1298/62/7/303">The
href="https://doi.org/10.1088/0370-1298/62/7/303">The Basis of the Basis of the Electron Theory of Metals, with Special Reference to the
Electron Theory of Metals, with Special Reference to the Transition Transition Metals</a></em>, Proc. Phys. Soc. A <strong>62</strong>, 416
Metals</a>. <em>Proc. Phys. Soc. A</em> <strong>62</strong>, 416–422
(1949).</div> (1949).</div>
</div> </div>
<div id="ref-fisherMottInsulatorsSpin1999" class="csl-entry" <div id="ref-fisherMottInsulatorsSpin1999" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div <div class="csl-left-margin">[16] </div><div class="csl-right-inline">M.
class="csl-right-inline">Fisher, M. P. A. <a P. A. Fisher, <em><a href="https://doi.org/10.1007/3-540-46637-1_8">Mott
href="https://doi.org/10.1007/3-540-46637-1_8">Mott insulators, Spin Insulators, Spin Liquids and Quantum Disordered
liquids and Quantum Disordered Superconductivity</a>. in <em>Aspects Superconductivity</a></em>, in <em>Aspects Topologiques de La Physique
topologiques de la physique en basse dimension. Topological aspects of En Basse Dimension. Topological Aspects of Low Dimensional Systems</em>,
low dimensional systems</em> (eds. Comtet, A., Jolicœur, T., Ouvry, S. edited by A. Comtet, T. Jolicœur, S. Ouvry, and F. David, Vol. 69
&amp; David, F.) vol. 69 575–641 (Springer Berlin Heidelberg, (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999), pp.
1999).</div> 575–641.</div>
</div> </div>
<div id="ref-boerSemiconductorsPartiallyCompletely1937" <div id="ref-boerSemiconductorsPartiallyCompletely1937"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div <div class="csl-left-margin">[17] </div><div class="csl-right-inline">J.
class="csl-right-inline">Boer, J. H. de &amp; Verwey, E. J. W. <a H. de Boer and E. J. W. Verwey, <em><a
href="https://doi.org/10.1088/0959-5309/49/4S/307">Semi-conductors with href="https://doi.org/10.1088/0959-5309/49/4S/307">Semi-Conductors with
partially and with completely filled &lt;script&gt;3d-lattice Partially and with Completely Filled &lt;Script&gt;3d-Lattice
bands&lt;/script&gt;</a>. <em>Proc. Phys. Soc.</em> <strong>49</strong>, Bands&lt;/Script&gt;</a></em>, Proc. Phys. Soc. <strong>49</strong>, 59
59–71 (1937).</div> (1937).</div>
</div> </div>
<div id="ref-mottDiscussionPaperBoer1937" class="csl-entry" <div id="ref-mottDiscussionPaperBoer1937" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div <div class="csl-left-margin">[18] </div><div class="csl-right-inline">N.
class="csl-right-inline">Mott, N. F. &amp; Peierls, R. <a F. Mott and R. Peierls, <em><a
href="https://doi.org/10.1088/0959-5309/49/4S/308">Discussion of the href="https://doi.org/10.1088/0959-5309/49/4S/308">Discussion of the
paper by de Boer and Verwey</a>. <em>Proc. Phys. Soc.</em> Paper by de Boer and Verwey</a></em>, Proc. Phys. Soc.
<strong>49</strong>, 72–73 (1937).</div> <strong>49</strong>, 72 (1937).</div>
</div> </div>
<div id="ref-bednorzPossibleHighTcSuperconductivity1986" <div id="ref-bednorzPossibleHighTcSuperconductivity1986"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">19. </div><div <div class="csl-left-margin">[19] </div><div class="csl-right-inline">J.
class="csl-right-inline">Bednorz, J. G. &amp; Müller, K. A. <a G. Bednorz and K. A. Müller, <em><a
href="https://doi.org/10.1007/BF01303701">Possible highTc href="https://doi.org/10.1007/BF01303701">Possible highTc
superconductivity in the Ba−La−Cu−O system</a>. <em>Z. Physik B - Superconductivity in the Ba−La−Cu−O System</a></em>, Z. Physik B -
Condensed Matter</em> <strong>64</strong>, 189–193 (1986).</div> Condensed Matter <strong>64</strong>, 189 (1986).</div>
</div> </div>
<div id="ref-leeDopingMottInsulator2006" class="csl-entry" <div id="ref-leeDopingMottInsulator2006" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">20. </div><div <div class="csl-left-margin">[20] </div><div class="csl-right-inline">P.
class="csl-right-inline">Lee, P. A., Nagaosa, N. &amp; Wen, X.-G. <a A. Lee, N. Nagaosa, and X.-G. Wen, <em><a
href="https://doi.org/10.1103/RevModPhys.78.17">Doping a Mott insulator: href="https://doi.org/10.1103/RevModPhys.78.17">Doping a Mott Insulator:
Physics of high-temperature superconductivity</a>. <em>Rev. Mod. Physics of High-Temperature Superconductivity</a></em>, Rev. Mod. Phys.
Phys.</em> <strong>78</strong>, 17–85 (2006).</div> <strong>78</strong>, 17 (2006).</div>
</div> </div>
<div id="ref-gutzwillerEffectCorrelationFerromagnetism1963" <div id="ref-gutzwillerEffectCorrelationFerromagnetism1963"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">21. </div><div <div class="csl-left-margin">[21] </div><div class="csl-right-inline">M.
class="csl-right-inline">Gutzwiller, M. C. <a C. Gutzwiller, <em><a
href="https://doi.org/10.1103/PhysRevLett.10.159">Effect of Correlation href="https://doi.org/10.1103/PhysRevLett.10.159">Effect of Correlation
on the Ferromagnetism of Transition Metals</a>. <em>Phys. Rev. on the Ferromagnetism of Transition Metals</a></em>, Phys. Rev. Lett.
Lett.</em> <strong>10</strong>, 159–162 (1963).</div> <strong>10</strong>, 159 (1963).</div>
</div> </div>
<div id="ref-kanamoriElectronCorrelationFerromagnetism1963" <div id="ref-kanamoriElectronCorrelationFerromagnetism1963"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">22. </div><div <div class="csl-left-margin">[22] </div><div class="csl-right-inline">J.
class="csl-right-inline">Kanamori, J. <a Kanamori, <em><a href="https://doi.org/10.1143/PTP.30.275">Electron
href="https://doi.org/10.1143/PTP.30.275">Electron Correlation and Correlation and Ferromagnetism of Transition Metals</a></em>, Progress
Ferromagnetism of Transition Metals</a>. <em>Progress of Theoretical of Theoretical Physics <strong>30</strong>, 275 (1963).</div>
Physics</em> <strong>30</strong>, 275–289 (1963).</div>
</div> </div>
<div id="ref-hubbardj.ElectronCorrelationsNarrow1963" class="csl-entry" <div id="ref-hubbardj.ElectronCorrelationsNarrow1963" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">23. </div><div <div class="csl-left-margin">[23] </div><div
class="csl-right-inline">Hubbard, J. <a class="csl-right-inline">Hubbard, J., <em><a
href="https://doi.org/10.1098/rspa.1963.0204">Electron correlations in href="https://doi.org/10.1098/rspa.1963.0204">Electron Correlations in
narrow energy bands</a>. <em>Proceedings of the Royal Society of London. Narrow Energy Bands</a></em>, Proceedings of the Royal Society of
Series A. Mathematical and Physical Sciences</em> <strong>276</strong>, London. Series A. Mathematical and Physical Sciences
238–257 (1963).</div> <strong>276</strong>, 238 (1963).</div>
</div> </div>
<div id="ref-hubbardElectronCorrelationsNarrow1964" class="csl-entry" <div id="ref-hubbardElectronCorrelationsNarrow1964" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">24. </div><div <div class="csl-left-margin">[24] </div><div class="csl-right-inline">J.
class="csl-right-inline">Hubbard, J. &amp; Flowers, B. H. <a Hubbard and B. H. Flowers, <em><a
href="https://doi.org/10.1098/rspa.1964.0190">Electron correlations in href="https://doi.org/10.1098/rspa.1964.0190">Electron Correlations in
narrow energy bands III. An improved solution</a>. <em>Proceedings of Narrow Energy Bands III. An Improved Solution</a></em>, Proceedings of
the Royal Society of London. Series A. Mathematical and Physical the Royal Society of London. Series A. Mathematical and Physical
Sciences</em> <strong>281</strong>, 401–419 (1964).</div> Sciences <strong>281</strong>, 401 (1964).</div>
</div> </div>
<div id="ref-mottMetalInsulatorTransitions1990" class="csl-entry" <div id="ref-mottMetalInsulatorTransitions1990" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">25. </div><div <div class="csl-left-margin">[25] </div><div class="csl-right-inline">N.
class="csl-right-inline">Mott, N. <em>Metal-Insulator Transitions</em>. Mott, <em><a href="https://doi.org/10.1201/b12795">Metal-Insulator
(CRC Press, 1990). doi:<a Transitions</a></em> (CRC Press, London, 1990).</div>
href="https://doi.org/10.1201/b12795">10.1201/b12795</a>.</div>
</div>
<div id="ref-slaterMagneticEffectsHartreeFock1951" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">26. </div><div
class="csl-right-inline">Slater, J. C. <a
href="https://doi.org/10.1103/PhysRev.82.538">Magnetic Effects and the
Hartree-Fock Equation</a>. <em>Phys. Rev.</em> <strong>82</strong>,
538–541 (1951).</div>
</div>
<div id="ref-greinerQuantumPhaseTransition2002" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">27. </div><div
class="csl-right-inline">Greiner, M., Mandel, O., Esslinger, T., Hänsch,
T. W. &amp; Bloch, I. <a href="https://doi.org/10.1038/415039a">Quantum
phase transition from a superfluid to a Mott insulator in a gas of
ultracold atoms</a>. <em>Nature</em> <strong>415</strong>, 39
(2002).</div>
</div>
<div id="ref-mazurenkoColdatomFermiHubbard2017" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">28. </div><div
class="csl-right-inline">Mazurenko, A. <em>et al.</em> <a
href="https://doi.org/10.1038/nature22362">A cold-atom Fermi–Hubbard
antiferromagnet</a>. <em>Nature</em> <strong>545</strong>, 462–466
(2017).</div>
</div> </div>
<div id="ref-law1TTaS2QuantumSpin2017" class="csl-entry" <div id="ref-law1TTaS2QuantumSpin2017" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">29. </div><div <div class="csl-left-margin">[26] </div><div class="csl-right-inline">K.
class="csl-right-inline">Law, K. T. &amp; Lee, P. A. <a T. Law and P. A. Lee, <em><a
href="https://doi.org/10.1073/pnas.1706769114">1T-TaS2 as a quantum spin href="https://doi.org/10.1073/pnas.1706769114">1t-TaS2 as a Quantum Spin
liquid</a>. <em>Proceedings of the National Academy of Sciences</em> Liquid</a></em>, Proceedings of the National Academy of Sciences
<strong>114</strong>, 6996–7000 (2017).</div> <strong>114</strong>, 6996 (2017).</div>
</div> </div>
<div id="ref-ribakGaplessExcitationsGround2017" class="csl-entry" <div id="ref-ribakGaplessExcitationsGround2017" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">30. </div><div <div class="csl-left-margin">[27] </div><div class="csl-right-inline">A.
class="csl-right-inline">Ribak, A. <em>et al.</em> <a Ribak, I. Silber, C. Baines, K. Chashka, Z. Salman, Y. Dagan, and A.
href="https://doi.org/10.1103/PhysRevB.96.195131">Gapless excitations in Kanigel, <em><a
the ground state of href="https://doi.org/10.1103/PhysRevB.96.195131">Gapless Excitations in
&lt;script&gt;$1T\text{\ensuremath{-}}{\mathrm{TaS}}_{2}$&lt;/script&gt;</a>. the Ground State of
<em>Phys. Rev. B</em> <strong>96</strong>, 195131 (2017).</div> &lt;Script&gt;$1T\text{\ensuremath{-}}{\Mathrm{TaS}}_{2}$&lt;/Script&gt;</a></em>,
Phys. Rev. B <strong>96</strong>, 195131 (2017).</div>
</div>
<div id="ref-slaterMagneticEffectsHartreeFock1951" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">J.
C. Slater, <em><a href="https://doi.org/10.1103/PhysRev.82.538">Magnetic
Effects and the Hartree-Fock Equation</a></em>, Phys. Rev.
<strong>82</strong>, 538 (1951).</div>
</div>
<div id="ref-greinerQuantumPhaseTransition2002" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">M.
Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, <em><a
href="https://doi.org/10.1038/415039a">Quantum Phase Transition from a
Superfluid to a Mott Insulator in a Gas of Ultracold Atoms</a></em>,
Nature <strong>415</strong>, 39 (2002).</div>
</div>
<div id="ref-mazurenkoColdatomFermiHubbard2017" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[30] </div><div class="csl-right-inline">A.
Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt,
F. Grusdt, E. Demler, D. Greif, and M. Greiner, <em><a
href="https://doi.org/10.1038/nature22362">A Cold-Atom Fermi–Hubbard
Antiferromagnet</a></em>, Nature <strong>545</strong>, 462 (2017).</div>
</div>
<div id="ref-belitzAndersonMottTransition1994" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">D.
Belitz and T. R. Kirkpatrick, <em><a
href="https://doi.org/10.1103/RevModPhys.66.261">The Anderson-Mott
Transition</a></em>, Rev. Mod. Phys. <strong>66</strong>, 261
(1994).</div>
</div>
<div id="ref-baskoMetalInsulatorTransition2006" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">D.
M. Basko, I. L. Aleiner, and B. L. Altshuler, <em><a
href="https://doi.org/10.1016/j.aop.2005.11.014">Metal–Insulator
Transition in a Weakly Interacting Many-Electron System with Localized
Single-Particle States</a></em>, Annals of Physics <strong>321</strong>,
1126 (2006).</div>
</div>
<div id="ref-brandtThermodynamicsCorrelationFunctions1989"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[33] </div><div class="csl-right-inline">U.
Brandt and C. Mielsch, <em><a
href="https://doi.org/10.1007/BF01321824">Thermodynamics and Correlation
Functions of the Falicov-Kimball Model in Large Dimensions</a></em>, Z.
Physik B - Condensed Matter <strong>75</strong>, 365 (1989).</div>
</div>
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[34] </div><div class="csl-right-inline">C.
Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a
href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the
Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B
<strong>41</strong>, 2198 (1990).</div>
</div>
<div id="ref-kennedyItinerantElectronModel1986" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[35] </div><div class="csl-right-inline">T.
Kennedy and E. H. Lieb, <em><a
href="https://doi.org/10.1016/0378-4371(86)90188-3">An Itinerant
Electron Model with Crystalline or Magnetic Long Range Order</a></em>,
Physica A: Statistical Mechanics and Its Applications
<strong>138</strong>, 320 (1986).</div>
</div>
<div id="ref-antipovCriticalExponentsStrongly2014" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[36] </div><div class="csl-right-inline">A.
E. Antipov, E. Gull, and S. Kirchner, <em><a
href="https://doi.org/10.1103/PhysRevLett.112.226401">Critical Exponents
of Strongly Correlated Fermion Systems from Diagrammatic Multiscale
Methods</a></em>, Phys. Rev. Lett. <strong>112</strong>, 226401
(2014).</div>
</div>
<div id="ref-ribicNonlocalCorrelationsSpectral2016" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[37] </div><div class="csl-right-inline">T.
Ribic, G. Rohringer, and K. Held, <em><a
href="https://doi.org/10.1103/PhysRevB.93.195105">Nonlocal Correlations
and Spectral Properties of the Falicov-Kimball Model</a></em>, Phys.
Rev. B <strong>93</strong>, 195105 (2016).</div>
</div>
<div id="ref-freericksExactDynamicalMeanfield2003" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[38] </div><div class="csl-right-inline">J.
K. Freericks and V. Zlatić, <em><a
href="https://doi.org/10.1103/RevModPhys.75.1333">Exact Dynamical
Mean-Field Theory of the Falicov-Kimball Model</a></em>, Rev. Mod. Phys.
<strong>75</strong>, 1333 (2003).</div>
</div>
<div id="ref-herrmannNonequilibriumDynamicalCluster2016"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[39] </div><div class="csl-right-inline">A.
J. Herrmann, N. Tsuji, M. Eckstein, and P. Werner, <em><a
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>
<div id="ref-andersonResonatingValenceBonds1973" class="csl-entry" <div id="ref-andersonResonatingValenceBonds1973" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">31. </div><div <div class="csl-left-margin">[40] </div><div class="csl-right-inline">P.
class="csl-right-inline">Anderson, P. W. <a W. Anderson, <em><a
href="https://doi.org/10.1016/0025-5408(73)90167-0">Resonating valence href="https://doi.org/10.1016/0025-5408(73)90167-0">Resonating Valence
bonds: A new kind of insulator?</a> <em>Materials Research Bulletin</em> Bonds: A New Kind of Insulator?</a></em>, Materials Research Bulletin
<strong>8</strong>, 153–160 (1973).</div> <strong>8</strong>, 153 (1973).</div>
</div> </div>
<div id="ref-TrebstPhysRep2022" class="csl-entry" <div id="ref-TrebstPhysRep2022" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">32. </div><div <div class="csl-left-margin">[41] </div><div class="csl-right-inline">S.
class="csl-right-inline">Trebst, S. &amp; Hickey, C. <a Trebst and C. Hickey, <em><a
href="https://doi.org/10.1016/j.physrep.2021.11.003">Kitaev href="https://doi.org/10.1016/j.physrep.2021.11.003">Kitaev
materials</a>. <em>Physics Reports</em> <strong>950</strong>, 1–37 Materials</a></em>, Physics Reports <strong>950</strong>, 1
(2022).</div> (2022).</div>
</div> </div>
</div> </div>

17
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@ -319,10 +319,10 @@ Majorana <span class="math inline">\(c_i\)</span> per site.</figcaption>
</div> </div>
<ul> <ul>
<li>strong spin orbit coupling yields spatial anisotropic spin exchange <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"><sup><a data-cites="kugelJahnTellerEffectMagnetism1982"> [<a
href="#ref-kugelJahnTellerEffectMagnetism1982" href="#ref-kugelJahnTellerEffectMagnetism1982"
role="doc-biblioref">1</a></sup></span></li> role="doc-biblioref">1</a>]</span></li>
<li>spin model of the Kitaev model is one</li> <li>spin model of the Kitaev model is one</li>
<li>has extensively many conserved fluxes</li> <li>has extensively many conserved fluxes</li>
<li></li> <li></li>
@ -335,15 +335,14 @@ Chern number</h2>
<h2 id="phase-diagram">Phase Diagram</h2> <h2 id="phase-diagram">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>
<div id="refs" class="references csl-bib-body" data-line-spacing="2" <div id="refs" class="references csl-bib-body" role="doc-bibliography">
role="doc-bibliography">
<div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry" <div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div <div class="csl-left-margin">[1] </div><div class="csl-right-inline">K.
class="csl-right-inline">Kugel’, K. I. &amp; Khomskiĭ, D. I. <a I. Kugel’ and D. I. Khomskiĭ, <em><a
href="https://doi.org/10.1070/PU1982v025n04ABEH004537">The Jahn-Teller href="https://doi.org/10.1070/PU1982v025n04ABEH004537">The Jahn-Teller
effect and magnetism: transition metal compounds</a>. <em>Sov. Phys. Effect and Magnetism: Transition Metal Compounds</a></em>, Sov. Phys.
Usp.</em> <strong>25</strong>, 231 (1982).</div> Usp. <strong>25</strong>, 231 (1982).</div>
</div> </div>
</div> </div>
</main> </main>

319
_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html
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@ -360,10 +360,10 @@ 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 direct sampling methods range from the trivial: take n random bits to
generate integers uniformly between 0 and <span generate integers uniformly between 0 and <span
class="math inline">\(2^n\)</span> to more complex methods such as 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"><sup><a data-cites="devroyeRandomSampling1986"> [<a
href="#ref-devroyeRandomSampling1986" href="#ref-devroyeRandomSampling1986"
role="doc-biblioref">1</a></sup></span>.</p> role="doc-biblioref">1</a>]</span>.</p>
<p>In physics the distribution we usually want to sample from is the <p>In physics the distribution we usually want to sample from is the
Boltzmann probability over states of the system <span Boltzmann probability over states of the system <span
class="math inline">\(S\)</span>: <span class="math display">\[ class="math inline">\(S\)</span>: <span class="math display">\[
@ -383,9 +383,9 @@ with system size. Even if we could calculate <span
class="math inline">\(\mathcal{Z}\)</span>, sampling from an class="math inline">\(\mathcal{Z}\)</span>, sampling from an
exponentially large number of options quickly become tricky. This kind exponentially large number of options quickly become tricky. This kind
of problem happens in many other disciplines too, particularly when 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"><sup><a href="#ref-BMCP2021" class="citation" data-cites="BMCP2021"> [<a href="#ref-BMCP2021"
role="doc-biblioref">2</a></sup></span>.</p> role="doc-biblioref">2</a>]</span>.</p>
<h2 id="markov-chains">Markov Chains</h2> <h2 id="markov-chains">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
@ -393,11 +393,11 @@ instead.</p>
<p>MCMC defines a weighted random walk over the states <span <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 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 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"><sup><a data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"> [<a
href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">3</a>–<a href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">3</a>–<a
href="#ref-wolffMonteCarloErrors2004" href="#ref-wolffMonteCarloErrors2004"
role="doc-biblioref">5</a></sup></span>.</p> role="doc-biblioref">5</a>]</span>.</p>
<p>In a physics context this lets us evaluate any observable with a mean <p>In a physics context this lets us evaluate any observable with a mean
over the states visited by the walk. <span over the states visited by the walk. <span
class="math display">\[\begin{aligned} class="math display">\[\begin{aligned}
@ -407,9 +407,9 @@ class="math display">\[\begin{aligned}
<p>The choice of the transition function for MCMC is under-determined as <p>The choice of the transition function for MCMC is under-determined as
one only needs to satisfy a set of balance conditions for which there one only needs to satisfy a set of balance conditions for which there
are many solutions <span class="citation" are many solutions <span class="citation"
data-cites="kellyReversibilityStochasticNetworks1981"><sup><a data-cites="kellyReversibilityStochasticNetworks1981"> [<a
href="#ref-kellyReversibilityStochasticNetworks1981" href="#ref-kellyReversibilityStochasticNetworks1981"
role="doc-biblioref">6</a></sup></span>.</p> role="doc-biblioref">6</a>]</span>.</p>
<h2 id="application-to-the-fk-model">Application to the FK Model</h2> <h2 id="application-to-the-fk-model">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>
@ -447,11 +447,11 @@ F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}
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
class="math inline">\(\{x: x \in S\}\)</span> and a free energy <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"><sup><a data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998" href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a></sup></span>. The thermal expectation value role="doc-biblioref">7</a>]</span>. The thermal expectation value is
is defined via a Boltzmann weighted sum over the entire states: <span defined via a Boltzmann weighted sum over the entire states: <span
class="math display">\[ class="math display">\[
\begin{aligned} \begin{aligned}
\expval{O} &amp;= \frac{1}{\mathcal{Z}} \sum_{x \in S} O(x) P(x) \\ \expval{O} &amp;= \frac{1}{\mathcal{Z}} \sum_{x \in S} O(x) P(x) \\
@ -526,10 +526,10 @@ P(x) \mathcal{T}(x \rightarrow x&#39;) = P(x&#39;) \mathcal{T}(x&#39;
\rightarrow x) \rightarrow x)
\]</span> % In practice most algorithms are constructed to satisfy \]</span> % In practice most algorithms are constructed to satisfy
detailed balance though there are arguments that relaxing the condition 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"><sup><a data-cites="kapferSamplingPolytopeHarddisk2013"> [<a
href="#ref-kapferSamplingPolytopeHarddisk2013" href="#ref-kapferSamplingPolytopeHarddisk2013"
role="doc-biblioref">8</a></sup></span>.</p> role="doc-biblioref">8</a>]</span>.</p>
<p>The goal of MCMC is then to choose <span <p>The goal of MCMC is then to choose <span
class="math inline">\(\mathcal{T}\)</span> so that it has the desired class="math inline">\(\mathcal{T}\)</span> so that it has the desired
thermal distribution <span class="math inline">\(P(x)\)</span> as its thermal distribution <span class="math inline">\(P(x)\)</span> as its
@ -558,10 +558,10 @@ 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 <p>The Metropolis-Hasting algorithm is a slight extension of the
original Metropolis algorithm that allows for non-symmetric proposal original Metropolis algorithm that allows for non-symmetric proposal
distributions $q(xx’) q(x’x) $. It can be derived starting from detailed distributions $q(xx’) q(x’x) $. It can be derived starting from detailed
balance<span class="citation" balance <span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"><sup><a data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998" href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a></sup></span>: <span role="doc-biblioref">7</a>]</span>: <span
class="math display">\[\begin{aligned} class="math display">\[\begin{aligned}
P(x)\mathcal{T}(x \to x&#39;) &amp;= P(x&#39;)\mathcal{T}(x&#39; \to x) P(x)\mathcal{T}(x \to x&#39;) &amp;= P(x&#39;)\mathcal{T}(x&#39; \to x)
\\ \\
@ -671,11 +671,11 @@ 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 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 then the walk will take longer to explore the state space and the
samples will be highly correlated. Ideal values for the acceptance rate 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"><sup><a data-cites="robertsWeakConvergenceOptimal1997"> [<a
href="#ref-robertsWeakConvergenceOptimal1997" href="#ref-robertsWeakConvergenceOptimal1997"
role="doc-biblioref">9</a></sup></span>. Here we monitor the acceptance role="doc-biblioref">9</a>]</span>. Here we monitor the acceptance rate
rate and if it is too high we re-run the MCMC with a modified proposal and if it is too high we re-run the MCMC with a modified proposal
distribution that has a chance to propose moves that flip multiple sites distribution that has a chance to propose moves that flip multiple sites
at a time.</p> at a time.</p>
<p>In addition we exploit the particle-hole symmetry of the problem by <p>In addition we exploit the particle-hole symmetry of the problem by
@ -686,10 +686,10 @@ produce a state at or near the energy of the current one.</p>
<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
gleaned from Ref.<span class="citation" gleaned from Ref. <span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"><sup><a data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998" href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a></sup></span>: <span class="math display">\[ role="doc-biblioref">7</a>]</span>: <span class="math display">\[
\begin{aligned} \begin{aligned}
q(k \to k&#39;) &amp;= \min\left(1, e^{\beta (H^{k&#39;} - H^k)}\right) q(k \to k&#39;) &amp;= \min\left(1, e^{\beta (H^{k&#39;} - H^k)}\right)
\\ \\
@ -700,12 +700,11 @@ without performing the diagonalisation at no cost to the accuracy of the
MCMC method.</p> MCMC method.</p>
<p>An extension of this idea is to try to define a classical model with <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 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"><sup><a data-cites="huangAcceleratedMonteCarlo2017"> [<a
href="#ref-huangAcceleratedMonteCarlo2017" href="#ref-huangAcceleratedMonteCarlo2017"
role="doc-biblioref">10</a></sup></span> does this with restricted role="doc-biblioref">10</a>]</span> does this with restricted Boltzmann
Boltzmann machines whose form is very similar to a classical spin machines whose form is very similar to a classical spin model.</p>
model.</p>
<h2 id="scaling">Scaling</h2> <h2 id="scaling">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
@ -726,12 +725,12 @@ 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 = \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 against temperature can be used as a diagnostic for the existence of a
phase transition. If multiple such curves are plotted for different phase transition. If multiple such curves are plotted for different
system sizes, a crossing indicates the location of a critical point<span system sizes, a crossing indicates the location of a critical point
class="citation" <span class="citation"
data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"><sup><a 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></sup></span>.</p> role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a>]</span>.</p>
<h2 id="markov-chain-monte-carlo-in-practice">Markov Chain Monte-Carlo <h2 id="markov-chain-monte-carlo-in-practice">Markov Chain Monte-Carlo
in Practice</h2> in Practice</h2>
<h3 id="quick-intro-to-mcmc">Quick Intro to MCMC</h3> <h3 id="quick-intro-to-mcmc">Quick Intro to MCMC</h3>
@ -758,13 +757,13 @@ very expensive operation!~\footnote{The effort involved in exact
diagonalisation scales like <span class="math inline">\(N^2\)</span> for diagonalisation scales like <span class="math inline">\(N^2\)</span> for
systems with a tri-diagonal matrix representation (open boundary systems with a tri-diagonal matrix representation (open boundary
conditions and nearest neighbour hopping) and like <span 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" class="citation"
data-cites="bolchQueueingNetworksMarkov2006 usmaniInversionTridiagonalJacobi1994"><sup><a data-cites="bolchQueueingNetworksMarkov2006 usmaniInversionTridiagonalJacobi1994"> [<a
href="#ref-bolchQueueingNetworksMarkov2006" href="#ref-bolchQueueingNetworksMarkov2006"
role="doc-biblioref">12</a>,<a role="doc-biblioref">12</a>,<a
href="#ref-usmaniInversionTridiagonalJacobi1994" href="#ref-usmaniInversionTridiagonalJacobi1994"
role="doc-biblioref">13</a></sup></span>.</p> role="doc-biblioref">13</a>]</span>.</p>
<p>c</p> <p>c</p>
<p>MCMC sidesteps these issues by defining a random walk that focuses on <p>MCMC sidesteps these issues by defining a random walk that focuses on
the states with the greatest Boltzmann weight. At low temperatures this the states with the greatest Boltzmann weight. At low temperatures this
@ -878,10 +877,10 @@ auto-correlation time <span class="math inline">\(\tau(O)\)</span>
informally as the number of MCMC samples of some observable O that are informally as the number of MCMC samples of some observable O that are
statistically equal to one independent sample or equivalently as the statistically equal to one independent sample or equivalently as the
number of MCMC steps after which the samples are correlated below some 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"><sup><a data-cites="krauthIntroductionMonteCarlo1996"> [<a
href="#ref-krauthIntroductionMonteCarlo1996" href="#ref-krauthIntroductionMonteCarlo1996"
role="doc-biblioref">14</a></sup></span> for a more rigorous definition role="doc-biblioref">14</a>]</span> for a more rigorous definition
involving a sum over the auto-correlation function. The auto-correlation involving a sum over the auto-correlation function. The auto-correlation
time is generally shorter than the convergence time so it therefore time is generally shorter than the convergence time so it therefore
makes sense from an efficiency standpoint to run a single walker for makes sense from an efficiency standpoint to run a single walker for
@ -960,28 +959,28 @@ than the current state.</p>
<h2 id="two-step-trick">Two Step Trick</h2> <h2 id="two-step-trick">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"><sup><a data-cites="hastingsMonteCarloSampling1970"> [<a
href="#ref-hastingsMonteCarloSampling1970" href="#ref-hastingsMonteCarloSampling1970"
role="doc-biblioref">15</a></sup></span> gleaned from Krauth <span role="doc-biblioref">15</a>]</span> gleaned from Krauth <span
class="citation" data-cites="krauthIntroductionMonteCarlo1998"><sup><a class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998" href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a></sup></span>.</p> role="doc-biblioref">7</a>]</span>.</p>
<p>In our computations <span class="citation" <p>In our computations <span class="citation"
data-cites="hodsonMCMCFKModel2021"><sup><a data-cites="hodsonMCMCFKModel2021"> [<a
href="#ref-hodsonMCMCFKModel2021" href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">16</a>]</span> we
role="doc-biblioref">16</a></sup></span> we employ a modification of the employ a modification of the algorithm which is based on the observation
algorithm which is based on the observation that the free energy of the that the free energy of the FK system is composed of a classical part
FK system is composed of a classical part which is much quicker to which is much quicker to compute than the quantum part. Hence, we can
compute than the quantum part. Hence, we can obtain a computational obtain a computational speedup by first considering the value of the
speedup by first considering the value of the classical energy classical energy difference <span class="math inline">\(\Delta
difference <span class="math inline">\(\Delta H_s\)</span> and rejecting H_s\)</span> and rejecting the transition if the former is too high. We
the transition if the former is too high. We only compute the quantum only compute the quantum energy difference <span
energy difference <span class="math inline">\(\Delta F_c\)</span> if the class="math inline">\(\Delta F_c\)</span> if the transition is accepted.
transition is accepted. We then perform a second rejection sampling step We then perform a second rejection sampling step based upon it. This
based upon it. This corresponds to two nested comparisons with the corresponds to two nested comparisons with the majority of the work only
majority of the work only occurring if the first test passes and has the occurring if the first test passes and has the acceptance function <span
acceptance function <span class="math display">\[\mathcal{A}(a \to b) = class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta
\min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta
F_c}\right)\;.\]</span></p> F_c}\right)\;.\]</span></p>
<p>For the model parameters used in Fig. <a href="#fig:indiv_IPR" <p>For the model parameters used in Fig. <a href="#fig:indiv_IPR"
data-reference-type="ref" data-reference="fig:indiv_IPR">2</a>, we find data-reference-type="ref" data-reference="fig:indiv_IPR">2</a>, we find
@ -1008,9 +1007,9 @@ distribution, a problem which MCMC was employed to solve in the first
place. For example, recent work trains restricted Boltzmann machines place. For example, recent work trains restricted Boltzmann machines
(RBMs) to generate samples for the proposal distribution of the FK (RBMs) to generate samples for the proposal distribution of the FK
model <span class="citation" model <span class="citation"
data-cites="huangAcceleratedMonteCarlo2017"><sup><a data-cites="huangAcceleratedMonteCarlo2017"> [<a
href="#ref-huangAcceleratedMonteCarlo2017" href="#ref-huangAcceleratedMonteCarlo2017"
role="doc-biblioref">10</a></sup></span>. The RBMs are chosen as a role="doc-biblioref">10</a>]</span>. The RBMs are chosen as a
parametrisation of the proposal distribution that can be efficiently parametrisation of the proposal distribution that can be efficiently
sampled from while offering sufficient flexibility that they can be 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
@ -1021,11 +1020,11 @@ 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
condition is sufficient (along with some technical constraints<span condition is sufficient (along with some technical constraints <span
class="citation" data-cites="wolffMonteCarloErrors2004"><sup><a class="citation" data-cites="wolffMonteCarloErrors2004"> [<a
href="#ref-wolffMonteCarloErrors2004" href="#ref-wolffMonteCarloErrors2004"
role="doc-biblioref">5</a></sup></span>) to guarantee that in the long role="doc-biblioref">5</a>]</span>) to guarantee that in the long time
time limit the algorithm produces samples from <span limit the algorithm produces samples from <span
class="math inline">\(\pi\)</span>. <span class="math inline">\(\pi\)</span>. <span
class="math display">\[\pi(a)\mathcal{T}(a \to b) = \pi(b)\mathcal{T}(b class="math display">\[\pi(a)\mathcal{T}(a \to b) = \pi(b)\mathcal{T}(b
\to a)\]</span></p> \to a)\]</span></p>
@ -1141,10 +1140,10 @@ for the additional complexity it would require.</p>
<h3 id="inverse-participation-ratio">Inverse Participation Ratio</h3> <h3 id="inverse-participation-ratio">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"
data-cites="kramerLocalizationTheoryExperiment1993"><sup><a data-cites="kramerLocalizationTheoryExperiment1993"> [<a
href="#ref-kramerLocalizationTheoryExperiment1993" href="#ref-kramerLocalizationTheoryExperiment1993"
role="doc-biblioref">17</a></sup></span>: <span class="math display">\[ role="doc-biblioref">17</a>]</span>: <span class="math display">\[
P^{-1} = \sum_i \abs{\psi_i}^4 P^{-1} = \sum_i \abs{\psi_i}^4
\]</span> % It acts as a measure of the portion of space occupied by the \]</span> % It acts as a measure of the portion of space occupied by the
wave function. For localised states it will be independent of system wave function. For localised states it will be independent of system
@ -1155,11 +1154,10 @@ fractal dimensionality <span class="math inline">\(d &gt; d* &gt;
P(L) \goeslike L^{d*} P(L) \goeslike L^{d*}
\]</span> % For extended states <span class="math inline">\(d* = \]</span> % For extended states <span class="math inline">\(d* =
0\)</span> while for localised ones <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" class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a
data-cites="andersonAbsenceDiffusionCertain1958"><sup><a
href="#ref-andersonAbsenceDiffusionCertain1958" href="#ref-andersonAbsenceDiffusionCertain1958"
role="doc-biblioref">18</a></sup></span>: <span class="math display">\[ role="doc-biblioref">18</a>]</span>: <span class="math display">\[
DOS(\omega) = \sum_n \delta(\omega - \epsilon_n) DOS(\omega) = \sum_n \delta(\omega - \epsilon_n)
IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n)
\abs{\psi_{n,i}}^4 \abs{\psi_{n,i}}^4
@ -1518,148 +1516,141 @@ class="sourceCode python"><code class="sourceCode python"><span id="cb6-1"><a hr
<div class="sourceCode" id="cb7"><pre <div class="sourceCode" id="cb7"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div> class="sourceCode python"><code class="sourceCode python"></code></pre></div>
<p></ij></ij></p> <p></ij></ij></p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2" <div id="refs" class="references csl-bib-body" role="doc-bibliography">
role="doc-bibliography">
<div id="ref-devroyeRandomSampling1986" class="csl-entry" <div id="ref-devroyeRandomSampling1986" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div <div class="csl-left-margin">[1] </div><div class="csl-right-inline">L.
class="csl-right-inline">Devroye, L. Random Sampling. in <em>Non-Uniform Devroye, <em><a
Random Variate Generation</em> (ed. Devroye, L.) 611–641 (Springer, href="https://doi.org/10.1007/978-1-4613-8643-8_12">Random
1986). doi:<a Sampling</a></em>, in <em>Non-Uniform Random Variate Generation</em>,
href="https://doi.org/10.1007/978-1-4613-8643-8_12">10.1007/978-1-4613-8643-8_12</a>.</div> edited by L. Devroye (Springer, New York, NY, 1986), pp. 611–641.</div>
</div> </div>
<div id="ref-BMCP2021" class="csl-entry" role="doc-biblioentry"> <div id="ref-BMCP2021" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div <div class="csl-left-margin">[2] </div><div class="csl-right-inline">O.
class="csl-right-inline">Martin, O. A., Kumar, R. &amp; Lao, J. A. Martin, R. Kumar, and J. Lao, <em>Bayesian Modeling and Computation
<em>Bayesian modeling and computation in python</em>. (2021).</div> in Python</em> (Boca Raton, 2021).</div>
</div> </div>
<div id="ref-binderGuidePracticalWork1988" class="csl-entry" <div id="ref-binderGuidePracticalWork1988" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div <div class="csl-left-margin">[3] </div><div class="csl-right-inline">K.
class="csl-right-inline">Binder, K. &amp; Heermann, D. W. Guide to Binder and D. W. Heermann, <em><a
Practical Work with the Monte Carlo Method. in <em>Monte Carlo href="https://doi.org/10.1007/978-3-662-08854-8_3">Guide to Practical
Simulation in Statistical Physics: An Introduction</em> (eds. Binder, K. Work with the Monte Carlo Method</a></em>, in <em>Monte Carlo Simulation
&amp; Heermann, D. W.) 68–112 (Springer Berlin Heidelberg, 1988). doi:<a in Statistical Physics: An Introduction</em>, edited by K. Binder and D.
href="https://doi.org/10.1007/978-3-662-08854-8_3">10.1007/978-3-662-08854-8_3</a>.</div> W. Heermann (Springer Berlin Heidelberg, Berlin, Heidelberg, 1988), pp.
68–112.</div>
</div> </div>
<div id="ref-kerteszAdvancesComputerSimulation1998" class="csl-entry" <div id="ref-kerteszAdvancesComputerSimulation1998" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div <div class="csl-left-margin">[4] </div><div class="csl-right-inline">J.
class="csl-right-inline"><em>Advances in Computer Simulation: Lectures Kertesz and I. Kondor, editors, <em><a
Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July href="https://doi.org/10.1007/BFb0105456">Advances in Computer
1996</em>. (Springer-Verlag, 1998). doi:<a Simulation: Lectures Held at the Eötvös Summer School in Budapest,
href="https://doi.org/10.1007/BFb0105456">10.1007/BFb0105456</a>.</div> Hungary, 16–20 July 1996</a></em> (Springer-Verlag, Berlin Heidelberg,
1998).</div>
</div> </div>
<div id="ref-wolffMonteCarloErrors2004" class="csl-entry" <div id="ref-wolffMonteCarloErrors2004" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div <div class="csl-left-margin">[5] </div><div class="csl-right-inline">U.
class="csl-right-inline">Wolff, U. <a Wolff, <em><a href="https://doi.org/10.1016/S0010-4655(03)00467-3">Monte
href="https://doi.org/10.1016/S0010-4655(03)00467-3">Monte Carlo errors Carlo Errors with Less Errors</a></em>, Computer Physics Communications
with less errors</a>. <em>Computer Physics Communications</em> <strong>156</strong>, 143 (2004).</div>
<strong>156</strong>, 143–153 (2004).</div>
</div> </div>
<div id="ref-kellyReversibilityStochasticNetworks1981" class="csl-entry" <div id="ref-kellyReversibilityStochasticNetworks1981" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div <div class="csl-left-margin">[6] </div><div class="csl-right-inline">F.
class="csl-right-inline">Kelly, F. P. <a P. Kelly, <em><a href="https://doi.org/10.2307/2287860">Reversibility
href="https://doi.org/10.2307/2287860">Reversibility and stochastic and Stochastic Networks / F.P. Kelly</a></em>, SERBIULA (Sistema Librum
networks / F.P. Kelly</a>. <em>SERBIULA (sistema Librum 2.0)</em> 2.0) <strong>76</strong>, (1981).</div>
<strong>76</strong>, (1981).</div>
</div> </div>
<div id="ref-krauthIntroductionMonteCarlo1998" class="csl-entry" <div id="ref-krauthIntroductionMonteCarlo1998" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div <div class="csl-left-margin">[7] </div><div class="csl-right-inline">W.
class="csl-right-inline">Krauth, W. Introduction To Monte Carlo Krauth, <em><a href="https://doi.org/10.1007/BFb0105456">Introduction To
Algorithms. in <em>Advances in Computer Simulation: Lectures Held at the Monte Carlo Algorithms</a></em>, in <em>Advances in Computer Simulation:
Eötvös Summer School in Budapest, Hungary, 16–20 July 1996</em> Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20
(Springer-Verlag, 1998). doi:<a July 1996</em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
href="https://doi.org/10.1007/BFb0105456">10.1007/BFb0105456</a>.</div>
</div> </div>
<div id="ref-kapferSamplingPolytopeHarddisk2013" class="csl-entry" <div id="ref-kapferSamplingPolytopeHarddisk2013" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div <div class="csl-left-margin">[8] </div><div class="csl-right-inline">S.
class="csl-right-inline">Kapfer, S. C. &amp; Krauth, W. <a C. Kapfer and W. Krauth, <em><a
href="https://doi.org/10.1088/1742-6596/454/1/012031">Sampling from a href="https://doi.org/10.1088/1742-6596/454/1/012031">Sampling from a
polytope and hard-disk Monte Carlo</a>. <em>J. Phys.: Conf. Ser.</em> Polytope and Hard-Disk Monte Carlo</a></em>, J. Phys.: Conf. Ser.
<strong>454</strong>, 012031 (2013).</div> <strong>454</strong>, 012031 (2013).</div>
</div> </div>
<div id="ref-robertsWeakConvergenceOptimal1997" class="csl-entry" <div id="ref-robertsWeakConvergenceOptimal1997" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div <div class="csl-left-margin">[9] </div><div class="csl-right-inline">G.
class="csl-right-inline">Roberts, G. O., Gelman, A. &amp; Gilks, W. R. O. Roberts, A. Gelman, and W. R. Gilks, <em><a
<a href="https://doi.org/10.1214/aoap/1034625254">Weak convergence and href="https://doi.org/10.1214/aoap/1034625254">Weak Convergence and
optimal scaling of random walk Metropolis algorithms</a>. <em>Ann. Appl. Optimal Scaling of Random Walk Metropolis Algorithms</a></em>, Ann.
Probab.</em> <strong>7</strong>, 110–120 (1997).</div> Appl. Probab. <strong>7</strong>, 110 (1997).</div>
</div> </div>
<div id="ref-huangAcceleratedMonteCarlo2017" class="csl-entry" <div id="ref-huangAcceleratedMonteCarlo2017" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div <div class="csl-left-margin">[10] </div><div class="csl-right-inline">L.
class="csl-right-inline">Huang, L. &amp; Wang, L. <a Huang and L. Wang, <em><a
href="https://doi.org/10.1103/PhysRevB.95.035105">Accelerated Monte href="https://doi.org/10.1103/PhysRevB.95.035105">Accelerated Monte
Carlo simulations with restricted Boltzmann machines</a>. <em>Phys. Rev. Carlo Simulations with Restricted Boltzmann Machines</a></em>, Phys.
B</em> <strong>95</strong>, 035105 (2017).</div> Rev. B <strong>95</strong>, 035105 (2017).</div>
</div> </div>
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" <div id="ref-binderFiniteSizeScaling1981" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div <div class="csl-left-margin">[11] </div><div class="csl-right-inline">K.
class="csl-right-inline">Binder, K. <a Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size
href="https://doi.org/10.1007/BF01293604">Finite size scaling analysis Scaling Analysis of Ising Model Block Distribution Functions</a></em>,
of ising model block distribution functions</a>. <em>Z. Physik B - Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
Condensed Matter</em> <strong>43</strong>, 119–140 (1981).</div>
</div> </div>
<div id="ref-bolchQueueingNetworksMarkov2006" class="csl-entry" <div id="ref-bolchQueueingNetworksMarkov2006" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div <div class="csl-left-margin">[12] </div><div class="csl-right-inline">G.
class="csl-right-inline">Bolch, G., Greiner, S., Meer, H. de &amp; Bolch, S. Greiner, H. de Meer, and K. S. Trivedi, <em>Queueing Networks
Trivedi, K. S. <em>Queueing Networks and Markov Chains: Modeling and and Markov Chains: Modeling and Performance Evaluation with Computer
Performance Evaluation with Computer Science Applications</em>. (John Science Applications</em> (John Wiley &amp; Sons, 2006).</div>
Wiley &amp; Sons, 2006).</div>
</div> </div>
<div id="ref-usmaniInversionTridiagonalJacobi1994" class="csl-entry" <div id="ref-usmaniInversionTridiagonalJacobi1994" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div <div class="csl-left-margin">[13] </div><div class="csl-right-inline">R.
class="csl-right-inline">Usmani, R. A. <a A. Usmani, <em><a
href="https://doi.org/10.1016/0024-3795(94)90414-6">Inversion of a href="https://doi.org/10.1016/0024-3795(94)90414-6">Inversion of a
tridiagonal jacobi matrix</a>. <em>Linear Algebra and its Tridiagonal Jacobi Matrix</a></em>, Linear Algebra and Its Applications
Applications</em> <strong>212-213</strong>, 413–414 (1994).</div> <strong>212-213</strong>, 413 (1994).</div>
</div> </div>
<div id="ref-krauthIntroductionMonteCarlo1996" class="csl-entry" <div id="ref-krauthIntroductionMonteCarlo1996" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div <div class="csl-left-margin">[14] </div><div class="csl-right-inline">W.
class="csl-right-inline">Krauth, W. <a Krauth, <em><a href="http://arxiv.org/abs/cond-mat/9612186">Introduction
href="http://arxiv.org/abs/cond-mat/9612186">Introduction To Monte Carlo To Monte Carlo Algorithms</a></em>, arXiv:cond-Mat/9612186 (1996).</div>
Algorithms</a>. <em>arXiv:cond-mat/9612186</em> (1996).</div>
</div> </div>
<div id="ref-hastingsMonteCarloSampling1970" class="csl-entry" <div id="ref-hastingsMonteCarloSampling1970" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div <div class="csl-left-margin">[15] </div><div class="csl-right-inline">W.
class="csl-right-inline">Hastings, W. K. <a K. Hastings, <em><a href="https://doi.org/10.1093/biomet/57.1.97">Monte
href="https://doi.org/10.1093/biomet/57.1.97">Monte Carlo sampling Carlo Sampling Methods Using Markov Chains and Their
methods using Markov chains and their applications</a>. Applications</a></em>, Biometrika <strong>57</strong>, 97 (1970).</div>
<em>Biometrika</em> <strong>57</strong>, 97–109 (1970).</div>
</div> </div>
<div id="ref-hodsonMCMCFKModel2021" class="csl-entry" <div id="ref-hodsonMCMCFKModel2021" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div <div class="csl-left-margin">[16] </div><div class="csl-right-inline">T.
class="csl-right-inline">Hodson, T. Markov Chain Monte Carlo for the Hodson, <em><a href="https://doi.org/10.5281/zenodo.4593904">Markov
Kitaev Model. (2021) doi:<a Chain Monte Carlo for the Kitaev Model</a></em>, (2021).</div>
href="https://doi.org/10.5281/zenodo.4593904">10.5281/zenodo.4593904</a>.</div>
</div> </div>
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry" <div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div <div class="csl-left-margin">[17] </div><div class="csl-right-inline">B.
class="csl-right-inline">Kramer, B. &amp; MacKinnon, A. <a Kramer and A. MacKinnon, <em><a
href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: theory href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory
and experiment</a>. <em>Rep. Prog. Phys.</em> <strong>56</strong>, and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469
1469–1564 (1993).</div> (1993).</div>
</div> </div>
<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry" <div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div <div class="csl-left-margin">[18] </div><div class="csl-right-inline">P.
class="csl-right-inline">Anderson, P. W. <a W. Anderson, <em><a
href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in
Certain Random Lattices</a>. <em>Phys. Rev.</em> <strong>109</strong>, Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492
1492–1505 (1958).</div> (1958).</div>
</div> </div>
</div> </div>
<section class="footnotes footnotes-end-of-document" <section class="footnotes footnotes-end-of-document"
@ -1675,11 +1666,11 @@ systems with a tri-diagonal matrix representation (open boundary
conditions and nearest neighbour hopping) and like <span 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" class="citation"
data-cites="bolchQueueingNetworksMarkov2006 usmaniInversionTridiagonalJacobi1994"><sup><a data-cites="bolchQueueingNetworksMarkov2006 usmaniInversionTridiagonalJacobi1994"> [<a
href="#ref-bolchQueueingNetworksMarkov2006" href="#ref-bolchQueueingNetworksMarkov2006"
role="doc-biblioref">12</a>,<a role="doc-biblioref">12</a>,<a
href="#ref-usmaniInversionTridiagonalJacobi1994" href="#ref-usmaniInversionTridiagonalJacobi1994"
role="doc-biblioref">13</a></sup></span>.<a href="#fnref2" role="doc-biblioref">13</a>]</span>.<a href="#fnref2"
class="footnote-back" role="doc-backlink">↩︎</a></p></li> class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn3" role="doc-endnote"><p>or, in the general case, any desired <li id="fn3" role="doc-endnote"><p>or, in the general case, any desired
distribution. MCMC has found a lot of use in sampling from the distribution. MCMC has found a lot of use in sampling from the
@ -1689,9 +1680,9 @@ role="doc-backlink">↩︎</a></p></li>
<li id="fn4" role="doc-endnote"><p>or equivalently as the number of MCMC <li id="fn4" role="doc-endnote"><p>or equivalently as the number of MCMC
steps after which the samples are correlated below some cutoff, steps after which the samples are correlated below some cutoff,
see <span class="citation" see <span class="citation"
data-cites="krauthIntroductionMonteCarlo1996"><sup><a data-cites="krauthIntroductionMonteCarlo1996"> [<a
href="#ref-krauthIntroductionMonteCarlo1996" href="#ref-krauthIntroductionMonteCarlo1996"
role="doc-biblioref">14</a></sup></span> for a more rigorous definition role="doc-biblioref">14</a>]</span> for a more rigorous definition
involving a sum over the auto-correlation function.<a href="#fnref4" 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>

198
_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html
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@ -327,9 +327,9 @@ constant <span class="math inline">\(U=5\)</span> and constant <span
class="math inline">\(J=5\)</span>, respectively. We determined the class="math inline">\(J=5\)</span>, respectively. We determined the
transition temperatures from the crossings of the Binder cumulants <span transition temperatures from the crossings of the Binder cumulants <span
class="math inline">\(B_4 = \tex{m^4}/\tex{m^2}^2\)</span> <span class="math inline">\(B_4 = \tex{m^4}/\tex{m^2}^2\)</span> <span
class="citation" data-cites="binderFiniteSizeScaling1981"><sup><a class="citation" data-cites="binderFiniteSizeScaling1981"> [<a
href="#ref-binderFiniteSizeScaling1981" href="#ref-binderFiniteSizeScaling1981"
role="doc-biblioref">1</a></sup></span>. For a representative set of role="doc-biblioref">1</a>]</span>. For a representative set of
parameters, Fig. [<a href="#fig:phase_diagram" data-reference-type="ref" parameters, Fig. [<a href="#fig:phase_diagram" data-reference-type="ref"
data-reference="fig:phase_diagram">1</a>c] shows the order parameter data-reference="fig:phase_diagram">1</a>c] shows the order parameter
<span class="math inline">\(\tex{m}^2\)</span>. Fig. [<a <span class="math inline">\(\tex{m}^2\)</span>. Fig. [<a
@ -350,12 +350,12 @@ fermion mediated RKKY interaction between the Ising spins is absent.</p>
<p>Our main interest concerns the additional structure of the fermionic <p>Our main interest concerns the additional structure of the fermionic
sector in the high temperature phase. Following Ref. <span sector in the high temperature phase. Following Ref. <span
class="citation" class="citation"
data-cites="antipovInteractionTunedAndersonMott2016"><sup><a data-cites="antipovInteractionTunedAndersonMott2016"> [<a
href="#ref-antipovInteractionTunedAndersonMott2016" href="#ref-antipovInteractionTunedAndersonMott2016"
role="doc-biblioref">2</a></sup></span>, we can distinguish between the role="doc-biblioref">2</a>]</span>, we can distinguish between the Mott
Mott and Anderson insulating phases. The former is characterised by a and Anderson insulating phases. The former is characterised by a gapped
gapped DOS in the absence of a CDW. Thus, the opening of a gap for large DOS in the absence of a CDW. Thus, the opening of a gap for large <span
<span class="math inline">\(U\)</span> is distinct from the gap-opening class="math inline">\(U\)</span> is distinct from the gap-opening
induced by the translational symmetry breaking in the CDW state below induced by the translational symmetry breaking in the CDW state below
<span class="math inline">\(T_c\)</span>, see also Fig. [<a <span class="math inline">\(T_c\)</span>, see also Fig. [<a
href="#fig:band_opening" data-reference-type="ref" href="#fig:band_opening" data-reference-type="ref"
@ -381,12 +381,11 @@ ka)^2}\;.\]</span></p>
<p>At infinite temperature, all the spin configurations become equally <p>At infinite temperature, all the spin configurations become equally
likely and the fermionic model reduces to one of binary uncorrelated likely and the fermionic model reduces to one of binary uncorrelated
disorder in which all eigenstates are Anderson localised <span disorder in which all eigenstates are Anderson localised <span
class="citation" class="citation" data-cites="abrahamsScalingTheoryLocalization1979"> [<a
data-cites="abrahamsScalingTheoryLocalization1979"><sup><a
href="#ref-abrahamsScalingTheoryLocalization1979" href="#ref-abrahamsScalingTheoryLocalization1979"
role="doc-biblioref">3</a></sup></span>. An Anderson localised state role="doc-biblioref">3</a>]</span>. An Anderson localised state centered
centered around <span class="math inline">\(r_0\)</span> has magnitude around <span class="math inline">\(r_0\)</span> has magnitude that drops
that drops exponentially over some localisation length <span exponentially over some localisation length <span
class="math inline">\(\xi\)</span> i.e <span class="math inline">\(\xi\)</span> i.e <span
class="math inline">\(|\psi(r)|^2 \sim \exp{-\abs{r - class="math inline">\(|\psi(r)|^2 \sim \exp{-\abs{r -
r_0}/\xi}\)</span>. Calculating <span class="math inline">\(\xi\)</span> r_0}/\xi}\)</span>. Calculating <span class="math inline">\(\xi\)</span>
@ -417,12 +416,12 @@ additional complication arises from the fact that the scaling exponent
may display intermediate behaviours for correlated disorder and in the may display intermediate behaviours for correlated disorder and in the
vicinity of a localisation-delocalisation transition <span vicinity of a localisation-delocalisation transition <span
class="citation" class="citation"
data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"><sup><a data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"> [<a
href="#ref-kramerLocalizationTheoryExperiment1993" href="#ref-kramerLocalizationTheoryExperiment1993"
role="doc-biblioref">4</a>,<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">4</a>,<a href="#ref-eversAndersonTransitions2008"
role="doc-biblioref">5</a></sup></span>. The thermal defects of the CDW role="doc-biblioref">5</a>]</span>. The thermal defects of the CDW phase
phase lead to a binary disorder potential with a finite correlation lead to a binary disorder potential with a finite correlation length,
length, which in principle could result in delocalized eigenstates.</p> which in principle could result in delocalized eigenstates.</p>
<p>The key question for our system is then: How is the <span <p>The key question for our system is then: How is the <span
class="math inline">\(T=0\)</span> CDW phase with fully delocalized class="math inline">\(T=0\)</span> CDW phase with fully delocalized
fermionic states connected to the fully localized phase at high fermionic states connected to the fully localized phase at high
@ -488,7 +487,7 @@ 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 4: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 &lt; \rho &lt; 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation  [2], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N &gt; 400" />
<figcaption aria-hidden="true"><span>Figure 4:</span> A comparison of <figcaption aria-hidden="true"><span>Figure 4:</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
@ -500,9 +499,9 @@ Energy resolved DOS(<span class="math inline">\(\omega\)</span>) and
and this data makes clear that the apparent scaling of IPR with system and this data makes clear that the apparent scaling of IPR with system
size is a finite size effect due to weak localisation <span size is a finite size effect due to weak localisation <span
class="citation" class="citation"
data-cites="antipovInteractionTunedAndersonMott2016"><sup><a data-cites="antipovInteractionTunedAndersonMott2016"> [<a
href="#ref-antipovInteractionTunedAndersonMott2016" href="#ref-antipovInteractionTunedAndersonMott2016"
role="doc-biblioref">2</a></sup></span>, hence all the states are indeed role="doc-biblioref">2</a>]</span>, hence all the states are indeed
localised as one would expect in 1D. The disorder model <span localised as one would expect in 1D. The disorder model <span
class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a)
<span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span <span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span
@ -534,22 +533,21 @@ class="math inline">\(\tau = 0.30\pm0.03\)</span> and <span
class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This
surprising finding suggests that the CDW bands are partially delocalised surprising finding suggests that the CDW bands are partially delocalised
with multi-fractal behaviour of the wavefunctions <span class="citation" with multi-fractal behaviour of the wavefunctions <span class="citation"
data-cites="eversAndersonTransitions2008"><sup><a data-cites="eversAndersonTransitions2008"> [<a
href="#ref-eversAndersonTransitions2008" href="#ref-eversAndersonTransitions2008"
role="doc-biblioref">5</a></sup></span>. This phenomenon would be role="doc-biblioref">5</a>]</span>. This phenomenon would be unexpected
unexpected in a 1D model as they generally do not support delocalisation in a 1D model as they generally do not support delocalisation in the
in the presence of disorder except as the result of correlations in the presence of disorder except as the result of correlations in the
emergent disorder potential <span class="citation" emergent disorder potential <span class="citation"
data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"><sup><a data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"> [<a
href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">6</a>,<a href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">6</a>,<a
href="#ref-goldshteinPurePointSpectrum1977" href="#ref-goldshteinPurePointSpectrum1977"
role="doc-biblioref">7</a></sup></span>. However, we later show by role="doc-biblioref">7</a>]</span>. However, we later show by comparison
comparison to an uncorrelated Anderson model that these nonzero to an uncorrelated Anderson model that these nonzero exponents are a
exponents are a finite size effect and the states are localised with a finite size effect and the states are localised with a finite <span
finite <span class="math inline">\(\xi\)</span> similar to the system class="math inline">\(\xi\)</span> similar to the system size. As a
size. As a result, the IPR does not scale correctly until the system result, the IPR does not scale correctly until the system size has grown
size has grown much larger than <span much larger than <span class="math inline">\(\xi\)</span>. Fig. [<a
class="math inline">\(\xi\)</span>. Fig. [<a
href="#fig:indiv_IPR_disorder" data-reference-type="ref" href="#fig:indiv_IPR_disorder" data-reference-type="ref"
data-reference="fig:indiv_IPR_disorder">4</a>] shows that the scaling of data-reference="fig:indiv_IPR_disorder">4</a>] shows that the scaling of
the IPR in the CDW phase does flatten out eventually.</p> the IPR in the CDW phase does flatten out eventually.</p>
@ -562,21 +560,21 @@ white, which highlights the distinction between the gapped Mott phase
and the ungapped Anderson phase. In-gap states appear just below the and the ungapped Anderson phase. In-gap states appear just below the
critical point, smoothly filling the bandgap in the Anderson phase and critical point, smoothly filling the bandgap in the Anderson phase and
forming islands in the Mott phase. As in the finite <span forming islands in the Mott phase. As in the finite <span
class="citation" data-cites="zondaGaplessRegimeCharge2019"><sup><a class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a
href="#ref-zondaGaplessRegimeCharge2019" href="#ref-zondaGaplessRegimeCharge2019"
role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a></sup></span> role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a>]</span>
and infinite dimensional <span class="citation" and infinite dimensional <span class="citation"
data-cites="hassanSpectralPropertiesChargedensitywave2007"><sup><a data-cites="hassanSpectralPropertiesChargedensitywave2007"> [<a
href="#ref-hassanSpectralPropertiesChargedensitywave2007" href="#ref-hassanSpectralPropertiesChargedensitywave2007"
role="doc-biblioref">8</a></sup></span> cases, the in-gap states merge role="doc-biblioref">8</a>]</span> cases, the in-gap states merge and
and are pushed to lower energy for decreasing U as the <span are pushed to lower energy for decreasing U as the <span
class="math inline">\(T=0\)</span> CDW gap closes. Intuitively, the class="math inline">\(T=0\)</span> CDW gap closes. Intuitively, the
presence of in-gap states can be understood as a result of domain wall presence of in-gap states can be understood as a result of domain wall
fluctuations away from the AFM ordered background. These domain walls fluctuations away from the AFM ordered background. These domain walls
act as local potentials for impurity-like bound states <span act as local potentials for impurity-like bound states <span
class="citation" data-cites="zondaGaplessRegimeCharge2019"><sup><a class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a
href="#ref-zondaGaplessRegimeCharge2019" href="#ref-zondaGaplessRegimeCharge2019"
role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a></sup></span>.</p> role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a>]</span>.</p>
<p>In order to understand the localization properties we can compare the <p>In order to understand the localization properties we can compare the
behaviour of our model with that of a simpler Anderson disorder model behaviour of our model with that of a simpler Anderson disorder model
(DM) in which the spins are replaced by a CDW background with (DM) in which the spins are replaced by a CDW background with
@ -623,18 +621,18 @@ modify the localisation behaviour? Similar to other soluble models of
disorder-free localisation, we expect intriguing out-of equilibrium disorder-free localisation, we expect intriguing out-of equilibrium
physics, for example slow entanglement dynamics akin to more generic physics, for example slow entanglement dynamics akin to more generic
interacting systems <span class="citation" interacting systems <span class="citation"
data-cites="hartLogarithmicEntanglementGrowth2020"><sup><a data-cites="hartLogarithmicEntanglementGrowth2020"> [<a
href="#ref-hartLogarithmicEntanglementGrowth2020" href="#ref-hartLogarithmicEntanglementGrowth2020"
role="doc-biblioref">9</a></sup></span>. One could also investigate role="doc-biblioref">9</a>]</span>. One could also investigate whether
whether the rich ground state phenomenology of the FK model as a the rich ground state phenomenology of the FK model as a function of
function of filling <span class="citation" filling <span class="citation"
data-cites="gruberGroundStatesSpinless1990"><sup><a data-cites="gruberGroundStatesSpinless1990"> [<a
href="#ref-gruberGroundStatesSpinless1990" href="#ref-gruberGroundStatesSpinless1990"
role="doc-biblioref">10</a></sup></span> such as the devil’s role="doc-biblioref">10</a>]</span> such as the devil’s staircase <span
staircase <span class="citation" class="citation"
data-cites="michelettiCompleteDevilTextquotesingles1997"><sup><a data-cites="michelettiCompleteDevilTextquotesingles1997"> [<a
href="#ref-michelettiCompleteDevilTextquotesingles1997" href="#ref-michelettiCompleteDevilTextquotesingles1997"
role="doc-biblioref">11</a></sup></span> could be stabilised at finite role="doc-biblioref">11</a>]</span> could be stabilised at finite
temperature. In a broader context, we envisage that long-range temperature. In a broader context, we envisage that long-range
interactions can also be used to gain a deeper understanding of the interactions can also be used to gain a deeper understanding of the
temperature evolution of topological phases. One example would be a temperature evolution of topological phases. One example would be a
@ -676,98 +674,94 @@ 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>
<div id="refs" class="references csl-bib-body" data-line-spacing="2" <div id="refs" class="references csl-bib-body" role="doc-bibliography">
role="doc-bibliography">
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" <div id="ref-binderFiniteSizeScaling1981" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div <div class="csl-left-margin">[1] </div><div class="csl-right-inline">K.
class="csl-right-inline">Binder, K. <a Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size
href="https://doi.org/10.1007/BF01293604">Finite size scaling analysis Scaling Analysis of Ising Model Block Distribution Functions</a></em>,
of ising model block distribution functions</a>. <em>Z. Physik B - Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
Condensed Matter</em> <strong>43</strong>, 119–140 (1981).</div>
</div> </div>
<div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry" <div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div <div class="csl-left-margin">[2] </div><div class="csl-right-inline">A.
class="csl-right-inline">Antipov, A. E., Javanmard, Y., Ribeiro, P. E. Antipov, Y. Javanmard, P. Ribeiro, and S. Kirchner, <em><a
&amp; Kirchner, S. <a
href="https://doi.org/10.1103/PhysRevLett.117.146601">Interaction-Tuned href="https://doi.org/10.1103/PhysRevLett.117.146601">Interaction-Tuned
Anderson versus Mott Localization</a>. <em>Phys. Rev. Lett.</em> Anderson Versus Mott Localization</a></em>, Phys. Rev. Lett.
<strong>117</strong>, 146601 (2016).</div> <strong>117</strong>, 146601 (2016).</div>
</div> </div>
<div id="ref-abrahamsScalingTheoryLocalization1979" class="csl-entry" <div id="ref-abrahamsScalingTheoryLocalization1979" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div <div class="csl-left-margin">[3] </div><div class="csl-right-inline">E.
class="csl-right-inline">Abrahams, E., Anderson, P. W., Licciardello, D. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan,
C. &amp; Ramakrishnan, T. V. <a <em><a href="https://doi.org/10.1103/PhysRevLett.42.673">Scaling Theory
href="https://doi.org/10.1103/PhysRevLett.42.673">Scaling Theory of of Localization: Absence of Quantum Diffusion in Two
Localization: Absence of Quantum Diffusion in Two Dimensions</a>. Dimensions</a></em>, Phys. Rev. Lett. <strong>42</strong>, 673
<em>Phys. Rev. Lett.</em> <strong>42</strong>, 673–676 (1979).</div> (1979).</div>
</div> </div>
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry" <div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div <div class="csl-left-margin">[4] </div><div class="csl-right-inline">B.
class="csl-right-inline">Kramer, B. &amp; MacKinnon, A. <a Kramer and A. MacKinnon, <em><a
href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: theory href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory
and experiment</a>. <em>Rep. Prog. Phys.</em> <strong>56</strong>, and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469
1469–1564 (1993).</div> (1993).</div>
</div> </div>
<div id="ref-eversAndersonTransitions2008" class="csl-entry" <div id="ref-eversAndersonTransitions2008" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div <div class="csl-left-margin">[5] </div><div class="csl-right-inline">F.
class="csl-right-inline">Evers, F. &amp; Mirlin, A. D. <a Evers and A. D. Mirlin, <em><a
href="https://doi.org/10.1103/RevModPhys.80.1355">Anderson href="https://doi.org/10.1103/RevModPhys.80.1355">Anderson
Transitions</a>. <em>Rev. Mod. Phys.</em> <strong>80</strong>, 1355–1417 Transitions</a></em>, Rev. Mod. Phys. <strong>80</strong>, 1355
(2008).</div> (2008).</div>
</div> </div>
<div id="ref-croyAndersonLocalization1D2011" class="csl-entry" <div id="ref-croyAndersonLocalization1D2011" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div <div class="csl-left-margin">[6] </div><div class="csl-right-inline">A.
class="csl-right-inline">Croy, A., Cain, P. &amp; Schreiber, M. <a Croy, P. Cain, and M. Schreiber, <em><a
href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson localization href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization
in 1D systems with correlated disorder</a>. <em>Eur. Phys. J. B</em> in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B
<strong>82</strong>, 107 (2011).</div> <strong>82</strong>, 107 (2011).</div>
</div> </div>
<div id="ref-goldshteinPurePointSpectrum1977" class="csl-entry" <div id="ref-goldshteinPurePointSpectrum1977" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div <div class="csl-left-margin">[7] </div><div class="csl-right-inline">I.
class="csl-right-inline">Gol’dshtein, I. Ya., Molchanov, S. A. &amp; Ya. Gol’dshtein, S. A. Molchanov, and L. A. Pastur, <em><a
Pastur, L. A. <a href="https://doi.org/10.1007/BF01135526">A pure point href="https://doi.org/10.1007/BF01135526">A Pure Point Spectrum of the
spectrum of the stochastic one-dimensional schrödinger operator</a>. Stochastic One-Dimensional Schrödinger Operator</a></em>, Funct Anal Its
<em>Funct Anal Its Appl</em> <strong>11</strong>, 1–8 (1977).</div> Appl <strong>11</strong>, 1 (1977).</div>
</div> </div>
<div id="ref-hassanSpectralPropertiesChargedensitywave2007" <div id="ref-hassanSpectralPropertiesChargedensitywave2007"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div <div class="csl-left-margin">[8] </div><div class="csl-right-inline">S.
class="csl-right-inline">Hassan, S. R. &amp; Krishnamurthy, H. R. <a R. Hassan and H. R. Krishnamurthy, <em><a
href="https://doi.org/10.1103/PhysRevB.76.205109">Spectral properties in href="https://doi.org/10.1103/PhysRevB.76.205109">Spectral Properties in
the charge-density-wave phase of the half-filled Falicov-Kimball the Charge-Density-Wave Phase of the Half-Filled Falicov-Kimball
model</a>. <em>Phys. Rev. B</em> <strong>76</strong>, 205109 Model</a></em>, Phys. Rev. B <strong>76</strong>, 205109 (2007).</div>
(2007).</div>
</div> </div>
<div id="ref-hartLogarithmicEntanglementGrowth2020" class="csl-entry" <div id="ref-hartLogarithmicEntanglementGrowth2020" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div <div class="csl-left-margin">[9] </div><div class="csl-right-inline">O.
class="csl-right-inline">Hart, O., Gopalakrishnan, S. &amp; Castelnovo, Hart, S. Gopalakrishnan, and C. Castelnovo, <em><a
C. <a href="http://arxiv.org/abs/2009.00618">Logarithmic entanglement href="http://arxiv.org/abs/2009.00618">Logarithmic Entanglement Growth
growth from disorder-free localisation in the two-leg compass from Disorder-Free Localisation in the Two-Leg Compass Ladder</a></em>,
ladder</a>. <em>arXiv:2009.00618 [cond-mat]</em> (2020).</div> arXiv:2009.00618 [Cond-Mat] (2020).</div>
</div> </div>
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry" <div id="ref-gruberGroundStatesSpinless1990" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div <div class="csl-left-margin">[10] </div><div class="csl-right-inline">C.
class="csl-right-inline">Gruber, C., Iwanski, J., Jedrzejewski, J. &amp; Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a
Lemberger, P. <a href="https://doi.org/10.1103/PhysRevB.41.2198">Ground href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the
states of the spinless Falicov-Kimball model</a>. <em>Phys. Rev. B</em> Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B
<strong>41</strong>, 2198–2209 (1990).</div> <strong>41</strong>, 2198 (1990).</div>
</div> </div>
<div id="ref-michelettiCompleteDevilTextquotesingles1997" <div id="ref-michelettiCompleteDevilTextquotesingles1997"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div <div class="csl-left-margin">[11] </div><div class="csl-right-inline">C.
class="csl-right-inline">Micheletti, C., Harris, A. B. &amp; Yeomans, J. Micheletti, A. B. Harris, and J. M. Yeomans, <em><a
M. <a href="https://doi.org/10.1088/0305-4470/30/21/002">A complete href="https://doi.org/10.1088/0305-4470/30/21/002">A Complete
devil\textquotesingles staircase in the Falicov - Kimball model</a>. Devil\textquotesingles Staircase in the Falicov - Kimball
<em>J. Phys. A: Math. Gen.</em> <strong>30</strong>, L711–L717 Model</a></em>, J. Phys. A: Math. Gen. <strong>30</strong>, L711
(1997).</div> (1997).</div>
</div> </div>
</div> </div>

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@ -539,10 +539,10 @@ symmetries</strong> and <strong><span class="math inline">\(2^2 =
4\)</span> topological sectors</strong>.</p> 4\)</span> topological sectors</strong>.</p>
<p>The topological sector forms the basis of proposals to construct <p>The topological sector forms the basis of proposals to construct
topologically protected qubits since the four sectors can only be mixed 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"><sup><a data-cites="kitaevFaulttolerantQuantumComputation2003"> [<a
href="#ref-kitaevFaulttolerantQuantumComputation2003" href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref">1</a></sup></span>.</p> role="doc-biblioref">1</a>]</span>.</p>
<p>Takeaway: The Extended Hilbert Space decomposes into a direct product <p>Takeaway: The Extended Hilbert Space decomposes into a direct product
of Flux Sectors, four Topological Sectors and a set of gauge of Flux Sectors, four Topological Sectors and a set of gauge
symmetries.</p> symmetries.</p>
@ -675,11 +675,11 @@ any information about the underlying lattice.</p>
<p><span class="math display">\[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i <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> \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 <p>The product over <span class="math inline">\(c_i\)</span> operators
reduces to a determinant of the Q matrix and the fermion parity, reduces to a determinant of the Q matrix and the fermion parity, see
see<span class="citation" <span class="citation"
data-cites="pedrocchiPhysicalSolutionsKitaev2011"><sup><a data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011" href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
role="doc-biblioref">2</a></sup></span>. The only difference from the role="doc-biblioref">2</a>]</span>. The only difference from the
honeycomb case is that we cannot explicitly compute the factors <span honeycomb case is that we cannot explicitly compute the factors <span
class="math inline">\(p_x,p_y,p_z = \pm\;1\)</span> that arise from class="math inline">\(p_x,p_y,p_z = \pm\;1\)</span> that arise from
reordering the b operators such that pairs of vertices linked by the reordering the b operators such that pairs of vertices linked by the
@ -702,20 +702,19 @@ depend only on the lattice structure.</p>
<p><span class="math inline">\(\hat{\pi} = \prod{i}^{N} (1 - <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 2\hat{n}_i)\)</span> is the parity of the particular many body state
determined by fermionic occupation numbers <span 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" class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
data-cites="pedrocchiPhysicalSolutionsKitaev2011"><sup><a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011" href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
role="doc-biblioref">2</a></sup></span>, <span role="doc-biblioref">2</a>]</span>, <span
class="math inline">\(\hat{\pi}\)</span> is gauge invariant in the sense class="math inline">\(\hat{\pi}\)</span> is gauge invariant in the sense
that <span class="math inline">\([\hat{\pi}, D_i] = 0\)</span>.</p> 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 <p>This implies that <span class="math inline">\(det(Q^u) \prod -i
u_{ij}\)</span> is also a gauge invariant quantity. In translation u_{ij}\)</span> is also a gauge invariant quantity. In translation
invariant models this quantity which can be related to the parity of the 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"><sup><a data-cites="yaoAlgebraicSpinLiquid2009"> [<a
href="#ref-yaoAlgebraicSpinLiquid2009" href="#ref-yaoAlgebraicSpinLiquid2009"
role="doc-biblioref">3</a></sup></span>.</p> role="doc-biblioref">3</a>]</span>.</p>
<p>All these factors take values <span class="math inline">\(\pm <p>All these factors take values <span class="math inline">\(\pm
1\)</span> so <span class="math inline">\(\mathcal{P}_0\)</span> is 0 or 1\)</span> so <span class="math inline">\(\mathcal{P}_0\)</span> is 0 or
1 for a particular state. Since <span 1 for a particular state. Since <span
@ -744,12 +743,12 @@ vortex pair, transporting one of them around the major or minor
diameters of the torus and, then, annihilating them again.</figcaption> diameters of the torus and, then, annihilating them again.</figcaption>
</figure> </figure>
</div> </div>
<p>More general arguments<span class="citation" <p>More general arguments <span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"><sup><a data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
href="#ref-chungExplicitMonodromyMoore2007" href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">4</a>,<a role="doc-biblioref">4</a>,<a
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007" href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
role="doc-biblioref">5</a></sup></span> imply that <span role="doc-biblioref">5</a>]</span> imply that <span
class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> has an class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> has an
interesting relationship to the topological fluxes. In the non-Abelian interesting relationship to the topological fluxes. In the non-Abelian
phase, we expect that it will change sign in exactly one of the four phase, we expect that it will change sign in exactly one of the four
@ -838,8 +837,8 @@ definition, the vortex free sector.</p>
<p>On the Honeycomb, Lieb’s theorem implies that the ground state <p>On the Honeycomb, Lieb’s theorem implies that the ground state
corresponds to the state where all <span class="math inline">\(u_{jk} = 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 1\)</span>. This implies that the flux free sector is the ground state
sector<span class="citation" data-cites="lieb_flux_1994"><sup><a sector <span class="citation" data-cites="lieb_flux_1994"> [<a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a></sup></span>.</p> href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span>.</p>
<p>Lieb’s theorem does not generalise easily to the amorphous case. <p>Lieb’s theorem does not generalise easily to the amorphous case.
However, we can get some intuition by examining the problem that will However, we can get some intuition by examining the problem that will
lead to a guess for the ground state. We will then provide numerical lead to a guess for the ground state. We will then provide numerical
@ -919,12 +918,11 @@ i)^{n_{\mathrm{sides}}},
class="math inline">\(n_{\mathrm{sides}}\)</span> is the number of edges 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 that form each plaquette and the choice of sign gives a twofold chiral
ground state degeneracy.</p> ground state degeneracy.</p>
<p>This conjecture is consistent with Lieb’s theorem on regular <p>This conjecture is consistent with Lieb’s theorem on regular lattices
lattices<span class="citation" data-cites="lieb_flux_1994"><sup><a <span class="citation" data-cites="lieb_flux_1994"> [<a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a></sup></span> and href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span> and is
is supported by numerical evidence. As noted before, any flux that supported by numerical evidence. As noted before, any flux that differs
differs from the ground state is an excitation which we call a from the ground state is an excitation which we call a vortex.</p>
vortex.</p>
<h3 id="finite-size-effects">Finite size effects</h3> <h3 id="finite-size-effects">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
@ -975,19 +973,18 @@ around the predicted ground state never yield a lower energy state.</p>
<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>
<p>This happens because we have broken the time reversal symmetry of the <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"><sup><a data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"> [<a
href="#ref-Chua2011" role="doc-biblioref">7</a>–<a href="#ref-Chua2011" role="doc-biblioref">7</a>–<a
href="#ref-WangHaoranPRB2021" href="#ref-WangHaoranPRB2021" role="doc-biblioref">14</a>]</span>.</p>
role="doc-biblioref">14</a></sup></span>.</p>
<p>Similarly to the behaviour of the original Kitaev model in response <p>Similarly to the behaviour of the original Kitaev model in response
to a magnetic field, we get two degenerate ground states of different to a magnetic field, we get two degenerate ground states of different
handedness. Practically speaking, one ground state is related to the handedness. Practically speaking, one ground state is related to the
other by inverting the imaginary <span 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"><sup><a data-cites="yaoExactChiralSpin2007"> [<a
href="#ref-yaoExactChiralSpin2007" href="#ref-yaoExactChiralSpin2007"
role="doc-biblioref">8</a></sup></span>.</p> role="doc-biblioref">8</a>]</span>.</p>
<h2 id="phases-of-the-kitaev-model">Phases of the Kitaev Model</h2> <h2 id="phases-of-the-kitaev-model">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>
@ -1114,190 +1111,185 @@ and construct the set <span class="math inline">\((+1, +1), (+1, -1),
<figure> <figure>
<img src="/assets/thesis/amk_chapter/topological_fluxes.png" <img src="/assets/thesis/amk_chapter/topological_fluxes.png"
data-short-caption="Topological Fluxes" style="width:57.0%" 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 make15." /> 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 <figcaption aria-hidden="true"><span>Figure 14:</span> Wilson loops that
wind the major or minor diameters of the torus measure flux winding 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 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 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"><sup><a href="#ref-parkerWhyDoesThis" data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis"
role="doc-biblioref">15</a></sup></span>.</figcaption> role="doc-biblioref">15</a>]</span>.</figcaption>
</figure> </figure>
</div> </div>
<p>However, in the non-Abelian phase we have to wrangle with <p>However, in the non-Abelian phase we have to wrangle with monodromy
monodromy<span class="citation" <span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"><sup><a data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
href="#ref-chungExplicitMonodromyMoore2007" href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">4</a>,<a role="doc-biblioref">4</a>,<a
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007" href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
role="doc-biblioref">5</a></sup></span>. Monodromy is the behaviour of role="doc-biblioref">5</a>]</span>. Monodromy is the behaviour of
objects as they move around a singularity. This manifests here in that objects as they move around a singularity. This manifests here in that
the identity of a vortex and cloud of Majoranas can change as we wind the identity of a vortex and cloud of Majoranas can change as we wind
them around the torus in such a way that, rather than annihilating to 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 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 state. This means that we end up with only three degenerate
ground states in the non-Abelian phase <span class="math inline">\((+1, 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"><sup><a data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"> [<a
href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">3</a>,<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">3</a>,<a
href="#ref-chungTopologicalQuantumPhase2010" href="#ref-chungTopologicalQuantumPhase2010"
role="doc-biblioref">16</a></sup></span>. Concretely, this is because role="doc-biblioref">16</a>]</span>. Concretely, this is because the
the projector enforces both flux and fermion parity. When we wind a projector enforces both flux and fermion parity. When we wind a vortex
vortex around both non-contractible loops of the torus, it flips the around both non-contractible loops of the torus, it flips the flux
flux parity. Therefore, we have to introduce a fermionic excitation to parity. Therefore, we have to introduce a fermionic excitation to make
make the state physical. Hence, the process does not give a fourth the state physical. Hence, the process does not give a fourth ground
ground state.</p> state.</p>
<p>Recently, the topology has notably gained interest because of <p>Recently, the topology has notably gained interest because of
proposals to use this ground state degeneracy to implement both proposals to use this ground state degeneracy to implement both
passively fault tolerant and actively stabilised quantum passively fault tolerant and actively stabilised quantum computations
computations<span class="citation" <span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"><sup><a data-cites="kitaevFaulttolerantQuantumComputation2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a
href="#ref-kitaevFaulttolerantQuantumComputation2003" href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref">1</a>,<a role="doc-biblioref">1</a>,<a
href="#ref-poulinStabilizerFormalismOperator2005" href="#ref-poulinStabilizerFormalismOperator2005"
role="doc-biblioref">17</a>,<a role="doc-biblioref">17</a>,<a
href="#ref-hastingsDynamicallyGeneratedLogical2021" href="#ref-hastingsDynamicallyGeneratedLogical2021"
role="doc-biblioref">18</a></sup></span>.</p> role="doc-biblioref">18</a>]</span>.</p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2" <div id="refs" class="references csl-bib-body" role="doc-bibliography">
role="doc-bibliography">
<div id="ref-kitaevFaulttolerantQuantumComputation2003" <div id="ref-kitaevFaulttolerantQuantumComputation2003"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div <div class="csl-left-margin">[1] </div><div class="csl-right-inline">A.
class="csl-right-inline">Kitaev, A. Yu. <a Yu. Kitaev, <em><a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-tolerant href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
quantum computation by anyons</a>. <em>Annals of Physics</em> Quantum Computation by Anyons</a></em>, Annals of Physics
<strong>303</strong>, 2–30 (2003).</div> <strong>303</strong>, 2 (2003).</div>
</div> </div>
<div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry" <div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div <div class="csl-left-margin">[2] </div><div class="csl-right-inline">F.
class="csl-right-inline">Pedrocchi, F. L., Chesi, S. &amp; Loss, D. <a L. Pedrocchi, S. Chesi, and D. Loss, <em><a
href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of
the Kitaev honeycomb model</a>. <em>Phys. Rev. B</em> the Kitaev honeycomb model</a></em>, Phys. Rev. B <strong>84</strong>,
<strong>84</strong>, 165414 (2011).</div> 165414 (2011).</div>
</div> </div>
<div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry" <div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div class="csl-right-inline">Yao, <div class="csl-left-margin">[3] </div><div class="csl-right-inline">H.
H., Zhang, S.-C. &amp; Kivelson, S. A. <a Yao, S.-C. Zhang, and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin
Liquid in an Exactly Solvable Spin Model</a>. <em>Phys. Rev. Lett.</em> Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett.
<strong>102</strong>, 217202 (2009).</div> <strong>102</strong>, 217202 (2009).</div>
</div> </div>
<div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry" <div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div <div class="csl-left-margin">[4] </div><div class="csl-right-inline">S.
class="csl-right-inline">Chung, S. B. &amp; Stone, M. <a B. Chung and M. Stone, <em><a
href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit monodromy of href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit Monodromy of
Moore–Read wavefunctions on a torus</a>. <em>J. Phys. A: Math. Moore–Read Wavefunctions on a Torus</a></em>, J. Phys. A: Math. Theor.
Theor.</em> <strong>40</strong>, 4923–4947 (2007).</div> <strong>40</strong>, 4923 (2007).</div>
</div> </div>
<div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007" <div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div <div class="csl-left-margin">[5] </div><div class="csl-right-inline">M.
class="csl-right-inline">Oshikawa, M., Kim, Y. B., Shtengel, K., Nayak, Oshikawa, Y. B. Kim, K. Shtengel, C. Nayak, and S. Tewari, <em><a
C. &amp; Tewari, S. <a href="https://doi.org/10.1016/j.aop.2006.08.001">Topological Degeneracy
href="https://doi.org/10.1016/j.aop.2006.08.001">Topological degeneracy of Non-Abelian States for Dummies</a></em>, Annals of Physics
of non-Abelian states for dummies</a>. <em>Annals of Physics</em> <strong>322</strong>, 1477 (2007).</div>
<strong>322</strong>, 1477–1498 (2007).</div>
</div> </div>
<div id="ref-lieb_flux_1994" class="csl-entry" role="doc-biblioentry"> <div id="ref-lieb_flux_1994" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div <div class="csl-left-margin">[6] </div><div class="csl-right-inline">E.
class="csl-right-inline">Lieb, E. H. <a H. Lieb, <em><a href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux
href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux Phase of the Phase of the Half-Filled Band</a></em>, Physical Review Letters
Half-Filled Band</a>. <em>Physical Review Letters</em> <strong>73</strong>, 2158 (1994).</div>
<strong>73</strong>, 2158–2161 (1994).</div>
</div> </div>
<div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry"> <div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div <div class="csl-left-margin">[7] </div><div class="csl-right-inline">V.
class="csl-right-inline">Chua, V., Yao, H. &amp; Fiete, G. A. <a Chua, H. Yao, and G. A. Fiete, <em><a
href="https://doi.org/10.1103/PhysRevB.83.180412">Exact chiral spin href="https://doi.org/10.1103/PhysRevB.83.180412">Exact Chiral Spin
liquid with stable spin Fermi surface on the kagome lattice</a>. Liquid with Stable Spin Fermi Surface on the Kagome Lattice</a></em>,
<em>Phys. Rev. B</em> <strong>83</strong>, 180412 (2011).</div> Phys. Rev. B <strong>83</strong>, 180412 (2011).</div>
</div> </div>
<div id="ref-yaoExactChiralSpin2007" class="csl-entry" <div id="ref-yaoExactChiralSpin2007" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div class="csl-right-inline">Yao, <div class="csl-left-margin">[8] </div><div class="csl-right-inline">H.
H. &amp; Kivelson, S. A. <a Yao and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.99.247203">An exact chiral href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral
spin liquid with non-Abelian anyons</a>. <em>Phys. Rev. Lett.</em> Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett.
<strong>99</strong>, 247203 (2007).</div> <strong>99</strong>, 247203 (2007).</div>
</div> </div>
<div id="ref-ChuaPRB2011" class="csl-entry" role="doc-biblioentry"> <div id="ref-ChuaPRB2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div <div class="csl-left-margin">[9] </div><div class="csl-right-inline">V.
class="csl-right-inline">Chua, V. &amp; Fiete, G. A. <a Chua and G. A. Fiete, <em><a
href="https://doi.org/10.1103/PhysRevB.84.195129">Exactly solvable href="https://doi.org/10.1103/PhysRevB.84.195129">Exactly Solvable
topological chiral spin liquid with random exchange</a>. <em>Phys. Rev. Topological Chiral Spin Liquid with Random Exchange</a></em>, Phys. Rev.
B</em> <strong>84</strong>, 195129 (2011).</div> B <strong>84</strong>, 195129 (2011).</div>
</div> </div>
<div id="ref-Fiete2012" class="csl-entry" role="doc-biblioentry"> <div id="ref-Fiete2012" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div <div class="csl-left-margin">[10] </div><div class="csl-right-inline">G.
class="csl-right-inline">Fiete, G. A. <em>et al.</em> <a A. Fiete, V. Chua, M. Kargarian, R. Lundgren, A. Rüegg, J. Wen, and V.
Zyuzin, <em><a
href="https://doi.org/10.1016/j.physe.2011.11.011">Topological href="https://doi.org/10.1016/j.physe.2011.11.011">Topological
insulators and quantum spin liquids</a>. <em>Physica E: Low-dimensional Insulators and Quantum Spin Liquids</a></em>, Physica E: Low-Dimensional
Systems and Nanostructures</em> <strong>44</strong>, 845–859 Systems and Nanostructures <strong>44</strong>, 845 (2012).</div>
(2012).</div>
</div> </div>
<div id="ref-Natori2016" class="csl-entry" role="doc-biblioentry"> <div id="ref-Natori2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div <div class="csl-left-margin">[11] </div><div class="csl-right-inline">W.
class="csl-right-inline">Natori, W. M. H., Andrade, E. C., Miranda, E. M. H. Natori, E. C. Andrade, E. Miranda, and R. G. Pereira, <em><a
&amp; Pereira, R. G. <a
href="https://link.aps.org/doi/10.1103/PhysRevLett.117.017204">Chiral href="https://link.aps.org/doi/10.1103/PhysRevLett.117.017204">Chiral
spin-orbital liquids with nodal lines</a>. <em>Phys. Rev. Lett.</em> Spin-Orbital Liquids with Nodal Lines</a></em>, Phys. Rev. Lett.
<strong>117</strong>, 017204 (2016).</div> <strong>117</strong>, 017204 (2016).</div>
</div> </div>
<div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div class="csl-right-inline">Wu, <div class="csl-left-margin">[12] </div><div class="csl-right-inline">C.
C., Arovas, D. &amp; Hung, H.-H. Γ-matrix generalization of the Kitaev Wu, D. Arovas, and H.-H. Hung, <em>Γ-Matrix Generalization of the Kitaev
model. <em>Physical Review B</em> <strong>79</strong>, 134427 Model</em>, Physical Review B <strong>79</strong>, 134427 (2009).</div>
(2009).</div>
</div> </div>
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry"> <div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div <div class="csl-left-margin">[13] </div><div class="csl-right-inline">V.
class="csl-right-inline">Peri, V. <em>et al.</em> <a Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R.
href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian chiral Moessner, and R. Thomale, <em><a
spin liquid on a simple non-Archimedean lattice</a>. <em>Phys. Rev. href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian Chiral
B</em> <strong>101</strong>, 041114 (2020).</div> Spin Liquid on a Simple Non-Archimedean Lattice</a></em>, Phys. Rev. B
<strong>101</strong>, 041114 (2020).</div>
</div> </div>
<div id="ref-WangHaoranPRB2021" class="csl-entry" <div id="ref-WangHaoranPRB2021" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div <div class="csl-left-margin">[14] </div><div class="csl-right-inline">H.
class="csl-right-inline">Wang, H. &amp; Principi, A. <a Wang and A. Principi, <em><a
href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana edge and href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana Edge and
corner states in square and kagome quantum spin-3/2 liquids</a>. Corner States in Square and Kagome Quantum Spin-3/2 Liquids</a></em>,
<em>Phys. Rev. B</em> <strong>104</strong>, 214422 (2021).</div> Phys. Rev. B <strong>104</strong>, 214422 (2021).</div>
</div> </div>
<div id="ref-parkerWhyDoesThis" class="csl-entry" <div id="ref-parkerWhyDoesThis" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div <div class="csl-left-margin">[15] </div><div
class="csl-right-inline"><em><a class="csl-right-inline"><em><a
href="https://www.youtube.com/watch?v=ymF1bp-qrjU">Why does this balloon href="https://www.youtube.com/watch?v=ymF1bp-qrjU">Why Does This Balloon
have -1 holes?</a></em></div> Have -1 Holes?</a></em> (n.d.).</div>
</div> </div>
<div id="ref-chungTopologicalQuantumPhase2010" class="csl-entry" <div id="ref-chungTopologicalQuantumPhase2010" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div <div class="csl-left-margin">[16] </div><div class="csl-right-inline">S.
class="csl-right-inline">Chung, S. B., Yao, H., Hughes, T. L. &amp; Kim, B. Chung, H. Yao, T. L. Hughes, and E.-A. Kim, <em><a
E.-A. <a href="https://doi.org/10.1103/PhysRevB.81.060403">Topological href="https://doi.org/10.1103/PhysRevB.81.060403">Topological Quantum
quantum phase transition in an exactly solvable model of a chiral spin Phase Transition in an Exactly Solvable Model of a Chiral Spin Liquid at
liquid at finite temperature</a>. <em>Phys. Rev. B</em> Finite Temperature</a></em>, Phys. Rev. B <strong>81</strong>, 060403
<strong>81</strong>, 060403 (2010).</div> (2010).</div>
</div> </div>
<div id="ref-poulinStabilizerFormalismOperator2005" class="csl-entry" <div id="ref-poulinStabilizerFormalismOperator2005" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div <div class="csl-left-margin">[17] </div><div class="csl-right-inline">D.
class="csl-right-inline">Poulin, D. <a Poulin, <em><a
href="https://doi.org/10.1103/PhysRevLett.95.230504">Stabilizer href="https://doi.org/10.1103/PhysRevLett.95.230504">Stabilizer
Formalism for Operator Quantum Error Correction</a>. <em>Phys. Rev. Formalism for Operator Quantum Error Correction</a></em>, Phys. Rev.
Lett.</em> <strong>95</strong>, 230504 (2005).</div> Lett. <strong>95</strong>, 230504 (2005).</div>
</div> </div>
<div id="ref-hastingsDynamicallyGeneratedLogical2021" class="csl-entry" <div id="ref-hastingsDynamicallyGeneratedLogical2021" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div <div class="csl-left-margin">[18] </div><div class="csl-right-inline">M.
class="csl-right-inline">Hastings, M. B. &amp; Haah, J. <a B. Hastings and J. Haah, <em><a
href="https://doi.org/10.22331/q-2021-10-19-564">Dynamically Generated href="https://doi.org/10.22331/q-2021-10-19-564">Dynamically Generated
Logical Qubits</a>. <em>Quantum</em> <strong>5</strong>, 564 Logical Qubits</a></em>, Quantum <strong>5</strong>, 564 (2021).</div>
(2021).</div>
</div> </div>
</div> </div>
</main> </main>

103
_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html
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@ -245,11 +245,11 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
guidance from Willian and Johannes. The project grew out of an interest guidance from Willian and Johannes. The project grew out of an interest
Gino, Peru and I had in studying amorphous systems, coupled with Gino, Peru and I had in studying amorphous systems, coupled with
Johannes’ expertise on the Kitaev model. The idea to use voronoi 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"><sup><a data-cites="marsalTopologicalWeaireThorpe2020"> [<a
href="#ref-marsalTopologicalWeaireThorpe2020" href="#ref-marsalTopologicalWeaireThorpe2020"
role="doc-biblioref">1</a></sup></span> and Gino did the implementation role="doc-biblioref">1</a>]</span> and Gino did the implementation of
of this. The idea and implementation of the edge colouring using SAT this. The idea and implementation of the edge colouring using SAT
solvers, the mapping from flux sector to bond sector using A* search solvers, the mapping from flux sector to bond sector using A* search
were both entirely my work. Peru came up with the ground state were both entirely my work. Peru came up with the ground state
conjecture and implemented the local markers. Gino and I did much of the conjecture and implemented the local markers. Gino and I did much of the
@ -289,11 +289,11 @@ material. Candidate materials, such as <span
class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have
sufficiently strong spin-orbit coupling and the correct lattice sufficiently strong spin-orbit coupling and the correct lattice
structure to behave according to the Kitaev Honeycomb model with small structure to behave according to the Kitaev Honeycomb model with small
corrections<span class="citation" corrections <span class="citation"
data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"><sup><a data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"> [<a
href="#ref-banerjeeProximateKitaevQuantum2016" href="#ref-banerjeeProximateKitaevQuantum2016"
role="doc-biblioref">2</a>,<a href="#ref-trebstKitaevMaterials2022" role="doc-biblioref">2</a>,<a href="#ref-trebstKitaevMaterials2022"
role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a></sup></span>.</p> role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a>]</span>.</p>
<p><strong>expand later: Why do we need spin orbit coupling and what <p><strong>expand later: Why do we need spin orbit coupling and what
will the corrections be?</strong></p> will the corrections be?</strong></p>
<p>Second, its ground state is the canonical example of the long sought <p>Second, its ground state is the canonical example of the long sought
@ -301,17 +301,17 @@ after quantum spin liquid state. Its excitations are anyons, particles
that can only exist in two dimensions that break the normal that can only exist in two dimensions that break the normal
fermion/boson dichotomy. Anyons have been the subject of much attention fermion/boson dichotomy. Anyons have been the subject of much attention
because, among other reasons, they can be braided through spacetime to 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"><sup><a data-cites="freedmanTopologicalQuantumComputation2003"> [<a
href="#ref-freedmanTopologicalQuantumComputation2003" href="#ref-freedmanTopologicalQuantumComputation2003"
role="doc-biblioref">3</a></sup></span>.</p> role="doc-biblioref">3</a>]</span>.</p>
<p>Third, and perhaps most importantly, this model is a rare many body <p>Third, and perhaps most importantly, this model is a rare many body
interacting quantum system that can be treated analytically. It is interacting quantum system that can be treated analytically. It is
exactly solvable. We can explicitly write down its many body ground 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"><sup><a data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006" href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a></sup></span>. The solubility of the Kitaev role="doc-biblioref">4</a>]</span>. The solubility of the Kitaev
Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes
about because the model has extensively many conserved degrees of 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
@ -326,9 +326,9 @@ lattices.</p>
look at the gauge symmetries of the model as well as its solution via a look at the gauge symmetries of the model as well as its solution via a
transformation to a Majorana hamiltonian. This discussion shows that, transformation to a Majorana hamiltonian. This discussion shows that,
for the the model to be solvable, it needs only be defined on a 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"><sup><a href="#ref-Nussinov2009" data-cites="Nussinov2009"> [<a href="#ref-Nussinov2009"
role="doc-biblioref">5</a></sup></span>.</p> role="doc-biblioref">5</a>]</span>.</p>
<p>The methods section discusses how to generate such lattices and <p>The methods section discusses how to generate such lattices and
colour them. It also explain how to map back and forth between colour them. It also explain how to map back and forth between
configurations of the gauge field and configurations of the gauge configurations of the gauge field and configurations of the gauge
@ -512,12 +512,11 @@ on site <span class="math inline">\(j\)</span> and <span
class="math inline">\(\langle j,k\rangle_\alpha\)</span> is a pair of class="math inline">\(\langle j,k\rangle_\alpha\)</span> is a pair of
nearest-neighbour indices connected by an <span nearest-neighbour indices connected by an <span
class="math inline">\(\alpha\)</span>-bond with exchange coupling <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"><sup><a data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006" href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a></sup></span>. For notational brevity, it is role="doc-biblioref">4</a>]</span>. For notational brevity, it is useful
useful to introduce the bond operators <span to introduce the bond operators <span class="math inline">\(K_{ij} =
class="math inline">\(K_{ij} =
\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> where <span \sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> where <span
class="math inline">\(\alpha\)</span> is a function of <span class="math inline">\(\alpha\)</span> is a function of <span
class="math inline">\(i,j\)</span> that picks the correct bond type.</p> class="math inline">\(i,j\)</span> that picks the correct bond type.</p>
@ -744,10 +743,9 @@ theory of the Majorana Hamiltonian further.</p>
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
have paired along bonds to become a classical gauge field <span have paired along bonds to become a classical gauge field <span
class="math inline">\(u_{ij}\)</span>. What follows is relatively 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"><sup><a class="citation" data-cites="BlaizotRipka1986"> [<a
href="#ref-BlaizotRipka1986" href="#ref-BlaizotRipka1986" role="doc-biblioref">6</a>]</span>.</p>
role="doc-biblioref">6</a></sup></span>.</p>
<p>Because of the antisymmetry of the matrix with entries <span <p>Because of the antisymmetry of the matrix with entries <span
class="math inline">\(J^{\alpha} u_{ij}\)</span>, the eigenvalues of the class="math inline">\(J^{\alpha} u_{ij}\)</span>, the eigenvalues of the
Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> come in Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> come in
@ -865,52 +863,49 @@ 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>
<div id="refs" class="references csl-bib-body" data-line-spacing="2" <div id="refs" class="references csl-bib-body" role="doc-bibliography">
role="doc-bibliography">
<div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry" <div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div <div class="csl-left-margin">[1] </div><div class="csl-right-inline">Q.
class="csl-right-inline">Marsal, Q., Varjas, D. &amp; Grushin, A. G. <a Marsal, D. Varjas, and A. G. Grushin, <em><a
href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire–Thorpe href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire–Thorpe
models of amorphous matter</a>. <em>Proceedings of the National Academy Models of Amorphous Matter</a></em>, Proceedings of the National Academy
of Sciences</em> <strong>117</strong>, 30260–30265 (2020).</div> of Sciences <strong>117</strong>, 30260 (2020).</div>
</div> </div>
<div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry" <div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div <div class="csl-left-margin">[2] </div><div class="csl-right-inline">A.
class="csl-right-inline">Banerjee, A. <em>et al.</em> <a Banerjee et al., <em><a
href="https://doi.org/10.1038/nmat4604">Proximate Kitaev Quantum Spin href="https://doi.org/10.1038/nmat4604">Proximate Kitaev Quantum Spin
Liquid Behaviour in {\alpha}-RuCl$_3$</a>. <em>Nature Mater</em> Liquid Behaviour in {\Alpha}-RuCl$_3$</a></em>, Nature Mater
<strong>15</strong>, 733–740 (2016).</div> <strong>15</strong>, 733 (2016).</div>
</div> </div>
<div id="ref-freedmanTopologicalQuantumComputation2003" <div id="ref-freedmanTopologicalQuantumComputation2003"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div <div class="csl-left-margin">[3] </div><div class="csl-right-inline">M.
class="csl-right-inline">Freedman, M., Kitaev, A., Larsen, M. &amp; Freedman, A. Kitaev, M. Larsen, and Z. Wang, <em><a
Wang, Z. <a href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological Quantum
href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological quantum Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31
computation</a>. <em>Bull. Amer. Math. Soc.</em> <strong>40</strong>, (2003).</div>
31–38 (2003).</div>
</div> </div>
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry" <div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div <div class="csl-left-margin">[4] </div><div class="csl-right-inline">A.
class="csl-right-inline">Kitaev, A. <a Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons
href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons in an exactly in an Exactly Solved Model and Beyond</a></em>, Annals of Physics
solved model and beyond</a>. <em>Annals of Physics</em> <strong>321</strong>, 2 (2006).</div>
<strong>321</strong>, 2–111 (2006).</div>
</div> </div>
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div <div class="csl-left-margin">[5] </div><div class="csl-right-inline">Z.
class="csl-right-inline">Nussinov, Z. &amp; Ortiz, G. <a Nussinov and G. Ortiz, <em><a
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond algebras and href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and
exact solvability of Hamiltonians: spin S=½ multilayer systems</a>. Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>,
<em>Physical Review B</em> <strong>79</strong>, 214440 (2009).</div> Physical Review B <strong>79</strong>, 214440 (2009).</div>
</div> </div>
<div id="ref-BlaizotRipka1986" class="csl-entry" role="doc-biblioentry"> <div id="ref-BlaizotRipka1986" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div <div class="csl-left-margin">[6] </div><div
class="csl-right-inline">Blaizot, J.-P. &amp; Ripka, G. <em>Quantum class="csl-right-inline">J.-P. Blaizot and G. Ripka, <em>Quantum Theory
theory of finite systems</em>. (The MIT Press, 1986).</div> of Finite Systems</em> (The MIT Press, 1986).</div>
</div> </div>
</div> </div>
</main> </main>

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@ -234,20 +234,20 @@ Markers</a></li>
<h1 id="methods">Methods</h1> <h1 id="methods">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"
data-cites="tomImperialCMTHKoalaFirst2022"><sup><a data-cites="tomImperialCMTHKoalaFirst2022"> [<a
href="#ref-tomImperialCMTHKoalaFirst2022" href="#ref-tomImperialCMTHKoalaFirst2022"
role="doc-biblioref"><strong>tomImperialCMTHKoalaFirst2022?</strong></a></sup></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> <h2 id="voronisation">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
class="citation" class="citation"
data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020 florescu_designer_2009"><sup><a data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020 florescu_designer_2009"> [<a
href="#ref-mitchellAmorphousTopologicalInsulators2018" href="#ref-mitchellAmorphousTopologicalInsulators2018"
role="doc-biblioref">1</a>–<a href="#ref-florescu_designer_2009" role="doc-biblioref">1</a>–<a href="#ref-florescu_designer_2009"
role="doc-biblioref">3</a></sup></span>. We start by sampling <em>seed role="doc-biblioref">3</a>]</span>. We start by sampling <em>seed
points</em> uniformly (or otherwise) on the torus. Then, we compute the points</em> uniformly (or otherwise) on the torus. Then, we compute the
partition of the torus into regions closest (with a Euclidean metric) to partition of the torus into regions closest (with a Euclidean metric) to
each seed point. The straight lines (if the torus is flattened out) at each seed point. The straight lines (if the torus is flattened out) at
@ -259,23 +259,23 @@ 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> vertex is connected to exactly three edges <strong>cite</strong>.</p>
<p>Ideally, we would sample uniformly from the space of possible <p>Ideally, we would sample uniformly from the space of possible
trivalent graphs. Indeed, there has been some work on how to do this 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"><sup><a data-cites="alyamiUniformSamplingDirected2016"> [<a
href="#ref-alyamiUniformSamplingDirected2016" href="#ref-alyamiUniformSamplingDirected2016"
role="doc-biblioref">4</a></sup></span>. However, it does not guarantee role="doc-biblioref">4</a>]</span>. However, it does not guarantee that
that the resulting graph is planar, which we must ensure so that the the resulting graph is planar, which we must ensure so that the edges
edges can be 3-coloured.</p> 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"><sup><a data-cites="barberQuickhullAlgorithmConvex1996"> [<a
href="#ref-barberQuickhullAlgorithmConvex1996" href="#ref-barberQuickhullAlgorithmConvex1996"
role="doc-biblioref">5</a></sup></span> from Scipy<span class="citation" role="doc-biblioref">5</a>]</span> from Scipy <span class="citation"
data-cites="virtanenSciPyFundamentalAlgorithms2020"><sup><a data-cites="virtanenSciPyFundamentalAlgorithms2020"> [<a
href="#ref-virtanenSciPyFundamentalAlgorithms2020" href="#ref-virtanenSciPyFundamentalAlgorithms2020"
role="doc-biblioref">6</a></sup></span> which computes the Voronoi role="doc-biblioref">6</a>]</span> which computes the Voronoi partition
partition of the plane. To compute the Voronoi partition of the torus, of the plane. To compute the Voronoi partition of the torus, we take the
we take the seed points and replicate them into a repeating grid. This seed points and replicate them into a repeating grid. This will be
will be either 3x3 or, for very small numbers of seed points, 5x5. Then, either 3x3 or, for very small numbers of seed points, 5x5. Then, we
we identify edges in the output to construct a lattice on the torus.</p> identify edges in the output to construct a lattice on the torus.</p>
<div id="fig:lattice_construction_animated" class="fignos"> <div id="fig:lattice_construction_animated" class="fignos">
<figure> <figure>
<img <img
@ -368,47 +368,46 @@ onto the plane without any edges crossing. Bridgeless graphs do not
contain any edges that, when removed, would partition the graph into contain any edges that, when removed, would partition the graph into
disconnected components.</p> disconnected components.</p>
<p>This problem must be distinguished from that considered by the famous <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"><sup><a data-cites="appelEveryPlanarMap1989"> [<a
href="#ref-appelEveryPlanarMap1989" href="#ref-appelEveryPlanarMap1989" role="doc-biblioref">7</a>]</span>.
role="doc-biblioref">7</a></sup></span>. The 4-colour theorem is The 4-colour theorem is concerned with assigning colours to the
concerned with assigning colours to the <strong>vertices</strong> of a <strong>vertices</strong> of a graph, such that no vertices that share
graph, such that no vertices that share an edge have the same colour. an edge have the same colour. Here we are concerned with an edge
Here we are concerned with an edge colouring.</p> colouring.</p>
<p>The four-colour theorem applies to planar graphs, those that can be <p>The four-colour theorem applies to planar graphs, those that can be
embedded onto the plane without any edges crossing. Here we are embedded onto the plane without any edges crossing. Here we are
concerned with Toroidal graphs, which can be embedded onto the torus concerned with Toroidal graphs, which can be embedded onto the torus
without any edges crossing. In fact, toroidal graphs require up to seven without any edges crossing. In fact, toroidal graphs require up to seven
colours<span class="citation" colours <span class="citation"
data-cites="heawoodMapColouringTheorems"><sup><a data-cites="heawoodMapColouringTheorems"> [<a
href="#ref-heawoodMapColouringTheorems" href="#ref-heawoodMapColouringTheorems"
role="doc-biblioref">8</a></sup></span>. The complete graph <span role="doc-biblioref">8</a>]</span>. The complete graph <span
class="math inline">\(K_7\)</span> is a good example of a toroidal graph class="math inline">\(K_7\)</span> is a good example of a toroidal graph
that requires seven colours.</p> that requires seven colours.</p>
<p><span class="math inline">\(\Delta + 1\)</span> colours are enough to <p><span class="math inline">\(\Delta + 1\)</span> colours are enough to
edge-colour any graph. An <span edge-colour any graph. An <span
class="math inline">\(\mathcal{O}(mn)\)</span> algorithm exists to do it 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 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"><sup><a data-cites="gEstimateChromaticClass1964"> [<a
href="#ref-gEstimateChromaticClass1964" href="#ref-gEstimateChromaticClass1964"
role="doc-biblioref">9</a></sup></span>. Restricting ourselves to graphs role="doc-biblioref">9</a>]</span>. Restricting ourselves to graphs with
with <span class="math inline">\(\Delta = 3\)</span> like ours, those <span class="math inline">\(\Delta = 3\)</span> like ours, those can be
can be four-edge-coloured in linear time<span class="citation" four-edge-coloured in linear time <span class="citation"
data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"><sup><a data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"> [<a
href="#ref-skulrattanakulchai4edgecoloringGraphsMaximum2002" href="#ref-skulrattanakulchai4edgecoloringGraphsMaximum2002"
role="doc-biblioref">10</a></sup></span>.</p> role="doc-biblioref">10</a>]</span>.</p>
<p>However, three-edge-colouring them is more difficult. Cubic, planar, <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 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"><sup><a href="#ref-tait1880remarks" data-cites="tait1880remarks"> [<a href="#ref-tait1880remarks"
role="doc-biblioref">11</a></sup></span>. An <span role="doc-biblioref">11</a>]</span>. An <span
class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists here
here<span class="citation" data-cites="robertson1996efficiently"><sup><a <span class="citation" data-cites="robertson1996efficiently"> [<a
href="#ref-robertson1996efficiently" href="#ref-robertson1996efficiently"
role="doc-biblioref">12</a></sup></span>. However, it is not clear role="doc-biblioref">12</a>]</span>. However, it is not clear whether
whether this extends to cubic, <strong>toroidal</strong> bridgeless this extends to cubic, <strong>toroidal</strong> bridgeless graphs.</p>
graphs.</p>
<div id="fig:multiple_colourings" class="fignos"> <div id="fig:multiple_colourings" class="fignos">
<figure> <figure>
<img <img
@ -467,22 +466,22 @@ 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 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 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 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"><sup><a href="#ref-Karp1972" data-cites="Karp1972"> [<a href="#ref-Karp1972"
role="doc-biblioref">13</a></sup></span>.</p> role="doc-biblioref">13</a>]</span>.</p>
<p>General purpose, high performance programs for solving SAT problems <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"><sup><a data-cites="alounehComprehensiveStudyAnalysis2019"> [<a
href="#ref-alounehComprehensiveStudyAnalysis2019" href="#ref-alounehComprehensiveStudyAnalysis2019"
role="doc-biblioref">14</a></sup></span>. Such programs are useful role="doc-biblioref">14</a>]</span>. Such programs are useful because,
because, by the Cook-Levin theorem, any NP problem can be encoded in by the Cook-Levin theorem, any NP problem can be encoded in polynomial
polynomial time as an instance of a SAT problem . This property is what time as an instance of a SAT problem . This property is what makes SAT
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" class="citation"
data-cites="cookComplexityTheoremprovingProcedures1971 levin1973universal"><sup><a data-cites="cookComplexityTheoremprovingProcedures1971 levin1973universal"> [<a
href="#ref-cookComplexityTheoremprovingProcedures1971" href="#ref-cookComplexityTheoremprovingProcedures1971"
role="doc-biblioref">15</a>,<a href="#ref-levin1973universal" role="doc-biblioref">15</a>,<a href="#ref-levin1973universal"
role="doc-biblioref">16</a></sup></span>.</p> role="doc-biblioref">16</a>]</span>.</p>
<p>Thus, it is a relatively standard technique in the computer science <p>Thus, it is a relatively standard technique in the computer science
community to solve NP problems by first transforming them to SAT community to solve NP problems by first transforming them to SAT
instances and then using an off the shelf SAT solver. The output of this instances and then using an off the shelf SAT solver. The output of this
@ -495,9 +494,9 @@ 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 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 turned out to work well enough and looking for a better solution was not
necessary.</p> 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"><sup><a href="#ref-imms-sat18" data-cites="imms-sat18"> [<a href="#ref-imms-sat18"
role="doc-biblioref">17</a></sup></span>. Like most modern SAT solvers, role="doc-biblioref">17</a>]</span>. Like most modern SAT solvers,
<code>MiniSAT</code> requires the input problem to be specified in <code>MiniSAT</code> requires the input problem to be specified in
Conjunctive Normal Form (CNF). CNF requires that the constraints be Conjunctive Normal Form (CNF). CNF requires that the constraints be
encoded as a set of <em>clauses</em> of the form <span encoded as a set of <em>clauses</em> of the form <span
@ -555,11 +554,11 @@ 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 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 particular n-colouring. However, it does include the option of using a
heuristic function that determine the order in which vertices will be heuristic function that determine the order in which vertices will be
coloured<span class="citation" coloured <span class="citation"
data-cites="kosowski2004classical matulaSmallestlastOrderingClustering1983"><sup><a data-cites="kosowski2004classical matulaSmallestlastOrderingClustering1983"> [<a
href="#ref-kosowski2004classical" role="doc-biblioref">18</a>,<a href="#ref-kosowski2004classical" role="doc-biblioref">18</a>,<a
href="#ref-matulaSmallestlastOrderingClustering1983" href="#ref-matulaSmallestlastOrderingClustering1983"
role="doc-biblioref">19</a></sup></span>. Perhaps</p> role="doc-biblioref">19</a>]</span>. Perhaps</p>
<div id="fig:times" class="fignos"> <div id="fig:times" class="fignos">
<figure> <figure>
<img src="/assets/thesis/amk_chapter/methods/times/times.svg" <img src="/assets/thesis/amk_chapter/methods/times/times.svg"
@ -658,154 +657,147 @@ 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>
<div id="refs" class="references csl-bib-body" data-line-spacing="2" <div id="refs" class="references csl-bib-body" role="doc-bibliography">
role="doc-bibliography">
<div id="ref-mitchellAmorphousTopologicalInsulators2018" <div id="ref-mitchellAmorphousTopologicalInsulators2018"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div <div class="csl-left-margin">[1] </div><div class="csl-right-inline">N.
class="csl-right-inline">Mitchell, N. P., Nash, L. M., Hexner, D., P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine,
Turner, A. M. &amp; Irvine, W. T. M. <a <em><a href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous
href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous topological topological insulators constructed from random point sets</a></em>,
insulators constructed from random point sets</a>. <em>Nature Phys</em> Nature Phys <strong>14</strong>, 380 (2018).</div>
<strong>14</strong>, 380–385 (2018).</div>
</div> </div>
<div id="ref-marsalTopologicalWeaireThorpeModels2020" class="csl-entry" <div id="ref-marsalTopologicalWeaireThorpeModels2020" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div <div class="csl-left-margin">[2] </div><div class="csl-right-inline">Q.
class="csl-right-inline">Marsal, Q., Varjas, D. &amp; Grushin, A. G. <a Marsal, D. Varjas, and A. G. Grushin, <em><a
href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire-Thorpe href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire-Thorpe
models of amorphous matter</a>. <em>Proc. Natl. Acad. Sci. U.S.A.</em> Models of Amorphous Matter</a></em>, Proc. Natl. Acad. Sci. U.S.A.
<strong>117</strong>, 30260–30265 (2020).</div> <strong>117</strong>, 30260 (2020).</div>
</div> </div>
<div id="ref-florescu_designer_2009" class="csl-entry" <div id="ref-florescu_designer_2009" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div <div class="csl-left-margin">[3] </div><div class="csl-right-inline">M.
class="csl-right-inline">Florescu, M., Torquato, S. &amp; Steinhardt, P. Florescu, S. Torquato, and P. J. Steinhardt, <em><a
J. <a href="https://doi.org/10.1073/pnas.0907744106">Designer disordered href="https://doi.org/10.1073/pnas.0907744106">Designer Disordered
materials with large, complete photonic band gaps</a>. <em>Proceedings Materials with Large, Complete Photonic Band Gaps</a></em>, Proceedings
of the National Academy of Sciences</em> <strong>106</strong>, of the National Academy of Sciences <strong>106</strong>, 20658
20658–20663 (2009).</div> (2009).</div>
</div> </div>
<div id="ref-alyamiUniformSamplingDirected2016" class="csl-entry" <div id="ref-alyamiUniformSamplingDirected2016" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div <div class="csl-left-margin">[4] </div><div class="csl-right-inline">S.
class="csl-right-inline">Alyami, S. A., Azad, A. K. M. &amp; Keith, J. A. Alyami, A. K. M. Azad, and J. M. Keith, <em><a
M. <a href="https://doi.org/10.1016/j.endm.2016.05.005">Uniform Sampling href="https://doi.org/10.1016/j.endm.2016.05.005">Uniform Sampling of
of Directed and Undirected Graphs Conditional on Vertex Directed and Undirected Graphs Conditional on Vertex
Connectivity</a>. <em>Electronic Notes in Discrete Mathematics</em> Connectivity</a></em>, Electronic Notes in Discrete Mathematics
<strong>53</strong>, 43–55 (2016).</div> <strong>53</strong>, 43 (2016).</div>
</div> </div>
<div id="ref-barberQuickhullAlgorithmConvex1996" class="csl-entry" <div id="ref-barberQuickhullAlgorithmConvex1996" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div <div class="csl-left-margin">[5] </div><div class="csl-right-inline">C.
class="csl-right-inline">Barber, C. B., Dobkin, D. P. &amp; Huhdanpaa, B. Barber, D. P. Dobkin, and H. Huhdanpaa, <em><a
H. <a href="https://doi.org/10.1145/235815.235821">The quickhull href="https://doi.org/10.1145/235815.235821">The Quickhull Algorithm for
algorithm for convex hulls</a>. <em>ACM Trans. Math. Softw.</em> Convex Hulls</a></em>, ACM Trans. Math. Softw. <strong>22</strong>, 469
<strong>22</strong>, 469–483 (1996).</div> (1996).</div>
</div> </div>
<div id="ref-virtanenSciPyFundamentalAlgorithms2020" class="csl-entry" <div id="ref-virtanenSciPyFundamentalAlgorithms2020" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div <div class="csl-left-margin">[6] </div><div class="csl-right-inline">P.
class="csl-right-inline">Virtanen, P. <em>et al.</em> <a Virtanen et al., <em><a
href="https://doi.org/10.1038/s41592-019-0686-2">SciPy 1.0: fundamental href="https://doi.org/10.1038/s41592-019-0686-2">SciPy 1.0: Fundamental
algorithms for scientific computing in Python</a>. <em>Nature Algorithms for Scientific Computing in Python</a></em>, Nature Methods
Methods</em> <strong>17</strong>, 261–272 (2020).</div> <strong>17</strong>, 3 (2020).</div>
</div> </div>
<div id="ref-appelEveryPlanarMap1989" class="csl-entry" <div id="ref-appelEveryPlanarMap1989" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div <div class="csl-left-margin">[7] </div><div class="csl-right-inline">K.
class="csl-right-inline">Appel, K. &amp; Haken, W. Every Planar Map Is Appel and W. Haken, <em><a href="https://doi.org/10.1090/conm/098">Every
Four Colorable. in (1989). doi:<a Planar Map Is Four Colorable</a></em>, in (1989).</div>
href="https://doi.org/10.1090/conm/098">10.1090/conm/098</a>.</div>
</div> </div>
<div id="ref-heawoodMapColouringTheorems" class="csl-entry" <div id="ref-heawoodMapColouringTheorems" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div <div class="csl-left-margin">[8] </div><div class="csl-right-inline">P.
class="csl-right-inline">Heawood, P. J. Map colouring theorems. J. Heawood, <em>Map Colouring Theorems</em>, Quarterly Journal of
<em>Quarterly Journal of Mathematics</em> 322–339.</div> Mathematics 322 (n.d.).</div>
</div> </div>
<div id="ref-gEstimateChromaticClass1964" class="csl-entry" <div id="ref-gEstimateChromaticClass1964" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div class="csl-right-inline">G, <div class="csl-left-margin">[9] </div><div class="csl-right-inline">V.
V. V. <a href="https://cir.nii.ac.jp/crid/1571980075458819456">On an V. G, <em><a href="https://cir.nii.ac.jp/crid/1571980075458819456">On an
estimate of the chromatic class of a p-graph</a>. <em>Discret Estimate of the Chromatic Class of a p-Graph</a></em>, Discret Analiz
Analiz</em> <strong>3</strong>, 25–30 (1964).</div> <strong>3</strong>, 25 (1964).</div>
</div> </div>
<div id="ref-skulrattanakulchai4edgecoloringGraphsMaximum2002" <div id="ref-skulrattanakulchai4edgecoloringGraphsMaximum2002"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div <div class="csl-left-margin">[10] </div><div class="csl-right-inline">S.
class="csl-right-inline">Skulrattanakulchai, S. <a Skulrattanakulchai, <em><a
href="https://doi.org/10.1016/S0020-0190(01)00221-6">4-edge-coloring href="https://doi.org/10.1016/S0020-0190(01)00221-6">4-edge-coloring
graphs of maximum degree 3 in linear time</a>. <em>Inf. Process. graphs of maximum degree 3 in linear time</a></em>, Inf. Process. Lett.
Lett.</em> <strong>81</strong>, 191–195 (2002).</div> <strong>81</strong>, 191 (2002).</div>
</div> </div>
<div id="ref-tait1880remarks" class="csl-entry" role="doc-biblioentry"> <div id="ref-tait1880remarks" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div <div class="csl-left-margin">[11] </div><div class="csl-right-inline">P.
class="csl-right-inline">Tait, P. G. Remarks on the colouring of maps. G. Tait, <em>Remarks on the Colouring of Maps</em>, in <em>Proc. Roy.
in <em>Proc. Roy. Soc. Edinburgh</em> vol. 10 501–503 (1880).</div> Soc. Edinburgh</em>, Vol. 10 (1880), pp. 501–503.</div>
</div> </div>
<div id="ref-robertson1996efficiently" class="csl-entry" <div id="ref-robertson1996efficiently" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div <div class="csl-left-margin">[12] </div><div class="csl-right-inline">N.
class="csl-right-inline">Robertson, N., Sanders, D. P., Seymour, P. Robertson, D. P. Sanders, P. Seymour, and R. Thomas, <em>Efficiently
&amp; Thomas, R. Efficiently four-coloring planar graphs. in Four-Coloring Planar Graphs</em>, in <em>Proceedings of the
<em>Proceedings of the twenty-eighth annual ACM symposium on Theory of Twenty-Eighth Annual ACM Symposium on Theory of Computing</em> (1996),
computing</em> 571–575 (1996).</div> pp. 571–575.</div>
</div> </div>
<div id="ref-Karp1972" class="csl-entry" role="doc-biblioentry"> <div id="ref-Karp1972" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div <div class="csl-left-margin">[13] </div><div class="csl-right-inline">R.
class="csl-right-inline">Karp, R. M. Reducibility among combinatorial M. Karp, <em><a
problems. in <em>Complexity of computer computations</em> (eds. Miller, href="https://doi.org/10.1007/978-1-4684-2001-2_9">Reducibility Among
R. E., Thatcher, J. W. &amp; Bohlinger, J. D.) 85–103 (Springer US, Combinatorial Problems</a></em>, in <em>Complexity of Computer
1972). doi:<a Computations</em>, edited by R. E. Miller, J. W. Thatcher, and J. D.
href="https://doi.org/10.1007/978-1-4684-2001-2_9">10.1007/978-1-4684-2001-2_9</a>.</div> Bohlinger (Springer US, Boston, MA, 1972), pp. 85–103.</div>
</div> </div>
<div id="ref-alounehComprehensiveStudyAnalysis2019" class="csl-entry" <div id="ref-alounehComprehensiveStudyAnalysis2019" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div <div class="csl-left-margin">[14] </div><div class="csl-right-inline">S.
class="csl-right-inline">Alouneh, S., Abed, S., Al Shayeji, M. H. &amp; Alouneh, S. Abed, M. H. Al Shayeji, and R. Mesleh, <em><a
Mesleh, R. <a href="https://doi.org/10.1007/s10462-018-9628-0">A href="https://doi.org/10.1007/s10462-018-9628-0">A Comprehensive Study
comprehensive study and analysis on SAT-solvers: advances, usages and and Analysis on SAT-Solvers: Advances, Usages and Achievements</a></em>,
achievements</a>. <em>Artif Intell Rev</em> <strong>52</strong>, Artif Intell Rev <strong>52</strong>, 2575 (2019).</div>
2575–2601 (2019).</div>
</div> </div>
<div id="ref-cookComplexityTheoremprovingProcedures1971" <div id="ref-cookComplexityTheoremprovingProcedures1971"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div <div class="csl-left-margin">[15] </div><div class="csl-right-inline">S.
class="csl-right-inline">Cook, S. A. The complexity of theorem-proving A. Cook, <em><a href="https://doi.org/10.1145/800157.805047">The
procedures. in <em>Proceedings of the third annual ACM symposium on Complexity of Theorem-Proving Procedures</a></em>, in <em>Proceedings of
Theory of computing</em> 151–158 (Association for Computing Machinery, the Third Annual ACM Symposium on Theory of Computing</em> (Association
1971). doi:<a for Computing Machinery, New York, NY, USA, 1971), pp. 151–158.</div>
href="https://doi.org/10.1145/800157.805047">10.1145/800157.805047</a>.</div>
</div> </div>
<div id="ref-levin1973universal" class="csl-entry" <div id="ref-levin1973universal" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div <div class="csl-left-margin">[16] </div><div class="csl-right-inline">L.
class="csl-right-inline">Levin, L. A. Universal sequential search A. Levin, <em>Universal Sequential Search Problems</em>, Problemy
problems. <em>Problemy peredachi informatsii</em> <strong>9</strong>, Peredachi Informatsii <strong>9</strong>, 115 (1973).</div>
115–116 (1973).</div>
</div> </div>
<div id="ref-imms-sat18" class="csl-entry" role="doc-biblioentry"> <div id="ref-imms-sat18" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div <div class="csl-left-margin">[17] </div><div class="csl-right-inline">A.
class="csl-right-inline">Ignatiev, A., Morgado, A. &amp; Marques-Silva, Ignatiev, A. Morgado, and J. Marques-Silva, <em><a
J. PySAT: A Python toolkit for prototyping with SAT oracles. in href="https://doi.org/10.1007/978-3-319-94144-8_26">PySAT: A Python
<em>SAT</em> 428–437 (2018). doi:<a Toolkit for Prototyping with SAT Oracles</a></em>, in <em>SAT</em>
href="https://doi.org/10.1007/978-3-319-94144-8_26">10.1007/978-3-319-94144-8_26</a>.</div> (2018), pp. 428–437.</div>
</div> </div>
<div id="ref-kosowski2004classical" class="csl-entry" <div id="ref-kosowski2004classical" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div <div class="csl-left-margin">[18] </div><div class="csl-right-inline">A.
class="csl-right-inline">Kosowski, A. &amp; Manuszewski, K. Classical Kosowski and K. Manuszewski, <em>Classical Coloring of Graphs</em>,
coloring of graphs. <em>Contemporary Mathematics</em> Contemporary Mathematics <strong>352</strong>, 1 (2004).</div>
<strong>352</strong>, 1–20 (2004).</div>
</div> </div>
<div id="ref-matulaSmallestlastOrderingClustering1983" class="csl-entry" <div id="ref-matulaSmallestlastOrderingClustering1983" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">19. </div><div <div class="csl-left-margin">[19] </div><div class="csl-right-inline">D.
class="csl-right-inline">Matula, D. W. &amp; Beck, L. L. <a W. Matula and L. L. Beck, <em><a
href="https://doi.org/10.1145/2402.322385">Smallest-last ordering and href="https://doi.org/10.1145/2402.322385">Smallest-Last Ordering and
clustering and graph coloring algorithms</a>. <em>J. ACM</em> Clustering and Graph Coloring Algorithms</a></em>, J. ACM
<strong>30</strong>, 417–427 (1983).</div> <strong>30</strong>, 417 (1983).</div>
</div> </div>
</div> </div>
</main> </main>

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@ -249,10 +249,10 @@ 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
some issues we will need to address to make this argument work.</p> some issues we will need to address to make this argument work.</p>
<p>We have two seemingly irreconcilable problems. Finite size effects <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"><sup><a class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006" href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">1</a></sup></span> so we would like to perform our role="doc-biblioref">1</a>]</span> so we would like to perform our
analysis for very large lattices. However for an amorphous system with analysis for very large lattices. However for an amorphous system with
<span class="math inline">\(N\)</span> plaquettes, <span <span class="math inline">\(N\)</span> plaquettes, <span
class="math inline">\(2N\)</span> edges and <span class="math inline">\(2N\)</span> edges and <span
@ -308,15 +308,15 @@ relatively regular pattern for the imaginary fluxes with only a global
two-fold chiral degeneracy.</p> two-fold chiral degeneracy.</p>
<p>Thus, states with a fixed flux sector spontaneously break time <p>Thus, states with a fixed flux sector spontaneously break time
reversal symmetry. This was first described by Yao and Kivelson for a 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"><sup><a href="#ref-Yao2011" class="citation" data-cites="Yao2011"> [<a href="#ref-Yao2011"
role="doc-biblioref">2</a></sup></span>.</p> role="doc-biblioref">2</a>]</span>.</p>
<p>So we have flux sectors that come in degenerate pairs, where time <p>So we have flux sectors that come in degenerate pairs, where time
reversal is equivalent to inverting the flux through every odd reversal is equivalent to inverting the flux through every odd
plaquette, a general feature for lattices with odd plaquettes <span plaquette, a general feature for lattices with odd plaquettes <span
class="citation" data-cites="yaoExactChiralSpin2007 Peri2020"><sup><a class="citation" data-cites="yaoExactChiralSpin2007 Peri2020"> [<a
href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>,<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>,<a
href="#ref-Peri2020" role="doc-biblioref">4</a></sup></span>. This 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>
@ -348,12 +348,11 @@ straight lines <span class="math inline">\(|J^x| = |J^y| +
class="math inline">\(x,y,z\)</span>, shown as dotted line on ~<a 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 href="#fig:phase_diagram">1</a> (Right). We find that on the amorphous
lattice these boundaries exhibit an inward curvature, similar to 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"><sup><a href="#ref-Nasu_Thermal_2015" data-cites="Nasu_Thermal_2015"> [<a href="#ref-Nasu_Thermal_2015"
role="doc-biblioref">5</a></sup></span> or bond<span class="citation" role="doc-biblioref">5</a>]</span> or bond <span class="citation"
data-cites="knolle_dynamics_2016"><sup><a data-cites="knolle_dynamics_2016"> [<a href="#ref-knolle_dynamics_2016"
href="#ref-knolle_dynamics_2016" role="doc-biblioref">6</a></sup></span> role="doc-biblioref">6</a>]</span> disorder.</p>
disorder.</p>
<div id="fig:phase_diagram" class="fignos"> <div id="fig:phase_diagram" class="fignos">
<figure> <figure>
<img <img
@ -388,11 +387,11 @@ class="math inline">\(0\)</span> to <span class="math inline">\(\pm
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
Abelian or non-Abelian statistics? To answer that we turn to Chern 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"><sup><a data-cites="berryQuantalPhaseFactors1984 simonHolonomyQuantumAdiabatic1983 thoulessQuantizedHallConductance1982"> [<a
href="#ref-berryQuantalPhaseFactors1984" role="doc-biblioref">7</a>–<a href="#ref-berryQuantalPhaseFactors1984" role="doc-biblioref">7</a>–<a
href="#ref-thoulessQuantizedHallConductance1982" href="#ref-thoulessQuantizedHallConductance1982"
role="doc-biblioref">9</a></sup></span>. As discussed earlier the Chern role="doc-biblioref">9</a>]</span>. As discussed earlier the Chern
number is a quantity intimately linked to both the topological number is a quantity intimately linked to both the topological
properties and the anyonic statistics of a model. Here we will make use properties and the anyonic statistics of a model. Here we will make use
of the fact that the Abelian/non-Abelian character of a model is linked of the fact that the Abelian/non-Abelian character of a model is linked
@ -400,28 +399,27 @@ to its Chern number <strong>[citation]</strong>. However the Chern
number is only defined for the translation invariant case because it number is only defined for the translation invariant case because it
relies on integrals defined in k-space.</p> relies on integrals defined in k-space.</p>
<p>A family of real space generalisations of the Chern number that work <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" class="citation"
data-cites="bianco_mapping_2011 Hastings_Almost_2010 mitchellAmorphousTopologicalInsulators2018"><sup><a data-cites="bianco_mapping_2011 Hastings_Almost_2010 mitchellAmorphousTopologicalInsulators2018"> [<a
href="#ref-bianco_mapping_2011" role="doc-biblioref">10</a>–<a href="#ref-bianco_mapping_2011" role="doc-biblioref">10</a>–<a
href="#ref-mitchellAmorphousTopologicalInsulators2018" href="#ref-mitchellAmorphousTopologicalInsulators2018"
role="doc-biblioref">12</a></sup></span> and indeed Kitaev defines one role="doc-biblioref">12</a>]</span> and indeed Kitaev defines one in his
in his original paper on the model<span class="citation" original paper on the model <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"><sup><a data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006" href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">1</a></sup></span>.</p> 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"><sup><a href="#ref-peru_preprint" data-cites="peru_preprint"> [<a href="#ref-peru_preprint"
role="doc-biblioref">13</a></sup></span> because it works well on role="doc-biblioref">13</a>]</span> because it works well on smaller
smaller systems. We calculate the projector <span systems. We calculate the projector <span class="math inline">\(P =
class="math inline">\(P = \sum_i |\psi_i\rangle \langle \psi_i|\)</span> \sum_i |\psi_i\rangle \langle \psi_i|\)</span> onto the occupied fermion
onto the occupied fermion eigenstates of the system in open boundary eigenstates of the system in open boundary conditions. The projector
conditions. The projector encodes local information about the occupied encodes local information about the occupied eigenstates of the system
eigenstates of the system and is typically exponentially localised and is typically exponentially localised <strong>[cite]</strong>. The
<strong>[cite]</strong>. The name <em>crosshair</em> comes from the fact name <em>crosshair</em> comes from the fact that the marker is defined
that the marker is defined with respect to a particular point <span with respect to a particular point <span class="math inline">\((x_0,
class="math inline">\((x_0, y_0)\)</span> by step functions in x and y_0)\)</span> by step functions in x and y</p>
y</p>
<p><span class="math display">\[\begin{aligned} <p><span class="math display">\[\begin{aligned}
\nu (x, y) = 4\pi \; \Im\; \mathrm{Tr}_{\mathrm{B}} \nu (x, y) = 4\pi \; \Im\; \mathrm{Tr}_{\mathrm{B}}
\left ( \left (
@ -440,54 +438,51 @@ character of the phases.</p>
<p>In the A phase of the amorphous model we find that <span <p>In the A phase of the amorphous model we find that <span
class="math inline">\(\nu=0\)</span> and hence the excitations have class="math inline">\(\nu=0\)</span> and hence the excitations have
Abelian character, similar to the honeycomb model. This phase is thus Abelian character, similar to the honeycomb model. This phase is thus
the amorphous analogue of the Abelian toric-code quantum spin the amorphous analogue of the Abelian toric-code quantum spin liquid
liquid<span class="citation" <span class="citation" data-cites="kitaev_fault-tolerant_2003"> [<a
data-cites="kitaev_fault-tolerant_2003"><sup><a
href="#ref-kitaev_fault-tolerant_2003" href="#ref-kitaev_fault-tolerant_2003"
role="doc-biblioref">14</a></sup></span>.</p> role="doc-biblioref">14</a>]</span>.</p>
<p>The B phase has <span class="math inline">\(\nu=\pm1\)</span> so is a <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 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"><sup><a data-cites="yaoExactChiralSpin2007"> [<a
href="#ref-yaoExactChiralSpin2007" href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>]</span>.
role="doc-biblioref">3</a></sup></span>. The CSL state is the the The CSL state is the the magnetic analogue of the fractional quantum
magnetic analogue of the fractional quantum Hall state Hall state <strong>[cite]</strong>. Hereafter we focus our attention on
<strong>[cite]</strong>. Hereafter we focus our attention on this this phase.</p>
phase.</p>
<div id="fig:phase_diagram_chern" class="fignos"> <div id="fig:phase_diagram_chern" class="fignos">
<figure> <figure>
<img <img
src="/assets/thesis/amk_chapter/results/phase_diagram_chern/phase_diagram_chern.svg" src="/assets/thesis/amk_chapter/results/phase_diagram_chern/phase_diagram_chern.svg"
data-short-caption="Local Chern Markers" style="width:100.0%" data-short-caption="Local Chern Markers" style="width:100.0%"
alt="Figure 2: (Center) The crosshair marker13, 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 <figcaption aria-hidden="true"><span>Figure 2:</span> (Center) The
crosshair marker<span class="citation" crosshair marker <span class="citation" data-cites="peru_preprint"> [<a
data-cites="peru_preprint"><sup><a href="#ref-peru_preprint" href="#ref-peru_preprint" role="doc-biblioref">13</a>]</span>, a local
role="doc-biblioref">13</a></sup></span>, a local topological marker, topological marker, evaluated on the Amorphous Kitaev Model. The marker
evaluated on the Amorphous Kitaev Model. The marker is defined around a is defined around a point, denoted by the dotted crosshair. Information
point, denoted by the dotted crosshair. Information about the local about the local topological properties of the system are encoded within
topological properties of the system are encoded within a region around a region around that point. (Left) Summing these contributions up to
that point. (Left) Summing these contributions up to some finite radius some finite radius (dotted line here, dotted circle in the centre) gives
(dotted line here, dotted circle in the centre) gives a generalised a generalised version of the Chern number for the system which becomes
version of the Chern number for the system which becomes quantised in quantised in the thermodynamic limit. The radius must be chosen large
the thermodynamic limit. The radius must be chosen large enough to enough to capture information about the local properties of the lattice
capture information about the local properties of the lattice while not while not so large as to include contributions from the edge states. The
so large as to include contributions from the edge states. The isotropic isotropic regime <span class="math inline">\(J_\alpha = 1\)</span> in
regime <span class="math inline">\(J_\alpha = 1\)</span> in red has red has <span class="math inline">\(\nu = \pm 1\)</span> implying it
<span class="math inline">\(\nu = \pm 1\)</span> implying it supports supports excitations with non-Abelian statistics, while the anisotropic
excitations with non-Abelian statistics, while the anisotropic regime in regime in orange has <span class="math inline">\(\nu = \pm 0\)</span>
orange has <span class="math inline">\(\nu = \pm 0\)</span> implying it implying it has Abelian statistics. (Right) Extending this analysis to
has Abelian statistics. (Right) Extending this analysis to the whole the whole <span class="math inline">\(J_\alpha\)</span> phase diagram
<span class="math inline">\(J_\alpha\)</span> phase diagram with fixed with fixed <span class="math inline">\(r = 0.3\)</span> nicely confirms
<span class="math inline">\(r = 0.3\)</span> nicely confirms that the that the isotropic phase is non-Abelian.</figcaption>
isotropic phase is non-Abelian.</figcaption>
</figure> </figure>
</div> </div>
<h3 id="edge-modes">Edge Modes</h3> <h3 id="edge-modes">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"><sup><a href="#ref-qi_general_2006" data-cites="qi_general_2006"> [<a href="#ref-qi_general_2006"
role="doc-biblioref">15</a></sup></span>. fig. <a role="doc-biblioref">15</a>]</span>. fig. <a
href="#fig:edge_modes">3</a> shows the probability density of one such href="#fig:edge_modes">3</a> shows the probability density of one such
edge mode. It is near zero energy and exponentially localised to the edge mode. It is near zero energy and exponentially localised to the
boundary of the system. While the model is gapped in periodic boundary boundary of the system. While the model is gapped in periodic boundary
@ -522,35 +517,34 @@ states.</figcaption>
Thermal Metal</h2> 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
class="citation" data-cites="selfThermallyInducedMetallic2019"><sup><a class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019" href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a></sup></span>.</p> role="doc-biblioref">16</a>]</span>.</p>
<p>This happens because at finite temperature, thermal fluctuations lead <p>This happens because at finite temperature, thermal fluctuations lead
to spontaneous vortex-pair formation. As discussed previously these to spontaneous vortex-pair formation. As discussed previously these
fluxes are dressed by Majorana bounds states and the composite object is 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"><sup><a href="#ref-Beenakker2013" data-cites="Beenakker2013"> [<a href="#ref-Beenakker2013"
role="doc-biblioref">17</a></sup></span>. The interactions between these role="doc-biblioref">17</a>]</span>. The interactions between these
anyons are oscillatory similar to the RKKY exchange and decay 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"><sup><a data-cites="Laumann2012 Lahtinen_2011 lahtinenTopologicalLiquidNucleation2012"> [<a
href="#ref-Laumann2012" role="doc-biblioref">18</a>–<a href="#ref-Laumann2012" role="doc-biblioref">18</a>–<a
href="#ref-lahtinenTopologicalLiquidNucleation2012" href="#ref-lahtinenTopologicalLiquidNucleation2012"
role="doc-biblioref">20</a></sup></span>. At sufficient density, the role="doc-biblioref">20</a>]</span>. At sufficient density, the anyons
anyons hybridise to a macroscopically degenerate state known as hybridise to a macroscopically degenerate state known as <em>thermal
<em>thermal metal</em><span class="citation" metal</em> <span class="citation" data-cites="Laumann2012"> [<a
data-cites="Laumann2012"><sup><a href="#ref-Laumann2012" href="#ref-Laumann2012" role="doc-biblioref">18</a>]</span>. At close
role="doc-biblioref">18</a></sup></span>. At close range the oscillatory range the oscillatory behaviour of the interactions can be modelled by a
behaviour of the interactions can be modelled by a random sign which random sign which forms the basis for a random matrix theory description
forms the basis for a random matrix theory description of the thermal of the thermal metal state.</p>
metal state.</p>
<p>The amorphous chiral spin liquid undergoes the same form of Anderson <p>The amorphous chiral spin liquid undergoes the same form of Anderson
transition to a thermal metal state. Markov Chain Monte Carlo would be 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"><sup><a data-cites="selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019" href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a></sup></span> but in order to avoid that role="doc-biblioref">16</a>]</span> but in order to avoid that
complexity in the current work we instead opted to use vortex density complexity in the current work we instead opted to use vortex density
<span class="math inline">\(\rho\)</span> as a proxy for <span class="math inline">\(\rho\)</span> as a proxy for
temperature.</p> temperature.</p>
@ -641,11 +635,11 @@ model onto a Majorana model with interactions that take random signs
which can itself be mapped onto a coarser lattice with lower energy which can itself be mapped onto a coarser lattice with lower energy
excitations and so on. This can be repeating indefinitely, showing the excitations and so on. This can be repeating indefinitely, showing the
model must have excitations at arbitrarily low energies in 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"><sup><a data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019" href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000" role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000"
role="doc-biblioref">21</a></sup></span>.</p> role="doc-biblioref">21</a>]</span>.</p>
<p>These signatures for our model and for the honeycomb model are shown <p>These signatures for our model and for the honeycomb model are shown
in fig. <a href="#fig:DOS_oscillations">6</a>. They do not occur in the in fig. <a href="#fig:DOS_oscillations">6</a>. They do not occur in the
honeycomb model unless the chiral symmetry is broken by a magnetic honeycomb model unless the chiral symmetry is broken by a magnetic
@ -656,21 +650,21 @@ field.</p>
src="/assets/thesis/amk_chapter/results/DOS_oscillations/DOS_oscillations.svg" src="/assets/thesis/amk_chapter/results/DOS_oscillations/DOS_oscillations.svg"
data-short-caption="Distinctive Oscillations in the Density of States" data-short-caption="Distinctive Oscillations in the Density of States"
style="width:100.0%" 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 state16,21. (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 <figcaption aria-hidden="true"><span>Figure 6:</span> Density of states
at high temperature showing the logarithmic divergence at zero energy 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" class="citation"
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"><sup><a data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019" href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000" role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000"
role="doc-biblioref">21</a></sup></span>. (a) shows the honeycomb role="doc-biblioref">21</a>]</span>. (a) shows the honeycomb lattice
lattice model in the B phase with magnetic field, while (b) shows that model in the B phase with magnetic field, while (b) shows that our model
our model transitions to a thermal metal phase without an external transitions to a thermal metal phase without an external magnetic field
magnetic field but rather due to the spontaneous chiral symmetry but rather due to the spontaneous chiral symmetry breaking. In both
breaking. In both plots the density of vortices is <span plots the density of vortices is <span class="math inline">\(\rho =
class="math inline">\(\rho = 0.5\)</span> corresponding to the <span 0.5\)</span> corresponding to the <span class="math inline">\(T =
class="math inline">\(T = \infty\)</span> limit.</figcaption> \infty\)</span> limit.</figcaption>
</figure> </figure>
</div> </div>
<h1 id="conclusion">Conclusion</h1> <h1 id="conclusion">Conclusion</h1>
@ -719,46 +713,45 @@ 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
are cooled rapidly, freezing them into a disordered configuration<span are cooled rapidly, freezing them into a disordered configuration <span
class="citation" class="citation"
data-cites="Weaire1976 Petrakovski1981 Kaneyoshi2018"><sup><a data-cites="Weaire1976 Petrakovski1981 Kaneyoshi2018"> [<a
href="#ref-Weaire1976" role="doc-biblioref">22</a>–<a href="#ref-Weaire1976" role="doc-biblioref">22</a>–<a
href="#ref-Kaneyoshi2018" role="doc-biblioref">24</a></sup></span>. href="#ref-Kaneyoshi2018" role="doc-biblioref">24</a>]</span>. Indeed
Indeed quenching has been used by humans to control the hardness of quenching has been used by humans to control the hardness of steel or
steel or iron for thousands of years. It would therefore be interesting iron for thousands of years. It would therefore be interesting to study
to study amorphous version of candidate Kitaev materials<span amorphous version of candidate Kitaev materials <span class="citation"
class="citation" data-cites="trebstKitaevMaterials2022"><sup><a data-cites="trebstKitaevMaterials2022"> [<a
href="#ref-trebstKitaevMaterials2022" href="#ref-trebstKitaevMaterials2022"
role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a></sup></span> role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a>]</span>
such as <span class="math inline">\(\alpha-\textrm{RuCl}_3\)</span> to such as <span class="math inline">\(\alpha-\textrm{RuCl}_3\)</span> to
see whether they maintain even approximate fixed coordination number 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"><sup><a class="citation" data-cites="Weaire1971 betteridge1973possible"> [<a
href="#ref-Weaire1971" role="doc-biblioref">25</a>,<a href="#ref-Weaire1971" role="doc-biblioref">25</a>,<a
href="#ref-betteridge1973possible" href="#ref-betteridge1973possible"
role="doc-biblioref">26</a></sup></span>.</p> role="doc-biblioref">26</a>]</span>.</p>
<p>Looking instead at more engineered realisation, metal organic <p>Looking instead at more engineered realisation, metal organic
frameworks have been shown to be capable of forming amorphous frameworks have been shown to be capable of forming amorphous
lattices <span class="citation" lattices <span class="citation" data-cites="bennett2014amorphous"> [<a
data-cites="bennett2014amorphous"><sup><a href="#ref-bennett2014amorphous" role="doc-biblioref">27</a>]</span> and
href="#ref-bennett2014amorphous" there are recent proposals for realizing strong Kitaev
role="doc-biblioref">27</a></sup></span> and there are recent proposals interactions <span class="citation"
for realizing strong Kitaev interactions <span class="citation" data-cites="yamadaDesigningKitaevSpin2017"> [<a
data-cites="yamadaDesigningKitaevSpin2017"><sup><a
href="#ref-yamadaDesigningKitaevSpin2017" href="#ref-yamadaDesigningKitaevSpin2017"
role="doc-biblioref">28</a></sup></span> as well as reports of QSL role="doc-biblioref">28</a>]</span> as well as reports of QSL
behavior <span class="citation" behavior <span class="citation"
data-cites="misumiQuantumSpinLiquid2020"><sup><a data-cites="misumiQuantumSpinLiquid2020"> [<a
href="#ref-misumiQuantumSpinLiquid2020" href="#ref-misumiQuantumSpinLiquid2020"
role="doc-biblioref">29</a></sup></span>.</p> role="doc-biblioref">29</a>]</span>.</p>
<h2 id="generalisations">Generalisations</h2> <h2 id="generalisations">Generalisations</h2>
<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
class="citation" class="citation"
data-cites="Rau2014 Chaloupka2010 Chaloupka2013 Chaloupka2015 Winter2016"><sup><a data-cites="Rau2014 Chaloupka2010 Chaloupka2013 Chaloupka2015 Winter2016"> [<a
href="#ref-Rau2014" role="doc-biblioref">30</a>–<a href="#ref-Rau2014" role="doc-biblioref">30</a>–<a
href="#ref-Winter2016" role="doc-biblioref">34</a></sup></span>.</p> href="#ref-Winter2016" role="doc-biblioref">34</a>]</span>.</p>
<p>Second, one could investigate whether a QSL phase may exist for for <p>Second, one could investigate whether a QSL phase may exist for for
other models defined on amorphous lattices. For example, in real other models defined on amorphous lattices. For example, in real
materials, there will generally be an additional small Heisenberg term materials, there will generally be an additional small Heisenberg term
@ -767,398 +760,382 @@ 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 \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 observation, it would be interesting to see if the properties of the
Kitaev-Heisenberg model generalise from the honeycomb to the amorphous Kitaev-Heisenberg model generalise from the honeycomb to the amorphous
case[<span class="citation" data-cites="Chaloupka2010"><sup><a case[<span class="citation" data-cites="Chaloupka2010"> [<a
href="#ref-Chaloupka2010" role="doc-biblioref">31</a></sup></span>;<span href="#ref-Chaloupka2010" role="doc-biblioref">31</a>]</span>; <span
class="citation" data-cites="Chaloupka2015"><sup><a class="citation" data-cites="Chaloupka2015"> [<a
href="#ref-Chaloupka2015" role="doc-biblioref">33</a></sup></span>;<span href="#ref-Chaloupka2015" role="doc-biblioref">33</a>]</span>; <span
class="citation" data-cites="Jackeli2009"><sup><a class="citation" data-cites="Jackeli2009"> [<a href="#ref-Jackeli2009"
href="#ref-Jackeli2009" role="doc-biblioref">35</a></sup></span>;<span role="doc-biblioref">35</a>]</span>; <span class="citation"
class="citation" data-cites="Kalmeyer1989"><sup><a data-cites="Kalmeyer1989"> [<a href="#ref-Kalmeyer1989"
href="#ref-Kalmeyer1989" role="doc-biblioref">36</a></sup></span>;<span role="doc-biblioref">36</a>]</span>; <span class="citation"
class="citation" data-cites="manousakisSpinTextonehalfHeisenberg1991"> [<a
data-cites="manousakisSpinTextonehalfHeisenberg1991"><sup><a
href="#ref-manousakisSpinTextonehalfHeisenberg1991" href="#ref-manousakisSpinTextonehalfHeisenberg1991"
role="doc-biblioref">37</a></sup></span>;].</p> role="doc-biblioref">37</a>]</span>;].</p>
<p>Finally it might be possible to look at generalizations to <p>Finally it might be possible to look at generalizations to
higher-spin models or those on random networks with different 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"><sup><a 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-Yao2011" role="doc-biblioref">2</a>,<a
href="#ref-Baskaran2008" role="doc-biblioref">38</a>–<a href="#ref-Baskaran2008" role="doc-biblioref">38</a>–<a
href="#ref-Wu2009" role="doc-biblioref">47</a></sup></span></p> href="#ref-Wu2009" role="doc-biblioref">47</a>]</span></p>
<p>Overall, there has been surprisingly little research on amorphous <p>Overall, there has been surprisingly little research on amorphous
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>
<div id="refs" class="references csl-bib-body" data-line-spacing="2" <div id="refs" class="references csl-bib-body" role="doc-bibliography">
role="doc-bibliography">
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry" <div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div <div class="csl-left-margin">[1] </div><div class="csl-right-inline">A.
class="csl-right-inline">Kitaev, A. <a Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons
href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons in an exactly in an Exactly Solved Model and Beyond</a></em>, Annals of Physics
solved model and beyond</a>. <em>Annals of Physics</em> <strong>321</strong>, 2 (2006).</div>
<strong>321</strong>, 2–111 (2006).</div>
</div> </div>
<div id="ref-Yao2011" class="csl-entry" role="doc-biblioentry"> <div id="ref-Yao2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div class="csl-right-inline">Yao, <div class="csl-left-margin">[2] </div><div class="csl-right-inline">H.
H. &amp; Lee, D.-H. <a Yao and D.-H. Lee, <em><a
href="https://doi.org/10.1103/PhysRevLett.107.087205">Fermionic magnons, href="https://doi.org/10.1103/PhysRevLett.107.087205">Fermionic Magnons,
non-abelian spinons, and the spin quantum hall effect from an exactly Non-Abelian Spinons, and the Spin Quantum Hall Effect from an Exactly
solvable spin-1/2 kitaev model with SU(2) symmetry</a>. <em>Phys. Rev. Solvable Spin-1/2 Kitaev Model with SU(2) Symmetry</a></em>, Phys. Rev.
Lett.</em> <strong>107</strong>, 087205 (2011).</div> Lett. <strong>107</strong>, 087205 (2011).</div>
</div> </div>
<div id="ref-yaoExactChiralSpin2007" class="csl-entry" <div id="ref-yaoExactChiralSpin2007" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div class="csl-right-inline">Yao, <div class="csl-left-margin">[3] </div><div class="csl-right-inline">H.
H. &amp; Kivelson, S. A. <a Yao and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.99.247203">An exact chiral href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral
spin liquid with non-Abelian anyons</a>. <em>Phys. Rev. Lett.</em> Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett.
<strong>99</strong>, 247203 (2007).</div> <strong>99</strong>, 247203 (2007).</div>
</div> </div>
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry"> <div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div <div class="csl-left-margin">[4] </div><div class="csl-right-inline">V.
class="csl-right-inline">Peri, V. <em>et al.</em> <a Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R.
href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian chiral Moessner, and R. Thomale, <em><a
spin liquid on a simple non-Archimedean lattice</a>. <em>Phys. Rev. href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian Chiral
B</em> <strong>101</strong>, 041114 (2020).</div> Spin Liquid on a Simple Non-Archimedean Lattice</a></em>, Phys. Rev. B
<strong>101</strong>, 041114 (2020).</div>
</div> </div>
<div id="ref-Nasu_Thermal_2015" class="csl-entry" <div id="ref-Nasu_Thermal_2015" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div <div class="csl-left-margin">[5] </div><div class="csl-right-inline">J.
class="csl-right-inline">Nasu, J., Udagawa, M. &amp; Motome, Y. <a Nasu, M. Udagawa, and Y. Motome, <em><a
href="https://doi.org/10.1103/PhysRevB.92.115122">Thermal href="https://doi.org/10.1103/PhysRevB.92.115122">Thermal
fractionalization of quantum spins in a Kitaev model: Temperature-linear Fractionalization of Quantum Spins in a Kitaev Model: Temperature-Linear
specific heat and coherent transport of Majorana fermions</a>. <em>Phys. Specific Heat and Coherent Transport of Majorana Fermions</a></em>,
Rev. B</em> <strong>92</strong>, 115122 (2015).</div> Phys. Rev. B <strong>92</strong>, 115122 (2015).</div>
</div> </div>
<div id="ref-knolle_dynamics_2016" class="csl-entry" <div id="ref-knolle_dynamics_2016" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div <div class="csl-left-margin">[6] </div><div class="csl-right-inline">J.
class="csl-right-inline">Knolle, J. Dynamics of a quantum spin liquid. Knolle, Dynamics of a Quantum Spin Liquid, Max Planck Institute for the
(Max Planck Institute for the Physics of Complex Systems, Dresden, Physics of Complex Systems, Dresden, 2016.</div>
2016).</div>
</div> </div>
<div id="ref-berryQuantalPhaseFactors1984" class="csl-entry" <div id="ref-berryQuantalPhaseFactors1984" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div <div class="csl-left-margin">[7] </div><div class="csl-right-inline">M.
class="csl-right-inline">Berry, M. V. <a V. Berry, <em><a href="https://doi.org/10.1098/rspa.1984.0023">Quantal
href="https://doi.org/10.1098/rspa.1984.0023">Quantal phase factors Phase Factors Accompanying Adiabatic Changes</a></em>, Proceedings of
accompanying adiabatic changes</a>. <em>Proceedings of the Royal Society the Royal Society of London. A. Mathematical and Physical Sciences
of London. A. Mathematical and Physical Sciences</em> <strong>392</strong>, 45 (1984).</div>
<strong>392</strong>, 45–57 (1984).</div>
</div> </div>
<div id="ref-simonHolonomyQuantumAdiabatic1983" class="csl-entry" <div id="ref-simonHolonomyQuantumAdiabatic1983" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div <div class="csl-left-margin">[8] </div><div class="csl-right-inline">B.
class="csl-right-inline">Simon, B. <a Simon, <em><a
href="https://doi.org/10.1103/PhysRevLett.51.2167">Holonomy, the Quantum href="https://doi.org/10.1103/PhysRevLett.51.2167">Holonomy, the Quantum
Adiabatic Theorem, and Berry’s Phase</a>. <em>Phys. Rev. Lett.</em> Adiabatic Theorem, and Berry’s Phase</a></em>, Phys. Rev. Lett.
<strong>51</strong>, 2167–2170 (1983).</div> <strong>51</strong>, 2167 (1983).</div>
</div> </div>
<div id="ref-thoulessQuantizedHallConductance1982" class="csl-entry" <div id="ref-thoulessQuantizedHallConductance1982" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div <div class="csl-left-margin">[9] </div><div class="csl-right-inline">D.
class="csl-right-inline">Thouless, D. J., Kohmoto, M., Nightingale, M. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, <em><a
P. &amp; den Nijs, M. <a
href="https://doi.org/10.1103/PhysRevLett.49.405">Quantized Hall href="https://doi.org/10.1103/PhysRevLett.49.405">Quantized Hall
Conductance in a Two-Dimensional Periodic Potential</a>. <em>Phys. Rev. Conductance in a Two-Dimensional Periodic Potential</a></em>, Phys. Rev.
Lett.</em> <strong>49</strong>, 405–408 (1982).</div> Lett. <strong>49</strong>, 405 (1982).</div>
</div> </div>
<div id="ref-bianco_mapping_2011" class="csl-entry" <div id="ref-bianco_mapping_2011" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div <div class="csl-left-margin">[10] </div><div class="csl-right-inline">R.
class="csl-right-inline">Bianco, R. &amp; Resta, R. <a Bianco and R. Resta, <em><a
href="https://doi.org/10.1103/PhysRevB.84.241106">Mapping topological href="https://doi.org/10.1103/PhysRevB.84.241106">Mapping Topological
order in coordinate space</a>. <em>Physical Review B</em> Order in Coordinate Space</a></em>, Physical Review B
<strong>84</strong>, 241106 (2011).</div> <strong>84</strong>, 241106 (2011).</div>
</div> </div>
<div id="ref-Hastings_Almost_2010" class="csl-entry" <div id="ref-Hastings_Almost_2010" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div <div class="csl-left-margin">[11] </div><div class="csl-right-inline">M.
class="csl-right-inline">Hastings, M. B. &amp; Loring, T. A. <a B. Hastings and T. A. Loring, <em><a
href="https://doi.org/10.1063/1.3274817">Almost commuting matrices, href="https://doi.org/10.1063/1.3274817">Almost Commuting Matrices,
localized Wannier functions, and the quantum Hall effect</a>. Localized Wannier Functions, and the Quantum Hall Effect</a></em>,
<em>Journal of Mathematical Physics</em> <strong>51</strong>, 015214 Journal of Mathematical Physics <strong>51</strong>, 015214
(2010).</div> (2010).</div>
</div> </div>
<div id="ref-mitchellAmorphousTopologicalInsulators2018" <div id="ref-mitchellAmorphousTopologicalInsulators2018"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div <div class="csl-left-margin">[12] </div><div class="csl-right-inline">N.
class="csl-right-inline">Mitchell, N. P., Nash, L. M., Hexner, D., P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine,
Turner, A. M. &amp; Irvine, W. T. M. <a <em><a href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous
href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous topological topological insulators constructed from random point sets</a></em>,
insulators constructed from random point sets</a>. <em>Nature Phys</em> Nature Phys <strong>14</strong>, 380 (2018).</div>
<strong>14</strong>, 380–385 (2018).</div>
</div> </div>
<div id="ref-peru_preprint" class="csl-entry" role="doc-biblioentry"> <div id="ref-peru_preprint" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div <div class="csl-left-margin">[13] </div><div class="csl-right-inline">P.
class="csl-right-inline">d’Ornellas, P., Barnett, R. &amp; Lee, D. K. K. d’Ornellas, R. Barnett, and D. K. K. Lee, <em><a
Quantised bulk conductivity as a local chern marker. <em>arXiv href="https://doi.org/10.48550/ARXIV.2207.01389">Quantised Bulk
preprint</em> (2022) doi:<a Conductivity as a Local Chern Marker</a></em>, arXiv Preprint
href="https://doi.org/10.48550/ARXIV.2207.01389">10.48550/ARXIV.2207.01389</a>.</div> (2022).</div>
</div> </div>
<div id="ref-kitaev_fault-tolerant_2003" class="csl-entry" <div id="ref-kitaev_fault-tolerant_2003" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div <div class="csl-left-margin">[14] </div><div class="csl-right-inline">A.
class="csl-right-inline">Kitaev, A. Y. <a Y. Kitaev, <em><a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-tolerant href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
quantum computation by anyons</a>. <em>Annals of Physics</em> Quantum Computation by Anyons</a></em>, Annals of Physics
<strong>303</strong>, 2–30 (2003).</div> <strong>303</strong>, 2 (2003).</div>
</div> </div>
<div id="ref-qi_general_2006" class="csl-entry" role="doc-biblioentry"> <div id="ref-qi_general_2006" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div class="csl-right-inline">Qi, <div class="csl-left-margin">[15] </div><div
X.-L., Wu, Y.-S. &amp; Zhang, S.-C. <a class="csl-right-inline">X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, <em><a
href="https://doi.org/10.1103/PhysRevB.74.045125">General theorem href="https://doi.org/10.1103/PhysRevB.74.045125">General Theorem
relating the bulk topological number to edge states in two-dimensional Relating the Bulk Topological Number to Edge States in Two-Dimensional
insulators</a>. <em>Physical Review B</em> <strong>74</strong>, 045125 Insulators</a></em>, Physical Review B <strong>74</strong>, 045125
(2006).</div> (2006).</div>
</div> </div>
<div id="ref-selfThermallyInducedMetallic2019" class="csl-entry" <div id="ref-selfThermallyInducedMetallic2019" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div <div class="csl-left-margin">[16] </div><div class="csl-right-inline">C.
class="csl-right-inline">Self, C. N., Knolle, J., Iblisdir, S. &amp; N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, <em><a
Pachos, J. K. <a href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally Induced
href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally induced Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of
metallic phase in a gapped quantum spin liquid - a Monte Carlo study of the Kitaev Model with Parity Projection</a></em>, Phys. Rev. B
the Kitaev model with parity projection</a>. <em>Phys. Rev. B</em>
<strong>99</strong>, 045142 (2019).</div> <strong>99</strong>, 045142 (2019).</div>
</div> </div>
<div id="ref-Beenakker2013" class="csl-entry" role="doc-biblioentry"> <div id="ref-Beenakker2013" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div <div class="csl-left-margin">[17] </div><div class="csl-right-inline">C.
class="csl-right-inline">Beenakker, C. W. J. <a W. J. Beenakker, <em><a
href="https://doi.org/10.1146/annurev-conmatphys-030212-184337">Search href="https://doi.org/10.1146/annurev-conmatphys-030212-184337">Search
for majorana fermions in superconductors</a>. <em>Annual Review of for Majorana Fermions in Superconductors</a></em>, Annual Review of
Condensed Matter Physics</em> <strong>4</strong>, 113–136 (2013).</div> Condensed Matter Physics <strong>4</strong>, 113 (2013).</div>
</div> </div>
<div id="ref-Laumann2012" class="csl-entry" role="doc-biblioentry"> <div id="ref-Laumann2012" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div <div class="csl-left-margin">[18] </div><div class="csl-right-inline">C.
class="csl-right-inline">Laumann, C. R., Ludwig, A. W. W., Huse, D. A. R. Laumann, A. W. W. Ludwig, D. A. Huse, and S. Trebst, <em><a
&amp; Trebst, S. <a href="https://doi.org/10.1103/PhysRevB.85.161301">Disorder-Induced
href="https://doi.org/10.1103/PhysRevB.85.161301">Disorder-induced Majorana Metal in Interacting Non-Abelian Anyon Systems</a></em>, Phys.
Majorana metal in interacting non-Abelian anyon systems</a>. <em>Phys. Rev. B <strong>85</strong>, 161301 (2012).</div>
Rev. B</em> <strong>85</strong>, 161301 (2012).</div>
</div> </div>
<div id="ref-Lahtinen_2011" class="csl-entry" role="doc-biblioentry"> <div id="ref-Lahtinen_2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">19. </div><div <div class="csl-left-margin">[19] </div><div class="csl-right-inline">V.
class="csl-right-inline">Lahtinen, V. <a Lahtinen, <em><a
href="https://doi.org/10.1088/1367-2630/13/7/075009">Interacting href="https://doi.org/10.1088/1367-2630/13/7/075009">Interacting
non-Abelian anyons as Majorana fermions in the honeycomb lattice Non-Abelian Anyons as Majorana Fermions in the Honeycomb Lattice
model</a>. <em>New Journal of Physics</em> <strong>13</strong>, 075009 Model</a></em>, New Journal of Physics <strong>13</strong>, 075009
(2011).</div> (2011).</div>
</div> </div>
<div id="ref-lahtinenTopologicalLiquidNucleation2012" class="csl-entry" <div id="ref-lahtinenTopologicalLiquidNucleation2012" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">20. </div><div <div class="csl-left-margin">[20] </div><div class="csl-right-inline">V.
class="csl-right-inline">Lahtinen, V., Ludwig, A. W. W., Pachos, J. K. Lahtinen, A. W. W. Ludwig, J. K. Pachos, and S. Trebst, <em><a
&amp; Trebst, S. <a href="https://doi.org/10.1103/PhysRevB.86.075115">Topological Liquid
href="https://doi.org/10.1103/PhysRevB.86.075115">Topological liquid Nucleation Induced by Vortex-Vortex Interactions in Kitaev’s Honeycomb
nucleation induced by vortex-vortex interactions in Kitaev’s honeycomb Model</a></em>, Phys. Rev. B <strong>86</strong>, 075115 (2012).</div>
model</a>. <em>Phys. Rev. B</em> <strong>86</strong>, 075115
(2012).</div>
</div> </div>
<div id="ref-bocquet_disordered_2000" class="csl-entry" <div id="ref-bocquet_disordered_2000" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">21. </div><div <div class="csl-left-margin">[21] </div><div class="csl-right-inline">M.
class="csl-right-inline">Bocquet, M., Serban, D. &amp; Zirnbauer, M. R. Bocquet, D. Serban, and M. R. Zirnbauer, <em><a
<a href="https://doi.org/10.1016/S0550-3213(00)00208-X">Disordered 2d href="https://doi.org/10.1016/S0550-3213(00)00208-X">Disordered 2d
quasiparticles in class D: Dirac fermions with random mass, and dirty Quasiparticles in Class D: Dirac Fermions with Random Mass, and Dirty
superconductors</a>. <em>Nuclear Physics B</em> <strong>578</strong>, Superconductors</a></em>, Nuclear Physics B <strong>578</strong>, 628
628–680 (2000).</div> (2000).</div>
</div> </div>
<div id="ref-Weaire1976" class="csl-entry" role="doc-biblioentry"> <div id="ref-Weaire1976" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">22. </div><div <div class="csl-left-margin">[22] </div><div class="csl-right-inline">D.
class="csl-right-inline">Weaire, D. &amp; Thorpe, M. F. <a Weaire and M. F. Thorpe, <em><a
href="https://doi.org/10.1080/00107517608210851">The structure of href="https://doi.org/10.1080/00107517608210851">The Structure of
amorphous solids</a>. <em>Contemporary Physics</em> <strong>17</strong>, Amorphous Solids</a></em>, Contemporary Physics <strong>17</strong>, 173
173–191 (1976).</div> (1976).</div>
</div> </div>
<div id="ref-Petrakovski1981" class="csl-entry" role="doc-biblioentry"> <div id="ref-Petrakovski1981" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">23. </div><div <div class="csl-left-margin">[23] </div><div class="csl-right-inline">G.
class="csl-right-inline">Petrakovskii, G. A. <a A. Petrakovskii, <em><a
href="https://doi.org/10.1070/pu1981v024n06abeh004850">Amorphous href="https://doi.org/10.1070/pu1981v024n06abeh004850">Amorphous
magnetic materials</a>. <em>Soviet Physics Uspekhi</em> Magnetic Materials</a></em>, Soviet Physics Uspekhi <strong>24</strong>,
<strong>24</strong>, 511–525 (1981).</div> 511 (1981).</div>
</div> </div>
<div id="ref-Kaneyoshi2018" class="csl-entry" role="doc-biblioentry"> <div id="ref-Kaneyoshi2018" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">24. </div><div <div class="csl-left-margin">[24] </div><div class="csl-right-inline">T.
class="csl-right-inline"><em>Amorphous magnetism</em>. (CRC Press, Kaneyoshi, editor, <em>Amorphous Magnetism</em> (CRC Press, Boca Raton,
2018).</div> 2018).</div>
</div> </div>
<div id="ref-Weaire1971" class="csl-entry" role="doc-biblioentry"> <div id="ref-Weaire1971" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">25. </div><div <div class="csl-left-margin">[25] </div><div class="csl-right-inline">D.
class="csl-right-inline">Weaire, D. &amp; Thorpe, M. F. <a Weaire and M. F. Thorpe, <em><a
href="https://doi.org/10.1103/PhysRevB.4.2508">Electronic properties of href="https://doi.org/10.1103/PhysRevB.4.2508">Electronic Properties of
an amorphous solid. I. A simple tight-binding theory</a>. <em>Phys. Rev. an Amorphous Solid. I. A Simple Tight-Binding Theory</a></em>, Phys.
B</em> <strong>4</strong>, 2508–2520 (1971).</div> Rev. B <strong>4</strong>, 2508 (1971).</div>
</div> </div>
<div id="ref-betteridge1973possible" class="csl-entry" <div id="ref-betteridge1973possible" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">26. </div><div <div class="csl-left-margin">[26] </div><div class="csl-right-inline">G.
class="csl-right-inline">Betteridge, G. A possible model of amorphous Betteridge, <em>A Possible Model of Amorphous Silicon and
silicon and germanium. <em>Journal of Physics C: Solid State Germanium</em>, Journal of Physics C: Solid State Physics
Physics</em> <strong>6</strong>, L427 (1973).</div> <strong>6</strong>, L427 (1973).</div>
</div> </div>
<div id="ref-bennett2014amorphous" class="csl-entry" <div id="ref-bennett2014amorphous" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">27. </div><div <div class="csl-left-margin">[27] </div><div class="csl-right-inline">T.
class="csl-right-inline">Bennett, T. D. &amp; Cheetham, A. K. Amorphous D. Bennett and A. K. Cheetham, <em>Amorphous Metal–Organic
metal–organic frameworks. <em>Accounts of chemical research</em> Frameworks</em>, Accounts of Chemical Research <strong>47</strong>, 1555
<strong>47</strong>, 1555–1562 (2014).</div> (2014).</div>
</div> </div>
<div id="ref-yamadaDesigningKitaevSpin2017" class="csl-entry" <div id="ref-yamadaDesigningKitaevSpin2017" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">28. </div><div <div class="csl-left-margin">[28] </div><div class="csl-right-inline">M.
class="csl-right-inline">Yamada, M. G., Fujita, H. &amp; Oshikawa, M. <a G. Yamada, H. Fujita, and M. Oshikawa, <em><a
href="https://doi.org/10.1103/PhysRevLett.119.057202">Designing Kitaev href="https://doi.org/10.1103/PhysRevLett.119.057202">Designing Kitaev
Spin Liquids in Metal-Organic Frameworks</a>. <em>Phys. Rev. Lett.</em> Spin Liquids in Metal-Organic Frameworks</a></em>, Phys. Rev. Lett.
<strong>119</strong>, 057202 (2017).</div> <strong>119</strong>, 057202 (2017).</div>
</div> </div>
<div id="ref-misumiQuantumSpinLiquid2020" class="csl-entry" <div id="ref-misumiQuantumSpinLiquid2020" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">29. </div><div <div class="csl-left-margin">[29] </div><div class="csl-right-inline">Y.
class="csl-right-inline">Misumi, Y. <em>et al.</em> <a Misumi, A. Yamaguchi, Z. Zhang, T. Matsushita, N. Wada, M. Tsuchiizu,
href="https://doi.org/10.1021/jacs.0c05472">Quantum Spin Liquid State in and K. Awaga, <em><a href="https://doi.org/10.1021/jacs.0c05472">Quantum
a Two-Dimensional Semiconductive Metal–Organic Framework</a>. <em>J. Am. Spin Liquid State in a Two-Dimensional Semiconductive Metal–Organic
Chem. Soc.</em> <strong>142</strong>, 16513–16517 (2020).</div> Framework</a></em>, J. Am. Chem. Soc. <strong>142</strong>, 16513
(2020).</div>
</div> </div>
<div id="ref-Rau2014" class="csl-entry" role="doc-biblioentry"> <div id="ref-Rau2014" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">30. </div><div <div class="csl-left-margin">[30] </div><div class="csl-right-inline">J.
class="csl-right-inline">Rau, J. G., Lee, E. K.-H. &amp; Kee, H.-Y. <a G. Rau, E. K.-H. Lee, and H.-Y. Kee, <em><a
href="https://doi.org/10.1103/PhysRevLett.112.077204">Generic spin model href="https://doi.org/10.1103/PhysRevLett.112.077204">Generic Spin Model
for the honeycomb iridates beyond the kitaev limit</a>. <em>Phys. Rev. for the Honeycomb Iridates Beyond the Kitaev Limit</a></em>, Phys. Rev.
Lett.</em> <strong>112</strong>, 077204 (2014).</div> Lett. <strong>112</strong>, 077204 (2014).</div>
</div> </div>
<div id="ref-Chaloupka2010" class="csl-entry" role="doc-biblioentry"> <div id="ref-Chaloupka2010" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">31. </div><div <div class="csl-left-margin">[31] </div><div class="csl-right-inline">J.
class="csl-right-inline">Chaloupka, J., Jackeli, G. &amp; Khaliullin, G. Chaloupka, G. Jackeli, and G. Khaliullin, <em>Kitaev-Heisenberg Model on
Kitaev-Heisenberg model on a honeycomb lattice: possible exotic phases a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides
in iridium oxides A₂IrO₃. <em>Phys. Rev. Lett.</em> A₂IrO₃</em>, Phys. Rev. Lett. <strong>105</strong>, 027204 (2010).</div>
<strong>105</strong>, 027204 (2010).</div>
</div> </div>
<div id="ref-Chaloupka2013" class="csl-entry" role="doc-biblioentry"> <div id="ref-Chaloupka2013" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">32. </div><div <div class="csl-left-margin">[32] </div><div class="csl-right-inline">J.
class="csl-right-inline">Chaloupka, J., Jackeli, G. &amp; Khaliullin, G. Chaloupka, G. Jackeli, and G. Khaliullin, <em><a
<a href="https://doi.org/10.1103/PhysRevLett.110.097204">Zigzag magnetic href="https://doi.org/10.1103/PhysRevLett.110.097204">Zigzag Magnetic
order in the iridium oxide Na₂IrO₃</a>. <em>Phys. Rev. Lett.</em> Order in the Iridium Oxide Na₂IrO₃</a></em>, Phys. Rev. Lett.
<strong>110</strong>, 097204 (2013).</div> <strong>110</strong>, 097204 (2013).</div>
</div> </div>
<div id="ref-Chaloupka2015" class="csl-entry" role="doc-biblioentry"> <div id="ref-Chaloupka2015" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">33. </div><div <div class="csl-left-margin">[33] </div><div class="csl-right-inline">J.
class="csl-right-inline">Chaloupka, J. &amp; Khaliullin, G. Hidden Chaloupka and G. Khaliullin, <em>Hidden Symmetries of the Extended
symmetries of the extended Kitaev-Heisenberg model: Implications for Kitaev-Heisenberg Model: Implications for Honeycomb Lattice Iridates
honeycomb lattice iridates A₂IrO₃. <em>Phys. Rev. B</em> A₂IrO₃</em>, Phys. Rev. B <strong>92</strong>, 024413 (2015).</div>
<strong>92</strong>, 024413 (2015).</div>
</div> </div>
<div id="ref-Winter2016" class="csl-entry" role="doc-biblioentry"> <div id="ref-Winter2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">34. </div><div <div class="csl-left-margin">[34] </div><div class="csl-right-inline">S.
class="csl-right-inline">Winter, S. M., Li, Y., Jeschke, H. O. &amp; M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, <em><a
Valentí, R. <a href="https://doi.org/10.1103/PhysRevB.93.214431">Challenges in Design
href="https://doi.org/10.1103/PhysRevB.93.214431">Challenges in design of Kitaev Materials: Magnetic Interactions from Competing Energy
of Kitaev materials: Magnetic interactions from competing energy Scales</a></em>, Phys. Rev. B <strong>93</strong>, 214431 (2016).</div>
scales</a>. <em>Phys. Rev. B</em> <strong>93</strong>, 214431
(2016).</div>
</div> </div>
<div id="ref-Jackeli2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-Jackeli2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">35. </div><div <div class="csl-left-margin">[35] </div><div class="csl-right-inline">G.
class="csl-right-inline">Jackeli, G. &amp; Khaliullin, G. <a Jackeli and G. Khaliullin, <em><a
href="https://doi.org/10.1103/PhysRevLett.102.017205">Mott insulators in href="https://doi.org/10.1103/PhysRevLett.102.017205">Mott Insulators in
the strong spin-orbit coupling limit: from Heisenberg to a quantum the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum
compass and Kitaev models</a>. <em>Physical Review Letters</em> Compass and Kitaev Models</a></em>, Physical Review Letters
<strong>102</strong>, 017205 (2009).</div> <strong>102</strong>, 017205 (2009).</div>
</div> </div>
<div id="ref-Kalmeyer1989" class="csl-entry" role="doc-biblioentry"> <div id="ref-Kalmeyer1989" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">36. </div><div <div class="csl-left-margin">[36] </div><div class="csl-right-inline">V.
class="csl-right-inline">Kalmeyer, V. &amp; Laughlin, R. B. <a Kalmeyer and R. B. Laughlin, <em><a
href="https://doi.org/10.1103/PhysRevB.39.11879">Theory of the spin href="https://doi.org/10.1103/PhysRevB.39.11879">Theory of the Spin
liquid state of the Heisenberg antiferromagnet</a>. <em>Phys. Rev. Liquid State of the Heisenberg Antiferromagnet</a></em>, Phys. Rev. B
B</em> <strong>39</strong>, 11879–11899 (1989).</div> <strong>39</strong>, 11879 (1989).</div>
</div> </div>
<div id="ref-manousakisSpinTextonehalfHeisenberg1991" class="csl-entry" <div id="ref-manousakisSpinTextonehalfHeisenberg1991" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">37. </div><div <div class="csl-left-margin">[37] </div><div class="csl-right-inline">E.
class="csl-right-inline">Manousakis, E. <a Manousakis, <em><a href="https://doi.org/10.1103/RevModPhys.63.1">The
href="https://doi.org/10.1103/RevModPhys.63.1">The spin-\textonehalf{} Spin-\Textonehalf{} Heisenberg Antiferromagnet on a Square Lattice and
Heisenberg antiferromagnet on a square lattice and its application to Its Application to the Cuprous Oxides</a></em>, Rev. Mod. Phys.
the cuprous oxides</a>. <em>Rev. Mod. Phys.</em> <strong>63</strong>, <strong>63</strong>, 1 (1991).</div>
1–62 (1991).</div>
</div> </div>
<div id="ref-Baskaran2008" class="csl-entry" role="doc-biblioentry"> <div id="ref-Baskaran2008" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">38. </div><div <div class="csl-left-margin">[38] </div><div class="csl-right-inline">G.
class="csl-right-inline">Baskaran, G., Sen, D. &amp; Shankar, R. <a Baskaran, D. Sen, and R. Shankar, <em><a
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev model: href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev Model:
Classical ground states, order from disorder, and exact correlation Classical Ground States, Order from Disorder, and Exact Correlation
functions</a>. <em>Phys. Rev. B</em> <strong>78</strong>, 115116 Functions</a></em>, Phys. Rev. B <strong>78</strong>, 115116
(2008).</div> (2008).</div>
</div> </div>
<div id="ref-Yao2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-Yao2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">39. </div><div <div class="csl-left-margin">[39] </div><div class="csl-right-inline">H.
class="csl-right-inline">Yao, H., Zhang, S.-C. &amp; Kivelson, S. A. <a Yao, S.-C. Zhang, and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic spin href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin
liquid in an exactly solvable spin model</a>. <em>Phys. Rev. Lett.</em> Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett.
<strong>102</strong>, 217202 (2009).</div> <strong>102</strong>, 217202 (2009).</div>
</div> </div>
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">40. </div><div <div class="csl-left-margin">[40] </div><div class="csl-right-inline">Z.
class="csl-right-inline">Nussinov, Z. &amp; Ortiz, G. <a Nussinov and G. Ortiz, <em><a
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond algebras and href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and
exact solvability of Hamiltonians: spin S=½ multilayer systems</a>. Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>,
<em>Physical Review B</em> <strong>79</strong>, 214440 (2009).</div> Physical Review B <strong>79</strong>, 214440 (2009).</div>
</div> </div>
<div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry"> <div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">41. </div><div <div class="csl-left-margin">[41] </div><div class="csl-right-inline">V.
class="csl-right-inline">Chua, V., Yao, H. &amp; Fiete, G. A. <a Chua, H. Yao, and G. A. Fiete, <em><a
href="https://doi.org/10.1103/PhysRevB.83.180412">Exact chiral spin href="https://doi.org/10.1103/PhysRevB.83.180412">Exact Chiral Spin
liquid with stable spin Fermi surface on the kagome lattice</a>. Liquid with Stable Spin Fermi Surface on the Kagome Lattice</a></em>,
<em>Phys. Rev. B</em> <strong>83</strong>, 180412 (2011).</div> Phys. Rev. B <strong>83</strong>, 180412 (2011).</div>
</div> </div>
<div id="ref-Natori2020" class="csl-entry" role="doc-biblioentry"> <div id="ref-Natori2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">42. </div><div <div class="csl-left-margin">[42] </div><div class="csl-right-inline">W.
class="csl-right-inline">Natori, W. M. H. &amp; Knolle, J. <a M. H. Natori and J. Knolle, <em><a
href="https://doi.org/10.1103/PhysRevLett.125.067201">Dynamics of a href="https://doi.org/10.1103/PhysRevLett.125.067201">Dynamics of a
two-dimensional quantum spin-orbital liquid: Spectroscopic signatures of Two-Dimensional Quantum Spin-Orbital Liquid: Spectroscopic Signatures of
fermionic magnons</a>. <em>Phys. Rev. Lett.</em> <strong>125</strong>, Fermionic Magnons</a></em>, Phys. Rev. Lett. <strong>125</strong>,
067201 (2020).</div> 067201 (2020).</div>
</div> </div>
<div id="ref-Chulliparambil2020" class="csl-entry" <div id="ref-Chulliparambil2020" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">43. </div><div <div class="csl-left-margin">[43] </div><div class="csl-right-inline">S.
class="csl-right-inline">Chulliparambil, S., Seifert, U. F. P., Vojta, Chulliparambil, U. F. P. Seifert, M. Vojta, L. Janssen, and H.-H. Tu,
M., Janssen, L. &amp; Tu, H.-H. <a <em><a href="https://doi.org/10.1103/PhysRevB.102.201111">Microscopic
href="https://doi.org/10.1103/PhysRevB.102.201111">Microscopic models Models for Kitaev’s Sixteenfold Way of Anyon Theories</a></em>, Phys.
for Kitaev’s sixteenfold way of anyon theories</a>. <em>Phys. Rev. Rev. B <strong>102</strong>, 201111 (2020).</div>
B</em> <strong>102</strong>, 201111 (2020).</div>
</div> </div>
<div id="ref-Chulliparambil2021" class="csl-entry" <div id="ref-Chulliparambil2021" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">44. </div><div <div class="csl-left-margin">[44] </div><div class="csl-right-inline">S.
class="csl-right-inline">Chulliparambil, S., Janssen, L., Vojta, M., Tu, Chulliparambil, L. Janssen, M. Vojta, H.-H. Tu, and U. F. P. Seifert,
H.-H. &amp; Seifert, U. F. P. <a <em><a href="https://doi.org/10.1103/PhysRevB.103.075144">Flux Crystals,
href="https://doi.org/10.1103/PhysRevB.103.075144">Flux crystals, Majorana Metals, and Flat Bands in Exactly Solvable Spin-Orbital
Majorana metals, and flat bands in exactly solvable spin-orbital Liquids</a></em>, Phys. Rev. B <strong>103</strong>, 075144
liquids</a>. <em>Phys. Rev. B</em> <strong>103</strong>, 075144
(2021).</div> (2021).</div>
</div> </div>
<div id="ref-Seifert2020" class="csl-entry" role="doc-biblioentry"> <div id="ref-Seifert2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">45. </div><div <div class="csl-left-margin">[45] </div><div class="csl-right-inline">U.
class="csl-right-inline">Seifert, U. F. P. <em>et al.</em> <a F. P. Seifert, X.-Y. Dong, S. Chulliparambil, M. Vojta, H.-H. Tu, and L.
Janssen, <em><a
href="https://doi.org/10.1103/PhysRevLett.125.257202">Fractionalized href="https://doi.org/10.1103/PhysRevLett.125.257202">Fractionalized
fermionic quantum criticality in spin-orbital mott insulators</a>. Fermionic Quantum Criticality in Spin-Orbital Mott Insulators</a></em>,
<em>Phys. Rev. Lett.</em> <strong>125</strong>, 257202 (2020).</div> Phys. Rev. Lett. <strong>125</strong>, 257202 (2020).</div>
</div> </div>
<div id="ref-WangHaoranPRB2021" class="csl-entry" <div id="ref-WangHaoranPRB2021" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">46. </div><div <div class="csl-left-margin">[46] </div><div class="csl-right-inline">H.
class="csl-right-inline">Wang, H. &amp; Principi, A. <a Wang and A. Principi, <em><a
href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana edge and href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana Edge and
corner states in square and kagome quantum spin-3/2 liquids</a>. Corner States in Square and Kagome Quantum Spin-3/2 Liquids</a></em>,
<em>Phys. Rev. B</em> <strong>104</strong>, 214422 (2021).</div> Phys. Rev. B <strong>104</strong>, 214422 (2021).</div>
</div> </div>
<div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">47. </div><div class="csl-right-inline">Wu, <div class="csl-left-margin">[47] </div><div class="csl-right-inline">C.
C., Arovas, D. &amp; Hung, H.-H. Γ-matrix generalization of the Kitaev Wu, D. Arovas, and H.-H. Hung, <em>Γ-Matrix Generalization of the Kitaev
model. <em>Physical Review B</em> <strong>79</strong>, 134427 Model</em>, Physical Review B <strong>79</strong>, 134427 (2009).</div>
(2009).</div>
</div> </div>
</div> </div>
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