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lots of LRFK figures!
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title: "The one-dimensional Long-Range Falikov-Kimball Model: Thermal Phase Transition and Disorder-Free Localisation"
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collection: publications
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permalink: /publication/2021-03-22-the-long-range-falikov-kimball-model
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permalink: /publication/2021-03-22-the-long-range-falicov-kimball-model
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excerpt: 'Disorder or interactions can turn metals into insulators. One of the simplest settings to study this physics is given by the Falikov-Kimball model, which describes itinerant fermions interacting with a classical Ising background field.'
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date: 2021-03-22
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venue: 'Physics Review B'
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H_{\mathrm{FK}} = & -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}). \\
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\end{aligned}\]</span></p>
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<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">\(T > 0\)</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>
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<p>In chapter <a href="../3_Long_Range_Falikov_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>
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<p>In chapter <a href="#chap:3-the-long-range-falicov-kimball-model">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>
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</section>
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<section id="quantum-spin-liquids" class="level1">
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<h1>Quantum Spin Liquids</h1>
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@ -118,9 +118,9 @@ H_{\mathrm{FK}} = & -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U
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<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>
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<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>
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<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>
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<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 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>
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<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>
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<p>In chapter <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">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>
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<p>The next chapter, Chapter <a href="../2_Background/2.1_FK_Model.html">2</a>, will introduce some necessary background to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and localisation. Then chapter <a href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3</a> introduces and studies the Long Range Falikov-Kimball Model in one dimension while chapter <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">4</a> focusses on the Amorphous Kitaev Model.</p>
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<p>The next chapter, Chapter <a href="../2_Background/2.1_FK_Model.html">2</a>, will introduce some necessary background to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and localisation. Then chapter <a href="#chap:3-the-long-range-falicov-kimball-model">3</a> introduces and studies the Long Range Falikov-Kimball Model in one dimension while chapter <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">4</a> focusses on the Amorphous Kitaev Model.</p>
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<p>Next Chapter: <a href="../2_Background/2.1_FK_Model.html">2 Background</a></p>
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</section>
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<section id="bibliography" class="level1 unnumbered">
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@ -99,7 +99,7 @@ H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\
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<p>In the disordered region above <span class="math inline">\(T_c(U)\)</span> there are two insulating phases. For weak interactions <span class="math inline">\(U << t\)</span>, thermal fluctuations in the spins act as an effective disorder potential for the fermions, causing them to localise and giving rise to an Anderson insulating state <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">16</a>]</span> which we will discuss more in section <a href="../2_Background/2.4_Disorder.html#bg-disorder-and-localisation">2.3</a>. For strong interactions <span class="math inline">\(U >> t\)</span>, the spins are not ordered but nevertheless their interaction with the electrons opens a gap, leading a Mott insulator analogous to that of the Hubbard model <span class="citation" data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a href="#ref-brandtThermodynamicsCorrelationFunctions1989" role="doc-biblioref">17</a>]</span>.</p>
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<p>By contrast, in the one dimensional FK model there is no finite-temperature phase transition (FTPT) to an ordered CDW phase <span class="citation" data-cites="liebAbsenceMottTransition1968"> [<a href="#ref-liebAbsenceMottTransition1968" role="doc-biblioref">18</a>]</span>. Indeed dimensionality is crucial for the physics of both localisation and FTPTs. In one dimension, disorder generally dominates: even the weakest disorder exponentially localises <em>all</em> single particle eigenstates. Only longer-range correlations of the disorder potential can potentially induce localisation-delocalisation transitions in one dimension <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">19</a>–<a href="#ref-dunlapAbsenceLocalizationRandomdimer1990" role="doc-biblioref">21</a>]</span>. Thermodynamically, short-range interactions cannot overcome thermal defects in one dimension which prevents ordered phases at non-zero temperature <span class="citation" data-cites="goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">22</a>–<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">24</a>]</span>.</p>
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<p>However, the absence of an FTPT in the short ranged FK chain is far from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the fermions <span class="citation" data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">25</a>–<a href="#ref-yosidaMagneticPropertiesCuMn1957" role="doc-biblioref">28</a>]</span> decays as <span class="math inline">\(r^{-1}\)</span> in one dimension <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">29</a>]</span>. This could in principle induce the necessary long-range interactions for the classical Ising background to order at low temperatures <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">30</a>,<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">31</a>]</span>. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at <span class="math inline">\(T = 0\)</span> for the one dimensional FK model <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">32</a>]</span>.</p>
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<p>Based on this primacy of dimensionality, we will go digging into the one dimensional case. In chapter <a href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3</a> we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long Range Ising Model which is the subject of the next section.</p>
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<p>Based on this primacy of dimensionality, we will go digging into the one dimensional case. In chapter <a href="#chap:3-the-long-range-falicov-kimball-model">3</a> we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long Range Ising Model which is the subject of the next section.</p>
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</section>
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<section id="long-ranged-ising-model" class="level2">
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<h2>Long Ranged Ising model</h2>
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@ -36,7 +36,6 @@ image:
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<li><a href="#an-emergent-gauge-field" id="toc-an-emergent-gauge-field">An Emergent Gauge Field</a></li>
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<li><a href="#anyons-topology-and-the-chern-number" id="toc-anyons-topology-and-the-chern-number">Anyons, Topology and the Chern number</a></li>
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<li><a href="#ground-state-phases" id="toc-ground-state-phases">Ground State Phases</a></li>
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<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
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</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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</ul>
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<li><a href="#an-emergent-gauge-field" id="toc-an-emergent-gauge-field">An Emergent Gauge Field</a></li>
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<li><a href="#anyons-topology-and-the-chern-number" id="toc-anyons-topology-and-the-chern-number">Anyons, Topology and the Chern number</a></li>
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<li><a href="#ground-state-phases" id="toc-ground-state-phases">Ground State Phases</a></li>
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<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
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</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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</ul>
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</section>
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<section id="the-spin-model" class="level2">
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<h2>The Spin Model</h2>
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<figure>
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<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." />
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<figcaption aria-hidden="true">Figure 2: A visual introduction to the Kitaev Model.</figcaption>
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</figure>
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<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>
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<p>In the spin hamiltonian eq. <a href="#eq:bg-kh-model">1</a> we can already tease out a set of conserved fluxes that will be key to the model’s solution. These fluxes are the expectations of Wilson loop operators</p>
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<p><span class="math display">\[\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha},\]</span></p>
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<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">2</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>
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<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>
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<figure>
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<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 2: 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 2: 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>
|
||||
<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>
|
||||
<p>As with the Falikov-Kimball model, the KH model has a extensive number of conserved quantities, the fluxes. As with the FK model it will make sense to work in the simultaneous eigenbasis of the fluxes and the Hamiltonian so that we can treat the fluxes like a classical degree of freedom. This is part of what makes the model tractable. We will find that the ground state of the model corresponds to some particular choice of fluxes. We will refer to local excitations away from the flux ground state as <strong>vortices</strong>. In order to fully solve the model however, we must first move to a Majorana picture.</p>
|
||||
</section>
|
||||
<section id="the-majorana-model" class="level2">
|
||||
<h2>The Majorana 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 3: A visual introduction to the Kitaev Model." />
|
||||
<figcaption aria-hidden="true">Figure 3: A visual introduction to the Kitaev Model.</figcaption>
|
||||
</figure>
|
||||
<p>Majorana fermions are something like ‘half of a complex fermion’ and are their own antiparticle. From a set of <span class="math inline">\(N\)</span> fermionic creation <span class="math inline">\(f_i^\dagger\)</span> and anhilation <span class="math inline">\(f_i\)</span> operators we can construct <span class="math inline">\(2N\)</span> Majorana operators <span class="math inline">\(c_m\)</span>. We can do this construction in multiple ways subject to only mild constraints required to keep the overall commutations relations correct <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. Majorana operators square to one but otherwise have standard fermionic commutation relations.</p>
|
||||
<p><span class="math inline">\(N\)</span> spins can be mapped to <span class="math inline">\(N\)</span> fermions with the well known Jordan-Wigner transformation and indeed this approach can be used to solve the Kitaev model <span class="citation" data-cites="chenExactResultsKitaev2008"> [<a href="#ref-chenExactResultsKitaev2008" role="doc-biblioref">15</a>]</span>. Here I will introduce the method Kitaev used in the original paper as this forms the basis for the results that will be presented in this thesis. Rather than mapping to <span class="math inline">\(N\)</span> fermions, Kitaev maps to <span class="math inline">\(4N\)</span> Majoranas, effectively <span class="math inline">\(2N\)</span> fermions. In contrast to the Jordan-Wigner approach which makes fermions out of strings of spin operators in order to correctly produce fermionic commutation relations, the Kitaev transformation maps each spin locally to four Majoranas. The downside is that this enlarges the Hilbert space from <span class="math inline">\(2^N\)</span> to <span class="math inline">\(4^N\)</span>. We will have to employ a projector <span class="math inline">\(\hat{P}\)</span> to come back down to the physical Hilbert space later. As everything is local, I will drop the site indices <span class="math inline">\(ijk\)</span> in expressions that refer to only a single site.</p>
|
||||
<p>The mapping is defined in terms of four Majoranas per site <span class="math inline">\(b_i^x,\;b_i^y,\;b_i^z,\;c_i\)</span> such that</p>
|
||||
<p><span id="eq:bg-kh-mapping"><span class="math display">\[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^z = i b^z c\qquad{(2)}\]</span></span></p>
|
||||
<p>The tildes on the spin operators <span class="math inline">\(\tilde{\sigma_i^\alpha}\)</span> emphasis that they live in this new extended Hilbert space and are only equivalent to the original spin operators after applying a projector <span class="math inline">\(\hat{P}\)</span>. The form of the projection operator can be understood in a few ways. From a group-theoretic perspective, before projection, the operators <span class="math inline">\(\{\tilde{\sigma}^x, \tilde{\sigma}^y, \tilde{\sigma}^z\}\)</span> form a representation of the gamma group <span class="math inline">\(G_{3,0}\)</span>. The gamma groups <span class="math inline">\(G_{p,q}\)</span> have <span class="math inline">\(p\)</span> generators that square to the identity and <span class="math inline">\(q\)</span> that square (roughly) to <span class="math inline">\(-1\)</span>. The generators otherwise obey standard anticommutation relations. The well known gamma matrices <span class="math inline">\(\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}\)</span> represent <span class="math inline">\(G_{1,3}\)</span> the quaternions <span class="math inline">\(G_{0,3}\)</span> and the Pauli matrices <span class="math inline">\(G_{3,0}\)</span>.</p>
|
||||
<p>The Pauli matrices, however, have the additional property that the <em>chiral element</em> <span class="math inline">\(\sigma^x \sigma^y \sigma^z = i\)</span>, this relation is not determined by the group properties of <span class="math inline">\(G_{3,0}\)</span>. Therefore to fully reproduce the algebra of the Pauli matrices we must project into the subspace where <span class="math inline">\(\tilde{\sigma}^x \tilde{\sigma}^y \tilde{\sigma}^z = i\)</span>. The chiral element of the gamma matrices for instance <span class="math inline">\(\gamma_5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3\)</span> is of central importance in quantum field theory. See <span class="citation" data-cites="petitjeanChiralityDiracSpinors2020"> [<a href="#ref-petitjeanChiralityDiracSpinors2020" role="doc-biblioref">16</a>]</span> for more discussion of this group theoretic view.</p>
|
||||
<p>The projector must project onto the subspace where <span class="math inline">\(\tilde{\sigma}^x \tilde{\sigma}^y \tilde{\sigma}^z = i\)</span>. If we work this through we find that in general $<sup>x</sup>y^z = iD $ where <span class="math inline">\(D = b^x b^y b^z c\)</span> must be the identity for every site. In other words, we can only work with <em>physical states</em> <span class="math inline">\(|\phi\rangle\)</span> that satisfy $ D_i|= |$ for all sites <span class="math inline">\(i\)</span>. From this we construct an on-site projector <span class="math inline">\(P_i = \frac{1 + D_i}{2}\)</span> and the overall projector is simply <span class="math inline">\(P = \prod_i P_i\)</span>.</p>
|
||||
<p>The projector must project onto the subspace where <span class="math inline">\(\tilde \sigma^x \tilde \sigma^y \tilde \sigma^z = i\)</span>. If we work this through we find that in general $^x ^y ^z = iD $ where <span class="math inline">\(D = b^x b^y b^z c\)</span> must be the identity for every site. In other words, we can only work with <em>physical states</em> <span class="math inline">\(|\phi\rangle\)</span> that satisfy $ D_i|= |$ for all sites <span class="math inline">\(i\)</span>. From this we construct an on-site projector <span class="math inline">\(P_i = \frac{1 + D_i}{2}\)</span> and the overall projector is simply <span class="math inline">\(P = \prod_i P_i\)</span>.</p>
|
||||
<p>Another way to see what this is doing physically is to explicitly construct the two intermediate fermionic operators <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> that give rise to these four Majoranas. Working through the algebra we see that the <span class="math inline">\(D\)</span> operator corresponds to the fermion parity <span class="math inline">\(D = -(2n_f - 1)(2n_g - 1)\)</span> where <span class="math inline">\(n_f,\; n_g\)</span> are the number operators. Expanding the product <span class="math inline">\(\prod_i P_i\)</span> out, we find that the projector corresponds to a symmetrisation over <span class="math inline">\(\{u_{ij}\}\)</span> states within a flux sector and and overall fermion parity <span class="math inline">\(\prod_i D_i\)</span>. This tells us that any arbitrary state can be made to have non-zero overlap with the physical subspace via the addition or removal of a single fermion. This implies that in the thermodynamic limit the projection step is not generally necessary to extract physical results, see <span class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a href="#ref-pedrocchiPhysicalSolutionsKitaev2011" role="doc-biblioref">17</a>]</span> or <a href="../6_Appendices/A.5_The_Projector.html#app-the-projector">appendix A.5</a> for more details.</p>
|
||||
<p>We can now rewrite the spin hamiltonian in Majorana form with caveat that they are only strictly equivalent after projection. The Ising interactions <span class="math inline">\(\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> decouple into the form <span class="math inline">\(-i (i b^\alpha_i b^\alpha_j) c_i c_j\)</span>. We factor out the <em>bond operators</em> <span class="math inline">\(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\)</span> which are Hermitian and, remarkably, commute with the Hamiltonian and each other.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
@ -184,6 +182,8 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}
|
||||
\sum_{(i,j,k)} \sigma_i^{\alpha} \sigma_j^{\beta} \sigma_k^{\gamma}
|
||||
\]</span> where the sum <span class="math inline">\((i,j,k)\)</span> runs over consecutive indices around plaquettes. The addition of this to the spin model leads to two bond terms in the corresponding Majorana model. The effect of breaking chiral symmetry is to open a gap in the B phase. The vortices of the gapped B phase are non-Abelian anyons.</p>
|
||||
<p>At finite temperatures, recent work has shown that the KH model undergoes a transition to a thermal metal phase.</p>
|
||||
<p><strong>Summary</strong></p>
|
||||
<p>We have seen that…</p>
|
||||
<p><strong>Summary</strong> The Kitaev Honeycomb model is remarkable because it combines three key properties. First, the form of the Hamiltonian plausibly be realised by a real material. Candidate materials, such as <span class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have sufficiently strong spin-orbit coupling and the correct lattice structure to behave according to the Kitaev Honeycomb model with small corrections <span class="citation" data-cites="banerjeeProximateKitaevQuantum2016 TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>,<a href="#ref-banerjeeProximateKitaevQuantum2016" role="doc-biblioref">34</a>]</span>. Second, its ground state is the canonical example of the long sought after quantum spin liquid state. Its excitations are anyons, particles that can only exist in two dimensions that break the normal fermion/boson dichotomy.</p>
|
||||
<p>Third, and perhaps most importantly, this model is a rare many body interacting quantum system that can be treated analytically. It is exactly solvable. We can explicitly write down its many body ground states in terms of single particle states <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. The solubility of the Kitaev Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes about because the model has extensively many conserved degrees of freedom. These conserved quantities can be factored out as classical degrees of freedom, leaving behind a non-interacting quantum model that is easy to solve.</p>
|
||||
<p>“dynamical two spin correlation functions are identically zero beyond nearest neighbor separation in the Kitaev Model” <span class="citation" data-cites="baskaranExactResultsSpin2007"> [<a href="#ref-baskaranExactResultsSpin2007" role="doc-biblioref">35</a>]</span></p>
|
||||
@ -192,57 +192,6 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}
|
||||
<li>really follows the Kitaev-Heisenberg model</li>
|
||||
<li>experimental probes include inelastic neutron scattering, Raman scattering</li>
|
||||
</ul>
|
||||
</section>
|
||||
<section id="glossary" class="level2">
|
||||
<h2>Glossary</h2>
|
||||
<ul>
|
||||
<li><p>Lattice: The underlying graph on which the models are defined. Composed of sites (vertices), bonds (edges) and plaquettes (faces).</p></li>
|
||||
<li><p>The model : Used when I refer to properties of the the Kitaev model that do not depend on the particular lattice.</p></li>
|
||||
<li><p>The Honeycomb model : The Kitaev Model defined on the honeycomb lattice.</p></li>
|
||||
<li><p>The Amorphous model : The Kitaev Model defined on the amorphous lattices described here.</p></li>
|
||||
</ul>
|
||||
<p><strong>The Spin Hamiltonian</strong></p>
|
||||
<ul>
|
||||
<li>Spin Bond Operators: <span class="math inline">\(\hat{k}_{ij} = \sigma_i^\alpha \sigma_j^\alpha\)</span></li>
|
||||
<li>Loop Operators: <span class="math inline">\(\hat{W_p} = \prod_{<i,j>} k_{ij}\)</span></li>
|
||||
<li>Plaquette Operators: Loops that enclose a single plaquette.</li>
|
||||
</ul>
|
||||
<p><strong>The Majorana Model</strong></p>
|
||||
<ul>
|
||||
<li>Majorana Operators on site <span class="math inline">\(i\)</span>: <span class="math inline">\(\hat{b}^x_i, \hat{b}^y_i, \hat{b}^z_i, \hat{c}_i\)</span></li>
|
||||
<li>Majorana Bond Operators: <span class="math inline">\(\hat{u}_{ij} = i b_i^\alpha b_j^\alpha\)</span></li>
|
||||
<li>Loop Operators: <span class="math inline">\(\hat{W_p} = \prod_{<i,j>} u_{ij}\)</span></li>
|
||||
<li>Plaquette Operators: Loops that enclose a single plaquette.</li>
|
||||
<li>Gauge Operators: <span class="math inline">\(D_i = \hat{b}^x_i \hat{b}^y_i \hat{b}^z_i \hat{c}_i\)</span></li>
|
||||
<li>The Extended Hilbert space: The larger Hilbert space spanned by the Majorana operators.</li>
|
||||
<li>The physical subspace: The subspace of the extended Hilbert space that we identify with the Hilbert space of the original spin model.</li>
|
||||
<li>The Projector <span class="math inline">\(\hat{P}\)</span>: The projector onto the physical subspace.</li>
|
||||
</ul>
|
||||
<p><strong>Flux Sectors</strong></p>
|
||||
<ul>
|
||||
<li><p>Odd/Even Plaquettes: Plaquettes with an odd/even number of sides.</p></li>
|
||||
<li><p>Fluxes <span class="math inline">\(\phi_i\)</span>: The expectation values of the plaquette operators <span class="math inline">\(\pm 1\)</span> for even and <span class="math inline">\(\pm i\)</span> for odd plaquettes.</p></li>
|
||||
<li><p>Flux Sector: A subspace of Hilbert space in which the fluxes take particular values.</p></li>
|
||||
<li><p>Ground state flux sector: The Flux Sector containing the lowest energy many body state.</p></li>
|
||||
<li><p>Vortices: Flux excitations away from the ground state flux sector.</p></li>
|
||||
<li><p>Dual Loops: A set of <span class="math inline">\(u_{jk}\)</span> that correspond to loops on the dual lattice.</p></li>
|
||||
<li><p>non-contractible loops or dual loops: The two loops topologically distinct loops on the torus that cannot be smoothly deformed to a point.</p></li>
|
||||
<li><p>Topological Fluxes <span class="math inline">\(\Phi_{x}, \Phi_{y}\)</span>: The two fluxes associated with the two non-contractible loops.</p></li>
|
||||
<li><p>Topological Transport Operators: <span class="math inline">\(\mathcal{T}_{x}, \mathcal{T}_{y}\)</span>: The two vortex-pair operations associated with the non-contractible <em>dual</em> loops.</p></li>
|
||||
</ul>
|
||||
<p><strong>Phases</strong></p>
|
||||
<ul>
|
||||
<li>The A phase: The three anisotropic regions of the phase diagram <span class="math inline">\(A_x, A_y, A_z\)</span> where <span class="math inline">\(A_\alpha\)</span> means <span class="math inline">\(J_\alpha >> J_\beta, J_\gamma\)</span>.</li>
|
||||
<li>The B phase: The roughly isotropic region of the phase diagram.</li>
|
||||
</ul>
|
||||
<p><strong>Vortices and their movements</strong></p>
|
||||
<p>See fig. <a href="#fig:types_of_dual_loops_animated">6</a> for a diagram of the next three paragraphs.</p>
|
||||
<p>We started from the ground state of the model and flipped the sign of a single bond (fig. <a href="#fig:types_of_dual_loops_animated">6</a> (a)). In doing so, we will flip the sign of the two plaquettes adjacent to that bond. We will call these disturbed plaquettes <em>vortices</em>. We will refer to a particular choice values for the plaquette operators as a <em>vortex sector</em>.</p>
|
||||
<p>If we chain multiple bond flips, we can create a pair of vortices at arbitrary locations (fig. <a href="#fig:types_of_dual_loops_animated">6</a> (b)). The chain of bonds that we must flip corresponds to a path on the dual of the lattice.</p>
|
||||
<p>We can also create a pair of vortices, move one around a loop and finally annihilate it with its partner (fig. <a href="#fig:types_of_dual_loops_animated">6</a> (c)). This corresponds to a closed loop on the dual lattice. Applying such a bond flip leaves the vortex sector unchanged. We can also do the same thing but move the vortex around one the non-contractible loops of the lattice (fig. <a href="#fig:types_of_dual_loops_animated">6</a> (d)).</p>
|
||||
<p>There is one kind of dual loop that we cannot build out of <span class="math inline">\(D_j\)</span>s, the non-contractible loops.</p>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p></i,j></i,j></p>
|
||||
<p>Next Section: <a href="../2_Background/2.4_Disorder.html">Disorder and Localisation</a></p>
|
||||
</section>
|
||||
</section>
|
||||
|
||||
@ -29,10 +29,8 @@ image:
|
||||
<ul>
|
||||
<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
|
||||
<ul>
|
||||
<li><a href="#localisation-anderson-many-body-and-disorder-free" id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: Anderson, Many Body and Disorder-Free</a></li>
|
||||
<li><a href="#disorder-and-spin-liquids" id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
|
||||
<li><a href="#amorphous-magnetism" id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
|
||||
<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -49,10 +47,8 @@ image:
|
||||
<ul>
|
||||
<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
|
||||
<ul>
|
||||
<li><a href="#localisation-anderson-many-body-and-disorder-free" id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: Anderson, Many Body and Disorder-Free</a></li>
|
||||
<li><a href="#disorder-and-spin-liquids" id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
|
||||
<li><a href="#amorphous-magnetism" id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
|
||||
<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -66,73 +62,127 @@ image:
|
||||
</div>
|
||||
<section id="bg-disorder-and-localisation" class="level1">
|
||||
<h1>Disorder and Localisation</h1>
|
||||
<section id="localisation-anderson-many-body-and-disorder-free" class="level2">
|
||||
<h2>Localisation: Anderson, Many Body and Disorder-Free</h2>
|
||||
</section>
|
||||
<ul>
|
||||
<li><p>disorder starts with the very simple anderson model</p></li>
|
||||
<li><p>Quenched vs Annealed disorder</p></li>
|
||||
<li></li>
|
||||
</ul>
|
||||
<p>Disorder is a fact of life for the condensed matter physicist. No sample will ever be completely free of contamination or of structural defects. The classical Drude theory of electron conductivity envisages electrons as scattering off impurities. Hence we would expect the electrical conductivity to be proportional to the mean free path <span class="citation" data-cites="lagendijkFiftyYearsAnderson2009"> [<a href="#ref-lagendijkFiftyYearsAnderson2009" role="doc-biblioref">1</a>]</span>, decreasing smoothly as the number of defects increases. However, Anderson showed in 1958 <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">2</a>]</span> that at some critical level of disorder <strong>all</strong> single particle eigenstates localise. What would later be known as Anderson localisation is characterised by exponentially localised eigenfunctions <span class="math inline">\(\psi(x) \sim e^{-x/\lambda}\)</span> which cannot contribute to transport processes. The localisation length <span class="math inline">\(\lambda\)</span> is the typical scale of localised state and can be extracted with transmission matrix methods <span class="citation" data-cites="pendrySymmetryTransportWaves1994"> [<a href="#ref-pendrySymmetryTransportWaves1994" role="doc-biblioref">3</a>]</span>. Anderson localisation provided a different kind of insulator to that of the band insulator.</p>
|
||||
<p>The Anderson model is about the simplest model of disorder one could imagine <span id="eq:bg-anderson-model"><span class="math display">\[
|
||||
H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j
|
||||
\qquad{(1)}\]</span></span></p>
|
||||
<p>It is one of non-interacting fermions subject to a disorder potential <span class="math inline">\(V_j\)</span> drawn uniformly from the interval <span class="math inline">\([-W,W]\)</span>. The discovery of localisation in quantum systems was surprising at the time given the seeming ubiquity of extended Bloch states. Within the Anderson model, all the states localise at the same disorder strength <span class="math inline">\(W\)</span> but later Mott showed that in other contexts extended Bloch states and localised states could coexist at the same disorder strength but different energies. The transition in energy between localised and extended states is known as a mobility edge <span class="citation" data-cites="mottMetalInsulatorTransitions1978"> [<a href="#ref-mottMetalInsulatorTransitions1978" role="doc-biblioref">4</a>]</span>.</p>
|
||||
<p>Localisation phenomena are strongly dimension dependent. In three dimensions the scaling theory of localisation <span class="citation" data-cites="edwardsNumericalStudiesLocalization1972 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-edwardsNumericalStudiesLocalization1972" role="doc-biblioref">5</a>,<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span> shows that Anderson localisation is a critical phenomenon with critical exponents both for how the conductivity vanishes with energy when approaching the mobility edge and for how the localisation length increases below it. By contrast, in one dimension disorder generally dominates. Even the weakest disorder exponentially localises <em>all</em> single particle eigenstates in the one dimensional Anderson model. Only long-range spatial correlations of the disorder potential can induce delocalisation <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990 izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">7</a>–<a href="#ref-izrailevAnomalousLocalizationLowDimensional2012" role="doc-biblioref">12</a>]</span>.</p>
|
||||
<p>Later localisation was found in disordered interacting many-body systems:</p>
|
||||
<p><span class="math display">\[
|
||||
H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U\sum_{jk} n_j n_k
|
||||
\]</span> Here, in contrast to the Anderson model, localisation phenomena are robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL) <span class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016"> [<a href="#ref-imbrieManyBodyLocalizationQuantum2016" role="doc-biblioref">13</a>]</span>.</p>
|
||||
<p>Both MBL and Anderson localisation depend crucially on the presence of <em>quenched</em> disorder. Quenched disorder takes the form a static background field drawn from an arbitrary probability distribution to which the model is coupled. Disorder may also be introduced into the initial state of the system rather than the Hamiltonian. This has led to ongoing interest in the possibility of disorder-free localisation where the disorder is instead <em>annealed</em>. In this scenario the disorder necessary to generate localisation is generated entirely from the thermal fluctuations of the model.</p>
|
||||
<p>The concept of disorder-free localisation was first proposed in the context of Helium mixtures <span class="citation" data-cites="kagan1984localization"> [<a href="#ref-kagan1984localization" role="doc-biblioref">14</a>]</span> and then extended to heavy-light mixtures in which multiple species with large mass ratios interact. The idea is that the heavier particles act as an effective disorder potential for the lighter ones, inducing localisation. Two such models <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">15</a>,<a href="#ref-schiulazDynamicsManybodyLocalized2015" role="doc-biblioref">16</a>]</span> instead find that the models thermalise exponentially slowly in system size, which Ref. <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">15</a>]</span> dubs Quasi-MBL.</p>
|
||||
<p>True disorder-free localisation does occur in exactly solvable models with extensively many conserved quantities <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">17</a>]</span>. As conserved quantities have no time dynamics this can be thought of as taking the separation of timescales to the infinite limit. The localisation phenomena present in the Falikov-Kimball model are instead the result of annealed disorder. A strong separation of timescales means that the heavy species is approximated as immobile with respect to the lighter itinerant species. At finite temperature the heavy species acts as a disorder potential for the lighter one. However, in contract to quenched disorder, the probability distribution of annealed disorder is entirely determined by the thermodynamics of the Hamiltonian. In the two dimensional FK model this leads to multiple phases where localisation effects are relevant. At low temperatures the heavy species orders leading to a traditional band gap insulator. At higher temperatures however thermal disorder causes the light species to localise. At weak coupling, the localisation length can be very large, so finite sized systems may still conduct, an effect known as weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">18</a>]</span>.</p>
|
||||
<p>In Chapter 3 we will consider a generalised FK model in one dimension and how the disorder generated near a one dimensional thermodynamic phase transition interacts with localisation physics.</p>
|
||||
<p>So far we have considered disorder as a static or dynamic field coupled to a model defined on a translation invariant lattice. Another kind of disordered system that worthy of study are amorphous systems.</p>
|
||||
<p>Amorphous systems have disordered bond connectivity, so called <em>topological disorder</em>. As discussed in the introduction these include amorphous semiconductors such as amorphous Germanium and Silicon <span class="citation" data-cites="Yonezawa1983 zallen2008physics Weaire1971 betteridge1973possible"> [<a href="#ref-Yonezawa1983" role="doc-biblioref">19</a>–<a href="#ref-betteridge1973possible" role="doc-biblioref">22</a>]</span>. While materials do not have long range lattice structure they can enforce local constraints such as the approximate coordination number <span class="math inline">\(z = 4\)</span> of silicon.</p>
|
||||
<p>Topological disorder can be qualitatively different from other disordered systems. Disordered graphs are constrained by fixed coordination number and the Euler equation. The Harris <span class="citation" data-cites="harrisEffectRandomDefects1974"> [<a href="#ref-harrisEffectRandomDefects1974" role="doc-biblioref">23</a>]</span> and the Imry-Mar <span class="citation" data-cites="imryRandomFieldInstabilityOrdered1975"> [<a href="#ref-imryRandomFieldInstabilityOrdered1975" role="doc-biblioref">24</a>]</span> criteria are key results on the effect of disorder on thermodynamic phase transitions. The Harris criterion signals when disorder will affect the universal of a thermodynamic critical point. It states that for a critical point in a <span class="math inline">\(d\)</span>-dimensional system with correlation length scaling exponent, disorder will be relevant if <span class="math inline">\(\nu\)</span> if <span class="math inline">\(d\nu < 2\)</span>. The Imry-Ma criterion simply forbids the formation of long range ordered states in <span class="math inline">\(d \leq 2\)</span> dimensions in the presence of disorder. The latter criteria is violated in the presence of correlated disorder <span class="citation" data-cites="changlaniChargeDensityWaves2016"> [<a href="#ref-changlaniChargeDensityWaves2016" role="doc-biblioref">25</a>]</span> and both are modified for topological disorder. In chapter 4 we will put the Kitaev model onto two dimensional Voronoi lattices. These lattices are have fixed coordination number <span class="math inline">\(z=3\)</span> and must satisfy the Euler equation for the plane, this leads to strong anticorrelations which mean that topological disorder is effectively weaker than standard disorder here <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">26</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">27</a>]</span>]. This does not apply to the three dimensional Voronoi lattices where the Euler equation is a weaker constraint.</p>
|
||||
<p>Lastly it is worth exploring how quantum spin liquids and disorder interact. The KH model has been studied subject to both bond and site disorder <strong>cite</strong>. In some instances it seems that disorder can even promote the formation of a QSL ground state <span class="citation" data-cites="wenDisorderedRouteCoulomb2017"> [<a href="#ref-wenDisorderedRouteCoulomb2017" role="doc-biblioref">28</a>]</span>.</p>
|
||||
<ul>
|
||||
<li>localisation length and IPR scaling</li>
|
||||
<li>multifractality</li>
|
||||
</ul>
|
||||
<section id="disorder-and-spin-liquids" class="level2">
|
||||
<h2>Disorder and Spin liquids</h2>
|
||||
</section>
|
||||
<section id="amorphous-magnetism" class="level2">
|
||||
<h2>Amorphous Magnetism</h2>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
</section>
|
||||
<section id="localisation" class="level2">
|
||||
<h2>Localisation</h2>
|
||||
<p>The discovery of localisation in quantum systems surprising at the time given the seeming ubiquity of extended Bloch states. Later, when thermalisation in quantum systems gained interest, localisation phenomena again stood out as counterexamples to the eigenstate thermalisation hypothesis <span class="citation" data-cites="abaninRecentProgressManybody2017 srednickiChaosQuantumThermalization1994"> [<a href="#ref-abaninRecentProgressManybody2017" role="doc-biblioref">1</a>,<a href="#ref-srednickiChaosQuantumThermalization1994" role="doc-biblioref">2</a>]</span>, allowing quantum systems to avoid to retain memory of their initial conditions in the face of thermal noise.</p>
|
||||
<p>The simplest and first discovered kind is Anderson localisation, first studied in 1958 <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">3</a>]</span> in the context of non-interacting fermions subject to a static or quenched disorder potential <span class="math inline">\(V_j\)</span> drawn uniformly from the interval <span class="math inline">\([-W,W]\)</span></p>
|
||||
<p><span class="math display">\[
|
||||
H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j
|
||||
\]</span></p>
|
||||
<p>this model exhibits exponentially localised eigenfunctions <span class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\)</span> which cannot contribute to transport processes. Initially it was thought that in one dimensional disordered models, all states would be localised, however it was later shown that in the presence of correlated disorder, bands of extended states can exist <span class="citation" data-cites="izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a href="#ref-izrailevLocalizationMobilityEdge1999" role="doc-biblioref">4</a>–<a href="#ref-izrailevAnomalousLocalizationLowDimensional2012" role="doc-biblioref">6</a>]</span>.</p>
|
||||
<p>Later localisation was found in interacting many-body systems with quenched disorder:</p>
|
||||
<p><span class="math display">\[
|
||||
H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U\sum_{jk} n_j n_k
|
||||
\]</span></p>
|
||||
<p>where the number operators <span class="math inline">\(n_j = c^\dagger_j c_j\)</span>. Here, in contrast to the Anderson model, localisation phenomena can be proven robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL) <span class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016"> [<a href="#ref-imbrieManyBodyLocalizationQuantum2016" role="doc-biblioref">7</a>]</span>.</p>
|
||||
<p>Both MBL and Anderson localisation depend crucially on the presence of quenched disorder. This has led to ongoing interest in the possibility of disorder-free localisation, in which the disorder necessary to generate localisation is generated entirely from the dynamics of the model. This contracts with typical models of disordered systems in which disorder is explicitly introduced into the Hamilton or the initial state.</p>
|
||||
<p>The concept of disorder-free localisation was first proposed in the context of Helium mixtures <span class="citation" data-cites="kagan1984localization"> [<a href="#ref-kagan1984localization" role="doc-biblioref">8</a>]</span> and then extended to heavy-light mixtures in which multiple species with large mass ratios interact. The idea is that the heavier particles act as an effective disorder potential for the lighter ones, inducing localisation. Two such models <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">9</a>,<a href="#ref-schiulazDynamicsManybodyLocalized2015" role="doc-biblioref">10</a>]</span> instead find that the models thermalise exponentially slowly in system size, which Ref. <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">9</a>]</span> dubs Quasi-MBL.</p>
|
||||
<p>True disorder-free localisation does occur in exactly solvable models with extensively many conserved quantities <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">11</a>]</span>. As conserved quantities have no time dynamics this can be thought of as taking the separation of timescales to the infinite limit.</p>
|
||||
<p>-link to the FK model</p>
|
||||
<p>-link to the Kitaev Model</p>
|
||||
<p>-link to the physics of amorphous systems</p>
|
||||
<p>Next Chapter: <a href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falikov-Kimball Model</a></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
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<div class="csl-left-margin">[2] </div><div class="csl-right-inline">M. Srednicki, <em><a href="https://doi.org/10.1103/PhysRevE.50.888">Chaos and Quantum Thermalization</a></em>, Phys. Rev. E <strong>50</strong>, 888 (1994).</div>
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<div class="csl-left-margin">[3] </div><div class="csl-right-inline">P. W. Anderson, <em><a href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492 (1958).</div>
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||||
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||||
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||||
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<div class="csl-left-margin">[7] </div><div class="csl-right-inline">S. Aubry and G. André, <em>Analyticity Breaking and Anderson Localization in Incommensurate Lattices</em>, Proceedings, VIII International Colloquium on Group-Theoretical Methods in Physics <strong>3</strong>, 18 (1980).</div>
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||||
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||||
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<div class="csl-left-margin">[8] </div><div class="csl-right-inline">S. Das Sarma, S. He, and X. C. Xie, <em><a href="https://doi.org/10.1103/PhysRevB.41.5544">Localization, Mobility Edges, and Metal-Insulator Transition in a Class of One-Dimensional Slowly Varying Deterministic Potentials</a></em>, Phys. Rev. B <strong>41</strong>, 5544 (1990).</div>
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||||
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||||
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||||
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<div class="csl-left-margin">[4] </div><div class="csl-right-inline">F. M. Izrailev and A. A. Krokhin, <em><a href="https://doi.org/10.1103/PhysRevLett.82.4062">Localization and the Mobility Edge in One-Dimensional Potentials with Correlated Disorder</a></em>, Phys. Rev. Lett. <strong>82</strong>, 4062 (1999).</div>
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<div class="csl-left-margin">[10] </div><div class="csl-right-inline">F. M. Izrailev and A. A. Krokhin, <em><a href="https://doi.org/10.1103/PhysRevLett.82.4062">Localization and the Mobility Edge in One-Dimensional Potentials with Correlated Disorder</a></em>, Phys. Rev. Lett. <strong>82</strong>, 4062 (1999).</div>
|
||||
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|
||||
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<div class="csl-left-margin">[5] </div><div class="csl-right-inline">A. Croy, P. Cain, and M. Schreiber, <em><a href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B <strong>82</strong>, 107 (2011).</div>
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<div class="csl-left-margin">[11] </div><div class="csl-right-inline">A. Croy, P. Cain, and M. Schreiber, <em><a href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B <strong>82</strong>, 107 (2011).</div>
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||||
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|
||||
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<div class="csl-left-margin">[6] </div><div class="csl-right-inline">F. M. Izrailev, A. A. Krokhin, and N. M. Makarov, <em><a href="https://doi.org/10.1016/j.physrep.2011.11.002">Anomalous Localization in Low-Dimensional Systems with Correlated Disorder</a></em>, Physics Reports <strong>512</strong>, 125 (2012).</div>
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||||
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||||
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<div class="csl-left-margin">[7] </div><div class="csl-right-inline">J. Z. Imbrie, <em><a href="https://doi.org/10.1007/s10955-016-1508-x">On Many-Body Localization for Quantum Spin Chains</a></em>, J Stat Phys <strong>163</strong>, 998 (2016).</div>
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<div class="csl-left-margin">[8] </div><div class="csl-right-inline">Y. Kagan and L. Maksimov, <em>Localization in a System of Interacting Particles Diffusing in a Regular Crystal</em>, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki <strong>87</strong>, 348 (1984).</div>
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<div class="csl-left-margin">[14] </div><div class="csl-right-inline">Y. Kagan and L. Maksimov, <em>Localization in a System of Interacting Particles Diffusing in a Regular Crystal</em>, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki <strong>87</strong>, 348 (1984).</div>
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<div class="csl-left-margin">[11] </div><div class="csl-right-inline">A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, <em><a href="https://doi.org/10.1103/PhysRevLett.118.266601">Disorder-Free Localization</a></em>, Phys. Rev. Lett. <strong>118</strong>, 266601 (2017).</div>
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||||
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, <em><a href="https://doi.org/10.1103/PhysRevLett.118.266601">Disorder-Free Localization</a></em>, Phys. Rev. Lett. <strong>118</strong>, 266601 (2017).</div>
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||||
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||||
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">A. E. Antipov, Y. Javanmard, P. Ribeiro, and S. Kirchner, <em><a href="https://doi.org/10.1103/PhysRevLett.117.146601">Interaction-Tuned Anderson Versus Mott Localization</a></em>, Phys. Rev. Lett. <strong>117</strong>, 146601 (2016).</div>
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||||
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||||
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||||
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">F. Yonezawa and T. Ninomiya, editors, <em>Topological Disorder in Condensed Matter</em>, Vol. 46 (Springer-Verlag, Berlin Heidelberg, 1983).</div>
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||||
</div>
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<div id="ref-zallen2008physics" class="csl-entry" role="doc-biblioentry">
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||||
<div class="csl-left-margin">[20] </div><div class="csl-right-inline">R. Zallen, <em>The Physics of Amorphous Solids</em> (John Wiley & Sons, 2008).</div>
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||||
</div>
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<div class="csl-left-margin">[21] </div><div class="csl-right-inline">D. Weaire and M. F. Thorpe, <em><a href="https://doi.org/10.1103/PhysRevB.4.2508">Electronic Properties of an Amorphous Solid. I. A Simple Tight-Binding Theory</a></em>, Phys. Rev. B <strong>4</strong>, 2508 (1971).</div>
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||||
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<div id="ref-betteridge1973possible" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[22] </div><div class="csl-right-inline">G. Betteridge, <em>A Possible Model of Amorphous Silicon and Germanium</em>, Journal of Physics C: Solid State Physics <strong>6</strong>, L427 (1973).</div>
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||||
</div>
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||||
<div id="ref-harrisEffectRandomDefects1974" class="csl-entry" role="doc-biblioentry">
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||||
<div class="csl-left-margin">[23] </div><div class="csl-right-inline">A. B. Harris, <em><a href="https://doi.org/10.1088/0022-3719/7/9/009">Effect of Random Defects on the Critical Behaviour of Ising Models</a></em>, J. Phys. C: Solid State Phys. <strong>7</strong>, 1671 (1974).</div>
|
||||
</div>
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||||
<div id="ref-imryRandomFieldInstabilityOrdered1975" class="csl-entry" role="doc-biblioentry">
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||||
<div class="csl-left-margin">[24] </div><div class="csl-right-inline">Y. Imry and S. Ma, <em><a href="https://doi.org/10.1103/PhysRevLett.35.1399">Random-Field Instability of the Ordered State of Continuous Symmetry</a></em>, Phys. Rev. Lett. <strong>35</strong>, 1399 (1975).</div>
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||||
</div>
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||||
<div id="ref-changlaniChargeDensityWaves2016" class="csl-entry" role="doc-biblioentry">
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||||
<div class="csl-left-margin">[25] </div><div class="csl-right-inline">H. J. Changlani, N. M. Tubman, and T. L. Hughes, <em><a href="https://doi.org/10.1038/srep31897">Charge Density Waves in Disordered Media Circumventing the Imry-Ma Argument</a></em>, Sci Rep <strong>6</strong>, 1 (2016).</div>
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||||
</div>
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||||
<div id="ref-barghathiPhaseTransitionsRandom2014" class="csl-entry" role="doc-biblioentry">
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||||
<div class="csl-left-margin">[26] </div><div class="csl-right-inline">H. Barghathi and T. Vojta, <em><a href="https://doi.org/10.1103/PhysRevLett.113.120602">Phase Transitions on Random Lattices: How Random Is Topological Disorder?</a></em>, Phys. Rev. Lett. <strong>113</strong>, 120602 (2014).</div>
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||||
</div>
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||||
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<div class="csl-left-margin">[27] </div><div class="csl-right-inline">M. Schrauth, J. Portela, and F. Goth, <em><a href="https://doi.org/10.1103/PhysRevLett.121.100601">Violation of the Harris-Barghathi-Vojta Criterion</a></em>, Physical Review Letters <strong>121</strong>, (2018).</div>
|
||||
</div>
|
||||
<div id="ref-wenDisorderedRouteCoulomb2017" class="csl-entry" role="doc-biblioentry">
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||||
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">J.-J. Wen et al., <em><a href="https://doi.org/10.1103/PhysRevLett.118.107206">Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in ${\Mathrm{Pr}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$</a></em>, Phys. Rev. Lett. <strong>118</strong>, 107206 (2017).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -27,7 +27,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#chap:3-the-long-range-falikov-kimball-model" id="toc-chap:3-the-long-range-falikov-kimball-model">3 The Long Range Falikov-Kimball Model</a></li>
|
||||
<li><a href="#chap:3-the-long-range-falicov-kimball-model" id="toc-chap:3-the-long-range-falicov-kimball-model">3 The Long Range Falikov-Kimball Model</a></li>
|
||||
<li><a href="#fk-model" id="toc-fk-model">The Model</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -42,7 +42,7 @@ image:
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#chap:3-the-long-range-falikov-kimball-model" id="toc-chap:3-the-long-range-falikov-kimball-model">3 The Long Range Falikov-Kimball Model</a></li>
|
||||
<li><a href="#chap:3-the-long-range-falicov-kimball-model" id="toc-chap:3-the-long-range-falicov-kimball-model">3 The Long Range Falikov-Kimball Model</a></li>
|
||||
<li><a href="#fk-model" id="toc-fk-model">The Model</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -54,7 +54,7 @@ image:
|
||||
<p>3 The Long Range Falikov-Kimball Model</p>
|
||||
<hr />
|
||||
</div>
|
||||
<section id="chap:3-the-long-range-falikov-kimball-model" class="level1">
|
||||
<section id="chap:3-the-long-range-falicov-kimball-model" class="level1">
|
||||
<h1>3 The Long Range Falikov-Kimball Model</h1>
|
||||
<p>This chapter expands on work presented in</p>
|
||||
<p> <span class="citation" data-cites="hodsonOnedimensionalLongrangeFalikovKimball2021"> [<a href="#ref-hodsonOnedimensionalLongrangeFalikovKimball2021" role="doc-biblioref">1</a>]</span> <a href="https://link.aps.org/doi/10.1103/PhysRevB.104.045116">One-dimensional long-range Falikov-Kimball model: Thermal phase transition and disorder-free localization</a>, Hodson, T. and Willsher, J. and Knolle, J., Phys. Rev. B, <strong>104</strong>, 4, 2021,</p>
|
||||
|
||||
@ -156,21 +156,13 @@ P(L) \goeslike L^{d*}
|
||||
DOS(\omega) = \sum_n \delta(\omega - \epsilon_n)
|
||||
IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_{n,i}}^4
|
||||
\]</span> Where <span class="math inline">\(\psi_{n,i}\)</span> is the wavefunction corresponding to the energy <span class="math inline">\(\epsilon_n\)</span> at the ith site. In practice we bin the energies and IPRs into a fine energy grid and use Lorentzian smoothing if necessary.</p>
|
||||
<figure>
|
||||
<embed src="figs/lsr/raw_steps_single_flip.pdf" id="fig:raw" />
|
||||
<figcaption aria-hidden="true">Figure 1: An MCMC walk starting from the staggered charge density wave ground state for a system with <span class="math inline">\(N = 100\)</span> sites and 10,000 MCMC steps. 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 this temperature the thermal average of m is zero, while the initial state has m = 1. We see that it takes about 1000 steps for the system to converge, after which it moves about the m = 0 average with a finite auto-correlation time. <span class="math inline">\(t = 1, \alpha = 1.25, T = 3, J = U = 5\)</span> <span id="fig:raw" label="fig:raw">[fig:raw]</span></figcaption>
|
||||
</figure>
|
||||
<!-- ![An MCMC walk starting from the staggered charge density wave ground state for a system with $N = 100$ sites and 10,000 MCMC steps. 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 this temperature the thermal average of m is zero, while the initial state has m = 1. We see that it takes about 1000 steps for the system to converge, after which it moves about the m = 0 average with a finite auto-correlation time. $t = 1, \alpha = 1.25, T = 3, J = U = 5$ [\[fig:raw\]]{#fig:raw label="fig:raw"}](figs/lsr/raw_steps_single_flip.pdf){#fig:raw width="\\columnwidth"} -->
|
||||
<p><span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> sidesteps these issues by defining a random walk that focuses on the states with the greatest Boltzmann weight. At low temperatures this means we need only visit a few low energy states to make good estimates while at high temperatures the weights become uniform so a small number of samples distributed across the state space suffice. However we will see that the method is not without difficulties of its own.</p>
|
||||
<figure>
|
||||
<embed src="figs/lsr/single.pdf" id="fig:single" />
|
||||
<figcaption aria-hidden="true">Figure 2: Two MCMC chains starting from the same initial state for a system with <span class="math inline">\(N = 90\)</span> sites and 1000 MCMC steps. In this simulation the MCMC step is defined differently: an attempt is made to flip n spins, where n is drawn from Uniform(1,N). This is repeated <span class="math inline">\(N^2/100\)</span> times for each step. This trades off computation time for storage space, as it makes the samples less correlated, giving smaller statistical error for a given number of stored samples. These simulations therefore have the potential to necessitate <span class="math inline">\(N^2/100\)</span> matrix diagonalisations for every MCMC sample, though this can be cut down with caching and other tricks. <span class="math inline">\(t = 1, \alpha = 1.25, T = 2.2, J = U = 5\)</span> <span id="fig:single" label="fig:single">[fig:single]</span></figcaption>
|
||||
</figure>
|
||||
<!-- ![Two MCMC chains starting from the same initial state for a system with $N = 90$ sites and 1000 MCMC steps. In this simulation the MCMC step is defined differently: an attempt is made to flip n spins, where n is drawn from Uniform(1,N). This is repeated $N^2/100$ times for each step. This trades off computation time for storage space, as it makes the samples less correlated, giving smaller statistical error for a given number of stored samples. These simulations therefore have the potential to necessitate $N^2/100$ matrix diagonalisations for every MCMC sample, though this can be cut down with caching and other tricks. $t = 1, \alpha = 1.25, T = 2.2, J = U = 5$ [\[fig:single\]]{#fig:single label="fig:single"}](figs/lsr/single.pdf){#fig:single width="\\columnwidth"} -->
|
||||
<p>In implementation <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> can be boiled down to choosing a transition function <span class="math inline">\(\mathcal{T}(\s_{t} \rightarrow \s_t+1)\)</span> where <span class="math inline">\(\s\)</span> are vectors representing classical spin configurations. We start in some initial state <span class="math inline">\(\s_0\)</span> and then repeatedly jump to new states according to the probabilities given by <span class="math inline">\(\mathcal{T}\)</span>. This defines a set of random walks <span class="math inline">\(\{\s_0\ldots \s_i\ldots \s_N\}\)</span>. Fig. <a href="#fig:single" data-reference-type="ref" data-reference="fig:single">2</a> shows this in practice: we have a (rather small) ensemble of <span class="math inline">\(M = 2\)</span> walkers starting at the same point in state space and then spreading outwards by flipping spins along the way.</p>
|
||||
<p>In pseudo-code one could write the MCMC simulation for a single walker as:</p>
|
||||
<div class="markdown">
|
||||
<p>“’python current_state = initial_state</p>
|
||||
<p>for i in range(N_steps): new_state = sample_T(current_state) states[i] = current_state “’</p>
|
||||
</div>
|
||||
<p>```python current_state = initial_state</p>
|
||||
<p>for i in range(N_steps): new_state = sample_T(current_state) states[i] = current_state ```</p>
|
||||
<p>Where the <code>sample_T</code> function here produces a state with probability determined by the <code>current_state</code> and the transition function <span class="math inline">\(\mathcal{T}\)</span>.</p>
|
||||
<p>If we ran many such walkers in parallel we could then approximate the distribution <span class="math inline">\(p_t(\s; \s_0)\)</span> which tells us where the walkers are likely to be after they’ve evolved for <span class="math inline">\(t\)</span> steps from an initial state <span class="math inline">\(\s_0\)</span>. We need to carefully choose <span class="math inline">\(\mathcal{T}\)</span> such that after a large number of steps <span class="math inline">\(k\)</span> (the convergence time) the probability <span class="math inline">\(p_t(\s;\s_0)\)</span> approaches the thermal distribution <span class="math inline">\(P(\s; \beta) = \mathcal{Z}^{-1} e^{-\beta F(\s)}\)</span>. This turns out to be quite easy to achieve using the Metropolis-Hasting algorithm.</p>
|
||||
</section>
|
||||
@ -182,10 +174,7 @@ IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_
|
||||
</section>
|
||||
<section id="auto-correlation-time" class="level2">
|
||||
<h2>Auto-correlation Time</h2>
|
||||
<figure>
|
||||
<img src="figs/lsr/m_autocorr.png" id="fig:m_autocorr" alt="Figure 3: (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 (\expval{m_i m_{i-j}} - \expval{m_i}\expval{m_i}) / 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 [fig:m_autocorr]" />
|
||||
<figcaption aria-hidden="true">Figure 3: (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">\((\expval{m_i m_{i-j}} - \expval{m_i}\expval{m_i}) / Var(m_i))\)</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> <span id="fig:m_autocorr" label="fig:m_autocorr">[fig:m_autocorr]</span></figcaption>
|
||||
</figure>
|
||||
<!-- ![(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 $(\expval{m_i m_{i-j}} - \expval{m_i}\expval{m_i}) / Var(m_i))$ 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. $t = 1, \alpha = 1.25, T = 2.2, J = U = 5$ [\[fig:m_autocorr\]]{#fig:m_autocorr label="fig:m_autocorr"}](figs/lsr/m_autocorr.png){#fig:m_autocorr width="\\columnwidth"} -->
|
||||
<p>At this stage one might think we’re done. We can indeed draw independent samples from <span class="math inline">\(P(\s; \beta)\)</span> by starting from some arbitrary initial state and doing <span class="math inline">\(k\)</span> steps to arrive at a sample. However a key insight is that after the convergence time, every state generated is a sample from <span class="math inline">\(P(\s; \beta)\)</span>! They are not, however, independent samples. In Fig. <a href="#fig:raw" data-reference-type="ref" data-reference="fig:raw">1</a> it is already clear that the samples of the order parameter m have some auto-correlation because only a few spins are flipped each step but even when the number of spins flipped per step is increased, Fig. <a href="#fig:m_autocorr" data-reference-type="ref" data-reference="fig:m_autocorr">3</a> shows 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. [^4] 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>
|
||||
<p>Once the random walk has been carried out for many steps, the expectation values of <span class="math inline">\(O\)</span> can be estimated from the MCMC samples <span class="math inline">\(\s_i\)</span>: <span class="math display">\[\tex{O} = \sum_{i = 0}^{N} O(\s_i) + \mathcal{O}(\frac{1}{\sqrt{N}})\]</span> The the samples are correlated so the N of them effectively contains less information than <span class="math inline">\(N\)</span> independent samples would, in fact roughly <span class="math inline">\(N/\tau\)</span> effective samples. As a consequence the variance is larger than the <span class="math inline">\(\qex{O^2} - \qex{O}^2\)</span> form it would have if the estimates were uncorrelated. There are many methods in the literature for estimating the true variance of <span class="math inline">\(\qex{O}\)</span> and deciding how many steps are needed but my approach has been to run a small number of parallel chains, which are independent, in order to estimate the statistical error produced. This is a slightly less computationally efficient because it requires throwing away those <span class="math inline">\(k\)</span> steps generated before convergence multiple times but it is a conceptually simple workaround.</p>
|
||||
<p>In summary, to do efficient simulations we want to reduce both the convergence time and the auto-correlation time as much as possible. In order to explain how, we need to introduce the Metropolis-Hasting (MH) algorithm and how it gives an explicit form for the transition function.</p>
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: The Long Range Falikov-Kimball Model - Results
|
||||
title: The Long Range Falicov-Kimball Model - Results
|
||||
excerpt:
|
||||
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|
||||
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|
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|
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|
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|
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<title>The Long Range Falikov-Kimball Model - Results</title>
|
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<title>The Long Range Falicov-Kimball Model - Results</title>
|
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|
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|
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</div>
|
||||
<section id="fk-results" class="level1">
|
||||
<h1>Results</h1>
|
||||
<p>Phase diagrams of the long-range 1D FK model. (a) The TJ plane at <span class="math inline">\(U = 5\)</span>: the CDW ordered phase is separated from a disordered Mott insulating (MI) phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (b) 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. (c) 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 <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> phase of the long-range 1D <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> 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> . (d) 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 (a) and (b). 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 varied.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/binder.png" id="fig:binder" data-short-caption="no title" style="width:100.0%" alt="Figure 1: Hello I am the figure caption!" />
|
||||
<figcaption aria-hidden="true">Figure 1: Hello I am the figure caption!</figcaption>
|
||||
</figure>
|
||||
<section id="lrfk-results-phase-diagram" class="level2">
|
||||
<h2>Phase Diagram</h2>
|
||||
<p>Figs. [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>a] and [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>b] show the phase diagram for constant <span class="math inline">\(U=5\)</span> and constant <span class="math inline">\(J=5\)</span>, respectively. We determined the transition temperatures from the crossings of the Binder cumulants <span class="math inline">\(B_4 = \langle m^4 \rangle /\langle m^2 \rangle^2\)</span> <span class="citation" data-cites="binderFiniteSizeScaling1981"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">1</a>]</span>. For a representative set of parameters, Fig. [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>c] shows the order parameter <span class="math inline">\(\rangle m \langle^2\)</span>. Fig. [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>d] shows the Binder cumulants, both as functions of system size and temperature. The crossings confirm that the system has a FTPT and that the ordered phase is not a finite size effect.</p>
|
||||
<p>The CDW transition temperature is largely independent from the strength of the interaction <span class="math inline">\(U\)</span>. This demonstrates that the phase transition is driven by the long-range term <span class="math inline">\(J\)</span> with little effect from the coupling to the fermions <span class="math inline">\(U\)</span>. The physics of the spin sector in our long-range FK model mimics that of the LRI model and is not significantly altered by the presence of the fermions, which shows that the long range tail expected from a basic fermion mediated RKKY interaction between the Ising spins is absent.</p>
|
||||
<p>Our main interest concerns the additional structure of the fermionic sector in the high temperature phase. Following Ref. <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">2</a>]</span>, we can distinguish between the Mott and Anderson insulating phases. The former is characterised by a gapped DOS in the absence of a CDW. Thus, the opening of a gap for large <span class="math inline">\(U\)</span> is distinct from the gap-opening induced by the translational symmetry breaking in the CDW state below <span class="math inline">\(T_c\)</span>, see also Fig. [<a href="#fig:band_opening" data-reference-type="ref" data-reference="fig:band_opening">3</a>a]. The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates.</p>
|
||||
<p>Using the MCMC methods described in the previous section I will now discuss the results of extensive MCMC simulations of the model, starting with the phase diagram in the fermion spin coupling <span class="math inline">\(U\)</span>, the strength of the long range spin-spin coupling <span class="math inline">\(J\)</span> and the temperature <span class="math inline">\(T\)</span>.</p>
|
||||
<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 1: (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. 2 . 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 1: (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. <a href="#fig:phase_diagram">2</a> . 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>Fig fig. <a href="#fig:phase_diagram">2</a> shows the phase diagram for constant <span class="math inline">\(U=5\)</span> and constant <span class="math inline">\(J=5\)</span>, respectively. The transition temperatures were determined from the crossings of the Binder cumulants <span class="math inline">\(B_4 = \langle m^4 \rangle /\langle m^2 \rangle^2\)</span> <span class="citation" data-cites="binderFiniteSizeScaling1981"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">1</a>]</span>. For a representative set of parameters, fig. <a href="#fig:binder_cumulants">1</a> shows the order parameter <span class="math inline">\(\langle m \rangle^2\)</span> and the Binder cumulants, both as functions of system size and temperature. The crossings confirm that the system has a FTPT and that the ordered phase is not a finite size effect.</p>
|
||||
<p>The CDW transition temperature is largely independent from the strength of the interaction <span class="math inline">\(U\)</span>. This demonstrates that the phase transition is driven by the long-range term <span class="math inline">\(J\)</span> with little effect from the coupling to the fermions <span class="math inline">\(U\)</span>. The physics of the spin sector in the long-range FK model mimics that of the long range Ising (LRI) model and is not significantly altered by the presence of the fermions. In two dimensions the transition to the CDW phase is mediated by an RKYY-like interaction <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">2</a>]</span> but this is insufficient to stabilise long range order in one dimension. That the critical temperature <span class="math inline">\(T_c\)</span> does not depend on <span class="math inline">\(U\)</span> in our model further confirms this.</p>
|
||||
<p>The main order parameters for this model is the staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> that signals the onset of a charge density wave (CDW) phase at low temperature. However, my main interest concerns the additional structure of the fermionic sector in the high temperature phase. Following Ref. <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">3</a>]</span>, we can distinguish between the Mott and Anderson insulating phases. The Mott insulator is characterised by a gapped DOS in the absence of a CDW, instead the gap is driven entirely by interaction effect. Thus, the opening of a gap for large <span class="math inline">\(U\)</span> is distinct from the gap-opening induced by the translational symmetry breaking in the CDW state below <span class="math inline">\(T_c\)</span>, see also fig. <strong>¿fig:gap_opening?</strong>. The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates hence it also has insulating character.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg" id="fig:phase_diagram" data-short-caption="Long Range Falicov Kimball Model Phase Diagram" style="width:100.0%" alt="Figure 2: 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 2: 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>
|
||||
<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($\omega$) and $\tau$ (the scaling exponent of IPR($\omega$) against system size $N$)." style="width:100.0%" alt="Figure 3: 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 3: 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>
|
||||
<p>In the limits of zero and infinite temperature, our model becomes a simple tight-binding model for the fermions. At zero temperature, the spin background is in one of the two translation invariant AFM ground states with two gapped fermionic CDW bands at energies <span class="math display">\[E_{\pm} = \pm\sqrt{\frac{1}{4}U^2 + 2t^2(1 + \cos ka)^2}\;.\]</span></p>
|
||||
<p>At infinite temperature, all the spin configurations become equally likely and the fermionic model reduces to one of binary uncorrelated disorder in which all eigenstates are Anderson localised <span class="citation" data-cites="abrahamsScalingTheoryLocalization1979"> [<a href="#ref-abrahamsScalingTheoryLocalization1979" role="doc-biblioref">3</a>]</span>. An Anderson localised state centered around <span class="math inline">\(r_0\)</span> has magnitude that drops exponentially over some localisation length <span class="math inline">\(\xi\)</span> i.e <span class="math inline">\(|\psi(r)|^2 \sim \exp{-\abs{r - r_0}/\xi}\)</span>. Calculating <span class="math inline">\(\xi\)</span> directly is numerically demanding. Therefore, we determine if a given state is localised via the energy-resolved IPR and the DOS defined as <span class="math display">\[\begin{aligned}
|
||||
<p>At infinite temperature, all the spin configurations become equally likely and the fermionic model reduces to one of binary uncorrelated disorder in which all eigenstates are Anderson localised <span class="citation" data-cites="abrahamsScalingTheoryLocalization1979"> [<a href="#ref-abrahamsScalingTheoryLocalization1979" role="doc-biblioref">4</a>]</span>. An Anderson localised state centered around <span class="math inline">\(r_0\)</span> has magnitude that drops exponentially over some localisation length <span class="math inline">\(\xi\)</span> i.e <span class="math inline">\(|\psi(r)|^2 \sim \exp{-|r - r_0|/\xi}\)</span>. Calculating <span class="math inline">\(\xi\)</span> directly is numerically demanding. Therefore, we determine if a given state is localised via the energy-resolved IPR and the DOS defined as <span class="math display">\[\begin{aligned}
|
||||
\mathrm{DOS}(\vec{S}, \omega)& = N^{-1} \sum_{i} \delta(\epsilon_i - \omega)\\
|
||||
\mathrm{IPR}(\vec{S}, \omega)& = \; N^{-1} \mathrm{DOS}(\vec{S}, \omega)^{-1} \sum_{i,j} \delta(\epsilon_i - \omega)\;\psi^{4}_{i,j}\end{aligned}\]</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>
|
||||
<p>The scaling of the IPR with system size <span class="math display">\[\mathrm{IPR} \propto N^{-\tau}\]</span> depends on the localisation properties of states at that energy. For delocalised states, e.g. Bloch waves, <span class="math inline">\(\tau\)</span> is the physical dimension. For fully localised states <span class="math inline">\(\tau\)</span> goes to zero in the thermodynamic limit. However, for special types of disorder such as binary disorder, the localisation lengths can be large comparable to the system size at hand, which can make it difficult to extract the correct scaling. An additional complication arises from the fact that the scaling exponent may display intermediate behaviours for correlated disorder and in the vicinity of a localisation-delocalisation transition <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">4</a>,<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">5</a>]</span>. The thermal defects of the CDW phase lead to a binary disorder potential with a finite correlation length, which in principle could result in delocalized eigenstates.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/DOS/IPR_scaling.svg" id="fig:IPR_scaling" data-short-caption="Energy resolved DOS($\omega$) and $\tau$ (the scaling exponent of IPR($\omega$) against system size $N$)." style="width:100.0%" alt="Figure 4: 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 4: 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>
|
||||
<p><span class="math display">\[\mathrm{IPR} \propto N^{-\tau}\]</span></p>
|
||||
<p>depends on the localisation properties of states at that energy. For delocalised states, e.g. Bloch waves, <span class="math inline">\(\tau\)</span> is the physical dimension. For fully localised states <span class="math inline">\(\tau\)</span> goes to zero in the thermodynamic limit. However, for special types of disorder such as binary disorder, the localisation lengths can be large comparable to the system size at hand, which can make it difficult to extract the correct scaling. An additional complication arises from the fact that the scaling exponent may display intermediate behaviours for correlated disorder and in the vicinity of a localisation-delocalisation transition <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">5</a>,<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">6</a>]</span>. The thermal defects of the CDW phase lead to a binary disorder potential with a finite correlation length, which in principle could result in delocalized eigenstates.</p>
|
||||
<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">3</a> shows the density of states as function of energy while fig. <a href="#fig:IPR_scaling">4</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="pdf_figs/indiv_IPR.svg" id="fig:indiv_IPR" alt="Figure 2: Energy resolved DOS(\omega) and \tau (the scaling exponent of IPR(\omega) against system size N). The left column shows the Anderson phase U = 2 at high T = 2.5 and the CDW phase at low T = 1.5 temperature. IPRs are evaluated for one of the in-gap states \omega_0/U = 0.057 and the center of the band \omega_1 U = 0.81. The right column shows instead the Mott and CDW phases at U = 5 with \omega_0/U = 0.24 and \omega_1/U = 0.571. For all the plots J = 5,\;\alpha = 1.25 and the fits for \tau use system sizes greater than 60. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalisation of the states." />
|
||||
<figcaption aria-hidden="true">Figure 2: Energy resolved DOS(<span class="math inline">\(\omega\)</span>) and <span class="math inline">\(\tau\)</span> (the scaling exponent of IPR(<span class="math inline">\(\omega\)</span>) against system size <span class="math inline">\(N\)</span>). The left column shows the Anderson phase <span class="math inline">\(U = 2\)</span> at high <span class="math inline">\(T = 2.5\)</span> and the CDW phase at low <span class="math inline">\(T = 1.5\)</span> temperature. IPRs are evaluated for one of the in-gap states <span class="math inline">\(\omega_0/U = 0.057\)</span> and the center of the band <span class="math inline">\(\omega_1\)</span> <span class="math inline">\(U = 0.81\)</span>. The right column shows instead the Mott and CDW phases at <span class="math inline">\(U = 5\)</span> with <span class="math inline">\(\omega_0/U = 0.24\)</span> and <span class="math inline">\(\omega_1/U = 0.571\)</span>. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span> and the fits for <span class="math inline">\(\tau\)</span> use system sizes greater than 60. 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 scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalisation of the states.</figcaption>
|
||||
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U5.svg" id="fig:gap_opening_U5" data-short-caption="DOS and Scaling Exponents for the transition from CDW to the Mott phase" style="width:100.0%" alt="Figure 5: 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 5: 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. <strong>¿fig:DM_IPR_scaling?</strong> 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. <strong>¿fig:gap_opening?</strong>. 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="pdf_figs/gap_openingboth.svg" id="fig:band_opening" alt="Figure 3: The DOS (a and c) and scaling exponent \tau (b and d) as a function of energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at U=2 while (c) and (d) show the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown a white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. U = 5,\;J = 5,\;\alpha = 1.25" />
|
||||
<figcaption aria-hidden="true">Figure 3: The DOS (a and c) and scaling exponent <span class="math inline">\(\tau\)</span> (b and d) as a function of energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at <span class="math inline">\(U=2\)</span> while (c) and (d) show 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 a 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">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span></figcaption>
|
||||
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U2.svg" id="fig:gap_opening_U2" data-short-caption="DOS and Scaling Exponents for the transition from CDW to the Anderson Phase" style="width:100.0%" alt="Figure 6: 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 6: 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><img src="../figure_code/fk_chapter/gap_opening_high_U.svg" id="fig:gap_opening_high_U" data-short-caption="no title" style="width:100.0%" alt="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="../figure_code/fk_chapter/gap_opening_low_U.svg" id="fig:gap_opening_low_U" data-short-caption="no title" style="width:100.0%" alt="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" /></p>
|
||||
<figure>
|
||||
<img src="pdf_figs/indiv_IPR_disorder.svg" id="fig:indiv_IPR_disorder" alt="Figure 4: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation [2], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N > 400" />
|
||||
<figcaption aria-hidden="true">Figure 4: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 < \rho < 1\)</span> matched to the largest corresponding FK model. As in Fig <a href="#fig:indiv_IPR" data-reference-type="ref" data-reference="fig:indiv_IPR">2</a>, the Energy resolved DOS(<span class="math inline">\(\omega\)</span>) and <span class="math inline">\(\tau\)</span> are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">2</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">\(N > 400\)</span></figcaption>
|
||||
</figure>
|
||||
<p>Fig. <a href="#fig:indiv_IPR" data-reference-type="ref" data-reference="fig:indiv_IPR">2</a> shows the DOS and <span class="math inline">\(\tau\)</span>, the scaling exponent of the IPR with system size, for a representative set of parameters covering all three phases. 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>
|
||||
<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">5</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">6</a>,<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">7</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. 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:indiv_IPR_disorder" data-reference-type="ref" data-reference="fig:indiv_IPR_disorder">4</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:band_opening" data-reference-type="ref" data-reference="fig:band_opening">3</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">8</a>]</span> and infinite dimensional <span class="citation" data-cites="hassanSpectralPropertiesChargedensitywave2007"> [<a href="#ref-hassanSpectralPropertiesChargedensitywave2007" role="doc-biblioref">9</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">8</a>]</span>.</p>
|
||||
<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, see Appendix ??. Fig. [<a href="#fig:indiv_IPR_disorder" data-reference-type="ref" data-reference="fig:indiv_IPR_disorder">4</a>] compares 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. 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. Overall, we see that the interplay of interactions, here manifest as a peculiar binary potential, and localization can be very intricate and the added advantage of our 1D model is that we can explore very large system sizes for a complete understanding.</p>
|
||||
<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>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) \\
|
||||
& -\;t \sum_{i} c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i}
|
||||
\end{aligned}\]</span></p>
|
||||
<p>fig. <strong>¿fig:DM_DOS?</strong> and fig. <strong>¿fig:DM_IPR_scaling?</strong> 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>
|
||||
<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>
|
||||
<p><img src="/assets/thesis/fk_chapter/disorder_model/DM_DOS.svg" id="fig:DM_DOS" data-short-caption="A comparison of the full FK model to a simple binary disorder model." style="width:100.0%" alt="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 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 3, 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_IPR_scaling.svg" id="fig:DM_IPR_scaling" data-short-caption="A comparison of the full FK model to a simple binary disorder model." style="width:100.0%" alt="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 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 4 \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 [3], 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 > 400" /></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="fk-conclusion" class="level1">
|
||||
<h1>Discussion and Conclusion</h1>
|
||||
<p>The FK model is one of the simplest non-trivial models of interacting fermions. We studied its thermodynamic and localisation properties brought down in dimensionality to 1D by adding a novel long-ranged coupling designed to stabilise the CDW phase present in dimension two and above. Our hybrid MCMC approach elucidates a disorder-free localization mechanism within our translationally invariant system. Further, we demonstrate a significant speedup over the naive method. We show that our long-range FK in 1D retains much of the rich phase diagram of its higher dimensional cousins. Careful scaling analysis indicates that all the single particle eigenstates eventually localise at nonzero temperature albeit only for very large system sizes of several thousand.</p>
|
||||
<p>Our work raises a number of interesting questions for future research. A straightforward but numerically challenging problem is to pin down the model’s behaviour closer to the critical point where correlations in the spin sector would become significant. Would this modify the localisation behaviour? Similar to other soluble models of disorder-free localisation, we expect intriguing out-of equilibrium physics, for example slow entanglement dynamics akin to more generic interacting systems <span class="citation" data-cites="hartLogarithmicEntanglementGrowth2020"> [<a href="#ref-hartLogarithmicEntanglementGrowth2020" role="doc-biblioref">10</a>]</span>. One could also investigate whether the rich ground state phenomenology of the FK model as a function of filling <span class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">11</a>]</span> such as the devil’s staircase <span class="citation" data-cites="michelettiCompleteDevilTextquotesingles1997"> [<a href="#ref-michelettiCompleteDevilTextquotesingles1997" role="doc-biblioref">12</a>]</span> could be stabilised at finite temperature. In a broader context, we envisage that long-range interactions can also be used to gain a deeper understanding of the temperature evolution of topological phases. One example would be a long-ranged FK version of the celebrated Su-Schrieffer-Heeger model where one could explore the interplay of topological bound states and thermal domain wall defects. Finally, the rich physics of our model should be realizable in systems with long-range interactions, such as trapped ion quantum simulators, where one can also explore the fully interacting regime with a dynamical background field.</p>
|
||||
<p><strong>UNCORRELATED DISORDER MODEL</strong></p>
|
||||
<p>The disorder model referred to in the main text is defined by replacing the spin degree of freedom in the FK model <span class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder 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. <span class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} - \tfrac{1}{2}) \\
|
||||
& -\;t \sum_{i} c^\dag_{i}c_{i+1} + c^\dag_{i+1}c_{i} \nonumber\end{aligned}\]</span></p>
|
||||
<p>Could look at doping the mott insulating phase, see results like <span class="citation" data-cites="caiVisualizingEvolutionMott2016"> [<a href="#ref-caiVisualizingEvolutionMott2016" role="doc-biblioref">13</a>]</span></p>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p>Our work raises a number of interesting questions for future research. A straightforward but numerically challenging problem is to pin down the model’s behaviour closer to the critical point where correlations in the spin sector would become significant. Would this modify the localisation behaviour? Similar to other soluble models of disorder-free localisation, we expect intriguing out-of equilibrium physics, for example slow entanglement dynamics akin to more generic interacting systems <span class="citation" data-cites="hartLogarithmicEntanglementGrowth2020"> [<a href="#ref-hartLogarithmicEntanglementGrowth2020" role="doc-biblioref">11</a>]</span>. One could also investigate whether the rich ground state phenomenology of the FK model as a function of filling <span class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">12</a>]</span> such as the devil’s staircase <span class="citation" data-cites="michelettiCompleteDevilTextquotesingles1997"> [<a href="#ref-michelettiCompleteDevilTextquotesingles1997" role="doc-biblioref">13</a>]</span> could be stabilised at finite temperature. In a broader context, we envisage that long-range interactions can also be used to gain a deeper understanding of the temperature evolution of topological phases. One example would be a long-ranged FK version of the celebrated Su-Schrieffer-Heeger model where one could explore the interplay of topological bound states and thermal domain wall defects. Finally, the rich physics of our model should be realizable in systems with long-range interactions, such as trapped ion quantum simulators, where one can also explore the fully interacting regime with a dynamical background field.</p>
|
||||
<p>Could look at doping the mott insulating phase, see results like <span class="citation" data-cites="caiVisualizingEvolutionMott2016"> [<a href="#ref-caiVisualizingEvolutionMott2016" role="doc-biblioref">14</a>]</span></p>
|
||||
<p>Next Chapter: <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">4 The Amorphous Kitaev Model</a></p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
@ -121,41 +132,44 @@ H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} - \tfrac{1
|
||||
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|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">K. Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size Scaling Analysis of Ising Model Block Distribution Functions</a></em>, Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
|
||||
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|
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|
||||
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|
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<div class="csl-left-margin">[2] </div><div class="csl-right-inline">A. E. Antipov, Y. Javanmard, P. Ribeiro, and S. Kirchner, <em><a href="https://doi.org/10.1103/PhysRevLett.117.146601">Interaction-Tuned Anderson Versus Mott Localization</a></em>, Phys. Rev. Lett. <strong>117</strong>, 146601 (2016).</div>
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<div class="csl-left-margin">[5] </div><div class="csl-right-inline">B. Kramer and A. MacKinnon, <em><a href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469 (1993).</div>
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<div class="csl-left-margin">[6] </div><div class="csl-right-inline">F. Evers and A. D. Mirlin, <em><a href="https://doi.org/10.1103/RevModPhys.80.1355">Anderson Transitions</a></em>, Rev. Mod. Phys. <strong>80</strong>, 1355 (2008).</div>
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|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -72,7 +72,7 @@ image:
|
||||
<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/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="(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>
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