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</section>
<section id="interacting-quantum-many-body-systems" class="level1">
<h1>Interacting Quantum Many Body Systems</h1>
<p>When you take many objects and let them interact together, it is often simpler to describe the behaviour of the group differently from the way one would describe the individual objects. Consider a flock of starlings like that of fig. <a href="#fig:Studland_Starlings">1</a>. Watching the flock you’ll see that it has a distinct outline, that waves of density will sometimes propagate through the closely packed birds and that the flock seems to respond to predators as a distinct object. The natural description of this phenomena is couched in terms of the flock rather than of the individual birds.</p>
<p>The behaviours of the flock are an <em>emergent phenomena</em>. The starlings are only interacting with their immediate six or seven neighbours <span class="citation" data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"> [<a href="#ref-king2012murmurations" role="doc-biblioref">1</a>,<a href="#ref-balleriniInteractionRulingAnimal2008" role="doc-biblioref">2</a>]</span>, what a physicist would call a <em>local interaction</em>. There is much philosophical debate about how exactly to define emergence <span class="citation" data-cites="andersonMoreDifferent1972 kivelsonDefiningEmergencePhysics2016"> [<a href="#ref-andersonMoreDifferent1972" role="doc-biblioref">3</a>,<a href="#ref-kivelsonDefiningEmergencePhysics2016" role="doc-biblioref">4</a>]</span> but for our purposes it enough to say that emergence is the fact that the aggregate behaviour of many interacting objects may necessitate a description very different from that of the individual objects.</p>
<p>When you take many objects and let them interact together, it is often simpler to describe the behaviour of the group in a different way to how one would describe the individual objects. Consider a flock of starlings like that of fig. <a href="#fig:Studland_Starlings">1</a>. Watching the flock you’ll see that it has a distinct outline, that waves of density will sometimes propagate through the closely packed birds and that the flock seems to respond to predators as a distinct object. The natural description of this phenomena is in terms of the flock, not the individual birds.</p>
<p>The behaviours of the flock are an <em>emergent phenomenon</em>. The starlings are only interacting with their immediate six or seven neighbours <span class="citation" data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"> [<a href="#ref-king2012murmurations" role="doc-biblioref">1</a>,<a href="#ref-balleriniInteractionRulingAnimal2008" role="doc-biblioref">2</a>]</span>, what a physicist would call a <em>local interaction</em>. There is much philosophical debate about how exactly to define emergence <span class="citation" data-cites="andersonMoreDifferent1972 kivelsonDefiningEmergencePhysics2016"> [<a href="#ref-andersonMoreDifferent1972" role="doc-biblioref">3</a>,<a href="#ref-kivelsonDefiningEmergencePhysics2016" role="doc-biblioref">4</a>]</span> but for our purposes it enough to say that emergence is the fact that the aggregate behaviour of many interacting objects may necessitate a description very different from that of the individual objects.</p>
<figure>
<img src="/assets/thesis/intro_chapter/Studland_Starlings.jpeg" id="fig:Studland_Starlings" data-short-caption="A murmuration of Starlings" style="width:100.0%" alt="Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0" />
<img src="/assets/thesis/intro_chapter/Studland_Starlings.jpeg" id="fig-Studland_Starlings" data-short-caption="A murmuration of Starlings" style="width:100.0%" alt="Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0" />
<figcaption aria-hidden="true">Figure 1: A murmuration of starlings. Dorset, UK. Credit <a href="https://twitter.com/arripay">Tanya Hart</a>, “Studland Starlings”, 2017, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a></figcaption>
</figure>
<p>To give an example closer to the topic at hand, our understanding of thermodynamics began with bulk properties like heat, energy, pressure and temperature <span class="citation" data-cites="saslowHistoryThermodynamicsMissing2020"> [<a href="#ref-saslowHistoryThermodynamicsMissing2020" role="doc-biblioref">5</a>]</span>. It was only later that we gained an understanding of how these properties emerge from microscopic interactions between very large numbers of particles <span class="citation" data-cites="flammHistoryOutlookStatistical1998"> [<a href="#ref-flammHistoryOutlookStatistical1998" role="doc-biblioref">6</a>]</span>.</p>
<p>Condensed Matter is, at its heart, the study of what behaviours emerge from large numbers of interacting quantum objects at low energy. When these three properties are present together: a large number of objects, those objects being quantum and there are interaction between the objects, we call it an interacting quantum many body system. From these three ingredients nature builds all manner of weird and wonderful materials.</p>
<p>Historically, we made initial headway in the study of many-body systems, ignoring interactions and quantum properties. The ideal gas law and the Drude classical electron gas <span class="citation" data-cites="ashcroftSolidStatePhysics1976"> [<a href="#ref-ashcroftSolidStatePhysics1976" role="doc-biblioref">7</a>]</span> are good examples. Including interactions into many-body physics leads to the Ising model <span class="citation" data-cites="isingBeitragZurTheorie1925"> [<a href="#ref-isingBeitragZurTheorie1925" role="doc-biblioref">8</a>]</span>, Landau theory <span class="citation" data-cites="landau2013fluid"> [<a href="#ref-landau2013fluid" role="doc-biblioref">9</a>]</span> and the classical theory of phase transitions <span class="citation" data-cites="jaegerEhrenfestClassificationPhase1998"> [<a href="#ref-jaegerEhrenfestClassificationPhase1998" role="doc-biblioref">10</a>]</span>. In contrast, condensed matter theory got it state in quantum many-body theory. Bloch’s theorem <span class="citation" data-cites="blochÜberQuantenmechanikElektronen1929"> [<a href="#ref-blochÜberQuantenmechanikElektronen1929" role="doc-biblioref">11</a>]</span> predicted the properties of non-interacting electrons in crystal lattices, leading to band theory. In the same vein, advances were made in understanding the quantum origins of magnetism, including ferromagnetism and antiferromagnetism <span class="citation" data-cites="MagnetismCondensedMatter"> [<a href="#ref-MagnetismCondensedMatter" role="doc-biblioref">12</a>]</span>.</p>
<p>The development of Landau-Fermi Liquid theory explained why band theory works so well even in cases where an analysis of the relevant energies suggests that it should not <span class="citation" data-cites="wenQuantumFieldTheory2007"> [<a href="#ref-wenQuantumFieldTheory2007" role="doc-biblioref">13</a>]</span>. Landau Fermi Liquid theory demonstrates that in many cases where electron-electron interactions are significant, the system can still be described in terms on generalised non-interacting quasiparticles.</p>
<p>However there are systems where even Landau Fermi Liquid theory fails. An effective theoretical description of these systems must include electron-electron correlations and they are thus called Strongly Correlated Materials <span class="citation" data-cites="morosanStronglyCorrelatedMaterials2012"> [<a href="#ref-morosanStronglyCorrelatedMaterials2012" role="doc-biblioref">14</a>]</span>, Correlated Electron systems or Quantum Materials. The canonical examples are superconductivity <span class="citation" data-cites="MicroscopicTheorySuperconductivity"> [<a href="#ref-MicroscopicTheorySuperconductivity" role="doc-biblioref">15</a>]</span>, the fractional quantum hall effect <span class="citation" data-cites="feldmanFractionalChargeFractional2021"> [<a href="#ref-feldmanFractionalChargeFractional2021" role="doc-biblioref">16</a>]</span> and the Mott insulators <span class="citation" data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">17</a>,<a href="#ref-fisherMottInsulatorsSpin1999" role="doc-biblioref">18</a>]</span>. We’ll start by looking at the latter but shall see that there are many links between three topics.</p>
<p>Condensed Matter is, at its heart, the study of what behaviours emerge from large numbers of interacting quantum objects at low energy. When these three properties are present together: a large number of objects, those objects being quantum and the presence interactions between the objects, we call it an interacting quantum many body system. From these three ingredients nature builds all manner of weird and wonderful materials.</p>
<p>Historically, we first made headway in the study of many-body systems, ignoring interactions and quantum properties. The ideal gas law and the Drude classical electron gas <span class="citation" data-cites="ashcroftSolidStatePhysics1976"> [<a href="#ref-ashcroftSolidStatePhysics1976" role="doc-biblioref">7</a>]</span> are good examples. Including interactions too leads to the Ising model <span class="citation" data-cites="isingBeitragZurTheorie1925"> [<a href="#ref-isingBeitragZurTheorie1925" role="doc-biblioref">8</a>]</span>, Landau theory <span class="citation" data-cites="landau2013fluid"> [<a href="#ref-landau2013fluid" role="doc-biblioref">9</a>]</span> and the classical theory of phase transitions <span class="citation" data-cites="jaegerEhrenfestClassificationPhase1998"> [<a href="#ref-jaegerEhrenfestClassificationPhase1998" role="doc-biblioref">10</a>]</span>. In contrast, condensed matter theory got its start in quantum many-body theory where the only electron-electron interaction considered is the Pauli exclusion principle. Bloch’s theorem <span class="citation" data-cites="blochÜberQuantenmechanikElektronen1929"> [<a href="#ref-blochÜberQuantenmechanikElektronen1929" role="doc-biblioref">11</a>]</span>, the core result of band theory, predicted the properties of non-interacting electrons in crystal lattices, in particular that band insulators arise when the electrons bands are filled, leaving the fermi level in a bandgap <span class="citation" data-cites="ashcroftSolidStatePhysics1976"> [<a href="#ref-ashcroftSolidStatePhysics1976" role="doc-biblioref">7</a>]</span>. In the same vein, advances were made in understanding the quantum origins of magnetism, including ferromagnetism and antiferromagnetism <span class="citation" data-cites="MagnetismCondensedMatter"> [<a href="#ref-MagnetismCondensedMatter" role="doc-biblioref">12</a>]</span>.</p>
<p>The development of Landau-Fermi Liquid theory explained why band theory works so well even in cases where an analysis of the relevant energies suggests that it should not <span class="citation" data-cites="wenQuantumFieldTheory2007"> [<a href="#ref-wenQuantumFieldTheory2007" role="doc-biblioref">13</a>]</span>. Landau Fermi Liquid theory demonstrates that in many cases where electron-electron interactions are significant, the system can still be described in terms of generalised non-interacting quasiparticles. This happens when the properties of the quasiparticles in the interacting system can be smoothly connected to the free fermions of the non-interacting system.</p>
<p>However there are systems where even Landau Fermi Liquid theory fails. An effective theoretical description of these systems must include electron-electron correlations and they are thus called Strongly Correlated Materials <span class="citation" data-cites="morosanStronglyCorrelatedMaterials2012"> [<a href="#ref-morosanStronglyCorrelatedMaterials2012" role="doc-biblioref">14</a>]</span>. The canonical examples are superconductivity <span class="citation" data-cites="MicroscopicTheorySuperconductivity"> [<a href="#ref-MicroscopicTheorySuperconductivity" role="doc-biblioref">15</a>]</span>, the fractional quantum hall effect <span class="citation" data-cites="feldmanFractionalChargeFractional2021"> [<a href="#ref-feldmanFractionalChargeFractional2021" role="doc-biblioref">16</a>]</span> and the Mott insulators <span class="citation" data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">17</a>,<a href="#ref-fisherMottInsulatorsSpin1999" role="doc-biblioref">18</a>]</span>. We’ll start by looking at the latter but shall see that there are many links between the three topics.</p>
</section>
<section id="mott-insulators" class="level1">
<h1>Mott Insulators</h1>
<p>Mott Insulators are remarkable because their electrical insulator properties come from electron-electron interactions. Electrical conductivity, the bulk movement of electrons, requires both that there are electronic states very close in energy to the ground state and that those states are delocalised so that they can contribute to macroscopic transport. Band insulators are systems whose Fermi level falls within a 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>Mott Insulators are remarkable because their electrical insulator properties come not from having filled bands but from electron-electron interactions other than Pauli exclusion. Electrical conductivity, the bulk movement of electrons, requires both that there are electronic states very close in energy to the ground state and that those states are delocalised so that they can contribute to macroscopic transport. Band insulators are systems whose Fermi level falls within a gap in the density of states and thus fail the first criteria. Band insulators derive their character from the characteristics of the underlying lattice. A third kind of insulator, the Anderson Insulators, have only localised electronic states near the fermi level and therefore fail the second criteria. We will discuss Anderson insulators and the disorder that drives them, in a later section.</p>
<p>Both band and Anderson insulators occur without electron-electron interactions. Mott insulators, by contrast, require a many body picture to understand and thus elude band theory and single-particle methods.</p>
<figure>
<img src="/assets/thesis/intro_chapter/venn_diagram.svg" id="fig:venn_diagram" data-short-caption="Interacting Quantum Many Body Systems Venn Diagram" style="width:100.0%" alt="Figure 2: Three key adjectives. Many Body, the fact of describing systems in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials." />
<figcaption aria-hidden="true">Figure 2: Three key adjectives. Many Body, the fact of describing systems in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials.</figcaption>
<img src="/assets/thesis/intro_chapter/venn_diagram.svg" id="fig-venn_diagram" data-short-caption="Interacting Quantum Many Body Systems Venn Diagram" style="width:100.0%" alt="Figure 2: Three key adjectives. Many Body refers to systems considered in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials." />
<figcaption aria-hidden="true">Figure 2: Three key adjectives. <em>Many Body</em> refers to systems considered in the limit of large numbers of particles. <em>Quantum</em>, objects whose behaviour requires quantum mechanics to describe accurately. <em>Interacting</em>, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials.</figcaption>
</figure>
<p>The theory of Mott insulators developed out of the observation that many transition metal oxides are erroneously predicted by band theory to be conductive <span class="citation" data-cites="boerSemiconductorsPartiallyCompletely1937"> [<a href="#ref-boerSemiconductorsPartiallyCompletely1937" role="doc-biblioref">19</a>]</span> leading to the suggestion that electron-electron interactions were the cause of this effect <span class="citation" data-cites="mottDiscussionPaperBoer1937"> [<a href="#ref-mottDiscussionPaperBoer1937" role="doc-biblioref">20</a>]</span>. Interest grew with the discovery of high temperature superconductivity in the cuprates in 1986 <span class="citation" data-cites="bednorzPossibleHighTcSuperconductivity1986"> [<a href="#ref-bednorzPossibleHighTcSuperconductivity1986" role="doc-biblioref">21</a>]</span> which is believed to arise as the result of a doped Mott insulator state <span class="citation" data-cites="leeDopingMottInsulator2006"> [<a href="#ref-leeDopingMottInsulator2006" role="doc-biblioref">22</a>]</span>.</p>
<p>The canonical toy model of the Mott insulator is the Hubbard model <span class="citation" data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"> [<a href="#ref-gutzwillerEffectCorrelationFerromagnetism1963" role="doc-biblioref">23</a>–<a href="#ref-hubbardj.ElectronCorrelationsNarrow1963" role="doc-biblioref">25</a>]</span> of <span class="math inline">\(1/2\)</span> fermions hopping on the lattice with hopping parameter <span class="math inline">\(t\)</span> and electron-electron repulsion <span class="math inline">\(U\)</span></p>
<p><span class="math display">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow} n_{i\downarrow} - \mu \sum_{i,\alpha} n_{i\alpha}\]</span></p>
<p>The canonical toy model of the Mott insulator is the Hubbard model <span class="citation" data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"> [<a href="#ref-gutzwillerEffectCorrelationFerromagnetism1963" role="doc-biblioref">23</a>–<a href="#ref-hubbardj.ElectronCorrelationsNarrow1963" role="doc-biblioref">25</a>]</span> of spin-<span class="math inline">\(1/2\)</span> fermions hopping on the lattice with hopping parameter <span class="math inline">\(t\)</span> and electron-electron repulsion <span class="math inline">\(U\)</span></p>
<p><span class="math display">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow} n_{i\downarrow} - \mu \sum_{i,\alpha} n_{i\alpha},\]</span></p>
<p>where <span class="math inline">\(c^\dagger_{i\alpha}\)</span> creates a spin <span class="math inline">\(\alpha\)</span> electron at site <span class="math inline">\(i\)</span> and the number operator <span class="math inline">\(n_{i\alpha}\)</span> measures the number of electrons with spin <span class="math inline">\(\alpha\)</span> at site <span class="math inline">\(i\)</span>. The sum runs over lattice neighbours <span class="math inline">\(\langle i,j \rangle\)</span> including both <span class="math inline">\(\langle i,j \rangle\)</span> and <span class="math inline">\(\langle j,i \rangle\)</span> so that the model is Hermition.</p>
<p>In the non-interacting limit <span class="math inline">\(U &lt;&lt; t\)</span>, the model reduces to free fermions and the many-body ground state is a separable product of Bloch waves filled up to the Fermi 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 class="math inline">\(\mu = \tfrac{U}{2}\)</span> where there is one electron per lattice site <span class="citation" data-cites="hubbardElectronCorrelationsNarrow1964"> [<a href="#ref-hubbardElectronCorrelationsNarrow1964" role="doc-biblioref">26</a>]</span>. Here the model can be rewritten in a symmetric form <span class="math display">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2})\]</span></p>
<p>The basic reason that the half filled state is insulating seems is trivial. Any excitation must include states of double occupancy that cost energy <span class="math inline">\(U\)</span>, hence the system has a finite bandgap and is an interaction driven Mott insulator. Depending on the lattice, the local moments may then order antiferromagnetically. Originally it was proposed that this antiferromagnetic order was the cause of the gap opening <span class="citation" data-cites="mottMetalInsulatorTransitions1990"> [<a href="#ref-mottMetalInsulatorTransitions1990" role="doc-biblioref">27</a>]</span>. However, Mott insulators have been found <span class="citation" data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">28</a>,<a href="#ref-ribakGaplessExcitationsGround2017" role="doc-biblioref">29</a>]</span> without magnetic order. Instead the local moments may form a highly entangled state known as a quantum spin liquid, which will be discussed shortly.</p>
<p>Various theoretical treatments of the Hubbard model have been made, including those based on Fermi liquid theory, mean field treatments, the local density approximation (LDA) <span class="citation" data-cites="slaterMagneticEffectsHartreeFock1951"> [<a href="#ref-slaterMagneticEffectsHartreeFock1951" role="doc-biblioref">30</a>]</span> and dynamical mean-field theory <span class="citation" data-cites="greinerQuantumPhaseTransition2002"> [<a href="#ref-greinerQuantumPhaseTransition2002" role="doc-biblioref">31</a>]</span>. None of these approaches are perfect. Strong correlations are poorly described by the Fermi liquid theory and the LDA approaches while mean field approximations do poorly in low dimensional systems. This theoretical difficulty has made the Hubbard model a target for cold atom simulations <span class="citation" data-cites="mazurenkoColdatomFermiHubbard2017"> [<a href="#ref-mazurenkoColdatomFermiHubbard2017" role="doc-biblioref">32</a>]</span>.</p>
<p>From here the discussion will branch two directions. First, we will discuss a limit of the Hubbard model called the Falikov-Kimball Model. Second, we will look at quantum spin liquids and the Kitaev honeycomb model.</p>
<p>In the non-interacting limit <span class="math inline">\(U &lt;&lt; t\)</span>, the model reduces to free fermions and the many-body ground state is a separable product of Bloch waves filled up to the Fermi level. In the interacting limit <span class="math inline">\(U &gt;&gt; t\)</span> on the other hand, the ground state is a direct product of the local Hilbert spaces <span class="math inline">\(|0\rangle, |\uparrow\rangle, |\downarrow\rangle, |\uparrow\downarrow\rangle\)</span>. At half filling, one electron per site, each site becomes a <em>local moment</em> in the reduced Hilbert space <span class="math inline">\(|\uparrow\rangle, |\downarrow\rangle\)</span> and thus acts like a spin-<span class="math inline">\(1/2\)</span> <span class="citation" data-cites="hubbardElectronCorrelationsNarrow1964"> [<a href="#ref-hubbardElectronCorrelationsNarrow1964" role="doc-biblioref">26</a>]</span>.</p>
<p>The Mott insulating phase occurs at half filling <span class="math inline">\(\mu = \tfrac{U}{2}\)</span>. Here the model can be rewritten in a symmetric form <span class="math display">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2}).\]</span></p>
<p>The basic reason that the half filled state is insulating seems trivial. Any excitation must include states of double occupancy that cost energy <span class="math inline">\(U\)</span>, hence the system has a finite bandgap and is an interaction driven Mott insulator. Depending on the lattice, the local moments may then order antiferromagnetically. Originally it was proposed that this antiferromagnetic (AFM) order was actually the reason for the insulating behaviour. This would make sense since AFM order doubles the unit cell and can turn a system into a band insulator with an even number of electrons per unit cell <span class="citation" data-cites="mottMetalInsulatorTransitions1990"> [<a href="#ref-mottMetalInsulatorTransitions1990" role="doc-biblioref">27</a>]</span>. However, Mott insulators have been found <span class="citation" data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">28</a>,<a href="#ref-ribakGaplessExcitationsGround2017" role="doc-biblioref">29</a>]</span> without magnetic order. Instead the local moments may form a highly entangled state known as a quantum spin liquid, which will be discussed shortly.</p>
<p>Various theoretical treatments of the Hubbard model have been made, including those based on Fermi liquid theory, mean field treatments, the local density approximation (LDA) <span class="citation" data-cites="slaterMagneticEffectsHartreeFock1951"> [<a href="#ref-slaterMagneticEffectsHartreeFock1951" role="doc-biblioref">30</a>]</span>, dynamical mean-field theory <span class="citation" data-cites="greinerQuantumPhaseTransition2002"> [<a href="#ref-greinerQuantumPhaseTransition2002" role="doc-biblioref">31</a>]</span>, density matrix renormalisation group methods <span class="citation" data-cites="hallbergNewTrendsDensity2006 schollwöckDensitymatrixRenormalizationGroup2005 whiteDensityMatrixFormulation1992"> [<a href="#ref-hallbergNewTrendsDensity2006" role="doc-biblioref">32</a>–<a href="#ref-whiteDensityMatrixFormulation1992" role="doc-biblioref">34</a>]</span> and Markov chain Monte Carlo <span class="citation" data-cites="blankenbeclerMonteCarloCalculations1981 hirschDiscreteHubbardStratonovichTransformation1983 whiteNumericalStudyTwodimensional1989"> [<a href="#ref-blankenbeclerMonteCarloCalculations1981" role="doc-biblioref">35</a>–<a href="#ref-whiteNumericalStudyTwodimensional1989" role="doc-biblioref">37</a>]</span>. None of these approaches are perfect. Strong correlations are poorly described by the Fermi liquid theory and the LDA approaches while mean field approximations do poorly in low dimensional systems. This theoretical difficulty has made the Hubbard model a target for cold atom simulations <span class="citation" data-cites="mazurenkoColdatomFermiHubbard2017"> [<a href="#ref-mazurenkoColdatomFermiHubbard2017" role="doc-biblioref">38</a>]</span>.</p>
<p>From here the discussion will branch in two directions. First, we will discuss a limit of the Hubbard model called the Falikov-Kimball Model. Second, we will look at quantum spin liquids and the Kitaev honeycomb model.</p>
<p><strong>The Falikov-Kimball Model</strong></p>
<p>Though not the original reason for its introduction, the Falikov-Kimball (FK) model is the limit of the Hubbard model as the mass ratio of the spin up and spin down electron is taken to infinity. This gives a model with two fermion species, one itinerant and one entirely immobile. The number operators for the immobile fermions are therefore conserved quantities and can be be treated like classical degrees of freedom. For our purposes it will be useful to replace the immobile fermions with a classical Ising background field <span class="math inline">\(S_i = \pm1\)</span>.</p>
<figure>
<img src="/assets/thesis/intro_chapter/fk_schematic.svg" id="fig-fk_schematic" data-short-caption="Falicov-Kimball Model Diagram" style="width:100.0%" alt="Figure 3: The Falicov-Kimball Model can be viewed as a model of classical spins S_i coupled to spinless fermions \hat{c}_i where the fermions are mobile with hopping t and the fermions are coupled to the spins by an Ising type interaction with strength U." />
<figcaption aria-hidden="true">Figure 3: The Falicov-Kimball Model can be viewed as a model of classical spins <span class="math inline">\(S_i\)</span> coupled to spinless fermions <span class="math inline">\(\hat{c}_i\)</span> where the fermions are mobile with hopping <span class="math inline">\(t\)</span> and the fermions are coupled to the spins by an Ising type interaction with strength <span class="math inline">\(U\)</span>.</figcaption>
</figure>
<p>Originally introduced to describe the metal-insulator transition in f-electron system <span class="citation" data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicovSimpleModelSemiconductorMetal1969"> [<a href="#ref-hubbardj.ElectronCorrelationsNarrow1963" role="doc-biblioref">25</a>,<a href="#ref-falicovSimpleModelSemiconductorMetal1969" role="doc-biblioref">39</a>]</span>, the Falikov-Kimball (FK) model is the limit of the Hubbard model as the mass of one of the spins states of the electron is taken to infinity. This gives a model with two fermion species, one itinerant and one entirely immobile. The number operators for the immobile fermions are therefore conserved quantities and can be be treated like classical degrees of freedom. For our purposes it will be useful to replace the immobile fermions with a classical Ising background field <span class="math inline">\(S_i = \pm1\)</span>.</p>
<p><span class="math display">\[\begin{aligned}
H_{\mathrm{FK}} = &amp; -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}). \\
\end{aligned}\]</span></p>
<p>Given that the physics of states near the metal-insulator (MI) transition is still poorly understood <span class="citation" data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a href="#ref-belitzAndersonMottTransition1994" role="doc-biblioref">33</a>,<a href="#ref-baskoMetalInsulatorTransition2006" role="doc-biblioref">34</a>]</span> the FK model provides a rich test bed to explore interaction driven MI transition physics. Despite its simplicity, the model has a rich phase diagram in <span class="math inline">\(D \geq 2\)</span> dimensions. It shows an Mott 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">35</a>]</span>. In 1D, the ground state phenomenology as a function of filling can be rich <span class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">36</a>]</span> but the system is disordered for all <span class="math inline">\(T &gt; 0\)</span> <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">37</a>]</span>. The model has also been a test-bed for many-body methods, interest took off when an exact dynamical mean-field theory solution in the infinite dimensional case was found <span class="citation" data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">38</a>–<a href="#ref-herrmannNonequilibriumDynamicalCluster2016" role="doc-biblioref">41</a>]</span>.</p>
<p>In chapter <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3</a> I will introduce a generalized Falikov-Kimball model in one dimension I call the Long-Range Falikov-Kimball model. With the addition of long-range interactions in the background field, the model shows a similarly rich phase diagram its higher dimensional cousins. 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>Given that the physics of states near the metal-insulator (MI) transition is still poorly understood <span class="citation" data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a href="#ref-belitzAndersonMottTransition1994" role="doc-biblioref">40</a>,<a href="#ref-baskoMetalInsulatorTransition2006" role="doc-biblioref">41</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">42</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">43</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">44</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">45</a>–<a href="#ref-herrmannNonequilibriumDynamicalCluster2016" role="doc-biblioref">48</a>]</span>.</p>
<p>In chapter <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3</a> I will introduce a generalized Falikov-Kimball model in one dimension I call the Long-Range Falikov-Kimball model. With the addition of long-range interactions in the background field, the model shows a similarly rich phase diagram as its higher dimensional cousins. Our goal is to understand the Mott transition in more detail, the the phase transition into a charge density wave (CDW) state and how the localisation properties of the fermionic sector behave in one dimension. We were particularly interested to see if correlations in the disorder potential are enough to bring about localisation effects such mobility edges that are normally only seen in higher dimensions. I use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localization properties of the fermions. We observe what seems like a hint of coexisting localised and delocalised states. However after careful comparison to an Anderson model of uncorrelated binary disorder about a background charge density wave field, we confirm that the fermionic sector does fully localize at larger system sizes as expected for one dimensional systems.</p>
</section>
<section id="quantum-spin-liquids" class="level1">
<h1>Quantum Spin Liquids</h1>
<p>To turn to the other key topic of this thesis, we have already discussed the AFM ordering of local moments in the Mott insulating state. Landau-Ginzburg-Wilson theory characterises phases of matter as inextricably linked to the emergence of long range order via a spontaneously broken symmetry. So within this paradigm we would not expect any interesting phases of matter not associated with AFM or other long-range order. However, Anderson first proposed in 1973 <span class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a href="#ref-andersonResonatingValenceBonds1973" role="doc-biblioref">42</a>]</span> that if long range order is suppressed by some mechanism, it might lead to a liquid-like state even at zero temperature, the Quantum Spin Liquid (QSL).</p>
<p>This QSL state would exist at zero or very low temperatures, so we would expect quantum effects to be very strong, which will turn out to have far reaching consequences. It was the discovery of a different phase, however that really kickstarted interest in the topic. The fractional quantum Hall (FQH) state, discovered in the 1980s is an explicit example of an interacting electron system that falls outside of the Landau-Ginzburg-Wilson paradigm. It shares many phenomenological properties with the QSL state. They both exhibit fractionalised excitations, braiding statistics and non-trivial topological properties <span class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a href="#ref-broholmQuantumSpinLiquids2020" role="doc-biblioref">43</a>]</span>. The many-body ground state of such systems acts as a complex and highly entangled vacuum. This vacuum can support quasiparticle excitations with properties unbound from that of the Dirac fermions of the standard model.</p>
<p>How do we actually make a QSL? Frustration is one mechanism that we can use to suppress magnetic order in spin models <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>]</span>. Frustration can be geometric, triangular lattices for instance cannot support AFM order. It can also come about as a result of spin-orbit coupling or other physics. There are also other routes to QSLs besides frustrated spin systems that we will not discuss here <span class="citation" data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">45</a>–<a href="#ref-linExactSymmetryWeaklyinteracting1998" role="doc-biblioref">47</a>]</span>.</p>
<p>To turn to the other key topic of this thesis, we have already discussed the AFM ordering of local moments in the Mott insulating state. Landau-Ginzburg-Wilson theory characterises phases of matter as inextricably linked to the emergence of long range order via a spontaneously broken symmetry. So within this paradigm we would not expect any interesting phases of matter not associated with AFM or other long-range order. However, Anderson first proposed in 1973 <span class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a href="#ref-andersonResonatingValenceBonds1973" role="doc-biblioref">49</a>]</span> that if long range order is suppressed by some mechanism, it might lead to a liquid-like state even at zero temperature, the Quantum Spin Liquid (QSL).</p>
<p>This QSL state would exist at zero or very low temperatures, so we would expect quantum effects to be very strong, which will turn out to have far reaching consequences. It was the discovery of a different phase, however that really kickstarted interest in the topic. The fractional quantum Hall (FQH) state, discovered in the 1980s is an explicit example of an interacting electron system that falls outside of the Landau-Ginzburg-Wilson paradigm. It shares many phenomenological properties with the QSL state. They both exhibit fractionalised excitations, braiding statistics and non-trivial topological properties <span class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a href="#ref-broholmQuantumSpinLiquids2020" role="doc-biblioref">50</a>]</span>. The many-body ground state of such systems acts as a complex and highly entangled vacuum. This vacuum can support quasiparticle excitations with properties unbound from that of the Dirac fermions of the standard model.</p>
<p>How do we actually make a QSL? Frustration is one mechanism that we can use to suppress magnetic order in spin models <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">51</a>]</span>. Frustration can be geometric, triangular lattices for instance cannot support AFM order. It can also come about as a result of spin-orbit coupling or other physics. There are also other routes to QSLs besides frustrated spin systems that we will not discuss here <span class="citation" data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">52</a>–<a href="#ref-linExactSymmetryWeaklyinteracting1998" role="doc-biblioref">54</a>]</span>.</p>
<!-- Experimentally, Mott insulating systems without magnetic order have been proposed as QSL systems\ [@law1TTaS2QuantumSpin2017; @ribakGaplessExcitationsGround2017]. -->
<!-- Other exampels: Quantum spin liquids are the analogous phase of matter for spin systems. Spin ice support deconfined magnetic monopoles. -->
<figure>
<img src="/assets/thesis/intro_chapter/kitaev_material_phase_diagram.svg" id="fig:kitaev-material-phase-diagram" data-short-caption="Phase Diagram" style="width:100.0%" alt="Figure 3: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev Model. Reproduced from  [44]." />
<figcaption aria-hidden="true">Figure 3: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev Model. Reproduced from <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>]</span>.</figcaption>
<img src="/assets/thesis/intro_chapter/kitaev_material_phase_diagram.svg" id="fig-kitaev-material-phase-diagram" data-short-caption="Phase Diagram" style="width:100.0%" alt="Figure 4: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev Model. Reproduced from  [51]." />
<figcaption aria-hidden="true">Figure 4: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev Model. Reproduced from <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">51</a>]</span>.</figcaption>
</figure>
<p>Spin-orbit coupling is a relativistic effect, that very roughly corresponds to the fact that in the frame of reference of a moving electron, the electric field of nearby nuclei look like magnetic fields to which the electron spin couples. This effectively couples the spatial and spin parts of the electron wavefunction, meaning that the lattice structure can influence the form of the spin-spin interactions leading to spatial anisotropy. This anisotropy will be how we frustrate the Mott insulators <span class="citation" data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a href="#ref-jackeliMottInsulatorsStrong2009" role="doc-biblioref">48</a>,<a href="#ref-khaliullinOrbitalOrderFluctuations2005" role="doc-biblioref">49</a>]</span>. As we saw with the Hubbard model, interaction effects are only strong or weak in comparison to the bandwidth or hopping integral <span class="math inline">\(t\)</span> so what we need to see strong frustration is a material with strong spin-orbit coupling <span class="math inline">\(\lambda\)</span> relative to its bandwidth <span class="math inline">\(t\)</span>.</p>
<p>In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-<span class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known Kitaev Materials, draw their name from the celebrated Kitaev Honeycomb Model which is expected to model their low temperature behaviour <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>,<a href="#ref-Jackeli2009" role="doc-biblioref">50</a>–<a href="#ref-Takagi2019" role="doc-biblioref">53</a>]</span>.</p>
<p>At this point we can sketch out a phase diagram like that of fig. <a href="#fig:kitaev-material-phase-diagram">3</a>. When both electron-electron interactions <span class="math inline">\(U\)</span> and spin-orbit couplings <span class="math inline">\(\lambda\)</span> are small relative to the bandwidth <span class="math inline">\(t\)</span> we recover standard band theory of band insulators and metals. In the upper left we have the simple Mott insulating state as described by the Hubbard model. In the lower right, strong spin-orbit coupling gives rise to Topological insulators (TIs) characterised by symmetry protected edge modes and non-zero Chern number. Kitaev materials occur in the region where strong electron-electron interaction and spin-orbit coupling interact. See <span class="citation" data-cites="witczak-krempaCorrelatedQuantumPhenomena2014"> [<a href="#ref-witczak-krempaCorrelatedQuantumPhenomena2014" role="doc-biblioref">54</a>]</span> for a much more expansive version of this diagram.</p>
<p>The Kitaev Honeycomb model <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">55</a>]</span> was the first concrete spin model with a QSL ground state. It is defined on the two dimensional honeycomb lattice and provides an exactly solvable model that can be reduced to a free fermion problem via a mapping to Majorana fermions. This yields an extensive number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes tied to an emergent gauge field. The model is remarkable not only for its QSL ground state but also for its fractionalised excitations with non-trivial braiding statistics. It has a rich phase diagram hosting gapless, Abelian and non-Abelian phases <span class="citation" data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a href="#ref-knolleDynamicsFractionalizationQuantum2015" role="doc-biblioref">56</a>]</span> and a finite temperature phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">57</a>]</span>. It been proposed that its non-Abelian excitations could be used to support robust topological quantum computing <span class="citation" data-cites="kitaev_fault-tolerant_2003 freedmanTopologicalQuantumComputation2003 nayakNonAbelianAnyonsTopological2008"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">58</a>–<a href="#ref-nayakNonAbelianAnyonsTopological2008" role="doc-biblioref">60</a>]</span>.</p>
<p>As Kitaev points out in his original paper, the model remains solvable on any tri-coordinated <span class="math inline">\(z=3\)</span> graph which can be 3-edge-coloured. Indeed many generalisations of the model to  <span class="citation" data-cites="Baskaran2007 Baskaran2008 Nussinov2009 OBrienPRB2016 hermanns2015weyl"> [<a href="#ref-Baskaran2007" role="doc-biblioref">61</a>–<a href="#ref-hermanns2015weyl" role="doc-biblioref">65</a>]</span>. Notably, the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">66</a>]</span> introduces triangular plaquettes to the honeycomb lattice leading to spontaneous chiral symmetry breaking. These extensions all retain translation symmetry, likely because edge-colouring and finding the ground state become much harder without it. Finding the ground state flux sector and understanding the QSL properties can still be challenging <span class="citation" data-cites="eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmann2019thermodynamics" role="doc-biblioref">67</a>,<a href="#ref-Peri2020" role="doc-biblioref">68</a>]</span>. Undeterred, this gap lead us to wonder what might happen if we remove translation symmetry from the Kitaev Model. This might would be a model of a tri-coordinated, highly bond anisotropic but otherwise amorphous material.</p>
<p>Amorphous materials do no have long-range lattice regularities but covalent compounds can induce short-range regularities in the lattice structure such as fixed coordination number <span class="math inline">\(z\)</span>. The best examples being amorphous Silicon and Germanium with <span class="math inline">\(z=4\)</span> which are used to make thin-film solar cells <span class="citation" data-cites="Weaire1971 betteridge1973possible"> [<a href="#ref-Weaire1971" role="doc-biblioref">69</a>,<a href="#ref-betteridge1973possible" role="doc-biblioref">70</a>]</span>. Recently is has been shown that topological insulating (TI) phases can exist in amorphous systems. Amorphous TIs are characterized by similar protected edge states to their translation invariant cousins and generalised topological bulk invariants <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 agarwala2019topological marsalTopologicalWeaireThorpeModels2020 costa2019toward agarwala2020higher spring2021amorphous corbae2019evidence"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">71</a>–<a href="#ref-corbae2019evidence" role="doc-biblioref">77</a>]</span>. However, research on amorphous electronic systems has been mostly focused on non-interacting systems with a few exceptions, for example, to account for the observation of superconductivity <span class="citation" data-cites="buckel1954einfluss mcmillan1981electron meisel1981eliashberg bergmann1976amorphous mannaNoncrystallineTopologicalSuperconductors2022"> [<a href="#ref-buckel1954einfluss" role="doc-biblioref">78</a>–<a href="#ref-mannaNoncrystallineTopologicalSuperconductors2022" role="doc-biblioref">82</a>]</span> in amorphous materials or very recently to understand the effect of strong electron repulsion in TIs <span class="citation" data-cites="kim2022fractionalization"> [<a href="#ref-kim2022fractionalization" role="doc-biblioref">83</a>]</span>.</p>
<p>Amorphous <em>magnetic</em> systems has been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems <span class="citation" data-cites="aharony1975critical Petrakovski1981 kaneyoshi1992introduction Kaneyoshi2018"> [<a href="#ref-aharony1975critical" role="doc-biblioref">84</a>–<a href="#ref-Kaneyoshi2018" role="doc-biblioref">87</a>]</span> and with numerical methods <span class="citation" data-cites="fahnle1984monte plascak2000ising"> [<a href="#ref-fahnle1984monte" role="doc-biblioref">88</a>,<a href="#ref-plascak2000ising" role="doc-biblioref">89</a>]</span>. Research on classical Heisenberg and Ising models has been shown to account for observed behaviour of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour <span class="citation" data-cites="coey1978amorphous"> [<a href="#ref-coey1978amorphous" role="doc-biblioref">90</a>]</span>. However, the role of spin-anisotropic interactions and quantum effects in amorphous magnets has not been addressed. It is an open question whether frustrated magnetic interactions on amorphous lattices can give rise to genuine quantum phases, i.e. to long-range entangled quantum spin liquids (QSL) <span class="citation" data-cites="Anderson1973 Knolle2019 Savary2016 Lacroix2011"> [<a href="#ref-Anderson1973" role="doc-biblioref">91</a>–<a href="#ref-Lacroix2011" role="doc-biblioref">94</a>]</span>.</p>
<p>In <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">chapter 4</a> I will introduce the Amorphous Kitaev model, a generalisation of the Kitaev honeycomb model to random lattices with fixed coordination number three. We will show that this model is a soluble chiral amorphous quantum spin liquid. The model retains its exact solubility but, as with the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">66</a>]</span>, the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We will confirm prior observations that the form of the ground state can be written in terms of the number of sides of elementary plaquettes of the model <span class="citation" data-cites="OBrienPRB2016 eschmannThermodynamicClassificationThreedimensional2020"> [<a href="#ref-OBrienPRB2016" role="doc-biblioref">64</a>,<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">95</a>]</span>. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I show that the system undergoes a finite-temperature phase transition to a conducting thermal metal state and discuss possible experimental realisations.</p>
<p>Spin-orbit coupling is a relativistic effect that, very roughly, corresponds to the fact that in the frame of reference of a moving electron, the electric field of nearby nuclei looks like a magnetic field to which the electron spin couples. This effectively couples the spatial and spin parts of the electron wavefunction, meaning that the lattice structure can influence the form of the spin-spin interactions leading to spatial anisotropy in the effective interactions. This spatial anisotropy can frustrate the Mott insulators <span class="citation" data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a href="#ref-jackeliMottInsulatorsStrong2009" role="doc-biblioref">55</a>,<a href="#ref-khaliullinOrbitalOrderFluctuations2005" role="doc-biblioref">56</a>]</span> leading to more exotic ground states than the AFM order we have seen so far. As we saw with the Hubbard model, interaction effects are only strong or weak in comparison to the bandwidth or hopping integral <span class="math inline">\(t\)</span>. Hence, we will see strong frustration in materials with strong spin-orbit coupling <span class="math inline">\(\lambda\)</span> relative to their bandwidth <span class="math inline">\(t\)</span>.</p>
<p>In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-<span class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known as Kitaev Materials, draw their name from the celebrated Kitaev Honeycomb Model which is expected to model their low temperature behaviour <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">51</a>,<a href="#ref-Jackeli2009" role="doc-biblioref">57</a>–<a href="#ref-Takagi2019" role="doc-biblioref">60</a>]</span>.</p>
<p>At this point we can sketch out a phase diagram like that of fig. <a href="#fig:kitaev-material-phase-diagram">4</a>. When both electron-electron interactions <span class="math inline">\(U\)</span> and spin-orbit couplings <span class="math inline">\(\lambda\)</span> are small relative to the bandwidth <span class="math inline">\(t\)</span> we recover standard band theory of band insulators and metals. In the upper left we have the simple Mott insulating state as described by the Hubbard model. In the lower right, strong spin-orbit coupling gives rise to Topological insulators (TIs) characterised by symmetry protected edge modes and non-zero Chern number. Kitaev materials occur in the region where strong electron-electron interaction and spin-orbit coupling interact. See <span class="citation" data-cites="witczak-krempaCorrelatedQuantumPhenomena2014"> [<a href="#ref-witczak-krempaCorrelatedQuantumPhenomena2014" role="doc-biblioref">61</a>]</span> for a much more expansive version of this diagram.</p>
<p>The Kitaev Honeycomb model <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">62</a>]</span> was the first exactly solvable spin model with a QSL ground state. It is defined on the two dimensional honeycomb lattice and provides an exactly solvable model that can be reduced to a free fermion problem via a mapping to Majorana fermions. This yields an extensive number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes tied to an emergent gauge field. The model is remarkable not only for its QSL ground state but also for its fractionalised excitations with non-trivial braiding statistics. It has a rich phase diagram hosting gapless, Abelian and non-Abelian phases <span class="citation" data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a href="#ref-knolleDynamicsFractionalizationQuantum2015" role="doc-biblioref">63</a>]</span> and a finite temperature phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">64</a>]</span>. It been proposed that its non-Abelian excitations could be used to support robust topological quantum computing <span class="citation" data-cites="kitaev_fault-tolerant_2003 freedmanTopologicalQuantumComputation2003 nayakNonAbelianAnyonsTopological2008"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">65</a>–<a href="#ref-nayakNonAbelianAnyonsTopological2008" role="doc-biblioref">67</a>]</span>.</p>
<p>The Kitaev model shares are lot with the FK model, they are both effectively models of spinless fermions coupled to a classical Ising background field. This is what makes them exactly solvable. At finite temperatures, fluctuations in the background field provide an effective disorder potential for the fermionic sector, so both models can be studied at finite temperature with Markov chain Monte Carlo methods <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">64</a>,<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">68</a>]</span>.</p>
<p>As Kitaev points out in his original paper, the model remains solvable on any tri-coordinated <span class="math inline">\(z=3\)</span> graph which can be 3-edge-coloured. Indeed many generalisations of the model exist <span class="citation" data-cites="Baskaran2007 Baskaran2008 Nussinov2009 OBrienPRB2016 hermanns2015weyl"> [<a href="#ref-Baskaran2007" role="doc-biblioref">69</a>–<a href="#ref-hermanns2015weyl" role="doc-biblioref">73</a>]</span>. Notably, the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">74</a>]</span> introduces triangular plaquettes to the honeycomb lattice leading to spontaneous chiral symmetry breaking. These extensions all retain translation symmetry, likely because edge-colouring, finding the ground state and understanding the QSL properties are much harder without it <span class="citation" data-cites="eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmann2019thermodynamics" role="doc-biblioref">75</a>,<a href="#ref-Peri2020" role="doc-biblioref">76</a>]</span>. Undeterred, this gap lead us to wonder what might happen if we remove translation symmetry from the Kitaev Model. This might would be a model of a tri-coordinated, highly bond anisotropic but otherwise amorphous material.</p>
<p>Amorphous materials do not have long-range lattice regularities but covalent compounds can induce short-range regularities in the lattice structure such as fixed coordination number <span class="math inline">\(z\)</span>. The best examples being amorphous Silicon and Germanium with <span class="math inline">\(z=4\)</span> which are used to make thin-film solar cells <span class="citation" data-cites="Weaire1971 betteridge1973possible"> [<a href="#ref-Weaire1971" role="doc-biblioref">77</a>,<a href="#ref-betteridge1973possible" role="doc-biblioref">78</a>]</span>. Recently it has been shown that topological insulating (TI) phases can exist in amorphous systems. Amorphous TIs are characterized by similar protected edge states to their translation invariant cousins and generalised topological bulk invariants <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 agarwala2019topological marsalTopologicalWeaireThorpeModels2020 costa2019toward agarwala2020higher spring2021amorphous corbae2019evidence"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">79</a>–<a href="#ref-corbae2019evidence" role="doc-biblioref">85</a>]</span>. However, research on amorphous electronic systems has been mostly focused on non-interacting systems with a few exceptions, for example, to account for the observation of superconductivity <span class="citation" data-cites="buckel1954einfluss mcmillan1981electron meisel1981eliashberg bergmann1976amorphous mannaNoncrystallineTopologicalSuperconductors2022"> [<a href="#ref-buckel1954einfluss" role="doc-biblioref">86</a>–<a href="#ref-mannaNoncrystallineTopologicalSuperconductors2022" role="doc-biblioref">90</a>]</span> in amorphous materials or very recently to understand the effect of strong electron repulsion in TIs <span class="citation" data-cites="kim2022fractionalization"> [<a href="#ref-kim2022fractionalization" role="doc-biblioref">91</a>]</span>.</p>
<p>Amorphous <em>magnetic</em> systems have been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems <span class="citation" data-cites="aharony1975critical Petrakovski1981 kaneyoshi1992introduction Kaneyoshi2018"> [<a href="#ref-aharony1975critical" role="doc-biblioref">92</a>–<a href="#ref-Kaneyoshi2018" role="doc-biblioref">95</a>]</span> and with numerical methods <span class="citation" data-cites="fahnle1984monte plascak2000ising"> [<a href="#ref-fahnle1984monte" role="doc-biblioref">96</a>,<a href="#ref-plascak2000ising" role="doc-biblioref">97</a>]</span>. Research on classical Heisenberg and Ising models has been shown to account for observed behaviour of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour <span class="citation" data-cites="coey1978amorphous"> [<a href="#ref-coey1978amorphous" role="doc-biblioref">98</a>]</span>. However, the role of spin-anisotropic interactions and quantum effects in amorphous magnets has not been addressed. In this thesis I will address in the question of whether frustrated magnetic interactions on amorphous lattices can give rise to genuine quantum phases, i.e. to long-range entangled quantum spin liquids (QSL) <span class="citation" data-cites="Anderson1973 Knolle2019 Savary2016 Lacroix2011"> [<a href="#ref-Anderson1973" role="doc-biblioref">99</a>–<a href="#ref-Lacroix2011" role="doc-biblioref">102</a>]</span>. We will find that the answer is yes.</p>
<p>In <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">chapter 4</a> I will introduce the Amorphous Kitaev model, a generalisation of the Kitaev honeycomb model to random lattices with fixed coordination number three. We will show that this model is a soluble chiral amorphous quantum spin liquid. The model retains its exact solubility but, as with the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">74</a>]</span>, the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We will confirm prior observations that the form of the ground state can be written in terms of the number of sides of elementary plaquettes of the model <span class="citation" data-cites="OBrienPRB2016 eschmannThermodynamicClassificationThreedimensional2020"> [<a href="#ref-OBrienPRB2016" role="doc-biblioref">72</a>,<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">103</a>]</span>. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I show that the system undergoes a finite-temperature phase transition to a conducting thermal metal state and discuss possible experimental realisations.</p>
<p>The next chapter, <a href="../2_Background/2.1_FK_Model.html">Chapter 2</a>, will introduce some necessary background to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and localisation. Then <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">chapter 3</a> introduces and studies the Long Range Falicov-Kimball Model in one dimension. <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">Chapter 4</a> focusses on the Amorphous Kitaev Model.</p>
<p>Next Chapter: <a href="../2_Background/2.1_FK_Model.html">2 Background</a></p>
</section>
@ -219,197 +225,221 @@ H_{\mathrm{FK}} = &amp; -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U
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</section>

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@ -80,7 +81,7 @@ H_{\mathrm{FK}} = &amp; \;U \sum_{i} (d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\da
<p>Due to Pauli exclusion, maximum filling occurs when each lattice site is fully occupied, <span class="math inline">\(\langle n_c + n_d \rangle = 2\)</span>. Here we will focus on the half filled case <span class="math inline">\(\langle n_c + n_d \rangle = 1\)</span>. The ground state phenomenology as the model is doped away from the half-filled state can be rich <span class="citation" data-cites="jedrzejewskiFalicovKimballModels2001 gruberGroundStatesSpinless1990"> [<a href="#ref-jedrzejewskiFalicovKimballModels2001" role="doc-biblioref">7</a>,<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">8</a>]</span> but from this point we will only consider the half-filled point.</p>
<p>At half-filling and on bipartite lattices, FK the model is particle-hole (PH) symmetric. That is, the Hamiltonian anticommutes with the particle hole operator <span class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a consequence the energy spectrum is symmetric about <span class="math inline">\(E = 0\)</span> and this is the Fermi energy. The particle hole operator corresponds to the substitution <span class="math inline">\(c^\dagger_i \rightarrow \epsilon_i c_i, d^\dagger_i \rightarrow d_i\)</span> where <span class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and <span class="math inline">\(-1\)</span> for the even sublattice <span class="citation" data-cites="gruberFalicovKimballModel2005"> [<a href="#ref-gruberFalicovKimballModel2005" role="doc-biblioref">9</a>]</span>. The absence of a hopping term for the heavy electrons means they do not need the factor of <span class="math inline">\(\epsilon_i\)</span>. See appendix <a href="../6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">A.1</a> for a full derivation of the PH symmetry.</p>
<figure>
<img src="/assets/thesis/background_chapter/simple_DOS.svg" id="fig:simple_DOS" data-short-caption="Cubic Lattice dispersion with disorder" style="width:100.0%" alt="Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in one dimension. (a) With not external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder." />
<img src="/assets/thesis/background_chapter/simple_DOS.svg" id="fig-simple_DOS" data-short-caption="Cubic Lattice dispersion with disorder" style="width:100.0%" alt="Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in one dimension. (a) With not external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder." />
<figcaption aria-hidden="true">Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model <span class="math inline">\(H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}\)</span> in one dimension. (a) With not external potential. (b) With a static charge density wave background <span class="math inline">\(V_i = (-1)^i\)</span> (c) A static charge density wave background with 2% binary disorder.</figcaption>
</figure>
<p>We will later add a long range interaction between the localised electrons at which point we will replace the immobile fermions with a classical Ising field <span class="math inline">\(S_i = 1 - 2d^\dagger_id_i = \pm\tfrac{1}{2}\)</span> which I will refer to as the spins.</p>
@ -92,7 +93,7 @@ H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\
<section id="phase-diagrams" class="level2">
<h2>Phase Diagrams</h2>
<figure>
<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg" id="fig:fk_phase_diagram" data-short-caption="Falikov-Kimball Temperatue-Interaction Phase Diagrams" style="width:100.0%" alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from  [10,14]" />
<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg" id="fig-fk_phase_diagram" data-short-caption="Falikov-Kimball Temperatue-Interaction Phase Diagrams" style="width:100.0%" alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from  [10,14]" />
<figcaption aria-hidden="true">Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">10</a>,<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">14</a>]</span></figcaption>
</figure>
<p>In dimensions greater than one, the FK model exhibits a phase transition at some <span class="math inline">\(U\)</span> dependent critical temperature <span class="math inline">\(T_c(U)\)</span> to a low temperature ordered phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">15</a>]</span>. In terms of the immobile electrons this corresponds to them occupying only one of the two sublattices A and B this is known as a charge density wave (CDW) phase. In terms of spins this is an AFM phase.</p>
@ -121,7 +122,7 @@ H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\
<p>One final complexity is that for <span class="math inline">\(\tfrac{3}{2} &lt; \alpha &lt; 2\)</span> renormalisation group methods show that the critical point has non-universal critical exponents that depend on <span class="math inline">\(\alpha\)</span>  <span class="citation" data-cites="fisherCriticalExponentsLongRange1972"> [<a href="#ref-fisherCriticalExponentsLongRange1972" role="doc-biblioref">42</a>]</span>. To avoid this potential confounding factors we will park ourselves at <span class="math inline">\(\alpha = 1.25\)</span> when we apply these ideas to the FK model.</p>
<p>Were we to extend this to arbitrary dimension <span class="math inline">\(d\)</span> we would find that thermodynamics properties generally both <span class="math inline">\(d\)</span> and <span class="math inline">\(\alpha\)</span>, long range interactions can modify the ‘effective dimension’ of thermodynamic systems <span class="citation" data-cites="angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">43</a>]</span>.</p>
<figure>
<img src="/assets/thesis/background_chapter/alpha_diagram.svg" id="fig:alpha_diagram" data-short-caption="Long Range Ising Model Behaviour" style="width:100.0%" alt="Figure 3: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied." />
<img src="/assets/thesis/background_chapter/alpha_diagram.svg" id="fig-alpha_diagram" data-short-caption="Long Range Ising Model Behaviour" style="width:100.0%" alt="Figure 3: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied." />
<figcaption aria-hidden="true">Figure 3: The thermodynamic behaviour of the long range Ising model <span class="math inline">\(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span class="math inline">\(\alpha\)</span> is varied.</figcaption>
</figure>
<p>Next Section: <a href="../2_Background/2.2_HKM_Model.html">The Kitaev Honeycomb Model</a></p>

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<section id="bg-hkm-model" class="level1">
<h1>The Kitaev Honeycomb Model</h1>
<figure>
<img src="/assets/thesis/amk_chapter/intro/honeycomb_zoom/intro_figure_by_hand.svg" id="fig:intro_figure_by_hand" data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%" alt="Figure 1: (a) The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. (b). After transforming to the Majorana representation we get an emergent gauge degree of freedom u_{jk} = \pm 1 that lives on each bond, the bond variables. These are antisymmetric, u_{jk} = -u_{kj}, so we represent them graphically with arrows on each bond that point in the direction that u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas b_i^x,\;b_i^y,\;b_i^z,\;c_i. The x, y and z Majoranas then pair along the bonds forming conserved \mathbb{Z}_2 bond operators u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle. The remaining c_i operators form an effective quadratic Hamiltonian H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j." />
<img src="/assets/thesis/amk_chapter/intro/honeycomb_zoom/intro_figure_by_hand.svg" id="fig-intro_figure_by_hand" data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%" alt="Figure 1: (a) The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. (b). After transforming to the Majorana representation we get an emergent gauge degree of freedom u_{jk} = \pm 1 that lives on each bond, the bond variables. These are antisymmetric, u_{jk} = -u_{kj}, so we represent them graphically with arrows on each bond that point in the direction that u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas b_i^x,\;b_i^y,\;b_i^z,\;c_i. The x, y and z Majoranas then pair along the bonds forming conserved \mathbb{Z}_2 bond operators u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle. The remaining c_i operators form an effective quadratic Hamiltonian H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j." />
<figcaption aria-hidden="true">Figure 1: <strong>(a)</strong> The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. <strong>(b)</strong>. After transforming to the Majorana representation we get an emergent gauge degree of freedom <span class="math inline">\(u_{jk} = \pm 1\)</span> that lives on each bond, the bond variables. These are antisymmetric, <span class="math inline">\(u_{jk} = -u_{kj}\)</span>, so we represent them graphically with arrows on each bond that point in the direction that <span class="math inline">\(u_{jk} = +1\)</span> <strong>(c)</strong>. The Majorana transformation can be visualised as breaking each spin into four Majoranas <span class="math inline">\(b_i^x,\;b_i^y,\;b_i^z,\;c_i\)</span>. The x, y and z Majoranas then pair along the bonds forming conserved <span class="math inline">\(\mathbb{Z}_2\)</span> bond operators <span class="math inline">\(u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle\)</span>. The remaining <span class="math inline">\(c_i\)</span> operators form an effective quadratic Hamiltonian <span class="math inline">\(H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\)</span>.</figcaption>
</figure>
<section id="the-spin-hamiltonian" class="level2">
@ -86,7 +87,7 @@ image:
<section id="the-spin-model" class="level2">
<h2>The Spin Model</h2>
<figure>
<img src="/assets/thesis/amk_chapter/visual_kitaev_1.svg" id="fig:visual_kitaev_1" data-short-caption="A Visual Intro to the Kitaev Model" style="width:100.0%" alt="Figure 2: A visual introduction to the Kitaev Model." />
<img src="/assets/thesis/amk_chapter/visual_kitaev_1.svg" id="fig-visual_kitaev_1" data-short-caption="A Visual Intro to the Kitaev Model" style="width:100.0%" alt="Figure 2: A visual introduction to the Kitaev Model." />
<figcaption aria-hidden="true">Figure 2: A visual introduction to the Kitaev Model.</figcaption>
</figure>
<p>As discussed in the introduction, spin hamiltonians like that of the Kitaev model arise in electronic systems as the result the balance of multiple effects <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span>. For instance, in certain transition metal systems with <span class="math inline">\(d^5\)</span> valence electrons, crystal field and spin-orbit couplings conspire to shift and split the <span class="math inline">\(d\)</span> orbitals into moments with spin <span class="math inline">\(j = 1/2\)</span> and <span class="math inline">\(j = 3/2\)</span>. Of these, the bandwidth <span class="math inline">\(t\)</span> of the <span class="math inline">\(j= 1/2\)</span> band is small, meaning that even relatively meagre electron correlations (such those induced by the <span class="math inline">\(U\)</span> term in the Hubbard model) can lead to the opening of a Mott gap. From there we have a <span class="math inline">\(j = 1/2\)</span> Mott insulator whose effective spin-spin interactions are again shaped by the lattice geometry and spin-orbit coupling leading some materials to have strong bond-directional Ising-type interactions <span class="citation" data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a href="#ref-jackeliMottInsulatorsStrong2009" role="doc-biblioref">12</a>,<a href="#ref-khaliullinOrbitalOrderFluctuations2005" role="doc-biblioref">13</a>]</span>. In the Kitaev Model the bond directionality refers to the fact that the coupling axis <span class="math inline">\(\alpha\)</span> in terms like <span class="math inline">\(\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> is strongly bond dependent.</p>
@ -94,7 +95,7 @@ image:
<p><span class="math display">\[\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha},\]</span></p>
<p>the products of bonds winding around a closed path <span class="math inline">\(p\)</span> on the lattice. These operators commute with the Hamiltonian and so have no time dynamics. The winding direction does not matter so long as it is fixed. By convention we will always use clockwise. Each closed path on the lattice is associated with a flux. The number of conserved quantities grows linearly with system size and is thus extensive, this is a common property for exactly solvable systems and can be compared to the heavy electrons present in the Falikov-Kimball model. The square of two loop operators is one so any contractible loop can be expressed as a product of loops around plaquettes of the lattice, as in fig. <a href="#fig:stokes_theorem">3</a>. For the honeycomb lattice the plaquettes are the hexagons. The expectations of <span class="math inline">\(\hat{W}_p\)</span> through each plaquette, the fluxes, are therefore enough to describe the whole flux sector. We will focus on these fluxes, denoting them by <span class="math inline">\(\phi_i\)</span>. Once we have made the mapping to the Majorana Hamiltonian I will explain how these fluxes can be connected to an emergent <span class="math inline">\(B\)</span> field which makes their interpretation as fluxes clear.</p>
<figure>
<img src="/assets/thesis/amk_chapter/stokes_theorem/stokes_theorem.svg" id="fig:stokes_theorem" data-short-caption="We can construct arbitrary loops from plaquette fluxes." style="width:71.0%" alt="Figure 3: In the Kitaev Honeycomb model, Wilson loop operators \hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha} can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette \phi_i. This relationship between the u_{ij} around a region and fluxes with one is evocative of Stokes’ theorem for classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later." />
<img src="/assets/thesis/amk_chapter/stokes_theorem/stokes_theorem.svg" id="fig-stokes_theorem" data-short-caption="We can construct arbitrary loops from plaquette fluxes." style="width:71.0%" alt="Figure 3: In the Kitaev Honeycomb model, Wilson loop operators \hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha} can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette \phi_i. This relationship between the u_{ij} around a region and fluxes with one is evocative of Stokes’ theorem for classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later." />
<figcaption aria-hidden="true">Figure 3: In the Kitaev Honeycomb model, Wilson loop operators <span class="math inline">\(\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha}\)</span> can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette <span class="math inline">\(\phi_i\)</span>. This relationship between the <span class="math inline">\(u_{ij}\)</span> around a region and fluxes with one is evocative of Stokes’ theorem for classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later.</figcaption>
</figure>
<p>It is worth noting in passing that the effective Hamiltonian for many Kitaev materials incorporates a contribution from an isotropic Heisenberg term <span class="math inline">\(\sum_{i,j} \vec{\sigma}_i\cdot\vec{\sigma}_j\)</span>, this is referred to as the Heisenberg-Kitaev Model <span class="citation" data-cites="Chaloupka2010"> [<a href="#ref-Chaloupka2010" role="doc-biblioref">14</a>]</span>. Materials for which the Kitaev term dominates are generally known as Kitaev Materials. See <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span> for a full discussion of Kitaev Materials.</p>
@ -135,7 +136,7 @@ H &amp;= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}
<p>We have transformed the spin Hamiltonian into a Majorana hamiltonian <span class="math inline">\(H = i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\)</span> describing the dynamics of a classical field <span class="math inline">\(u_{ij}\)</span> and Majoranas <span class="math inline">\(c_i\)</span>. It is natural to ask how the classical field <span class="math inline">\(u_{ij}\)</span> relates to the fluxes of the original spin model. We can evaluate the fluxes <span class="math inline">\(\phi_i\)</span> in terms of the bond operators</p>
<p><span id="eq:flux-majorana"><span class="math display">\[\phi_i = \prod_{\langle j,k\rangle \in \mathcal{P}_i} i u_{jk}.\qquad{(4)}\]</span></span></p>
<figure>
<img src="/assets/thesis/amk_chapter/intro/gauge_symmetries/gauge_symmetries.svg" id="fig:gauge_symmetries" data-short-caption="Gauge Symmetries" style="width:100.0%" alt="Figure 4: A honeycomb lattice with edges in light grey, along with its dual, the triangle lattice in light blue. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator D_j at a vertex is to flip the value of the three u_{jk} variables (black lines) surrounding site j. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent D_j operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing D_j operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of u_{ij}s flipped on its edge. Therefore, all retain the same flux \phi_i." />
<img src="/assets/thesis/amk_chapter/intro/gauge_symmetries/gauge_symmetries.svg" id="fig-gauge_symmetries" data-short-caption="Gauge Symmetries" style="width:100.0%" alt="Figure 4: A honeycomb lattice with edges in light grey, along with its dual, the triangle lattice in light blue. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator D_j at a vertex is to flip the value of the three u_{jk} variables (black lines) surrounding site j. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent D_j operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing D_j operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of u_{ij}s flipped on its edge. Therefore, all retain the same flux \phi_i." />
<figcaption aria-hidden="true">Figure 4: A honeycomb lattice with edges in light grey, along with its dual, the triangle lattice in light blue. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator <span class="math inline">\(D_j\)</span> at a vertex is to flip the value of the three <span class="math inline">\(u_{jk}\)</span> variables (black lines) surrounding site <span class="math inline">\(j\)</span>. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent <span class="math inline">\(D_j\)</span> operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing <span class="math inline">\(D_j\)</span> operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of <span class="math inline">\(u_{ij}\)</span>s flipped on its edge. Therefore, all retain the same flux <span class="math inline">\(\phi_i\)</span>.</figcaption>
</figure>
<p>In addition, the bond operators form a highly degenerate description of the system. The operators <span class="math inline">\(D_i = b^x_i b^y_i b^z_i c_i\)</span> commute with <span class="math inline">\(H\)</span> so form a set of local symmetries. The action of <span class="math inline">\(D_i\)</span> on a state is to flip the values of the three <span class="math inline">\(u_{ij}\)</span> bonds that connect to site <span class="math inline">\(i\)</span>. This changes the bond configuration <span class="math inline">\(\{u_{ij}\}\)</span> but leaves the flux configuration <span class="math inline">\(\{\phi_i\}\)</span> unchanged. Physically, we interpret <span class="math inline">\(u_{ij}\)</span> as a gauge field with a high degree of degeneracy and <span class="math inline">\(\{D_i\}\)</span> as the set of gauge symmetries. The Majorana bond operators <span class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span class="math inline">\(\mathbb{Z_2}\)</span> gauge field! The flux configuration <span class="math inline">\(\{\phi_i\}\)</span> is what encodes physical information about the system.</p>
@ -157,7 +158,7 @@ A honeycomb lattice (in black) along with its dual (in red). (Left) The product
</figcaption>
</figure> -->
<figure>
<img src="/assets/thesis/amk_chapter/topological_fluxes.png" id="fig:topological_fluxes" data-short-caption="Topological Fluxes" style="width:57.0%" alt="Figure 5: 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 with both had a jam filling and a hole, this analogy would be a lot easier to make  [23]." />
<img src="/assets/thesis/amk_chapter/topological_fluxes.png" id="fig-topological_fluxes" data-short-caption="Topological Fluxes" style="width:57.0%" alt="Figure 5: 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 with both had a jam filling and a hole, this analogy would be a lot easier to make  [23]." />
<figcaption aria-hidden="true">Figure 5: 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 with both had a jam filling and a hole, this analogy would be a lot easier to make <span class="citation" data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis" role="doc-biblioref">23</a>]</span>.</figcaption>
</figure>
<p>A final but important point to mention is that is that the local fluxes <span class="math inline">\(\phi_i\)</span> are not quite all there is. We’ve seen that products of <span class="math inline">\(\phi_i\)</span> can be used to construct the flux associated with arbitrary contractible loops. On the plane contractible loops are all there are. However, on the torus we can construct two global fluxes <span class="math inline">\(\Phi_x\)</span> and <span class="math inline">\(\Phi_y\)</span> which correspond to paths tracing the major and minor axes. The four sectors spanned by the <span class="math inline">\(\pm1\)</span> values of these fluxes are gapped away from one another but only by virtual tunnelling processes so the gap decays exponentially with system size <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. Physically <span class="math inline">\(\Phi_x\)</span> and <span class="math inline">\(\Phi_y\)</span> could be thought of as measuring the flux that threads through the hole of the doughnut. In general, surfaces with genus <span class="math inline">\(g\)</span> have <span class="math inline">\(g\)</span> ‘handles’ and <span class="math inline">\(2g\)</span> of these global fluxes. At first glance it may seem this would not have much relevance to physical realisations of the Kitaev model that will likely have a planar geometry with open boundary conditions. However these fluxes are closely linked to topology and the existence of anyonic quasiparticle excitations in the model, which we will discuss next.</p>
@ -165,11 +166,11 @@ A honeycomb lattice (in black) along with its dual (in red). (Left) The product
<section id="sec:anyons" class="level2">
<h2>Anyons, Topology and the Chern number</h2>
<figure>
<img src="/assets/thesis/amk_chapter/braiding.png" id="fig:braiding" data-short-caption="Braiding in Two Dimensions" style="width:71.0%" alt="Figure 6: Worldlines of particles in two dimensions can become tangled or braided with one another." />
<img src="/assets/thesis/amk_chapter/braiding.png" id="fig-braiding" data-short-caption="Braiding in Two Dimensions" style="width:71.0%" alt="Figure 6: Worldlines of particles in two dimensions can become tangled or braided with one another." />
<figcaption aria-hidden="true">Figure 6: Worldlines of particles in two dimensions can become tangled or <em>braided</em> with one another.</figcaption>
</figure>
<p>To discuss different ground state phases of the KH model we must first review the topic of anyons and topology. The standard argument for the existence of Fermions and Bosons goes like this: the quantum state of a system must pick up a factor of <span class="math inline">\(\pm1\)</span> if two identical particles are swapped. Only <span class="math inline">\(\pm1\)</span> are allowed since swapping twice must correspond to the identity. This argument works in three dimensions for states without topological degeneracy, which seems to be true of the real world, but condensed matter systems are subject to no such constraints.</p>
<p>In gapped condensed matter systems, all equal time correlators decay exponentially with distance <span class="citation" data-cites="hastingsLiebSchultzMattisHigherDimensions2004"> [<a href="#ref-hastingsLiebSchultzMattisHigherDimensions2004" role="doc-biblioref">24</a>]</span>. Put another way, the system supports quasiparticles with a definite location in space and a finite extent. As such it is meaningful to consider what would happen to the overall quantum state if we were to adiabatically carry out a series of swaps as described above. This is known as braiding.</p>
<p>In gapped condensed matter systems, all equal time correlators decay exponentially with distance <span class="citation" data-cites="hastingsLiebSchultzMattisHigherDimensions2004"> [<a href="#ref-hastingsLiebSchultzMattisHigherDimensions2004" role="doc-biblioref">24</a>]</span>. Put another way, gapped systems support quasiparticles with a definite location in space and finite extent. As such it is meaningful to consider what would happen to the overall quantum state if we were to adiabatically carry out a series of swaps as described above. This is known as braiding.</p>
<p>First we realise that in two dimensions, swapping identical particles twice is not topologically equivalent to the identity, see fig. <a href="#fig:braiding">6</a>. Instead it corresponds to encircling one particle around the other. This means we can in general pick up any complex phase <span class="math inline">\(e^{i\theta}\)</span>, hence the name <strong>any</strong>-ons upon exchange. These are known as Abelian anyons because complex multiplication commutes and hence the group of braiding operations forms and Abelian group.</p>
<p>The KH model has a topologically degenerate ground state with sectors labelled by the values of the topological fluxes <span class="math inline">\((\Phi_x\)</span>, <span class="math inline">\(\Phi_y)\)</span>. Consider the operation in which a quasiparticle pair is created from the ground state, transported around one of the non-contractible loops and then annihilated together, call them <span class="math inline">\(\mathcal{T}_{x}\)</span> and <span class="math inline">\(\mathcal{T}_{y}\)</span>. These operations move us around within the ground state manifold and they need not commute. This leads to non-Abelian anyons. As Kitaev points out, these operations are not specific to the torus: the operation <span class="math inline">\(\mathcal{T}_{x}\mathcal{T}_{y}\mathcal{T}_{x}^{-1}\mathcal{T}_{y}^{-1}\)</span> corresponds to an operation in which none of the particles crosses the torus, rather one simply winds around the other, hence these effects of relevant even for the planar case.</p>
<!-- <figure>
@ -186,7 +187,7 @@ This all works the same way for the amorphous lattice but the diagram is a lot m
<section id="ground-state-phases" class="level2">
<h2>Ground State Phases</h2>
<figure>
<img src="/assets/thesis/background_chapter/KH_phase_diagram.svg" id="fig:KH_phase_diagram" data-short-caption="Kitaev Honeycomb Model Phase Diagram" style="width:100.0%" alt="Figure 7: Setting the energy scale of the Kitaev Model with the constraint that J_x + J_y + J_z = 1 yields a triangular phase diagram where each of the corners represents J_\alpha = 1. For each corner \alpha the region |J_\alpha &gt; |J_\beta| + |J_\gamma| supports a gapped non-Abelian phase equivalent to that of the Toric code  [30,33]. The point around equal coupling J_x = J_y = J_z, the B phase, is gapless. The B phase is known a Majorana metal and on the honeycomb lattice it has a Dirac cone dispersion similar to that of graphene." />
<img src="/assets/thesis/background_chapter/KH_phase_diagram.svg" id="fig-KH_phase_diagram" data-short-caption="Kitaev Honeycomb Model Phase Diagram" style="width:100.0%" alt="Figure 7: Setting the energy scale of the Kitaev Model with the constraint that J_x + J_y + J_z = 1 yields a triangular phase diagram where each of the corners represents J_\alpha = 1. For each corner \alpha the region |J_\alpha &gt; |J_\beta| + |J_\gamma| supports a gapped non-Abelian phase equivalent to that of the Toric code  [30,33]. The point around equal coupling J_x = J_y = J_z, the B phase, is gapless. The B phase is known a Majorana metal and on the honeycomb lattice it has a Dirac cone dispersion similar to that of graphene." />
<figcaption aria-hidden="true">Figure 7: Setting the energy scale of the Kitaev Model with the constraint that <span class="math inline">\(J_x + J_y + J_z = 1\)</span> yields a triangular phase diagram where each of the corners represents <span class="math inline">\(J_\alpha = 1\)</span>. For each corner <span class="math inline">\(\alpha\)</span> the region <span class="math inline">\(|J_\alpha &gt; |J_\beta| + |J_\gamma|\)</span> supports a gapped non-Abelian phase equivalent to that of the Toric code <span class="citation" data-cites="kitaev1997quantum kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">30</a>,<a href="#ref-kitaev1997quantum" role="doc-biblioref">33</a>]</span>. The point around equal coupling <span class="math inline">\(J_x = J_y = J_z\)</span>, the B phase, is gapless. The B phase is known a Majorana metal and on the honeycomb lattice it has a Dirac cone dispersion similar to that of graphene.</figcaption>
</figure>
<p>Setting the overall energy scale with the constraint <span class="math inline">\(J_x + J_y + J_z = 1\)</span> yields a triangular phase diagram. In each of the corners one of the spin-coupling directions dominates, <span class="math inline">\(|J_\alpha &gt; |J_\beta| + |J_\gamma|\)</span>, yielding three equivalent <span class="math inline">\(A_\alpha\)</span> phases while the central triangle around <span class="math inline">\(J_x = J_y = J_z\)</span> is called the B phase. Both phases support two kinds of quasiparticles, fermions and <span class="math inline">\(\mathbb{Z}_2\)</span>-vortices. In the A phases, the vortices have bosonic statistics with respect to themselves but act like fermions with respect to the fermions, hence they are Abelian anyons, This phase has the same anyonic structure as the Toric code <span class="citation" data-cites="kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">30</a>]</span>. Since the B phase is gapless, the quasiparticles aren’t localised and so don’t have braiding statistics.</p>

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@ -82,7 +83,7 @@ H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U
<section id="diagnosing-localisation-in-practice" class="level2">
<h2>Diagnosing Localisation in practice</h2>
<figure>
<img src="/assets/thesis/background_chapter/localisation_radius_vs_length.svg" id="fig:localisation_radius_vs_length" data-short-caption="Localisation length vs diameter" style="width:100.0%" alt="Figure 1: A localised state \psi in an potential well that has formed from random fluctuations in the disorder potential V(x). The localisation length \lambda governs how quickly the state decays away from the well while the diameter R of the state is controlled by the size of the well. Reproduced from  [6]." />
<img src="/assets/thesis/background_chapter/localisation_radius_vs_length.svg" id="fig-localisation_radius_vs_length" data-short-caption="Localisation length vs diameter" style="width:100.0%" alt="Figure 1: A localised state \psi in an potential well that has formed from random fluctuations in the disorder potential V(x). The localisation length \lambda governs how quickly the state decays away from the well while the diameter R of the state is controlled by the size of the well. Reproduced from  [6]." />
<figcaption aria-hidden="true">Figure 1: A localised state <span class="math inline">\(\psi\)</span> in an potential well that has formed from random fluctuations in the disorder potential <span class="math inline">\(V(x)\)</span>. The localisation length <span class="math inline">\(\lambda\)</span> governs how quickly the state decays away from the well while the diameter <span class="math inline">\(R\)</span> of the state is controlled by the size of the well. Reproduced from <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span>.</figcaption>
</figure>
<p>Looking at practical tools for diagnosing localisation, there are a few standard methods <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span>.</p>

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<h1>The Model</h1>
<p>Dimensionality is crucial for the physics of both localisation and phase transitions. We have already seen that the one dimensional standard FK model cannot support an ordered phase at finite temperatures and therefore has no finite temperature phase transition (FTPT).</p>
<p>On bipartite lattices in dimensions 2 and above the FK model exhibits a finite temperature phase transition to an ordered charge density wave (CDW) phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">3</a>]</span>. In this phase, the spins order anti-ferromagnetically, breaking the <span class="math inline">\(\mathbb{Z}_2\)</span> symmetry. In 1D, however, Periel’s argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">4</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">5</a>]</span> states that domain walls only introduce a constant energy penalty into the free energy while bringing a entropic contribution logarithmic in system size. Hence the 1D model does not have a finite temperature phase transition. However 1D systems are much easier to study numerically and admit simpler realisations experimentally. We therefore introduce a long-range coupling between the ions in order to stabilise a CDW phase in 1D.</p>
<figure>
<img src="/assets/thesis/intro_chapter/lrfk_schematic.svg" id="fig-lrfk_schematic" data-short-caption="Falicov-Kimball Model Diagram" style="width:100.0%" alt="Figure 1: The Long Range Falicov-Kimball (LRFK) Model is a model of classical spins S_i coupled to spinless fermions \hat{c}_i where the fermions are mobile with hopping t and the fermions are coupled to the spins by an Ising type interaction with strength U. The difference from the standard FK model is the presence of a long range interaction between the spins J_{ij}S_i S_j." />
<figcaption aria-hidden="true">Figure 1: The Long Range Falicov-Kimball (LRFK) Model is a model of classical spins <span class="math inline">\(S_i\)</span> coupled to spinless fermions <span class="math inline">\(\hat{c}_i\)</span> where the fermions are mobile with hopping <span class="math inline">\(t\)</span> and the fermions are coupled to the spins by an Ising type interaction with strength <span class="math inline">\(U\)</span>. The difference from the standard FK model is the presence of a long range interaction between the spins <span class="math inline">\(J_{ij}S_i S_j\)</span>.</figcaption>
</figure>
<p>We interpret the FK model as a model of spinless fermions, <span class="math inline">\(c^\dagger_{i}\)</span>, hopping on a 1D lattice against a classical Ising spin background, <span class="math inline">\(S_i \in {\pm \frac{1}{2}}\)</span>. The fermions couple to the spins via an onsite interaction with strength <span class="math inline">\(U\)</span> which we supplement by a long-range interaction, <span class="math display">\[
J_{ij} = 4\kappa J\; (-1)^{|i-j|} |i-j|^{-\alpha},
\]</span></p>

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<section id="thermodynamics-of-the-lrfk-model" class="level2">
<h2>Thermodynamics of the LRFK Model</h2>
<figure>
<img src="/assets/thesis/fk_chapter/lsr/pdf_figs/raw_steps_single_flip.svg" id="fig:raw_steps_single_flip" data-short-caption="Comparison of different proposal distributions" style="width:100.0%" alt="Figure 1: Two MCMC walks starting from the CDW state for a system with N = 100 sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation m = N^{-1} \sum_i (-1)^i \; S_i order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5" />
<img src="/assets/thesis/fk_chapter/lsr/pdf_figs/raw_steps_single_flip.svg" id="fig-raw_steps_single_flip" data-short-caption="Comparison of different proposal distributions" style="width:100.0%" alt="Figure 1: Two MCMC walks starting from the CDW state for a system with N = 100 sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation m = N^{-1} \sum_i (-1)^i \; S_i order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5" />
<figcaption aria-hidden="true">Figure 1: Two MCMC walks starting from the CDW state for a system with <span class="math inline">\(N = 100\)</span> sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. <span class="math inline">\(t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5\)</span></figcaption>
</figure>
<p>The classical Markov Chain Monte Carlo (MCMC) method which we discuss in the following allows us to solve our long-range FK model efficiently, yielding unbiased estimates of thermal expectation values.</p>
@ -107,7 +108,7 @@ H_c&amp; = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{
<section id="scaling" class="level2">
<h2>Scaling</h2>
<figure>
<img src="/assets/thesis/fk_chapter/binder_cumulants/binder_cumulants.svg" id="fig:binder_cumulants" data-short-caption="Binder Cumulants" style="width:100.0%" alt="Figure 2: (Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. ¿fig:phase-diagram-lrfk?. All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly mentioned." />
<img src="/assets/thesis/fk_chapter/binder_cumulants/binder_cumulants.svg" id="fig-binder_cumulants" data-short-caption="Binder Cumulants" style="width:100.0%" alt="Figure 2: (Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. ¿fig:phase-diagram-lrfk?. All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly mentioned." />
<figcaption aria-hidden="true">Figure 2: (Left) The order parameters, <span class="math inline">\(\langle m^2 \rangle\)</span>(solid) and <span class="math inline">\(1 - f\)</span> (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle\)</span> . (Right) The crossing of the Binder cumulant, <span class="math inline">\(B = \langle m^4 \rangle / \langle m^2 \rangle^2\)</span>, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. <strong>¿fig:phase-diagram-lrfk?</strong>. All plots use system sizes <span class="math inline">\(N = [10,20,30,50,70,110,160,250]\)</span> and lines are coloured from <span class="math inline">\(N = 10\)</span> in dark blue to <span class="math inline">\(N = 250\)</span> in yellow. The parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span> except where explicitly mentioned.</figcaption>
</figure>
<p>To improve the scaling of finite size effects, we make the replacement <span class="math inline">\(|i - j|^{-\alpha} \rightarrow |f(i - j)|^{-\alpha}\)</span>, in both <span class="math inline">\(J_{ij}\)</span> and <span class="math inline">\(\kappa\)</span>, where <span class="math inline">\(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\)</span>. <span class="math inline">\(f\)</span> is smooth across the circular boundary and its effect effect diminished for larger systems <span class="citation" data-cites="fukuiOrderNClusterMonte2009"> [<a href="#ref-fukuiOrderNClusterMonte2009" role="doc-biblioref">12</a>]</span>. We only consider even system sizes given that odd system sizes are not commensurate with a CDW state.</p>

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<h1>Results</h1>
<p>Looking at the results of our MCMC simulations, we find a rich phase diagram with a CDW FTPT and interaction-tuned Anderson versus Mott localized phases similar to the 2D FK model <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>. We explore the localization properties of the fermionic sector and find that the localisation lengths vary dramatically across the phases and for different energies. Although moderate system sizes indicate the coexistence of localized and delocalized states within the CDW phase, we find quantitatively similar behaviour in a model of uncorrelated binary disorder on a CDW background. For large system sizes, i.e. for our 1D disorder model we can treat linear sizes of several thousand sites, we find that all states are eventually localized with a localization length which diverges towards zero temperature.</p>
<figure>
<img src="/assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg" id="fig:phase-diagram-lrfk" data-short-caption="Long Range Falicov Kimball Model Phase Diagram" style="width:100.0%" alt="Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature T_c, linear in J. (Right) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas." />
<img src="/assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg" id="fig-phase-diagram-lrfk" data-short-caption="Long Range Falicov Kimball Model Phase Diagram" style="width:100.0%" alt="Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature T_c, linear in J. (Right) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas." />
<figcaption aria-hidden="true">Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at <span class="math inline">\(U = 5\)</span>: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (Right) The TU plane at <span class="math inline">\(J = 5\)</span>: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at <span class="math inline">\(E=0\)</span> in the single particle energy spectrum. <span class="math inline">\(U_c\)</span> is independent of temperature. At <span class="math inline">\(U = 0\)</span> the fermions are decoupled from the spins forming a simple Fermi gas.</figcaption>
</figure>
<section id="lrfk-results-phase-diagram" class="level2">
@ -79,7 +80,7 @@ image:
<section id="localisation-properties" class="level2">
<h2>Localisation Properties</h2>
<figure>
<img src="/assets/thesis/fk_chapter/DOS/DOS.svg" id="fig:DOS" data-short-caption="Energy resolved DOS($\omega$) in the difference phases." style="width:100.0%" alt="Figure 2: Energy resolved DOS(\omega) against system size N in all three phases. The charge density wave phase is shown in both the high and low U regime for completeness. The top left panel shows the Anderson phase at U = 2 and high T = 2.5, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at U = 2 and low T = 1.5, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at U = 5 and high T = 2.5 the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at U = 5 and T = 1.5 is qualitatively similar to the lower left panel except that the gap scales with U. For all the plots J = 5,\;\alpha = 1.25." />
<img src="/assets/thesis/fk_chapter/DOS/DOS.svg" id="fig-DOS" data-short-caption="Energy resolved DOS($\omega$) in the difference phases." style="width:100.0%" alt="Figure 2: Energy resolved DOS(\omega) against system size N in all three phases. The charge density wave phase is shown in both the high and low U regime for completeness. The top left panel shows the Anderson phase at U = 2 and high T = 2.5, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at U = 2 and low T = 1.5, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at U = 5 and high T = 2.5 the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at U = 5 and T = 1.5 is qualitatively similar to the lower left panel except that the gap scales with U. For all the plots J = 5,\;\alpha = 1.25." />
<figcaption aria-hidden="true">Figure 2: Energy resolved DOS(<span class="math inline">\(\omega\)</span>) against system size <span class="math inline">\(N\)</span> in all three phases. The charge density wave phase is shown in both the high and low <span class="math inline">\(U\)</span> regime for completeness. The top left panel shows the Anderson phase at <span class="math inline">\(U = 2\)</span> and high <span class="math inline">\(T = 2.5\)</span>, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at <span class="math inline">\(U = 2\)</span> and low <span class="math inline">\(T = 1.5\)</span>, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at <span class="math inline">\(U = 5\)</span> and high <span class="math inline">\(T = 2.5\)</span> the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at <span class="math inline">\(U = 5\)</span> and <span class="math inline">\(T = 1.5\)</span> is qualitatively similar to the lower left panel except that the gap scales with <span class="math inline">\(U\)</span>. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span>.</figcaption>
</figure>
<p>The MCMC formulation suggests viewing the spin configurations as a form of annealed binary disorder whose probability distribution is given by the Boltzmann weight <span class="math inline">\(e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]}\)</span>. This makes apparent the link to the study of disordered systems and Anderson localisation. While these systems are typically studied by defining the probability distribution for the quenched disorder potential externally, here we have a translation invariant system with disorder as a natural consequence of the Ising background field conserved under the dynamics.</p>
@ -88,7 +89,7 @@ image:
\mathrm{DOS}(\vec{S}, \omega)&amp; = N^{-1} \sum_{i} \delta(\epsilon_i - \omega)\\
\mathrm{IPR}(\vec{S}, \omega)&amp; = \; N^{-1} \mathrm{DOS}(\vec{S}, \omega)^{-1} \sum_{i,j} \delta(\epsilon_i - \omega)\;\psi^{4}_{i,j}\end{aligned}\]</span> where <span class="math inline">\(\epsilon_i\)</span> and <span class="math inline">\(\psi_{i,j}\)</span> are the <span class="math inline">\(i\)</span>th energy level and <span class="math inline">\(j\)</span>th element of the corresponding eigenfunction, both dependent on the background spin configuration <span class="math inline">\(\vec{S}\)</span>.</p>
<figure>
<img src="/assets/thesis/fk_chapter/DOS/IPR_scaling.svg" id="fig:IPR_scaling" data-short-caption="Scaling of IPR($\omega$) against system size $N$." style="width:100.0%" alt="Figure 3: The IPR(\omega) scaling with N at fixed energy for each phase and for points both in the gap (\omega_0) and in a band (\omega_1). The slope of the line yields the scaling exponent \tau defined by \mathrm{IPR} \propto N^{-\tau}). \tau close to zero implies that the states at that energy are localised while \tau = -d corresponds to extended states where d is the system dimension. All but the bands of the charge density wave phase are approximately localised with \tau is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots J = 5,\;\alpha = 1.25. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states." />
<img src="/assets/thesis/fk_chapter/DOS/IPR_scaling.svg" id="fig-IPR_scaling" data-short-caption="Scaling of IPR($\omega$) against system size $N$." style="width:100.0%" alt="Figure 3: The IPR(\omega) scaling with N at fixed energy for each phase and for points both in the gap (\omega_0) and in a band (\omega_1). The slope of the line yields the scaling exponent \tau defined by \mathrm{IPR} \propto N^{-\tau}). \tau close to zero implies that the states at that energy are localised while \tau = -d corresponds to extended states where d is the system dimension. All but the bands of the charge density wave phase are approximately localised with \tau is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots J = 5,\;\alpha = 1.25. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states." />
<figcaption aria-hidden="true">Figure 3: The IPR(<span class="math inline">\(\omega\)</span>) scaling with <span class="math inline">\(N\)</span> at fixed energy for each phase and for points both in the gap (<span class="math inline">\(\omega_0\)</span>) and in a band (<span class="math inline">\(\omega_1\)</span>). The slope of the line yields the scaling exponent <span class="math inline">\(\tau\)</span> defined by <span class="math inline">\(\mathrm{IPR} \propto N^{-\tau}\)</span>). <span class="math inline">\(\tau\)</span> close to zero implies that the states at that energy are localised while <span class="math inline">\(\tau = -d\)</span> corresponds to extended states where <span class="math inline">\(d\)</span> is the system dimension. All but the bands of the charge density wave phase are approximately localised with <span class="math inline">\(\tau\)</span> is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span>. The measured <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\((0.06\pm0.01, 0.02\pm0.01\)</span> (b) <span class="math inline">\(0.04\pm0.02, 0.00\pm0.01\)</span> (c) <span class="math inline">\(0.05\pm0.03, 0.30\pm0.03\)</span> (d) <span class="math inline">\(0.06\pm0.04, 0.15\pm0.05\)</span> We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states.</figcaption>
</figure>
<p>The scaling of the IPR with system size</p>
@ -97,13 +98,13 @@ image:
<p>The key question for our system is then: How is the <span class="math inline">\(T=0\)</span> CDW phase with fully delocalized fermionic states connected to the fully localized phase at high temperatures?</p>
<p>For a representative set of parameters covering all three phases fig. <a href="#fig:DOS">2</a> shows the density of states as function of energy while fig. <a href="#fig:IPR_scaling">3</a> shows <span class="math inline">\(\tau\)</span>, the scaling exponent of the IPR with system size, The DOS is symmetric about <span class="math inline">\(0\)</span> because of the particle hole symmetry of the model. At high temperatures, all of the eigenstates are localised in both the Mott and Anderson phases (with <span class="math inline">\(\tau \leq 0.07\)</span> for our system sizes). We also checked that the states are localised by direct inspection. Note that there are in-gap states for instance at <span class="math inline">\(\omega_0\)</span>, below the upper band which are localized and smoothly connected across the phase transition.</p>
<figure>
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U5.svg" id="fig:gap_opening_U5" data-short-caption="The transition from CDW to the Mott phase" style="width:100.0%" alt="Figure 4: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U5.svg" id="fig-gap_opening_U5" data-short-caption="The transition from CDW to the Mott phase" style="width:100.0%" alt="Figure 4: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
<figcaption aria-hidden="true">Figure 4: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW to gapped Mott phase transition at <span class="math inline">\(U=5\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
</figure>
<p>In the CDW phases at <span class="math inline">\(U=2\)</span> and <span class="math inline">\(U=5\)</span>, we find for the states within the gapped CDW bands, e.g. at <span class="math inline">\(\omega_1\)</span>, scaling exponents <span class="math inline">\(\tau = 0.30\pm0.03\)</span> and <span class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This surprising finding suggests that the CDW bands are partially delocalised with multi-fractal behaviour of the wavefunctions <span class="citation" data-cites="eversAndersonTransitions2008"> [<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">6</a>]</span>. This phenomenon would be unexpected in a 1D model as they generally do not support delocalisation in the presence of disorder except as the result of correlations in the emergent disorder potential <span class="citation" data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"> [<a href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">7</a>,<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">8</a>]</span>. However, we later show by comparison to an uncorrelated Anderson model that these nonzero exponents are a finite size effect and the states are localised with a finite <span class="math inline">\(\xi\)</span> similar to the system size, an example of weak localisation. As a result, the IPR does not scale correctly until the system size has grown much larger than <span class="math inline">\(\xi\)</span>. fig. <a href="#fig:DM_IPR_scaling">7</a> shows that the scaling of the IPR in the CDW phase does flatten out eventually.</p>
<p>Next, we use the DOS and the scaling exponent <span class="math inline">\(\tau\)</span> to explore the localisation properties over the energy-temperature plane in fig. <a href="#fig:gap_opening_U2">5</a> and fig. <a href="#fig:gap_opening_U5">4</a>. Gapped areas are shown in white, which highlights the distinction between the gapped Mott phase and the ungapped Anderson phase. In-gap states appear just below the critical point, smoothly filling the bandgap in the Anderson phase and forming islands in the Mott phase. As in the finite <span class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">9</a>]</span> and infinite dimensional <span class="citation" data-cites="hassanSpectralPropertiesChargedensitywave2007"> [<a href="#ref-hassanSpectralPropertiesChargedensitywave2007" role="doc-biblioref">10</a>]</span> cases, the in-gap states merge and are pushed to lower energy for decreasing U as the <span 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 fluctuations away from the AFM ordered background. These domain walls act as local potentials for impurity-like bound states <span class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">9</a>]</span>.</p>
<figure>
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U2.svg" id="fig:gap_opening_U2" data-short-caption="The transition from CDW to the Anderson Phase" style="width:100.0%" alt="Figure 5: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at U=2. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U2.svg" id="fig-gap_opening_U2" data-short-caption="The transition from CDW to the Anderson Phase" style="width:100.0%" alt="Figure 5: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at U=2. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
<figcaption aria-hidden="true">Figure 5: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at <span class="math inline">\(U=2\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
</figure>
<p>In order to understand the localization properties we can compare the behaviour of our model with that of a simpler Anderson disorder model (DM) in which the spins are replaced by a CDW background with uncorrelated binary defect potentials. This is defined by replacing the spin degree of freedom in the FK model <span class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder potential <span class="math inline">\(d_i = \pm \tfrac{1}{2}\)</span> controlled by a defect density <span class="math inline">\(\rho\)</span> such that <span class="math inline">\(d_i = -\tfrac{1}{2}\)</span> with probability <span class="math inline">\(\rho/2\)</span> and <span class="math inline">\(d_i = \tfrac{1}{2}\)</span> otherwise. <span class="math inline">\(\rho/2\)</span> is used rather than <span class="math inline">\(\rho\)</span> so that the disorder potential takes on the zero temperature CDW ground state at <span class="math inline">\(\rho = 0\)</span> and becomes a random choice over spin states at <span class="math inline">\(\rho = 1\)</span> i.e the infinite temperature limit.</p>
@ -113,12 +114,12 @@ H_{\mathrm{DM}} = &amp; \;U \sum_{i} (-1)^i \; d_i \;(c^\dagger_{i}c_{i} - \tfra
\end{aligned}\]</span></p>
<p>fig. <a href="#fig:DM_DOS">6</a> and fig. <a href="#fig:DM_IPR_scaling">7</a> compare the FK model to the disorder model at different system sizes, matching the defect densities of the disorder model to the FK model at <span class="math inline">\(N = 270\)</span> above and below the CDW transition. We find very good, even quantitative, agreement between the FK and disorder models, which suggests that correlations in the spin sector do not play a significant role.</p>
<figure>
<img src="/assets/thesis/fk_chapter/disorder_model/DM_DOS.svg" id="fig:DM_DOS" data-short-caption="FK model compared to binary disorder model: DOS" style="width:100.0%" alt="Figure 6: 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 \rho for the largest corresponding FK model. As in fig. 2, the Energy resolved DOS(\omega) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters." />
<img src="/assets/thesis/fk_chapter/disorder_model/DM_DOS.svg" id="fig-DM_DOS" data-short-caption="FK model compared to binary disorder model: DOS" style="width:100.0%" alt="Figure 6: 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 \rho for the largest corresponding FK model. As in fig. 2, the Energy resolved DOS(\omega) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters." />
<figcaption aria-hidden="true">Figure 6: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 &lt; \rho &lt; 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:DOS">2</a>, the Energy resolved DOS(<span class="math inline">\(\omega\)</span>) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters.</figcaption>
</figure>
<p>As we can sample directly from the disorder model, rather than through MCMC, the samples are uncorrelated. Hence we can evaluate much larger system sizes with the disorder model which enables us to pin down the correct localisation effects. In particular, what appear to be delocalized states for small system sizes eventually turn out to be states with large localization length. The localization length diverges towards the ordered zero temperature CDW state. The interplay of interactions, which here produce as peculiar binary potential, and localization can be very intricate and the added advantage of a 1D model is that we can explore very large system sizes.</p>
<figure>
<img src="/assets/thesis/fk_chapter/disorder_model/DM_IPR_scaling.svg" id="fig:DM_IPR_scaling" data-short-caption="FK model compared to binary disorder model: IPR Scaling" style="width:100.0%" alt="Figure 7: 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 \rho for the largest corresponding FK model. As in fig. 3 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation  [1], 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" />
<img src="/assets/thesis/fk_chapter/disorder_model/DM_IPR_scaling.svg" id="fig-DM_IPR_scaling" data-short-caption="FK model compared to binary disorder model: IPR Scaling" style="width:100.0%" alt="Figure 7: 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 \rho for the largest corresponding FK model. As in fig. 3 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation  [1], 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">Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 &lt; \rho &lt; 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:IPR_scaling">3</a> <span class="math inline">\(\tau(\omega)\)</span> the scaling of IPR(<span class="math inline">\(\omega\)</span>) with system size, , is shown both in gap (<span class="math inline">\(\omega_0\)</span>) and in the band (<span class="math inline">\(\omega_1\)</span>). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>, hence all the states are indeed localised as one would expect in 1D. The disorder model <span 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 class="math inline">\(0.01\pm0.04, -0.01\pm0.04\)</span> (c) <span class="math inline">\(0.05\pm0.06, 0.04\pm0.06\)</span> (d) <span class="math inline">\(-0.03\pm0.06, 0.01\pm0.06\)</span>. The lines are fit on system sizes <span class="math inline">\(N &gt; 400\)</span></figcaption>
</figure>
</section>

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<section id="gauge-degeneracy-and-the-euler-equation" class="level3">
<h3>Gauge Degeneracy and the Euler Equation</h3>
<figure>
<img src="/assets/thesis/amk_chapter/intro/state_decomposition_animated/state_decomposition_animated.gif" id="fig:state_decomposition_animated" data-short-caption="State Decomposition" style="width:100.0%" alt="Figure 1: (Bond Sector) A state in the bond sector is specified by assigning \pm 1 to each edge of the lattice. However, this description has a substantial gauge degeneracy. We can simplify things by decomposing each state into the product of three kinds of objects: (Vortex Sector) Only a small number of bonds need to be flipped (compared to some arbitrary reference) to reconstruct the vortex sector. Here, the edges are chosen from a spanning tree of the dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of the bond sector can be factored out. This gives a network of loops that can always be written as a product of the gauge operators D_j. (Topological Sector) Finally, there are two loops that have no effect on the vortex sector, nor can they be constructed from gauge symmetries. These can be thought of as two fluxes \Phi_{x/y} that thread through the major and minor axes of the torus. Measuring \Phi_{x/y} amounts to constructing Wilson loops around the axes of the torus. We can flip the value of \Phi_{x} by transporting a vortex pair around the torus in the y direction, as shown here. In each of the three figures on the right, black bonds correspond to those that must be flipped. Composing the three together gives back the original bond sector on the left." />
<img src="/assets/thesis/amk_chapter/intro/state_decomposition_animated/state_decomposition_animated.gif" id="fig-state_decomposition_animated" data-short-caption="State Decomposition" style="width:100.0%" alt="Figure 1: (Bond Sector) A state in the bond sector is specified by assigning \pm 1 to each edge of the lattice. However, this description has a substantial gauge degeneracy. We can simplify things by decomposing each state into the product of three kinds of objects: (Vortex Sector) Only a small number of bonds need to be flipped (compared to some arbitrary reference) to reconstruct the vortex sector. Here, the edges are chosen from a spanning tree of the dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of the bond sector can be factored out. This gives a network of loops that can always be written as a product of the gauge operators D_j. (Topological Sector) Finally, there are two loops that have no effect on the vortex sector, nor can they be constructed from gauge symmetries. These can be thought of as two fluxes \Phi_{x/y} that thread through the major and minor axes of the torus. Measuring \Phi_{x/y} amounts to constructing Wilson loops around the axes of the torus. We can flip the value of \Phi_{x} by transporting a vortex pair around the torus in the y direction, as shown here. In each of the three figures on the right, black bonds correspond to those that must be flipped. Composing the three together gives back the original bond sector on the left." />
<figcaption aria-hidden="true">Figure 1: (Bond Sector) A state in the bond sector is specified by assigning <span class="math inline">\(\pm 1\)</span> to each edge of the lattice. However, this description has a substantial gauge degeneracy. We can simplify things by decomposing each state into the product of three kinds of objects: (Vortex Sector) Only a small number of bonds need to be flipped (compared to some arbitrary reference) to reconstruct the vortex sector. Here, the edges are chosen from a spanning tree of the dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of the bond sector can be factored out. This gives a network of loops that can always be written as a product of the gauge operators <span class="math inline">\(D_j\)</span>. (Topological Sector) Finally, there are two loops that have no effect on the vortex sector, nor can they be constructed from gauge symmetries. These can be thought of as two fluxes <span class="math inline">\(\Phi_{x/y}\)</span> that thread through the major and minor axes of the torus. Measuring <span class="math inline">\(\Phi_{x/y}\)</span> amounts to constructing Wilson loops around the axes of the torus. We can flip the value of <span class="math inline">\(\Phi_{x}\)</span> by transporting a vortex pair around the torus in the <span class="math inline">\(y\)</span> direction, as shown here. In each of the three figures on the right, black bonds correspond to those that must be flipped. Composing the three together gives back the original bond sector on the left.</figcaption>
</figure>
<p>We can check this analysis with a counting argument. For a lattice with <span class="math inline">\(B\)</span> bonds, <span class="math inline">\(P\)</span> plaquettes and <span class="math inline">\(V\)</span> vertices, we can count the number of bond sectors, vortices sectors and gauge symmetries and check them against Euler’s polyhedra equation.</p>
@ -100,7 +101,7 @@ image:
<p>We can also consider the sum of the number of bonds in each plaquette <span class="math inline">\(S_p\)</span>, since each bond is a member of exactly two plaquettes <span class="math display">\[S_p = 2B = 6N\]</span></p>
<p>The mean size of a plaquette in a trivalent lattice on the torus is exactly six. As the sum is even, this also tells us that all odd plaquettes must come in pairs.</p>
<figure>
<img src="/assets/thesis/amk_chapter/intro/flood_fill_amorphous/flood_fill_amorphous.gif" id="fig:flood_fill_amorphous" data-short-caption="Gauge Operators on Amorphous Lattices" style="width:100.0%" alt="Figure 2: The same as fig. ¿fig:flood_fill? but for the amorphous lattice." />
<img src="/assets/thesis/amk_chapter/intro/flood_fill_amorphous/flood_fill_amorphous.gif" id="fig-flood_fill_amorphous" data-short-caption="Gauge Operators on Amorphous Lattices" style="width:100.0%" alt="Figure 2: The same as fig. ¿fig:flood_fill? but for the amorphous lattice." />
<figcaption aria-hidden="true">Figure 2: The same as fig. <strong>¿fig:flood_fill?</strong> but for the amorphous lattice.</figcaption>
</figure>
</section>
@ -150,7 +151,7 @@ image:
<p>This happens because we have broken the time reversal symmetry of the original model by adding odd plaquettes <span class="citation" data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"> [<a href="#ref-Chua2011" role="doc-biblioref">3</a>–<a href="#ref-WangHaoranPRB2021" role="doc-biblioref">10</a>]</span>.</p>
<p>Similarly to the behaviour of the original Kitaev model in response to a magnetic field, we get two degenerate ground states of different handedness. Practically speaking, one ground state is related to the other by inverting the imaginary <span class="math inline">\(\phi\)</span> fluxes <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">4</a>]</span>.</p>
<figure>
<img src="/assets/thesis/amk_chapter/majorana_bound_states/majorana_bound_states.svg" id="fig:majorana_bound_states" data-short-caption="Majorana Bound States" style="width:100.0%" alt="Figure 3: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object." />
<img src="/assets/thesis/amk_chapter/majorana_bound_states/majorana_bound_states.svg" id="fig-majorana_bound_states" data-short-caption="Majorana Bound States" style="width:100.0%" alt="Figure 3: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object." />
<figcaption aria-hidden="true">Figure 3: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object.</figcaption>
</figure>
<p>Next Section: <a href="../4_Amorphous_Kitaev_Model/4.1_AMK_Model.html">The Model</a></p>

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<p>This was a joint project of Gino, Peru and myself with advice and guidance from Willian and Johannes. The project grew out of an interest the three of us had in studying amorphous systems, coupled with Johannes’ expertise on the Kitaev model. The idea to use voronoi partitions came from <span class="citation" data-cites="marsalTopologicalWeaireThorpe2020"> [<a href="#ref-marsalTopologicalWeaireThorpe2020" role="doc-biblioref">3</a>]</span> and Gino did the implementation of this. The idea and implementation of the edge colouring using SAT solvers, the mapping from flux sector to bond sector using A* search were both entirely my work. Peru found the ground state and implemented the local markers. Gino and I did much of the rest of the programming for Koala while pair programming and ’whiteboard’ing, this included the phase diagram, edge mode and finite temperature analyses as well as the derivation of the projector in the amorphous case.</p>
<section id="amk-Model" class="level1">
<h1>The Model</h1>
<p><img src="/assets/thesis/amk_chapter/intro/amk_zoom/amk_zoom.svg" id="fig:amk_zoom" data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%" alt="(a) The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each (b). We represent the antisymmetric gauge degree of freedom u_{jk} = \pm 1 with arrows that point in the direction u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \mathbb{Z}_2 gauge field u_{ij}. This leavies a single Majorana c_i per site." /> <img src="/assets/thesis/amk_chapter/intro/regular_plaquettes/regular_plaquettes.svg" id="fig:regular_plaquettes" data-short-caption="Plaquettes in the Kitaev Model" style="width:86.0%" alt="The eigenvalues of a loop or plaquette operators depend on the number of bonds in its enclosing path." /></p>
<p><img src="/assets/thesis/amk_chapter/intro/amk_zoom/amk_zoom.svg" id="fig-amk_zoom" data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%" alt="(a) The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each (b). We represent the antisymmetric gauge degree of freedom u_{jk} = \pm 1 with arrows that point in the direction u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \mathbb{Z}_2 gauge field u_{ij}. This leavies a single Majorana c_i per site." /> <img src="/assets/thesis/amk_chapter/intro/regular_plaquettes/regular_plaquettes.svg" id="fig-regular_plaquettes" data-short-caption="Plaquettes in the Kitaev Model" style="width:86.0%" alt="The eigenvalues of a loop or plaquette operators depend on the number of bonds in its enclosing path." /></p>
<section id="amorphous-systems" class="level2">
<h2>Amorphous Systems</h2>
<p><strong>Insert discussion of why a generalisation to the amorphous case is interesting</strong></p>
@ -87,7 +88,7 @@ image:
<p>Care must be taken when defining open boundary conditions. Simply removing bonds from the lattice leaves behind unpaired <span class="math inline">\(b^\alpha\)</span> operators that must be paired in some way to arrive at fermionic modes. To fix a pairing, we always start from a lattice defined on the torus and generate a lattice with open boundary conditions by defining the bond coupling <span class="math inline">\(J^{\alpha}_{ij} = 0\)</span> for sites joined by bonds <span class="math inline">\((i,j)\)</span> that we want to remove. This creates fermionic zero modes <span class="math inline">\(u_{ij}\)</span> associated with these cut bonds which we set to 1 when calculating the projector.</p>
<p>Alternatively, since all the fermionic zero modes are degenerate anyway, an arbitrary pairing of the unpaired <span class="math inline">\(b^\alpha\)</span> operators could be performed.</p>
<figure>
<img src="/assets/thesis/amk_chapter/loops_and_dual_loops/loops_and_dual_loops.svg" id="fig:loops_and_dual_loops" data-short-caption="Topological Loops and Dual Loops" style="width:100.0%" alt="Figure 1: (Left) The two topological flux operators of the toroidal lattice. These do not correspond to any face of the lattice, but rather measure flux that threads through the major and minor axes of the torus. This shows a particular choice. Yet, any loop that crosses the boundary is gauge equivalent to one of or the sum of these two loop. (Right) The two ways to transport vortices around the diameters. These amount to creating a vortex pair, transporting one of them around the major or minor diameters of the torus and, then, annihilating them again." />
<img src="/assets/thesis/amk_chapter/loops_and_dual_loops/loops_and_dual_loops.svg" id="fig-loops_and_dual_loops" data-short-caption="Topological Loops and Dual Loops" style="width:100.0%" alt="Figure 1: (Left) The two topological flux operators of the toroidal lattice. These do not correspond to any face of the lattice, but rather measure flux that threads through the major and minor axes of the torus. This shows a particular choice. Yet, any loop that crosses the boundary is gauge equivalent to one of or the sum of these two loop. (Right) The two ways to transport vortices around the diameters. These amount to creating a vortex pair, transporting one of them around the major or minor diameters of the torus and, then, annihilating them again." />
<figcaption aria-hidden="true">Figure 1: (Left) The two topological flux operators of the toroidal lattice. These do not correspond to any face of the lattice, but rather measure flux that threads through the major and minor axes of the torus. This shows a particular choice. Yet, any loop that crosses the boundary is gauge equivalent to one of or the sum of these two loop. (Right) The two ways to transport vortices around the diameters. These amount to creating a vortex pair, transporting one of them around the major or minor diameters of the torus and, then, annihilating them again.</figcaption>
</figure>
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>

11
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<p>Ideally, we would sample uniformly from the space of possible trivalent graphs. Indeed, there has been some work on how to do this using a Markov Chain Monte Carlo approach <span class="citation" data-cites="alyamiUniformSamplingDirected2016"> [<a href="#ref-alyamiUniformSamplingDirected2016" role="doc-biblioref">5</a>]</span>. However, it does not guarantee that the resulting graph is planar, which we must ensure so that the edges can be 3-coloured.</p>
<p>In practice, we use a standard algorithm <span class="citation" data-cites="barberQuickhullAlgorithmConvex1996"> [<a href="#ref-barberQuickhullAlgorithmConvex1996" role="doc-biblioref">6</a>]</span> from Scipy <span class="citation" data-cites="virtanenSciPyFundamentalAlgorithms2020"> [<a href="#ref-virtanenSciPyFundamentalAlgorithms2020" role="doc-biblioref">7</a>]</span> which computes the Voronoi partition of the plane. To compute the Voronoi partition of the torus, we take the seed points and replicate them into a repeating grid. This will be either 3x3 or, for very small numbers of seed points, 5x5. Then, we identify edges in the output to construct a lattice on the torus.</p>
<figure>
<img src="/assets/thesis/amk_chapter/lattice_construction_animated/lattice_construction_animated.gif" id="fig:lattice_construction_animated" data-short-caption="Lattice Construction" style="width:100.0%" alt="Figure 1: (Left) Lattice construction begins with the Voronoi partition of the plane with respect to a set of seed points (black points) sampled uniformly from \mathbb{R}^2. (Center) However, we want the Voronoi partition of the torus, so we tile the seed points into a three by three grid. The boundaries of each tile are shown in light grey. (Right) Finally, we identify edges corresponding to each other across the boundaries to produce a graph on the torus. An edge colouring is shown here to help the reader identify corresponding edges." />
<img src="/assets/thesis/amk_chapter/lattice_construction_animated/lattice_construction_animated.gif" id="fig-lattice_construction_animated" data-short-caption="Lattice Construction" style="width:100.0%" alt="Figure 1: (Left) Lattice construction begins with the Voronoi partition of the plane with respect to a set of seed points (black points) sampled uniformly from \mathbb{R}^2. (Center) However, we want the Voronoi partition of the torus, so we tile the seed points into a three by three grid. The boundaries of each tile are shown in light grey. (Right) Finally, we identify edges corresponding to each other across the boundaries to produce a graph on the torus. An edge colouring is shown here to help the reader identify corresponding edges." />
<figcaption aria-hidden="true">Figure 1: (Left) Lattice construction begins with the Voronoi partition of the plane with respect to a set of seed points (black points) sampled uniformly from <span class="math inline">\(\mathbb{R}^2\)</span>. (Center) However, we want the Voronoi partition of the torus, so we tile the seed points into a three by three grid. The boundaries of each tile are shown in light grey. (Right) Finally, we identify edges corresponding to each other across the boundaries to produce a graph on the torus. An edge colouring is shown here to help the reader identify corresponding edges.</figcaption>
</figure>
</section>
@ -102,7 +103,7 @@ image:
<p>Finally, we need to encode the topology of the graph. This is necessary because, if we are simply given an edge <span class="math inline">\((i, j)\)</span> we do not know how the edge gets from vertex i to vertex j. One method would be taking the shortest path, but it could also ‘go the long way around’ by crossing one of the cuts. To encode this information, we store an additional vector <span class="math inline">\(\vec{r}\)</span> associated with each edge. <span class="math inline">\(r_i^x = 0\)</span> means that edge i does not cross the x. <span class="math inline">\(r_i^x = +1\)</span> (<span class="math inline">\(-1\)</span>) means it crossed the cut in a positive (negative) sense.</p>
<p>This description of the lattice has a very nice relationship to Bloch’s theorem. Applying Bloch’s theorem to a periodic lattice essentially means wrappping the unit cell onto a torus. Variations that happen at longer length scales than the size of the unit cell are captured by the crystal momentum. The crystal momentum inserts a phase factor <span class="math inline">\(e^{i \vec{q}\cdot\vec{r}}\)</span> onto bonds that cross to adjacent unit cells. The vector <span class="math inline">\(\vec{r}\)</span> is exactly what we use to encode the topology of our lattices.</p>
<figure>
<img src="/assets/thesis/amk_chapter/methods/bloch.png" id="fig:bloch" data-short-caption="Bloch&#39;s Theorem and the Torus" style="width:100.0%" alt="Figure 2: Bloch’s theorem can be thought of as transforming from a periodic Hamiltonian on the place to the unit cell defined an torus. In addition we get some phase factors e^{i\vec{k}\cdot\vec{r}} associated with bonds that cross unit cells that depend on the sense in which they do so \vec{r} = (\pm1, \pm1). Representing graphs on the torus turns out to require a similar idea, we unwrap the torus to one unit cell and keep track of which bonds cross the cell boundaries." />
<img src="/assets/thesis/amk_chapter/methods/bloch.png" id="fig-bloch" data-short-caption="Bloch&#39;s Theorem and the Torus" style="width:100.0%" alt="Figure 2: Bloch’s theorem can be thought of as transforming from a periodic Hamiltonian on the place to the unit cell defined an torus. In addition we get some phase factors e^{i\vec{k}\cdot\vec{r}} associated with bonds that cross unit cells that depend on the sense in which they do so \vec{r} = (\pm1, \pm1). Representing graphs on the torus turns out to require a similar idea, we unwrap the torus to one unit cell and keep track of which bonds cross the cell boundaries." />
<figcaption aria-hidden="true">Figure 2: Bloch’s theorem can be thought of as transforming from a periodic Hamiltonian on the place to the unit cell defined an torus. In addition we get some phase factors <span class="math inline">\(e^{i\vec{k}\cdot\vec{r}}\)</span> associated with bonds that cross unit cells that depend on the sense in which they do so <span class="math inline">\(\vec{r} = (\pm1, \pm1)\)</span>. Representing graphs on the torus turns out to require a similar idea, we unwrap the torus to one unit cell and keep track of which bonds cross the cell boundaries.</figcaption>
</figure>
</section>
@ -115,7 +116,7 @@ image:
<p><span class="math inline">\(\Delta + 1\)</span> colours are enough to edge-colour any graph. An <span class="math inline">\(\mathcal{O}(mn)\)</span> algorithm exists to do it for a graph with <span class="math inline">\(m\)</span> edges and <span class="math inline">\(n\)</span> vertices <span class="citation" data-cites="gEstimateChromaticClass1964"> [<a href="#ref-gEstimateChromaticClass1964" role="doc-biblioref">10</a>]</span>. Restricting ourselves to graphs with <span class="math inline">\(\Delta = 3\)</span> like ours, those can be four-edge-coloured in linear time <span class="citation" data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"> [<a href="#ref-skulrattanakulchai4edgecoloringGraphsMaximum2002" role="doc-biblioref">11</a>]</span>.</p>
<p>However, three-edge-colouring them is more difficult. Cubic, planar, bridgeless graphs can be three-edge-coloured if and only if they can be four-face-coloured <span class="citation" data-cites="tait1880remarks"> [<a href="#ref-tait1880remarks" role="doc-biblioref">12</a>]</span>. An <span class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists here <span class="citation" data-cites="robertson1996efficiently"> [<a href="#ref-robertson1996efficiently" role="doc-biblioref">13</a>]</span>. However, it is not clear whether this extends to cubic, <strong>toroidal</strong> bridgeless graphs.</p>
<figure>
<img src="/assets/thesis/amk_chapter/multiple_colourings/multiple_colourings.svg" id="fig:multiple_colourings" data-short-caption="Colourings of an Amorphous Lattice" style="width:100.0%" alt="Figure 3: Three different valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted." />
<img src="/assets/thesis/amk_chapter/multiple_colourings/multiple_colourings.svg" id="fig-multiple_colourings" data-short-caption="Colourings of an Amorphous Lattice" style="width:100.0%" alt="Figure 3: Three different valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted." />
<figcaption aria-hidden="true">Figure 3: Three different valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted.</figcaption>
</figure>
<section id="four-colourings-and-three-colourings" class="level3">
@ -156,7 +157,7 @@ image:
<p>When translating problems to CNF form, there is often some flexibility. For instance, we used three boolean variables to encode the colour of each edge and, then, additional constraints to require that only one of these variables be true. An alternative method which we did not try would be to encode the label of each edge using two variables, yielding four states per edge, and then add a constraint that one of the states, say (true, true) is disallowed. This would, however, have added some complexity to the encoding of the constraint that no adjacent edges can have the same colour.</p>
<p>The popular <em>Networkx</em> Python library uses a greedy graph colouring algorithm. It simply iterates over the vertices/edges/faces of a graph and assigns them a colour that is not already disallowed. This does not work for our purposes because it is not designed to look for a particular n-colouring. However, it does include the option of using a heuristic function that determine the order in which vertices will be coloured <span class="citation" data-cites="kosowski2004classical matulaSmallestlastOrderingClustering1983"> [<a href="#ref-kosowski2004classical" role="doc-biblioref">19</a>,<a href="#ref-matulaSmallestlastOrderingClustering1983" role="doc-biblioref">20</a>]</span>. Perhaps</p>
<figure>
<img src="/assets/thesis/amk_chapter/methods/times/times.svg" id="fig:times" data-short-caption="Computation Time Spent on Different Procedures." style="width:100.0%" alt="Figure 4: The proportion of computation time taken up by the four longest running steps when generating a lattice. For larger systems, the time taken to perform the diagonalisation dominates." />
<img src="/assets/thesis/amk_chapter/methods/times/times.svg" id="fig-times" data-short-caption="Computation Time Spent on Different Procedures." style="width:100.0%" alt="Figure 4: The proportion of computation time taken up by the four longest running steps when generating a lattice. For larger systems, the time taken to perform the diagonalisation dominates." />
<figcaption aria-hidden="true">Figure 4: The proportion of computation time taken up by the four longest running steps when generating a lattice. For larger systems, the time taken to perform the diagonalisation dominates.</figcaption>
</figure>
</section>
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<li><p>Compute paths along the dual lattice between each pair of plaquettes. Flipping the corresponding set of bonds transports one flux to the other and annihilates them.</p></li>
</ol>
<figure>
<img src="/assets/thesis/amk_chapter/flux_finding/flux_finding.svg" id="fig:flux_finding" data-short-caption="Finding Bond Sectors from Flux Sectors" style="width:100.0%" alt="Figure 5: (Left) The ground state flux sector and bond sector for an amorphous lattice. Bond arrows indicate the direction in which u_{jk} = +1. Plaquettes are coloured blue when \hat{\phi}_i = -1 (-i) for even/odd plaquettes and orange when \hat{\phi}_i = +1 (+i) for even/odd plaquettes. (Centre) To transform this to the target flux sector (all +1/+i), we first flip any u_{jk} that are between two fluxes. This leaves a set of isolated fluxes that must be annihilated. Then, these are paired up as indicated by the black lines. (Right) A* search is used to find paths (coloured plaquettes) on the dual lattice between each pair of fluxes and the corresponding u_{jk} (shown in black) are flipped. One flux will remain because the starting and target flux sectors differed by an odd number of fluxes." />
<img src="/assets/thesis/amk_chapter/flux_finding/flux_finding.svg" id="fig-flux_finding" data-short-caption="Finding Bond Sectors from Flux Sectors" style="width:100.0%" alt="Figure 5: (Left) The ground state flux sector and bond sector for an amorphous lattice. Bond arrows indicate the direction in which u_{jk} = +1. Plaquettes are coloured blue when \hat{\phi}_i = -1 (-i) for even/odd plaquettes and orange when \hat{\phi}_i = +1 (+i) for even/odd plaquettes. (Centre) To transform this to the target flux sector (all +1/+i), we first flip any u_{jk} that are between two fluxes. This leaves a set of isolated fluxes that must be annihilated. Then, these are paired up as indicated by the black lines. (Right) A* search is used to find paths (coloured plaquettes) on the dual lattice between each pair of fluxes and the corresponding u_{jk} (shown in black) are flipped. One flux will remain because the starting and target flux sectors differed by an odd number of fluxes." />
<figcaption aria-hidden="true">Figure 5: (Left) The ground state flux sector and bond sector for an amorphous lattice. Bond arrows indicate the direction in which <span class="math inline">\(u_{jk} = +1\)</span>. Plaquettes are coloured blue when <span class="math inline">\(\hat{\phi}_i = -1\)</span> (<span class="math inline">\(-i\)</span>) for even/odd plaquettes and orange when <span class="math inline">\(\hat{\phi}_i = +1\)</span> (<span class="math inline">\(+i\)</span>) for even/odd plaquettes. (Centre) To transform this to the target flux sector (all <span class="math inline">\(+1\)</span>/<span class="math inline">\(+i\)</span>), we first flip any <span class="math inline">\(u_{jk}\)</span> that are between two fluxes. This leaves a set of isolated fluxes that must be annihilated. Then, these are paired up as indicated by the black lines. (Right) A* search is used to find paths (coloured plaquettes) on the dual lattice between each pair of fluxes and the corresponding <span class="math inline">\(u_{jk}\)</span> (shown in black) are flipped. One flux will remain because the starting and target flux sectors differed by an odd number of fluxes.</figcaption>
</figure>
</section>

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@ -113,7 +114,7 @@ image:
<p>Interestingly, the gap closing exists in only one of the four topological sectors, though this is certainly a finite size effect as the sectors must become degenerate in the thermodynamic limit. Nevertheless this could be a useful way to define the (0, 0) topological flux sector for the amorphous model.</p>
<p>In the honeycomb model, the phase boundaries are located on the straight lines <span class="math inline">\(|J^x| = |J^y| + |J^x|\)</span> and permutations of <span class="math inline">\(x,y,z\)</span>, shown as dotted line on ~fig. <a href="#fig:phase_diagram">1</a> (Right). We find that on the amorphous lattice these boundaries exhibit an inward curvature, similar to honeycomb Kitaev models with flux <span class="citation" data-cites="Nasu_Thermal_2015"> [<a href="#ref-Nasu_Thermal_2015" role="doc-biblioref">5</a>]</span> or bond <span class="citation" data-cites="knolle_dynamics_2016"> [<a href="#ref-knolle_dynamics_2016" role="doc-biblioref">6</a>]</span> disorder.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/phase_diagram/phase_diagram.svg" id="fig:phase_diagram" data-short-caption="The Ground State Phase Diagram" style="width:100.0%" alt="Figure 1: (Center) We choose an energy scale for the Hamiltonian by setting J_x + J_y + J_z = 1. This intersects a plane with the unit cube spanned by J_\alpha \in [0,1], giving a triangle with corners (1,0,0), (0,1,0), (0,0,1). To compute critical lines efficiently in this space we evaluate the order parameter of interest along rays shooting from the corners. The ray highlighted in red defines the values of J used for the left figure. (Left) The fermion gap as a function of J for an amorphous system with 20 plaquettes, where the x axis is the position on the red line in the central figure from 0 to 1. For finite size systems the four topological sectors are not degenerate and only one of them has a true gap closing. (Right) The Abelian A_\alpha phases of the model and the non-Abelian B phase separated by critical lines where the fermion gap closes. Later we will show that the Chern number \nu changes from 0 to \pm 1 from the A phases to the B phase. Indeed the gap must close in order for the Chern number to change citation." />
<img src="/assets/thesis/amk_chapter/results/phase_diagram/phase_diagram.svg" id="fig-phase_diagram" data-short-caption="The Ground State Phase Diagram" style="width:100.0%" alt="Figure 1: (Center) We choose an energy scale for the Hamiltonian by setting J_x + J_y + J_z = 1. This intersects a plane with the unit cube spanned by J_\alpha \in [0,1], giving a triangle with corners (1,0,0), (0,1,0), (0,0,1). To compute critical lines efficiently in this space we evaluate the order parameter of interest along rays shooting from the corners. The ray highlighted in red defines the values of J used for the left figure. (Left) The fermion gap as a function of J for an amorphous system with 20 plaquettes, where the x axis is the position on the red line in the central figure from 0 to 1. For finite size systems the four topological sectors are not degenerate and only one of them has a true gap closing. (Right) The Abelian A_\alpha phases of the model and the non-Abelian B phase separated by critical lines where the fermion gap closes. Later we will show that the Chern number \nu changes from 0 to \pm 1 from the A phases to the B phase. Indeed the gap must close in order for the Chern number to change citation." />
<figcaption aria-hidden="true">Figure 1: (Center) We choose an energy scale for the Hamiltonian by setting <span class="math inline">\(J_x + J_y + J_z = 1\)</span>. This intersects a plane with the unit cube spanned by <span class="math inline">\(J_\alpha \in [0,1]\)</span>, giving a triangle with corners <span class="math inline">\((1,0,0), (0,1,0), (0,0,1)\)</span>. To compute critical lines efficiently in this space we evaluate the order parameter of interest along rays shooting from the corners. The ray highlighted in red defines the values of J used for the left figure. (Left) The fermion gap as a function of J for an amorphous system with 20 plaquettes, where the x axis is the position on the red line in the central figure from 0 to 1. For finite size systems the four topological sectors are not degenerate and only one of them has a true gap closing. (Right) The Abelian <span class="math inline">\(A_\alpha\)</span> phases of the model and the non-Abelian B phase separated by critical lines where the fermion gap closes. Later we will show that the Chern number <span class="math inline">\(\nu\)</span> changes from <span class="math inline">\(0\)</span> to <span class="math inline">\(\pm 1\)</span> from the A phases to the B phase. Indeed the gap <em>must</em> close in order for the Chern number to change <strong>citation</strong>.</figcaption>
</figure>
<section id="is-it-abelian-or-non-abelian" class="level3">
@ -133,7 +134,7 @@ image:
<p>In the A phase of the amorphous model we find that <span class="math inline">\(\nu=0\)</span> and hence the excitations have Abelian character, similar to the honeycomb model. This phase is thus the amorphous analogue of the Abelian toric-code quantum spin liquid <span class="citation" data-cites="kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">14</a>]</span>.</p>
<p>The B phase has <span class="math inline">\(\nu=\pm1\)</span> so is a non-Abelian <em>chiral spin liquid</em> (CSL) similar to that of the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>]</span>. The CSL state is the the magnetic analogue of the fractional quantum Hall state <strong>[cite]</strong>. Hereafter we focus our attention on this phase.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/phase_diagram_chern/phase_diagram_chern.svg" id="fig:phase_diagram_chern" data-short-caption="Local Chern Markers" style="width:100.0%" alt="Figure 2: (Center) The crosshair marker  [13], a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = \pm 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian." />
<img src="/assets/thesis/amk_chapter/results/phase_diagram_chern/phase_diagram_chern.svg" id="fig-phase_diagram_chern" data-short-caption="Local Chern Markers" style="width:100.0%" alt="Figure 2: (Center) The crosshair marker  [13], a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = \pm 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian." />
<figcaption aria-hidden="true">Figure 2: (Center) The crosshair marker <span class="citation" data-cites="peru_preprint"> [<a href="#ref-peru_preprint" role="doc-biblioref">13</a>]</span>, a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information 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 <span class="math inline">\(J_\alpha = 1\)</span> in red has <span class="math inline">\(\nu = \pm 1\)</span> implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has <span class="math inline">\(\nu = \pm 0\)</span> implying it has Abelian statistics. (Right) Extending this analysis to the whole <span class="math inline">\(J_\alpha\)</span> phase diagram with fixed <span class="math inline">\(r = 0.3\)</span> nicely confirms that the isotropic phase is non-Abelian.</figcaption>
</figure>
</section>
@ -142,7 +143,7 @@ image:
<p>Chiral Spin Liquids support topological protected edge modes on open boundary conditions <span class="citation" data-cites="qi_general_2006"> [<a href="#ref-qi_general_2006" role="doc-biblioref">15</a>]</span>. Fig. <a 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 boundary of the system. While the model is gapped in periodic boundary conditions (i.e on the torus) these edge modes appear in the gap when the boundary is cut.</p>
<p>The localization of the edge modes can be quantified by their inverse participation ratio (IPR), <span class="math display">\[\mathrm{IPR} = \int d^2r|\psi(\mathbf{r})|^4 \propto L^{-\tau},\]</span> where <span class="math inline">\(L\sim\sqrt{N}\)</span> is the linear dimension of the amorphous lattices and <span class="math inline">\(\tau\)</span> the dimensional scaling exponent of IPR. This is relevant because localised in-gap states do not participate in transport and hence do not turn band insulators into metals. It is only when the gap fills with extended states that we get a metallic state.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/edge_modes/edge_modes.svg" id="fig:edge_modes" data-short-caption="Edges States and Density of States" style="width:100.0%" alt="Figure 3: (a) The density of one of the topologically protected edge states in the B phase. (Below) the log density plotted along the black path showing that the state is exponentially localised. (a)/(b) The density of states of the corresponding lattice in (a) periodic boundary conditions, (b) open boundary conditions. The colour of the bars shows the mean log IPR for each energy window. Cutting the boundary fills the gap with localised states." />
<img src="/assets/thesis/amk_chapter/results/edge_modes/edge_modes.svg" id="fig-edge_modes" data-short-caption="Edges States and Density of States" style="width:100.0%" alt="Figure 3: (a) The density of one of the topologically protected edge states in the B phase. (Below) the log density plotted along the black path showing that the state is exponentially localised. (a)/(b) The density of states of the corresponding lattice in (a) periodic boundary conditions, (b) open boundary conditions. The colour of the bars shows the mean log IPR for each energy window. Cutting the boundary fills the gap with localised states." />
<figcaption aria-hidden="true">Figure 3: (a) The density of one of the topologically protected edge states in the B phase. (Below) the log density plotted along the black path showing that the state is exponentially localised. (a)/(b) The density of states of the corresponding lattice in (a) periodic boundary conditions, (b) open boundary conditions. The colour of the bars shows the mean log IPR for each energy window. Cutting the boundary fills the gap with localised states.</figcaption>
</figure>
</section>
@ -155,18 +156,18 @@ image:
<p>We simply give each plaquette probability <span class="math inline">\(\rho\)</span> of being a vortex, possibly with one additional adjustment to preserve overall vortex parity. This approximation is exact in the limits <span class="math inline">\(T = 0\)</span> (corresponding to <span class="math inline">\(\rho = 0\)</span>) and <span class="math inline">\(T \to \infty\)</span> (corresponding to <span class="math inline">\(\rho = 0.5\)</span>) while at intermediate temperatures there may be vortex-vortex correlations that are not captured by positioning vortices using uncorrelated random variables.</p>
<p>First we performed a finite size scaling to that the presence of a gap in the CSL ground state and absence of a gap in the thermal phase are both robust as we go to larger systems, see fig. <a href="#fig:fermion_gap_vs_L">4</a>.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/fermion_gap_vs_L/fermion_gap_vs_L.svg" id="fig:fermion_gap_vs_L" data-short-caption="Finite Size Scaing of the Fermion Gap" style="width:100.0%" alt="Figure 4: Within a flux sector, the fermion gap \Delta_f measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size L is defined \sqrt{N} where N is the number of plaquettes in the system. Error bars shown are 3 times the standard error of the mean. The lines shown are fits of \tfrac{\Delta_f}{J} = aL ^ b with fit parameters: Ground State: a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004 Random Flux Sector: a = 1.8 \pm 0.2, b = -2.21 \pm 0.03" />
<img src="/assets/thesis/amk_chapter/results/fermion_gap_vs_L/fermion_gap_vs_L.svg" id="fig-fermion_gap_vs_L" data-short-caption="Finite Size Scaing of the Fermion Gap" style="width:100.0%" alt="Figure 4: Within a flux sector, the fermion gap \Delta_f measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size L is defined \sqrt{N} where N is the number of plaquettes in the system. Error bars shown are 3 times the standard error of the mean. The lines shown are fits of \tfrac{\Delta_f}{J} = aL ^ b with fit parameters: Ground State: a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004 Random Flux Sector: a = 1.8 \pm 0.2, b = -2.21 \pm 0.03" />
<figcaption aria-hidden="true">Figure 4: Within a flux sector, the fermion gap <span class="math inline">\(\Delta_f\)</span> measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size <span class="math inline">\(L\)</span> is defined <span class="math inline">\(\sqrt{N}\)</span> where N is the number of plaquettes in the system. Error bars shown are <span class="math inline">\(3\)</span> times the standard error of the mean. The lines shown are fits of <span class="math inline">\(\tfrac{\Delta_f}{J} = aL ^ b\)</span> with fit parameters: Ground State: <span class="math inline">\(a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004\)</span> Random Flux Sector: <span class="math inline">\(a = 1.8 \pm 0.2, b = -2.21 \pm 0.03\)</span></figcaption>
</figure>
<p>Next we evaluated the fermionic density of states (DOS), Inverse Participation Ratio (IPR) and IPR scaling exponent <span class="math inline">\(\tau\)</span> as functions of the vortex density <span class="math inline">\(\rho\)</span>, see fig. <a href="#fig:DOS_vs_rho">5</a>. This leads to a nice picture of what happens as we raise the temperature of the system away from the gapped, insulating CSL phase. At small <span class="math inline">\(\rho\)</span>, states begin to populate the gap but they have <span class="math inline">\(\tau\approx0\)</span>, indicating that they are localised states pinned to the vortices, and the system remains insulating. At large <span class="math inline">\(\rho\)</span>, the in-gap states merge with the bulk band and become extensive, closing the gap, and the system transitions to the thermal metal phase.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/DOS_vs_rho/DOS_vs_rho.svg" id="fig:DOS_vs_rho" data-short-caption="Transition to a Thermal Metal" style="width:100.0%" alt="Figure 5: (Top) Density of states and (Bottom) scaling exponent \tau of the amorphous Kitaev model as a vortex density \rho is increased. The scaling exponent \tau is the exponent with which the inverse participation ratio scales with system size. It gives a measure of the degree of localisation of the states in each (E/J, \rho) bin. At zero \rho we have the gapped ground state. At small \rho, states begin to populate the gap. These states have \tau\approx0, indicating that they are localised states pinned to fluxes, and the system remains insulating. As \rho increases further, the in-gap states merge with the bulk band and become extensive, fully closing the gap, and the system transitions to a thermal metal phase." />
<img src="/assets/thesis/amk_chapter/results/DOS_vs_rho/DOS_vs_rho.svg" id="fig-DOS_vs_rho" data-short-caption="Transition to a Thermal Metal" style="width:100.0%" alt="Figure 5: (Top) Density of states and (Bottom) scaling exponent \tau of the amorphous Kitaev model as a vortex density \rho is increased. The scaling exponent \tau is the exponent with which the inverse participation ratio scales with system size. It gives a measure of the degree of localisation of the states in each (E/J, \rho) bin. At zero \rho we have the gapped ground state. At small \rho, states begin to populate the gap. These states have \tau\approx0, indicating that they are localised states pinned to fluxes, and the system remains insulating. As \rho increases further, the in-gap states merge with the bulk band and become extensive, fully closing the gap, and the system transitions to a thermal metal phase." />
<figcaption aria-hidden="true">Figure 5: (Top) Density of states and (Bottom) scaling exponent <span class="math inline">\(\tau\)</span> of the amorphous Kitaev model as a vortex density <span class="math inline">\(\rho\)</span> is increased. The scaling exponent <span class="math inline">\(\tau\)</span> is the exponent with which the inverse participation ratio scales with system size. It gives a measure of the degree of localisation of the states in each <span class="math inline">\((E/J, \rho)\)</span> bin. At zero <span class="math inline">\(\rho\)</span> we have the gapped ground state. At small <span class="math inline">\(\rho\)</span>, states begin to populate the gap. These states have <span class="math inline">\(\tau\approx0\)</span>, indicating that they are localised states pinned to fluxes, and the system remains insulating. As <span class="math inline">\(\rho\)</span> increases further, the in-gap states merge with the bulk band and become extensive, fully closing the gap, and the system transitions to a thermal metal phase.</figcaption>
</figure>
<p>The thermal metal phase has a signature logarithmic divergence at zero energy and oscillations in the DOS. These signatures can be shown to occur by a recursive argument that involves mapping the original model onto a Majorana model with interactions that take random signs which can itself be mapped onto a coarser lattice with lower energy excitations and so on. This can be repeating indefinitely, showing the model must have excitations at arbitrarily low energies in the thermodynamic limit <span class="citation" data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000" role="doc-biblioref">21</a>]</span>.</p>
<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 honeycomb model unless the chiral symmetry is broken by a magnetic field.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/DOS_oscillations/DOS_oscillations.svg" id="fig:DOS_oscillations" data-short-caption="Distinctive Oscillations in the Density of States" style="width:100.0%" alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state  [16,21]. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is \rho = 0.5 corresponding to the T = \infty limit." />
<img src="/assets/thesis/amk_chapter/results/DOS_oscillations/DOS_oscillations.svg" id="fig-DOS_oscillations" data-short-caption="Distinctive Oscillations in the Density of States" style="width:100.0%" alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state  [16,21]. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is \rho = 0.5 corresponding to the T = \infty limit." />
<figcaption aria-hidden="true">Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state <span class="citation" data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000" role="doc-biblioref">21</a>]</span>. (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 <span class="math inline">\(\rho = 0.5\)</span> corresponding to the <span class="math inline">\(T = \infty\)</span> limit.</figcaption>
</figure>
</section>

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@ -205,7 +206,7 @@ P(S)q(S \to S&#39;)\mathcal{A}(S \to S&#39;) = P(S&#39;)q(S&#39; \to S)\mathcal{
<section id="app-mcmc-autocorrelation" class="level3">
<h3>Auto-correlation Time</h3>
<figure>
<img src="/assets/thesis/fk_chapter/lsr/figs/m_autocorr.png" id="fig:m_autocorr" data-short-caption="Autocorrelation in MCMC" style="width:100.0%" alt="Figure 1: (Upper) 10 MCMC chains starting from the same initial state for a system with N = 150 sites and 3000 MCMC steps. At each MCMC step, n spins are flipped where n is drawn from Uniform(1,N) and this is repeated N^2/100 times. The simulations therefore have the potential to necessitate 10*N^2 matrix diagonalisations for each 100 MCMC steps. (Lower) The normalised auto-correlation (\langle m_i m_{i-j}\rangle - \langle m_i\rangle \langle m_i \rangle) / Var(m_i)) averaged over i. It can be seen that even with each MCMC step already being composed of many individual flip attempts, the auto-correlation is still non negligible and must be taken into account in the statistics. t = 1, \alpha = 1.25, T = 2.2, J = U = 5" />
<img src="/assets/thesis/fk_chapter/lsr/figs/m_autocorr.png" id="fig-m_autocorr" data-short-caption="Autocorrelation in MCMC" style="width:100.0%" alt="Figure 1: (Upper) 10 MCMC chains starting from the same initial state for a system with N = 150 sites and 3000 MCMC steps. At each MCMC step, n spins are flipped where n is drawn from Uniform(1,N) and this is repeated N^2/100 times. The simulations therefore have the potential to necessitate 10*N^2 matrix diagonalisations for each 100 MCMC steps. (Lower) The normalised auto-correlation (\langle m_i m_{i-j}\rangle - \langle m_i\rangle \langle m_i \rangle) / Var(m_i)) averaged over i. It can be seen that even with each MCMC step already being composed of many individual flip attempts, the auto-correlation is still non negligible and must be taken into account in the statistics. t = 1, \alpha = 1.25, T = 2.2, J = U = 5" />
<figcaption aria-hidden="true">Figure 1: (Upper) 10 MCMC chains starting from the same initial state for a system with <span class="math inline">\(N = 150\)</span> sites and 3000 MCMC steps. At each MCMC step, n spins are flipped where n is drawn from Uniform(1,N) and this is repeated <span class="math inline">\(N^2/100\)</span> times. The simulations therefore have the potential to necessitate <span class="math inline">\(10*N^2\)</span> matrix diagonalisations for each 100 MCMC steps. (Lower) The normalised auto-correlation <span class="math inline">\((\langle m_i m_{i-j}\rangle - \langle m_i\rangle \langle m_i \rangle) / Var(m_i))\)</span> averaged over <span class="math inline">\(i\)</span>. It can be seen that even with each MCMC step already being composed of many individual flip attempts, the auto-correlation is still non negligible and must be taken into account in the statistics. <span class="math inline">\(t = 1, \alpha = 1.25, T = 2.2, J = U = 5\)</span></figcaption>
</figure>
<p>At this stage one might think we’re done. We can indeed draw independent samples from our target Boltzmann distribution by starting from some arbitrary initial state and doing <span class="math inline">\(k\)</span> steps to arrive at a sample. These are not, however, independent samples. In fig. <a href="#fig:m_autocorr">1</a> it is already clear that the samples of the order parameter <span class="math inline">\(m\)</span> have some auto-correlation because only a few spins are flipped each step. Even when the number of spins flipped per step is increased that it can be an important effect near the phase transition. Let’s define the auto-correlation time <span class="math inline">\(\tau(O)\)</span> informally as the number of MCMC samples of some observable O that are statistically equal to one independent sample or equivalently as the number of MCMC steps after which the samples are correlated below some cut-off, see <span class="citation" data-cites="krauthIntroductionMonteCarlo1996"> [<a href="#ref-krauthIntroductionMonteCarlo1996" role="doc-biblioref">9</a>]</span>. The auto-correlation time is generally shorter than the convergence time so it therefore makes sense from an efficiency standpoint to run a single walker for many MCMC steps rather than to run a huge ensemble for <span class="math inline">\(k\)</span> steps each.</p>
@ -217,7 +218,7 @@ P(S)q(S \to S&#39;)\mathcal{A}(S \to S&#39;) = P(S&#39;)q(S&#39; \to S)\mathcal{
<section id="tuning-the-proposal-distribution" class="level3">
<h3>Tuning the proposal distribution</h3>
<figure>
<img src="/assets/thesis/fk_chapter/lsr/figs/autocorr_multiple_proposals.png" id="fig:autocorr_multiple_proposals" data-short-caption="Comparison of different proposal distributions" style="width:100.0%" alt="Figure 2: Simulations showing how the autocorrelation of the order parameter depends on the proposal distribution used at different temperatures, we see that at T = 1.5 &lt; T_c a single spin flip is likely the best choice, while at the high temperature T = 2.5 &gt; T_c flipping two sites or a mixture of flipping two and 1 sites is likely a better choice. $t = 1, = 1.25, J = U = 5 $" />
<img src="/assets/thesis/fk_chapter/lsr/figs/autocorr_multiple_proposals.png" id="fig-autocorr_multiple_proposals" data-short-caption="Comparison of different proposal distributions" style="width:100.0%" alt="Figure 2: Simulations showing how the autocorrelation of the order parameter depends on the proposal distribution used at different temperatures, we see that at T = 1.5 &lt; T_c a single spin flip is likely the best choice, while at the high temperature T = 2.5 &gt; T_c flipping two sites or a mixture of flipping two and 1 sites is likely a better choice. $t = 1, = 1.25, J = U = 5 $" />
<figcaption aria-hidden="true">Figure 2: Simulations showing how the autocorrelation of the order parameter depends on the proposal distribution used at different temperatures, we see that at <span class="math inline">\(T = 1.5 &lt; T_c\)</span> a single spin flip is likely the best choice, while at the high temperature <span class="math inline">\(T = 2.5 &gt; T_c\)</span> flipping two sites or a mixture of flipping two and 1 sites is likely a better choice. $t = 1, = 1.25, J = U = 5 $</figcaption>
</figure>
<p>Now we can discuss how to minimise the auto-correlations. The general principle is that one must balance the proposal distribution between two extremes. Choose overlay small steps, like flipping only a single spin and the acceptance rate will be high because <span class="math inline">\(\Delta F\)</span> will usually be small, but each state will be very similar to the previous and the auto-correlations will be high too, making sampling inefficient. On the other hand, overlay large steps, like randomising a large portion of the spins each step, will result in very frequent rejections, especially at low temperatures.</p>

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<h1>The Projector</h1>
<p>The projection from the extended space to the physical space will not be particularly important for the results presented here. However, the theory remains useful to explain why this is.</p>
<figure>
<img src="/assets/thesis/amk_chapter/hilbert_spaces.svg" id="fig:hilbert_spaces" data-short-caption="How the different Hilbert Spaces relate to one another" style="width:100.0%" alt="Figure 1: The relationship between the different Hilbert spaces used in the solution. needs updating" />
<img src="/assets/thesis/amk_chapter/hilbert_spaces.svg" id="fig-hilbert_spaces" data-short-caption="How the different Hilbert Spaces relate to one another" style="width:100.0%" alt="Figure 1: The relationship between the different Hilbert spaces used in the solution. needs updating" />
<figcaption aria-hidden="true">Figure 1: The relationship between the different Hilbert spaces used in the solution. <strong>needs updating</strong></figcaption>
</figure>
<p>The physical states are defined as those for which <span class="math inline">\(D_i |\phi\rangle = |\phi\rangle\)</span> for all <span class="math inline">\(D_i\)</span>. Since <span class="math inline">\(D_i\)</span> has eigenvalues <span class="math inline">\(\pm1\)</span>, the quantity <span class="math inline">\(\tfrac{(1+D_i)}{2}\)</span> has eigenvalue <span class="math inline">\(1\)</span> for physical states and <span class="math inline">\(0\)</span> for extended states so is the local projector onto the physical subspace.</p>

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