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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="#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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<p>In chapter <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3</a> I will introduce a generalized Falikov-Kimball model in one dimension I call the Long-Range Falikov-Kimball model. With the addition of long-range interactions in the background field, the model shows a similarly rich phase diagram its higher dimensional cousins. I use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localization properties of the fermions. I then compare the behaviour of this transitionally invariant model to an Anderson model of uncorrelated binary disorder about a background charge density wave field which confirms that the fermionic sector only fully localizes for very large system sizes.</p>
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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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@ -109,18 +109,18 @@ H_{\mathrm{FK}} = & -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U
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<!-- Experimentally, Mott insulating systems without magnetic order have been proposed as QSL systems\ [@law1TTaS2QuantumSpin2017; @ribakGaplessExcitationsGround2017]. -->
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<!-- Other exampels: Quantum spin liquids are the analogous phase of matter for spin systems. Spin ice support deconfined magnetic monopoles. -->
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<figure>
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<img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png" id="fig:correlation_spin_orbit_PT" data-short-caption="Phase Diagram" style="width:100.0%" alt="Figure 3: From [44]." />
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<figcaption aria-hidden="true">Figure 3: From <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>]</span>.</figcaption>
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<img src="/assets/thesis/intro_chapter/kitaev_material_phase_diagram.svg" id="fig:kitaev-material-phase-diagram" data-short-caption="Phase Diagram" style="width:100.0%" alt="Figure 3: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev Model. Reproduced from [44]." />
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<figcaption aria-hidden="true">Figure 3: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev Model. Reproduced from <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>]</span>.</figcaption>
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</figure>
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<p>Spin-orbit coupling is a relativistic effect, that very roughly corresponds to the fact that in the frame of reference of a moving electron, the electric field of nearby nuclei look like magnetic fields to which the electron spin couples. This effectively couples the spatial and spin parts of the electron wavefunction, meaning that the lattice structure can influence the form of the spin-spin interactions leading to spatial anisotropy. This anisotropy will be how we frustrate the Mott insulators <span class="citation" data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a href="#ref-jackeliMottInsulatorsStrong2009" role="doc-biblioref">48</a>,<a href="#ref-khaliullinOrbitalOrderFluctuations2005" role="doc-biblioref">49</a>]</span>. As we saw with the Hubbard model, interaction effects are only strong or weak in comparison to the bandwidth or hopping integral <span class="math inline">\(t\)</span> so what we need to see strong frustration is a material with strong spin-orbit coupling <span class="math inline">\(\lambda\)</span> relative to its bandwidth <span class="math inline">\(t\)</span>.</p>
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<p>In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-<span class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known Kitaev Materials, draw their name from the celebrated Kitaev Honeycomb Model which is expected to model their low temperature behaviour <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>,<a href="#ref-Jackeli2009" role="doc-biblioref">50</a>–<a href="#ref-Takagi2019" role="doc-biblioref">53</a>]</span>.</p>
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<p>At this point we can sketch out a phase diagram like that of fig. <a href="#fig:correlation_spin_orbit_PT">3</a>. When both electron-electron interactions <span class="math inline">\(U\)</span> and spin-orbit couplings <span class="math inline">\(\lambda\)</span> are small relative to the bandwidth <span class="math inline">\(t\)</span> we recover standard band theory of band insulators and metals. In the upper left we have the simple Mott insulating state as described by the Hubbard model. In the lower right, strong spin-orbit coupling gives rise to Topological insulators (TIs) characterised by symmetry protected edge modes and non-zero Chern number. Kitaev materials occur in the region where strong electron-electron interaction and spin-orbit coupling interact. See <span class="citation" data-cites="witczak-krempaCorrelatedQuantumPhenomena2014"> [<a href="#ref-witczak-krempaCorrelatedQuantumPhenomena2014" role="doc-biblioref">54</a>]</span> for a much more expansive version of this diagram.</p>
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<p>At this point we can sketch out a phase diagram like that of fig. <a href="#fig:kitaev-material-phase-diagram">3</a>. When both electron-electron interactions <span class="math inline">\(U\)</span> and spin-orbit couplings <span class="math inline">\(\lambda\)</span> are small relative to the bandwidth <span class="math inline">\(t\)</span> we recover standard band theory of band insulators and metals. In the upper left we have the simple Mott insulating state as described by the Hubbard model. In the lower right, strong spin-orbit coupling gives rise to Topological insulators (TIs) characterised by symmetry protected edge modes and non-zero Chern number. Kitaev materials occur in the region where strong electron-electron interaction and spin-orbit coupling interact. See <span class="citation" data-cites="witczak-krempaCorrelatedQuantumPhenomena2014"> [<a href="#ref-witczak-krempaCorrelatedQuantumPhenomena2014" role="doc-biblioref">54</a>]</span> for a much more expansive version of this diagram.</p>
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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 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="#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>In <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">chapter 4</a> I will introduce the Amorphous Kitaev model, a generalisation of the Kitaev honeycomb model to random lattices with fixed coordination number three. We will show that this model is a soluble chiral amorphous quantum spin liquid. The model retains its exact solubility but, as with the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">66</a>]</span>, the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We will confirm prior observations that the form of the ground state can be written in terms of the number of sides of elementary plaquettes of the model <span class="citation" data-cites="OBrienPRB2016 eschmannThermodynamicClassificationThreedimensional2020"> [<a href="#ref-OBrienPRB2016" role="doc-biblioref">64</a>,<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">95</a>]</span>. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I show that the system undergoes a finite-temperature phase transition to a conducting thermal metal state and discuss possible experimental realisations.</p>
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<p>The next chapter, <a href="../2_Background/2.1_FK_Model.html">Chapter 2</a>, will introduce some necessary background to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and localisation. Then <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">chapter 3</a> introduces and studies the Long Range Falicov-Kimball Model in one dimension. <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">Chapter 4</a> focusses on the Amorphous Kitaev Model.</p>
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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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@ -92,33 +92,34 @@ H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\
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<section id="phase-diagrams" class="level2">
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<h2>Phase Diagrams</h2>
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<figure>
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<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg" id="fig:fk_phase_diagram" data-short-caption="Fermi-Hubbard and Falikov-Kimball Temperatue-Interaction Phase Diagrams" style="width:100.0%" alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]" />
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<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg" id="fig:fk_phase_diagram" data-short-caption="Falikov-Kimball Temperatue-Interaction Phase Diagrams" style="width:100.0%" alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]" />
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<figcaption aria-hidden="true">Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">10</a>,<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">14</a>]</span></figcaption>
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</figure>
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<p>In dimensions greater than one, the FK model exhibits a phase transition at some <span class="math inline">\(U\)</span> dependent critical temperature <span class="math inline">\(T_c(U)\)</span> to a low temperature ordered phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">15</a>]</span>. In terms of the immobile electrons this corresponds to them occupying only one of the two sublattices A and B this is known as a charge density wave (CDW) phase. In terms of spins this is an AFM phase.</p>
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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="#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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<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. In the one dimensional Kitaev model this means the whole spectrum is localised at all finite temperatures, though at low temperatures the localisation length may be so large that the states appear extended in finite size systems. 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>The one dimensional FK model has been studied numerically, as a perturbation in interaction strength <span class="math inline">\(U\)</span> and in the continuum limit <span class="citation" data-cites="bursillOneDimensionalContinuum1994"> [<a href="#ref-bursillOneDimensionalContinuum1994" role="doc-biblioref">25</a>]</span> with the main results beings for attractive <span class="math inline">\(U > U_c\)</span> the system forms electron spin bound state ‘atoms’ which repel on another <span class="citation" data-cites="gruberGroundStateEnergyLowTemperature1993"> [<a href="#ref-gruberGroundStateEnergyLowTemperature1993" role="doc-biblioref">26</a>]</span> and that the ground state phase diagram has a has a fractal structure as a function of electron filling <span class="citation" data-cites="freericksTwostateOnedimensionalSpinless1990"> [<a href="#ref-freericksTwostateOnedimensionalSpinless1990" role="doc-biblioref">27</a>]</span>.</p>
|
||||
<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">28</a>–<a href="#ref-yosidaMagneticPropertiesCuMn1957" role="doc-biblioref">31</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">32</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">33</a>,<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">34</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">35</a>]</span>.</p>
|
||||
<p>Based on this primacy of dimensionality, we will go digging into the one dimensional case. In chapter <a href="../3_Long_Range_Falicov_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>
|
||||
</section>
|
||||
<section id="long-ranged-ising-model" class="level2">
|
||||
<h2>Long Ranged Ising model</h2>
|
||||
<p>The suppression of phase transitions is a common phenomena in one dimensional systems and the Ising model serves as a great illustration of this. In terms of classical spins <span class="math inline">\(S_i = \pm \frac{1}{2}\)</span> the standard Ising model reads</p>
|
||||
<p><span class="math display">\[H_{\mathrm{I}} = \sum_{\langle ij \rangle} S_i S_j\]</span></p>
|
||||
<p>Like the FK model, the Ising model shows an FTPT to an ordered state only in two dimensions and above. This can be understood via Peierls’ argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">31</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">32</a>]</span> to be a consequence of the low energy penalty for domain walls in one dimensional systems.</p>
|
||||
<p>Like the FK model, the Ising model shows an FTPT to an ordered state only in two dimensions and above. This can be understood via Peierls’ argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">34</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">35</a>]</span> to be a consequence of the low energy penalty for domain walls in one dimensional systems.</p>
|
||||
<p>Following Peierls’ argument, consider the difference in free energy <span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span> between an ordered state and a state with single domain wall in a discrete order parameter. If this value is negative it implies that the ordered state is unstable with respect to domain wall defects, and they will thus proliferate, destroying the ordered phase. If we consider the scaling of the two terms with system size <span class="math inline">\(L\)</span> we see that short range interactions produce a constant energy penalty <span class="math inline">\(\Delta E\)</span> for a domain wall. In contrast, the number of such single domain wall states scales linearly with system size so the entropy is <span class="math inline">\(\propto \ln L\)</span>. Thus the entropic contribution dominates (eventually) in the thermodynamic limit and no finite temperature order is possible. In two dimensions and above, the energy penalty of a domain wall scales like <span class="math inline">\(L^{d-1}\)</span> which is why they can support ordered phases. This argument does not quite apply to the FK model because of the aforementioned RKKY interaction. Instead this argument will give us insight into how to recover an ordered phase in the one dimensional FK model.</p>
|
||||
<p>In contrast the long range Ising (LRI) model <span class="math inline">\(H_{\mathrm{LRI}}\)</span> can have an FTPT in one dimension.</p>
|
||||
<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|) S_i S_j = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\]</span></p>
|
||||
<p>Renormalisation group analyses show that the LRI model has an ordered phase in 1D for <span class="math inline">\(1 < \alpha < 2\)</span> <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">33</a>]</span>. Peierls’ argument can be extended <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">30</a>]</span> to long range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state <span class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span> and a domain wall state <span class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\)</span>. In the case of the LRI model, careful counting shows that this energy penalty is <span class="math display">\[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\]</span></p>
|
||||
<p>because each interaction between spins separated across the domain by a bond length <span class="math inline">\(n\)</span> can be drawn between <span class="math inline">\(n\)</span> equivalent pairs of sites. The behaviour then depends crucially on the sum scales with system size. Ruelle proved rigorously for a very general class of 1D systems, that if <span class="math inline">\(\Delta E\)</span> or its many-body generalisation converges to a constant in the thermodynamic limit then the free energy is analytic <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">34</a>]</span>. This rules out a finite order phase transition, though not one of the Kosterlitz-Thouless type. Dyson also proves this though with a slightly different condition on <span class="math inline">\(J(n)\)</span> <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">33</a>]</span>.</p>
|
||||
<p>Renormalisation group analyses show that the LRI model has an ordered phase in 1D for <span class="math inline">\(1 < \alpha < 2\)</span> <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">36</a>]</span>. Peierls’ argument can be extended <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">33</a>]</span> to long range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state <span class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span> and a domain wall state <span class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\)</span>. In the case of the LRI model, careful counting shows that this energy penalty is <span class="math display">\[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\]</span></p>
|
||||
<p>because each interaction between spins separated across the domain by a bond length <span class="math inline">\(n\)</span> can be drawn between <span class="math inline">\(n\)</span> equivalent pairs of sites. The behaviour then depends crucially on the sum scales with system size. Ruelle proved rigorously for a very general class of 1D systems, that if <span class="math inline">\(\Delta E\)</span> or its many-body generalisation converges to a constant in the thermodynamic limit then the free energy is analytic <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">37</a>]</span>. This rules out a finite order phase transition, though not one of the Kosterlitz-Thouless type. Dyson also proves this though with a slightly different condition on <span class="math inline">\(J(n)\)</span> <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">36</a>]</span>.</p>
|
||||
<p>With a power law form for <span class="math inline">\(J(n)\)</span>, there are a few cases to consider:</p>
|
||||
<p>For <span class="math inline">\(\alpha = 0\)</span> i.e infinite range interactions, the Ising model is exactly solvable and mean field theory is exact <span class="citation" data-cites="lipkinValidityManybodyApproximation1965"> [<a href="#ref-lipkinValidityManybodyApproximation1965" role="doc-biblioref">35</a>]</span>. This limit is the same as the infinite dimensional limit.</p>
|
||||
<p>For <span class="math inline">\(\alpha \leq 1\)</span> we have very slowly decaying interactions. <span class="math inline">\(\Delta E\)</span> does not converge as a function of system size so the Hamiltonian is non-extensive, a topic not without some considerable controversy <span class="citation" data-cites="grossNonextensiveHamiltonianSystems2002 lutskoQuestioningValidityNonextensive2011 wangCommentNonextensiveHamiltonian2003"> [<a href="#ref-grossNonextensiveHamiltonianSystems2002" role="doc-biblioref">36</a>–<a href="#ref-wangCommentNonextensiveHamiltonian2003" role="doc-biblioref">38</a>]</span> that we will not consider further here.</p>
|
||||
<p>For <span class="math inline">\(\alpha = 0\)</span> i.e infinite range interactions, the Ising model is exactly solvable and mean field theory is exact <span class="citation" data-cites="lipkinValidityManybodyApproximation1965"> [<a href="#ref-lipkinValidityManybodyApproximation1965" role="doc-biblioref">38</a>]</span>. This limit is the same as the infinite dimensional limit.</p>
|
||||
<p>For <span class="math inline">\(\alpha \leq 1\)</span> we have very slowly decaying interactions. <span class="math inline">\(\Delta E\)</span> does not converge as a function of system size so the Hamiltonian is non-extensive, a topic not without some considerable controversy <span class="citation" data-cites="grossNonextensiveHamiltonianSystems2002 lutskoQuestioningValidityNonextensive2011 wangCommentNonextensiveHamiltonian2003"> [<a href="#ref-grossNonextensiveHamiltonianSystems2002" role="doc-biblioref">39</a>–<a href="#ref-wangCommentNonextensiveHamiltonian2003" role="doc-biblioref">41</a>]</span> that we will not consider further here.</p>
|
||||
<p>For <span class="math inline">\(1 < \alpha < 2\)</span>, we get a phase transition to an ordered state at a finite temperature, this is what we want!</p>
|
||||
<p>For <span class="math inline">\(\alpha = 2\)</span>, the energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transition <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">30</a>]</span>.</p>
|
||||
<p>For <span class="math inline">\(\alpha = 2\)</span>, the energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transition <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">33</a>]</span>.</p>
|
||||
<p>Finally, for <span class="math inline">\(2 < \alpha\)</span> we have very quickly decaying interactions and domain walls again have a finite energy penalty, hence Peirels’ argument holds and there is no phase transition.</p>
|
||||
<p>One final complexity is that for <span class="math inline">\(\tfrac{3}{2} < \alpha < 2\)</span> renormalisation group methods show that the critical point has non-universal critical exponents that depend on <span class="math inline">\(\alpha\)</span> <span class="citation" data-cites="fisherCriticalExponentsLongRange1972"> [<a href="#ref-fisherCriticalExponentsLongRange1972" role="doc-biblioref">39</a>]</span>. To avoid this potential confounding factors we will park ourselves at <span class="math inline">\(\alpha = 1.25\)</span> when we apply these ideas to the FK model.</p>
|
||||
<p>Were we to extend this to arbitrary dimension <span class="math inline">\(d\)</span> we would find that thermodynamics properties generally both <span class="math inline">\(d\)</span> and <span class="math inline">\(\alpha\)</span>, long range interactions can modify the ‘effective dimension’ of thermodynamic systems <span class="citation" data-cites="angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">40</a>]</span>.</p>
|
||||
<p>One final complexity is that for <span class="math inline">\(\tfrac{3}{2} < \alpha < 2\)</span> renormalisation group methods show that the critical point has non-universal critical exponents that depend on <span class="math inline">\(\alpha\)</span> <span class="citation" data-cites="fisherCriticalExponentsLongRange1972"> [<a href="#ref-fisherCriticalExponentsLongRange1972" role="doc-biblioref">42</a>]</span>. To avoid this potential confounding factors we will park ourselves at <span class="math inline">\(\alpha = 1.25\)</span> when we apply these ideas to the FK model.</p>
|
||||
<p>Were we to extend this to arbitrary dimension <span class="math inline">\(d\)</span> we would find that thermodynamics properties generally both <span class="math inline">\(d\)</span> and <span class="math inline">\(\alpha\)</span>, long range interactions can modify the ‘effective dimension’ of thermodynamic systems <span class="citation" data-cites="angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">43</a>]</span>.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/background_chapter/alpha_diagram.svg" id="fig:alpha_diagram" data-short-caption="Long Range Ising Model Behaviour" style="width:100.0%" alt="Figure 3: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied." />
|
||||
<figcaption aria-hidden="true">Figure 3: The thermodynamic behaviour of the long range Ising model <span class="math inline">\(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span class="math inline">\(\alpha\)</span> is varied.</figcaption>
|
||||
@ -201,53 +202,62 @@ H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\
|
||||
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[24] </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>
|
||||
</div>
|
||||
<div id="ref-bursillOneDimensionalContinuum1994" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[25] </div><div class="csl-right-inline">R. J. Bursill, <em><a href="https://doi.org/10.1016/0378-4371(94)90322-0">One Dimensional Continuum Falicov-Kimball Model in the Strongly Correlated Limit</a></em>, Physica A: Statistical Mechanics and Its Applications <strong>206</strong>, 521 (1994).</div>
|
||||
</div>
|
||||
<div id="ref-gruberGroundStateEnergyLowTemperature1993" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[26] </div><div class="csl-right-inline">C. Gruber, J. L. Lebowitz, and N. Macris, <em><a href="https://doi.org/10.1209/0295-5075/21/4/002">Ground-State Energy and Low-Temperature Behavior of the One-Dimensional Falicov-Kimball Model</a></em>, EPL <strong>21</strong>, 389 (1993).</div>
|
||||
</div>
|
||||
<div id="ref-freericksTwostateOnedimensionalSpinless1990" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[27] </div><div class="csl-right-inline">J. K. Freericks and L. M. Falicov, <em><a href="https://doi.org/10.1103/PhysRevB.41.2163">Two-State One-Dimensional Spinless Fermi Gas</a></em>, Phys. Rev. B <strong>41</strong>, 2163 (1990).</div>
|
||||
</div>
|
||||
<div id="ref-kasuyaTheoryMetallicFerro1956" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[25] </div><div class="csl-right-inline">T. Kasuya, <em><a href="https://doi.org/10.1143/PTP.16.45">A Theory of Metallic Ferro- and Antiferromagnetism on Zener’s Model</a></em>, Prog Theor Phys <strong>16</strong>, 45 (1956).</div>
|
||||
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">T. Kasuya, <em><a href="https://doi.org/10.1143/PTP.16.45">A Theory of Metallic Ferro- and Antiferromagnetism on Zener’s Model</a></em>, Prog Theor Phys <strong>16</strong>, 45 (1956).</div>
|
||||
</div>
|
||||
<div id="ref-rudermanIndirectExchangeCoupling1954" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[26] </div><div class="csl-right-inline">M. A. Ruderman and C. Kittel, <em><a href="https://doi.org/10.1103/PhysRev.96.99">Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons</a></em>, Phys. Rev. <strong>96</strong>, 99 (1954).</div>
|
||||
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">M. A. Ruderman and C. Kittel, <em><a href="https://doi.org/10.1103/PhysRev.96.99">Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons</a></em>, Phys. Rev. <strong>96</strong>, 99 (1954).</div>
|
||||
</div>
|
||||
<div id="ref-vanvleckNoteInteractionsSpins1962" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[27] </div><div class="csl-right-inline">J. H. Van Vleck, <em><a href="https://doi.org/10.1103/RevModPhys.34.681">Note on the Interactions Between the Spins of Magnetic Ions or Nuclei in Metals</a></em>, Rev. Mod. Phys. <strong>34</strong>, 681 (1962).</div>
|
||||
<div class="csl-left-margin">[30] </div><div class="csl-right-inline">J. H. Van Vleck, <em><a href="https://doi.org/10.1103/RevModPhys.34.681">Note on the Interactions Between the Spins of Magnetic Ions or Nuclei in Metals</a></em>, Rev. Mod. Phys. <strong>34</strong>, 681 (1962).</div>
|
||||
</div>
|
||||
<div id="ref-yosidaMagneticPropertiesCuMn1957" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">K. Yosida, <em><a href="https://doi.org/10.1103/PhysRev.106.893">Magnetic Properties of Cu-Mn Alloys</a></em>, Phys. Rev. <strong>106</strong>, 893 (1957).</div>
|
||||
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">K. Yosida, <em><a href="https://doi.org/10.1103/PhysRev.106.893">Magnetic Properties of Cu-Mn Alloys</a></em>, Phys. Rev. <strong>106</strong>, 893 (1957).</div>
|
||||
</div>
|
||||
<div id="ref-rusinCalculationRKKYRange2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">T. M. Rusin and W. Zawadzki, <em><a href="https://doi.org/10.1016/j.jmmm.2017.06.007">On Calculation of RKKY Range Function in One Dimension</a></em>, Journal of Magnetism and Magnetic Materials <strong>441</strong>, 387 (2017).</div>
|
||||
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">T. M. Rusin and W. Zawadzki, <em><a href="https://doi.org/10.1016/j.jmmm.2017.06.007">On Calculation of RKKY Range Function in One Dimension</a></em>, Journal of Magnetism and Magnetic Materials <strong>441</strong>, 387 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-thoulessLongRangeOrderOneDimensional1969" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[30] </div><div class="csl-right-inline">D. J. Thouless, <em><a href="https://doi.org/10.1103/PhysRev.187.732">Long-Range Order in One-Dimensional Ising Systems</a></em>, Phys. Rev. <strong>187</strong>, 732 (1969).</div>
|
||||
<div class="csl-left-margin">[33] </div><div class="csl-right-inline">D. J. Thouless, <em><a href="https://doi.org/10.1103/PhysRev.187.732">Long-Range Order in One-Dimensional Ising Systems</a></em>, Phys. Rev. <strong>187</strong>, 732 (1969).</div>
|
||||
</div>
|
||||
<div id="ref-peierlsIsingModelFerromagnetism1936" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">R. Peierls, <em><a href="https://doi.org/10.1017/S0305004100019174">On Ising’s Model of Ferromagnetism</a></em>, Mathematical Proceedings of the Cambridge Philosophical Society <strong>32</strong>, 477 (1936).</div>
|
||||
<div class="csl-left-margin">[34] </div><div class="csl-right-inline">R. Peierls, <em><a href="https://doi.org/10.1017/S0305004100019174">On Ising’s Model of Ferromagnetism</a></em>, Mathematical Proceedings of the Cambridge Philosophical Society <strong>32</strong>, 477 (1936).</div>
|
||||
</div>
|
||||
<div id="ref-kennedyItinerantElectronModel1986" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">T. Kennedy and E. H. Lieb, <em><a href="https://doi.org/10.1016/0378-4371(86)90188-3">An Itinerant Electron Model with Crystalline or Magnetic Long Range Order</a></em>, Physica A: Statistical Mechanics and Its Applications <strong>138</strong>, 320 (1986).</div>
|
||||
<div class="csl-left-margin">[35] </div><div class="csl-right-inline">T. Kennedy and E. H. Lieb, <em><a href="https://doi.org/10.1016/0378-4371(86)90188-3">An Itinerant Electron Model with Crystalline or Magnetic Long Range Order</a></em>, Physica A: Statistical Mechanics and Its Applications <strong>138</strong>, 320 (1986).</div>
|
||||
</div>
|
||||
<div id="ref-dysonExistencePhasetransitionOnedimensional1969" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[33] </div><div class="csl-right-inline">F. J. Dyson, <em><a href="https://doi.org/10.1007/BF01645907">Existence of a Phase-Transition in a One-Dimensional Ising Ferromagnet</a></em>, Commun.Math. Phys. <strong>12</strong>, 91 (1969).</div>
|
||||
<div class="csl-left-margin">[36] </div><div class="csl-right-inline">F. J. Dyson, <em><a href="https://doi.org/10.1007/BF01645907">Existence of a Phase-Transition in a One-Dimensional Ising Ferromagnet</a></em>, Commun.Math. Phys. <strong>12</strong>, 91 (1969).</div>
|
||||
</div>
|
||||
<div id="ref-ruelleStatisticalMechanicsOnedimensional1968" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[34] </div><div class="csl-right-inline">D. Ruelle, <em><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys. <strong>9</strong>, 267 (1968).</div>
|
||||
<div class="csl-left-margin">[37] </div><div class="csl-right-inline">D. Ruelle, <em><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys. <strong>9</strong>, 267 (1968).</div>
|
||||
</div>
|
||||
<div id="ref-lipkinValidityManybodyApproximation1965" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[35] </div><div class="csl-right-inline">H. J. Lipkin, N. Meshkov, and A. J. Glick, <em><a href="https://doi.org/10.1016/0029-5582(65)90862-X">Validity of Many-Body Approximation Methods for a Solvable Model. (I). Exact Solutions and Perturbation Theory</a></em>, Nuclear Physics <strong>62</strong>, 188 (1965).</div>
|
||||
<div class="csl-left-margin">[38] </div><div class="csl-right-inline">H. J. Lipkin, N. Meshkov, and A. J. Glick, <em><a href="https://doi.org/10.1016/0029-5582(65)90862-X">Validity of Many-Body Approximation Methods for a Solvable Model. (I). Exact Solutions and Perturbation Theory</a></em>, Nuclear Physics <strong>62</strong>, 188 (1965).</div>
|
||||
</div>
|
||||
<div id="ref-grossNonextensiveHamiltonianSystems2002" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[36] </div><div class="csl-right-inline">D. H. E. Gross, <em><a href="https://doi.org/10.1016/S0378-4371(01)00646-X">Non-Extensive Hamiltonian Systems Follow Boltzmann’s Principle Not Tsallis Statistics. -- Phase Transitions, Second Law of Thermodynamics</a></em>, Physica A: Statistical Mechanics and Its Applications <strong>305</strong>, 99 (2002).</div>
|
||||
<div class="csl-left-margin">[39] </div><div class="csl-right-inline">D. H. E. Gross, <em><a href="https://doi.org/10.1016/S0378-4371(01)00646-X">Non-Extensive Hamiltonian Systems Follow Boltzmann’s Principle Not Tsallis Statistics. -- Phase Transitions, Second Law of Thermodynamics</a></em>, Physica A: Statistical Mechanics and Its Applications <strong>305</strong>, 99 (2002).</div>
|
||||
</div>
|
||||
<div id="ref-lutskoQuestioningValidityNonextensive2011" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[37] </div><div class="csl-right-inline">J. F. Lutsko and J. P. Boon, <em><a href="https://doi.org/10.1209/0295-5075/95/20006">Questioning the Validity of Non-Extensive Thermodynamics for Classical Hamiltonian Systems</a></em>, EPL <strong>95</strong>, 20006 (2011).</div>
|
||||
<div class="csl-left-margin">[40] </div><div class="csl-right-inline">J. F. Lutsko and J. P. Boon, <em><a href="https://doi.org/10.1209/0295-5075/95/20006">Questioning the Validity of Non-Extensive Thermodynamics for Classical Hamiltonian Systems</a></em>, EPL <strong>95</strong>, 20006 (2011).</div>
|
||||
</div>
|
||||
<div id="ref-wangCommentNonextensiveHamiltonian2003" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[38] </div><div class="csl-right-inline">Q. A. Wang, <em><a href="https://hal.archives-ouvertes.fr/hal-00009462">Comment on <span>“Nonextensive Hamiltonian Systems Follow Boltzmann’s Principle Not Tsallis Statistics-Phase Transition, Second Law of Thermodynamics”</span> by Gross</a></em>, (2003).</div>
|
||||
<div class="csl-left-margin">[41] </div><div class="csl-right-inline">Q. A. Wang, <em><a href="https://hal.archives-ouvertes.fr/hal-00009462">Comment on <span>“Nonextensive Hamiltonian Systems Follow Boltzmann’s Principle Not Tsallis Statistics-Phase Transition, Second Law of Thermodynamics”</span> by Gross</a></em>, (2003).</div>
|
||||
</div>
|
||||
<div id="ref-fisherCriticalExponentsLongRange1972" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[39] </div><div class="csl-right-inline">M. E. Fisher, S. Ma, and B. G. Nickel, <em><a href="https://doi.org/10.1103/PhysRevLett.29.917">Critical Exponents for Long-Range Interactions</a></em>, Phys. Rev. Lett. <strong>29</strong>, 917 (1972).</div>
|
||||
<div class="csl-left-margin">[42] </div><div class="csl-right-inline">M. E. Fisher, S. Ma, and B. G. Nickel, <em><a href="https://doi.org/10.1103/PhysRevLett.29.917">Critical Exponents for Long-Range Interactions</a></em>, Phys. Rev. Lett. <strong>29</strong>, 917 (1972).</div>
|
||||
</div>
|
||||
<div id="ref-angeliniRelationsShortrangeLongrange2014" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[40] </div><div class="csl-right-inline">M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E <strong>89</strong>, 062120 (2014).</div>
|
||||
<div class="csl-left-margin">[43] </div><div class="csl-right-inline">M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E <strong>89</strong>, 062120 (2014).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -34,7 +34,7 @@ image:
|
||||
<li><a href="#the-majorana-model" id="toc-the-majorana-model">The Majorana Model</a></li>
|
||||
<li><a href="#the-fermion-problem" id="toc-the-fermion-problem">The Fermion Problem</a></li>
|
||||
<li><a href="#an-emergent-gauge-field" id="toc-an-emergent-gauge-field">An Emergent Gauge Field</a></li>
|
||||
<li><a href="#anyons-topology-and-the-chern-number" id="toc-anyons-topology-and-the-chern-number">Anyons, Topology and the Chern number</a></li>
|
||||
<li><a href="#sec:anyons" id="toc-sec:anyons">Anyons, Topology and the Chern number</a></li>
|
||||
<li><a href="#ground-state-phases" id="toc-ground-state-phases">Ground State Phases</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
@ -57,7 +57,7 @@ image:
|
||||
<li><a href="#the-majorana-model" id="toc-the-majorana-model">The Majorana Model</a></li>
|
||||
<li><a href="#the-fermion-problem" id="toc-the-fermion-problem">The Fermion Problem</a></li>
|
||||
<li><a href="#an-emergent-gauge-field" id="toc-an-emergent-gauge-field">An Emergent Gauge Field</a></li>
|
||||
<li><a href="#anyons-topology-and-the-chern-number" id="toc-anyons-topology-and-the-chern-number">Anyons, Topology and the Chern number</a></li>
|
||||
<li><a href="#sec:anyons" id="toc-sec:anyons">Anyons, Topology and the Chern number</a></li>
|
||||
<li><a href="#ground-state-phases" id="toc-ground-state-phases">Ground State Phases</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
@ -81,8 +81,7 @@ image:
|
||||
<p>This section introduces the Kitaev honeycomb (KH) model. The KH model is an exactly solvable model of interacting spin<span class="math inline">\(-1/2\)</span> spins on the vertices of a honeycomb lattice. Each bond in the lattice is assigned a label <span class="math inline">\(\alpha \in \{ x, y, z\}\)</span> and that bond couple two spins along the <span class="math inline">\(\alpha\)</span> axis. See fig. <a href="#fig:intro_figure_by_hand">1</a> for a diagram of the setup.</p>
|
||||
<p>This gives us the Hamiltonian <span id="eq:bg-kh-model"><span class="math display">\[ H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha}, \qquad{(1)}\]</span></span> where <span class="math inline">\(\sigma^\alpha_j\)</span> is the <span class="math inline">\(\alpha\)</span> component of a Pauli matrix acting on site <span class="math inline">\(j\)</span> and <span class="math inline">\(\langle j,k\rangle_\alpha\)</span> is a pair of nearest-neighbour indices connected by an <span class="math inline">\(\alpha\)</span>-bond with exchange coupling <span class="math inline">\(J^\alpha\)</span>. Kitaev introduced this model in his seminal 2006 paper <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>.</p>
|
||||
<p>The KH can arise as the result of strong spin-orbit couplings in, for example, the transition metal based compounds <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-Jackeli2009" role="doc-biblioref">2</a>–<a href="#ref-Takagi2019" role="doc-biblioref">6</a>]</span>. The model is highly frustrated: each spin would like to align along a different direction with each of its three neighbours. This cannot be achieved even classically <span class="citation" data-cites="chandraClassicalHeisenbergSpins2010 selaOrderbydisorderSpinorbitalLiquids2014"> [<a href="#ref-chandraClassicalHeisenbergSpins2010" role="doc-biblioref">7</a>,<a href="#ref-selaOrderbydisorderSpinorbitalLiquids2014" role="doc-biblioref">8</a>]</span>. This frustration leads the the model to have a quantum spin liquid (QSL) ground state, a complex many body state with a high degree of entanglement but no long range magnetic order even at zero temperature. While the possibility of a QSL ground state was suggested much earlier <span class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a href="#ref-andersonResonatingValenceBonds1973" role="doc-biblioref">9</a>]</span>, the KH model was one of the first concrete models of the QSL state. The KH model has a rich ground state phase diagram with gapless and gapped phases, the latter supporting fractionalised quasiparticles with both Abelian and non-Abelian quasiparticle excitations. Anyons have been the subject of much attention because, among other reasons, they can be braided through spacetime to achieve noise tolerant quantum computations <span class="citation" data-cites="freedmanTopologicalQuantumComputation2003"> [<a href="#ref-freedmanTopologicalQuantumComputation2003" role="doc-biblioref">10</a>]</span>. At finite temperature the KH model undergoes a phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">11</a>]</span>. The KH model can be solved exactly via a mapping to Majorana fermions. This mapping yields an extensive number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes tied to an emergent gauge field with the remaining fermions are governed by a free fermion hamiltonian.</p>
|
||||
<p>This section will go over the standard model in detail, first discussing <a href="../2_Background/2.2_HKM_Model.html#the-spin-model">the spin model</a>, then detailing the transformation to a <a href="../2_Background/2.2_HKM_Model.html#the-majorana-model">Majorana hamiltonian</a> that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the <a href="#emergent-gauge-fields">emergent gauge fields</a> and the projector. We will then discuss the <a href="#bg-the-ground-state">ground state</a> found via Lieb’s theorem as well as work on generalisations of the ground state to other lattices. Finally we will look at the phase diagram.</p>
|
||||
<p>The <a href="#anyonic-statistics">next section</a> will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics.</p>
|
||||
<p>This section will go over the standard model in detail, first discussing <a href="../2_Background/2.2_HKM_Model.html#the-spin-model">the spin model</a>, then detailing the transformation to a <a href="../2_Background/2.2_HKM_Model.html#the-majorana-model">Majorana hamiltonian</a> that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the <a href="../2_Background/2.2_HKM_Model.html#an-emergent-gauge-field">emergent gauge fields</a> and the projector. The <a href="../2_Background/2.2_HKM_Model.html#sec:anyons">next section</a> will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics. Finally will then discuss the ground state found via Lieb’s theorem as well as work on generalisations of the ground state to other lattices. Finally we will look at the <a href="../2_Background/2.2_HKM_Model.html#ground-state-phases">phase diagram</a>.</p>
|
||||
</section>
|
||||
<section id="the-spin-model" class="level2">
|
||||
<h2>The Spin Model</h2>
|
||||
@ -109,7 +108,7 @@ image:
|
||||
<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 $^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>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 <span class="math inline">\(\tilde \sigma^x \tilde \sigma^y \tilde\sigma^z = iD\)</span> 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 <span class="math inline">\(D_i|\phi\rangle = |\phi\rangle\)</span> 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">\[\begin{aligned}
|
||||
@ -151,23 +150,36 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}
|
||||
\end{aligned}\qquad{(5)}\]</span></span></p>
|
||||
<p>Thus we can interpret the fluxes <span class="math inline">\(\phi_i\)</span> as the exponential of magnetic fluxes of some fictitious gauge field <span class="math inline">\(\vec{A}\)</span> and the bond operators as <span class="math inline">\(u_{ij} = \exp i \int_i^j \vec{A} \cdot d\vec{l}\)</span>. In this analogy to classical electromagnetism, the sets <span class="math inline">\(\{u_ij\}\)</span> that correspond to the same <span class="math inline">\(\{\phi_i\}\)</span> are all gauge equivalent. The fact that fluxes can be written as products of bond operators and composed is a consequence of (the exponential of) Stokes’ theorem. The additional phase factors of <span class="math inline">\(i^n\)</span> can be incorporated as additional <span class="math inline">\(\tfrac{\pi}{2}\)</span> phases but then make little difference when all the plaquettes are hexagons or have an even number of sides. However if the lattice contains odd plaquettes, as in the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">22</a>]</span>, the complex fluxes that appear are a sign that chiral symmetry has been broken.</p>
|
||||
<p>Understanding <span class="math inline">\(u_{ij}\)</span> as a gauge field provides another way to understand the action of the projector. The local projector <span class="math inline">\(P_i = \frac{1 + D_i}{2}\)</span> applied to a state constructs a superposition of the original state and the gauge equivalent state linked to it by flipping the three <span class="math inline">\(u_{ij}\)</span> around site <span class="math inline">\(i\)</span>. The overall projector <span class="math inline">\(P = \prod_i P_i\)</span> can thus be understood as a symmetrisation over all gauge equivalent states, removing the gauge degeneracy introduced by the mapping from spins to Majoranas.</p>
|
||||
<p><img src="/assets/thesis/amk_chapter/intro/flood_fill/flood_fill.gif" id="fig:flood_fill" data-short-caption="Gauge Operators" style="width:100.0%" alt="A honeycomb lattice (in black) along with its dual (in red). (Left) The product of sets of D_j operators (Bold Vertices) can be used to construct arbitrary contractible loops that flip u_{ij} values. If we take the product of every D_j the boundary contracts to a point and disappears. This is a visual proof that \prod_i D_i \propto \mathbb{1}. This observation forms a key part of constructing an explicit expression for the projector, see appendix A.5. (Right) In black and red the edges and dual edges that must be flipped to add vortices at the sites highlighted in orange. Flipping all the plaquettes in the system is not equivalent to the identity. Not that the edges that must be flipped can always be chosen from a spanning tree since loops can always be removed by a gauge transformation." /> <img src="/assets/thesis/amk_chapter/topological_fluxes.png" id="fig:topological_fluxes" data-short-caption="Topological Fluxes" style="width:57.0%" alt="Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts with both had a jam filling and a hole, this analogy would be a lot easier to make [23]." /></p>
|
||||
<!-- <figure>
|
||||
<img src="../../figure_code/amk_chapter/intro/flood_fill/flood_fill.gif" style="max-width:700px;" title="Gauge Operators">
|
||||
<figcaption>
|
||||
A honeycomb lattice (in black) along with its dual (in red). (Left) The product of sets of $D_j$ operators (Bold Vertices) can be used to construct arbitrary contractible loops that flip $u_{ij}$ values. If we take the product of _every_ $D_j$ the boundary contracts to a point and disappears. This is a visual proof that $\prod_i D_i \propto \mathbb{1}$. This observation forms a key part of constructing an explicit expression for the projector, see [appendix A.5](#app-the-projector). (Right) In black and red the edges and dual edges that must be flipped to add vortices at the sites highlighted in orange. Flipping all the _plaquettes_ in the system is __not__ equivalent to the identity. Not that the edges that must be flipped can always be chosen from a spanning tree since loops can always be removed by a gauge transformation.
|
||||
</figcaption>
|
||||
</figure> -->
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/topological_fluxes.png" id="fig:topological_fluxes" data-short-caption="Topological Fluxes" style="width:57.0%" alt="Figure 5: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts with both had a jam filling and a hole, this analogy would be a lot easier to make [23]." />
|
||||
<figcaption aria-hidden="true">Figure 5: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts with both had a jam filling and a hole, this analogy would be a lot easier to make <span class="citation" data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis" role="doc-biblioref">23</a>]</span>.</figcaption>
|
||||
</figure>
|
||||
<p>A final but important point to mention is that is that the local fluxes <span class="math inline">\(\phi_i\)</span> are not quite all there is. We’ve seen that products of <span class="math inline">\(\phi_i\)</span> can be used to construct the flux associated with arbitrary contractible loops. On the plane contractible loops are all there are. However, on the torus we can construct two global fluxes <span class="math inline">\(\Phi_x\)</span> and <span class="math inline">\(\Phi_y\)</span> which correspond to paths tracing the major and minor axes. The four sectors spanned by the <span class="math inline">\(\pm1\)</span> values of these fluxes are gapped away from one another but only by virtual tunnelling processes so the gap decays exponentially with system size <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. Physically <span class="math inline">\(\Phi_x\)</span> and <span class="math inline">\(\Phi_y\)</span> could be thought of as measuring the flux that threads through the hole of the doughnut. In general, surfaces with genus <span class="math inline">\(g\)</span> have <span class="math inline">\(g\)</span> ‘handles’ and <span class="math inline">\(2g\)</span> of these global fluxes. At first glance it may seem this would not have much relevance to physical realisations of the Kitaev model that will likely have a planar geometry with open boundary conditions. However these fluxes are closely linked to topology and the existence of anyonic quasiparticle excitations in the model, which we will discuss next.</p>
|
||||
</section>
|
||||
<section id="anyons-topology-and-the-chern-number" class="level2">
|
||||
<section id="sec:anyons" class="level2">
|
||||
<h2>Anyons, Topology and the Chern number</h2>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/braiding.png" id="fig:braiding" data-short-caption="Braiding in Two Dimensions" style="width:71.0%" alt="Figure 5: Worldlines of particles in two dimensions can become tangled or braided with one another." />
|
||||
<figcaption aria-hidden="true">Figure 5: Worldlines of particles in two dimensions can become tangled or <em>braided</em> with one another.</figcaption>
|
||||
<img src="/assets/thesis/amk_chapter/braiding.png" id="fig:braiding" data-short-caption="Braiding in Two Dimensions" style="width:71.0%" alt="Figure 6: Worldlines of particles in two dimensions can become tangled or braided with one another." />
|
||||
<figcaption aria-hidden="true">Figure 6: Worldlines of particles in two dimensions can become tangled or <em>braided</em> with one another.</figcaption>
|
||||
</figure>
|
||||
<p>To discuss different ground state phases of the KH model we must first review the topic of anyons and topology. The standard argument for the existence of Fermions and Bosons goes like this: the quantum state of a system must pick up a factor of <span class="math inline">\(\pm1\)</span> if two identical particles are swapped. Only <span class="math inline">\(\pm1\)</span> are allowed since swapping twice must correspond to the identity. This argument works in three dimensions for states without topological degeneracy, which seems to be true of the real world, but condensed matter systems are subject to no such constraints.</p>
|
||||
<p>In gapped condensed matter systems, all equal time correlators decay exponentially with distance <span class="citation" data-cites="hastingsLiebSchultzMattisHigherDimensions2004"> [<a href="#ref-hastingsLiebSchultzMattisHigherDimensions2004" role="doc-biblioref">24</a>]</span>. Put another way, the system supports quasiparticles with a definite location in space and a finite extent. As such it is meaningful to consider what would happen to the overall quantum state if we were to adiabatically carry out a series of swaps as described above. This is known as braiding.</p>
|
||||
<p>First we realise that in two dimensions, swapping identical particles twice is not topologically equivalent to the identity, see fig. <a href="#fig:braiding">5</a>. Instead it corresponds to encircling one particle around the other. This means we can in general pick up any complex phase <span class="math inline">\(e^{i\theta}\)</span>, hence the name <strong>any</strong>-ons upon exchange. These are known as Abelian anyons because complex multiplication commutes and hence the group of braiding operations forms and Abelian group.</p>
|
||||
<p>First we realise that in two dimensions, swapping identical particles twice is not topologically equivalent to the identity, see fig. <a href="#fig:braiding">6</a>. Instead it corresponds to encircling one particle around the other. This means we can in general pick up any complex phase <span class="math inline">\(e^{i\theta}\)</span>, hence the name <strong>any</strong>-ons upon exchange. These are known as Abelian anyons because complex multiplication commutes and hence the group of braiding operations forms and Abelian group.</p>
|
||||
<p>The KH model has a topologically degenerate ground state with sectors labelled by the values of the topological fluxes <span class="math inline">\((\Phi_x\)</span>, <span class="math inline">\(\Phi_y)\)</span>. Consider the operation in which a quasiparticle pair is created from the ground state, transported around one of the non-contractible loops and then annihilated together, call them <span class="math inline">\(\mathcal{T}_{x}\)</span> and <span class="math inline">\(\mathcal{T}_{y}\)</span>. These operations move us around within the ground state manifold and they need not commute. This leads to non-Abelian anyons. As Kitaev points out, these operations are not specific to the torus: the operation <span class="math inline">\(\mathcal{T}_{x}\mathcal{T}_{y}\mathcal{T}_{x}^{-1}\mathcal{T}_{y}^{-1}\)</span> corresponds to an operation in which none of the particles crosses the torus, rather one simply winds around the other, hence these effects of relevant even for the planar case.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/intro/types_of_dual_loops_animated/types_of_dual_loops_animated.gif" id="fig:types_of_dual_loops_animated" data-short-caption="Dual Loops and Vortex Pairs" style="width:100.0%" alt="Figure 6: The different kinds of strings and loops that we can make by flipping bond variables or transporting vortices around. (a) Flipping a single bond u_{ij} makes a pair of vortices on either side. (b) Flipping a string of bonds separates the vortex pair spatially. The flipped bonds form a path (in red) on the dual lattice. (c) If we create a vortex-vortex pair, transport one of them around a loop and then annihilate them, we can change the bond sector without changing the vortex sector. This is a manifestation of the gauge symmetry of the bond sector. (d) If we transport a vortex around the major or minor axes of the torus, we create a non-contractable loop of bonds \hat{\mathcal{T}}_{x/y}. Unlike all the other dual loops, These operators cannot be constructed from the contractable loops created by D_j. operators and they flip the value of the topological fluxes." />
|
||||
<figcaption aria-hidden="true">Figure 6: The different kinds of strings and loops that we can make by flipping bond variables or transporting vortices around. (a) Flipping a single bond <span class="math inline">\(u_{ij}\)</span> makes a pair of vortices on either side. (b) Flipping a string of bonds separates the vortex pair spatially. The flipped bonds form a path (in red) on the dual lattice. (c) If we create a vortex-vortex pair, transport one of them around a loop and then annihilate them, we can change the bond sector without changing the vortex sector. This is a manifestation of the gauge symmetry of the bond sector. (d) If we transport a vortex around the major or minor axes of the torus, we create a non-contractable loop of bonds <span class="math inline">\(\hat{\mathcal{T}}_{x/y}\)</span>. Unlike all the other dual loops, These operators cannot be constructed from the contractable loops created by <span class="math inline">\(D_j\)</span>. operators and they flip the value of the topological fluxes.</figcaption>
|
||||
</figure>
|
||||
<!-- <figure>
|
||||
<img src="../../figure_code/amk_chapter/intro/types_of_dual_loops_animated/types_of_dual_loops_animated.gif" style="max-width:700px;" title="Dual Loops and Vortex Pairs">
|
||||
<figcaption>
|
||||
The different kinds of strings and loops that we can make by flipping bond variables or transporting vortices around. (a) Flipping a single bond $u_{ij}$ makes a pair of vortices on either side. (b) Flipping a string of bonds separates the vortex pair spatially. The flipped bonds form a path (in red) on the dual lattice. (c) If we create a vortex-vortex pair, transport one of them around a loop and then annihilate them, we can change the bond sector without changing the vortex sector. This is a manifestation of the gauge symmetry of the bond sector. (d) If we transport a vortex around the major or minor axes of the torus, we create a non-contractable loop of bonds $\hat{\mathcal{T}}_{x/y}$. Unlike all the other dual loops, These operators cannot be constructed from the contractable loops created by $D_j$. operators and they flip the value of the topological fluxes.
|
||||
|
||||
This all works the same way for the amorphous lattice but the diagram is a lot messier so I've stuck with the honeycomb here.
|
||||
</figcaption>
|
||||
</figure> -->
|
||||
<p>In condensed matter systems, the existence of anyonic excitations automatically implies the system has topological ground state degeneracy on the torus <span class="citation" data-cites="einarssonFractionalStatisticsTorus1990"> [<a href="#ref-einarssonFractionalStatisticsTorus1990" role="doc-biblioref">25</a>]</span> and indeed anyons and topology are intimately linked <span class="citation" data-cites="oshikawaTopologicalDegeneracyNonAbelian2007 Chung_Topological_quantum2010 yaoAlgebraicSpinLiquid2009"> [<a href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007" role="doc-biblioref">26</a>–<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">28</a>]</span>. Originally a concept used to describe complex vector bundles in algebraic topology <span class="citation" data-cites="chernCharacteristicClassesHermitian1946"> [<a href="#ref-chernCharacteristicClassesHermitian1946" role="doc-biblioref">29</a>]</span>, the Chern number has found use in physics as a powerful tool diagnostic tool for topological systems. Kitaev showed that vortices in the KH model are Abelian when the Chern number is even and non-Abelian when the Chern number is odd. In the odd case the non-Abelian statistics of the vortices arise due to unpaired Majorana modes that are bound to them.</p>
|
||||
<p>Recently, topological systems have gained interest because of proposals to use their ground state degeneracy to implement both passively fault tolerant and actively stabilised quantum computations <span class="citation" data-cites="kitaev_fault-tolerant_2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">30</a>–<a href="#ref-hastingsDynamicallyGeneratedLogical2021" role="doc-biblioref">32</a>]</span>.</p>
|
||||
</section>
|
||||
@ -182,16 +194,8 @@ 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>To surmise, 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 dynamical spin-spin correlation functions are zero beyond nearest neighbour separation <span class="citation" data-cites="baskaranExactResultsSpin2007"> [<a href="#ref-baskaranExactResultsSpin2007" role="doc-biblioref">35</a>]</span>. 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>
|
||||
<ul>
|
||||
<li>RuCl_3 is the classic QSL candidate material</li>
|
||||
<li>really follows the Kitaev-Heisenberg model</li>
|
||||
<li>experimental probes include inelastic neutron scattering, Raman scattering</li>
|
||||
</ul>
|
||||
<p>Next Section: <a href="../2_Background/2.4_Disorder.html">Disorder and Localisation</a></p>
|
||||
</section>
|
||||
</section>
|
||||
|
||||
@ -60,24 +60,25 @@ image:
|
||||
</div>
|
||||
<section id="bg-disorder-and-localisation" class="level1">
|
||||
<h1>Disorder and Localisation</h1>
|
||||
<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>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. In this model one 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 in 1958 <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">2</a>]</span> showed that in a simple model, there is some critical level of disorder at which <strong>all</strong> single particle eigenstates localise.</p>
|
||||
<p>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>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>. Later Mott showed that in other contexts extended Bloch states and localised states can 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>
|
||||
\]</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 gogolinEquilibrationThermalisationEmergence2016"> [<a href="#ref-imbrieManyBodyLocalizationQuantum2016" role="doc-biblioref">13</a>,<a href="#ref-gogolinEquilibrationThermalisationEmergence2016" role="doc-biblioref">14</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>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">15</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">16</a>,<a href="#ref-schiulazDynamicsManybodyLocalized2015" role="doc-biblioref">17</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">16</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">18</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">19</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 anti-correlations 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 flux <span class="citation" data-cites="Nasu_Thermal_2015"> [<a href="#ref-Nasu_Thermal_2015" role="doc-biblioref">28</a>]</span> and bond <span class="citation" data-cites="knolle_dynamics_2016"> [<a href="#ref-knolle_dynamics_2016" role="doc-biblioref">29</a>]</span> disorder. 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">30</a>]</span>. I will look at how adding lattice disorder to the mix affects the picture. It has also been shown that the KH model exhibits disorder free localisation after a quantum quench <span class="citation" data-cites="zhuSubdiffusiveDynamicsCritical2021"> [<a href="#ref-zhuSubdiffusiveDynamicsCritical2021" role="doc-biblioref">31</a>]</span>.</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. 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">20</a>–<a href="#ref-betteridge1973possible" role="doc-biblioref">23</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. A standard method for generating such graphs with coordination number <span class="math inline">\(d+1\)</span> is Voronoi tessellation <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">24</a>,<a href="#ref-marsalTopologicalWeaireThorpeModels2020" role="doc-biblioref">25</a>]</span>. The Harris <span class="citation" data-cites="harrisEffectRandomDefects1974"> [<a href="#ref-harrisEffectRandomDefects1974" role="doc-biblioref">26</a>]</span> and the Imry-Mar <span class="citation" data-cites="imryRandomFieldInstabilityOrdered1975"> [<a href="#ref-imryRandomFieldInstabilityOrdered1975" role="doc-biblioref">27</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 while 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. Both these criteria are modified for the case of topological disorder where the Euler equation an vertex degree constraints lead to strong anti-correlations which mean that topological disorder is effectively weaker than standard disorder in two dimensions <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">28</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">29</a>]</span>. This does not apply to the three dimensional Voronoi lattices where the Euler equation contains an extra volume term and so is effectively 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 flux <span class="citation" data-cites="Nasu_Thermal_2015"> [<a href="#ref-Nasu_Thermal_2015" role="doc-biblioref">30</a>]</span> and bond <span class="citation" data-cites="knolle_dynamics_2016"> [<a href="#ref-knolle_dynamics_2016" role="doc-biblioref">31</a>]</span> disorder. 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">32</a>]</span>. It has also been shown that the KH model exhibits disorder free localisation after a quantum quench <span class="citation" data-cites="zhuSubdiffusiveDynamicsCritical2021"> [<a href="#ref-zhuSubdiffusiveDynamicsCritical2021" role="doc-biblioref">33</a>]</span>.</p>
|
||||
<p>In chapter 4 we will put the Kitaev model onto two dimensional Voronoi lattices and show that much of the rich character of the model is preserved despite the lack of long range order.</p>
|
||||
<section id="diagnosing-localisation-in-practice" class="level2">
|
||||
<h2>Diagnosing Localisation in practice</h2>
|
||||
<figure>
|
||||
@ -91,17 +92,19 @@ H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U
|
||||
<p><span class="math display">\[
|
||||
P^{-1} = \sum_i |\psi_i|^4
|
||||
\]</span></p>
|
||||
<p>The name derive from the fact that this operator acts as a measure of the volume where the wavefunction is significantly different from zero. They can alternatively be thougt of as providing a measure of the average diameter <span class="math inline">\(R\)</span> from <span class="math inline">\(R = P^{1/d}\)</span>, see fig. <a href="#fig:localisation_radius_vs_length">1</a> for the distinction between <span class="math inline">\(R\)</span> and <span class="math inline">\(\lambda\)</span>.</p>
|
||||
<p>The name derive from the fact that this operator acts as a measure of the volume where the wavefunction is significantly different from zero. They can alternatively be thought of as providing a measure of the average diameter <span class="math inline">\(R\)</span> from <span class="math inline">\(R = P^{1/d}\)</span>. See fig. <a href="#fig:localisation_radius_vs_length">1</a> for the distinction between <span class="math inline">\(R\)</span> and <span class="math inline">\(\lambda\)</span>.</p>
|
||||
<p>For localised states, the <em>inverse</em> participation ratio <span class="math inline">\(P^{-1}\)</span> is independent of system size while for plane wave states in <span class="math inline">\(d\)</span> dimensions <span class="math inline">\(P^{-1} = L^{-d}\)</span>. States may also be intermediate between localised and extended, described by their fractal dimensionality <span class="math inline">\(d > d* > 0\)</span>:</p>
|
||||
<p><span class="math display">\[
|
||||
P(L)^{-1} \sim L^{-d*}
|
||||
\]</span></p>
|
||||
<p>For finite size systems, these relations only hold once the system size <span class="math inline">\(L\)</span> is much greater than the localisation length. When the localisation length is comparable to the system size the states contribute to transport. This is called weak localisation <span class="citation" data-cites="altshulerMagnetoresistanceHallEffect1980 dattaElectronicTransportMesoscopic1995"> [<a href="#ref-altshulerMagnetoresistanceHallEffect1980" role="doc-biblioref">32</a>,<a href="#ref-dattaElectronicTransportMesoscopic1995" role="doc-biblioref">33</a>]</span>.</p>
|
||||
<p>Such intermediate states tend to appear as critical phenomena near mobility edges <span class="citation" data-cites="eversAndersonTransitions2008"> [<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">34</a>]</span>. For finite size systems, these relations only hold once the system size <span class="math inline">\(L\)</span> is much greater than the localisation length. When the localisation length is comparable to the system size the states contribute to transport. This is called weak localisation <span class="citation" data-cites="altshulerMagnetoresistanceHallEffect1980 dattaElectronicTransportMesoscopic1995"> [<a href="#ref-altshulerMagnetoresistanceHallEffect1980" role="doc-biblioref">35</a>,<a href="#ref-dattaElectronicTransportMesoscopic1995" role="doc-biblioref">36</a>]</span>.</p>
|
||||
<p>For extended states <span class="math inline">\(d* = 0\)</span> while for localised ones <span class="math inline">\(d* = 0\)</span>. In both chapters I will use an energy resolved IPR <span class="math display">\[
|
||||
DOS(\omega) = \sum_n \delta(\omega - \epsilon_n)\\
|
||||
IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) |\psi_{n,i}|^4
|
||||
\]</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 I bin the energies and IPRs into a fine energy grid and use the mean within each bin.</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>
|
||||
<p><strong>Chapter Summary</strong></p>
|
||||
<p>In this chapter we have covered the Falicov-Kimball model, the Kitaev Honeycomb model and the theory of disorder and localisation. We saw that the FK model is one of immobile species (spins) interacting with an itinerant quantum species (electrons). While the KH model is specified in terms of spins on a honeycomb lattice interacting via a highly anisotropic Ising coupling, it can be transformed into one of Majorana fermions interacting with a classical gauge field that supports immobile flux excitations. In each case it is the immobile species that makes each model exactly solvable. Both models have rich ground state and thermodynamic phase diagrams. The last part of this chapter dealt with disorder and how it almost inevitably leads to localisation. Both the FK and KH models are effectively disordered at finite temperatures by their immobile species. In the next chapter we will look at a version of the FK model in one dimension augmented with long range interactions in order to retain its ordered phase. The model is translation invariant but we will see that it exhibits disorder free localisation. After that we will look at the KH model defined on an amorphous lattice with vertex degree <span class="math inline">\(z=3\)</span>.</p>
|
||||
<p>Next Chapter: <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falicov-Kimball Model</a></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
@ -146,65 +149,74 @@ IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) |\psi_{n,i
|
||||
<div id="ref-imbrieManyBodyLocalizationQuantum2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">J. Z. Imbrie, <em><a href="https://doi.org/10.1007/s10955-016-1508-x">On Many-Body Localization for Quantum Spin Chains</a></em>, J Stat Phys <strong>163</strong>, 998 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-gogolinEquilibrationThermalisationEmergence2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">C. Gogolin and J. Eisert, <em><a href="https://doi.org/10.1088/0034-4885/79/5/056001">Equilibration, Thermalisation, and the Emergence of Statistical Mechanics in Closed Quantum Systems</a></em>, Rep. Prog. Phys. <strong>79</strong>, 056001 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-kagan1984localization" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">Y. Kagan and L. Maksimov, <em>Localization in a System of Interacting Particles Diffusing in a Regular Crystal</em>, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki <strong>87</strong>, 348 (1984).</div>
|
||||
</div>
|
||||
<div id="ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">N. Y. Yao, C. R. Laumann, J. I. Cirac, M. D. Lukin, and J. E. Moore, <em><a href="https://doi.org/10.1103/PhysRevLett.117.240601">Quasi-Many-Body Localization in Translation-Invariant Systems</a></em>, Phys. Rev. Lett. <strong>117</strong>, 240601 (2016).</div>
|
||||
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">N. Y. Yao, C. R. Laumann, J. I. Cirac, M. D. Lukin, and J. E. Moore, <em><a href="https://doi.org/10.1103/PhysRevLett.117.240601">Quasi-Many-Body Localization in Translation-Invariant Systems</a></em>, Phys. Rev. Lett. <strong>117</strong>, 240601 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-schiulazDynamicsManybodyLocalized2015" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">M. Schiulaz, A. Silva, and M. Müller, <em><a href="https://doi.org/10.1103/PhysRevB.91.184202">Dynamics in Many-Body Localized Quantum Systems Without Disorder</a></em>, Phys. Rev. B <strong>91</strong>, 184202 (2015).</div>
|
||||
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">M. Schiulaz, A. Silva, and M. Müller, <em><a href="https://doi.org/10.1103/PhysRevB.91.184202">Dynamics in Many-Body Localized Quantum Systems Without Disorder</a></em>, Phys. Rev. B <strong>91</strong>, 184202 (2015).</div>
|
||||
</div>
|
||||
<div id="ref-smithDisorderFreeLocalization2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[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>
|
||||
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, <em><a href="https://doi.org/10.1103/PhysRevLett.118.266601">Disorder-Free Localization</a></em>, Phys. Rev. Lett. <strong>118</strong>, 266601 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
<div class="csl-left-margin">[19] </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>
|
||||
</div>
|
||||
<div id="ref-Yonezawa1983" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
<div class="csl-left-margin">[20] </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>
|
||||
</div>
|
||||
<div id="ref-zallen2008physics" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
<div class="csl-left-margin">[21] </div><div class="csl-right-inline">R. Zallen, <em>The Physics of Amorphous Solids</em> (John Wiley & Sons, 2008).</div>
|
||||
</div>
|
||||
<div id="ref-Weaire1971" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
<div class="csl-left-margin">[22] </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>
|
||||
</div>
|
||||
<div id="ref-betteridge1973possible" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
<div class="csl-left-margin">[23] </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>
|
||||
</div>
|
||||
<div id="ref-mitchellAmorphousTopologicalInsulators2018" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[24] </div><div class="csl-right-inline">N. P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine, <em><a href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous topological insulators constructed from random point sets</a></em>, Nature Phys <strong>14</strong>, 380 (2018).</div>
|
||||
</div>
|
||||
<div id="ref-marsalTopologicalWeaireThorpeModels2020" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[25] </div><div class="csl-right-inline">Q. Marsal, D. Varjas, and A. G. Grushin, <em><a href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire-Thorpe Models of Amorphous Matter</a></em>, Proc. Natl. Acad. Sci. U.S.A. <strong>117</strong>, 30260 (2020).</div>
|
||||
</div>
|
||||
<div id="ref-harrisEffectRandomDefects1974" class="csl-entry" role="doc-biblioentry">
|
||||
<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 class="csl-left-margin">[26] </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>
|
||||
<div id="ref-imryRandomFieldInstabilityOrdered1975" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
</div>
|
||||
<div id="ref-changlaniChargeDensityWaves2016" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
<div class="csl-left-margin">[27] </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>
|
||||
</div>
|
||||
<div id="ref-barghathiPhaseTransitionsRandom2014" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
<div class="csl-left-margin">[28] </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>
|
||||
</div>
|
||||
<div id="ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" class="csl-entry" role="doc-biblioentry">
|
||||
<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 class="csl-left-margin">[29] </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-Nasu_Thermal_2015" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">J. Nasu, M. Udagawa, and Y. Motome, <em><a href="https://doi.org/10.1103/PhysRevB.92.115122">Thermal Fractionalization of Quantum Spins in a Kitaev Model: Temperature-Linear Specific Heat and Coherent Transport of Majorana Fermions</a></em>, Phys. Rev. B <strong>92</strong>, 115122 (2015).</div>
|
||||
<div class="csl-left-margin">[30] </div><div class="csl-right-inline">J. Nasu, M. Udagawa, and Y. Motome, <em><a href="https://doi.org/10.1103/PhysRevB.92.115122">Thermal Fractionalization of Quantum Spins in a Kitaev Model: Temperature-Linear Specific Heat and Coherent Transport of Majorana Fermions</a></em>, Phys. Rev. B <strong>92</strong>, 115122 (2015).</div>
|
||||
</div>
|
||||
<div id="ref-knolle_dynamics_2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">J. Knolle, Dynamics of a Quantum Spin Liquid, Max Planck Institute for the Physics of Complex Systems, Dresden, 2016.</div>
|
||||
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">J. Knolle, Dynamics of a Quantum Spin Liquid, Max Planck Institute for the Physics of Complex Systems, Dresden, 2016.</div>
|
||||
</div>
|
||||
<div id="ref-wenDisorderedRouteCoulomb2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[30] </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 class="csl-left-margin">[32] </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 id="ref-zhuSubdiffusiveDynamicsCritical2021" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">G.-Y. Zhu and M. Heyl, <em><a href="https://doi.org/10.1103/PhysRevResearch.3.L032069">Subdiffusive Dynamics and Critical Quantum Correlations in a Disorder-Free Localized Kitaev Honeycomb Model Out of Equilibrium</a></em>, Phys. Rev. Research <strong>3</strong>, L032069 (2021).</div>
|
||||
<div class="csl-left-margin">[33] </div><div class="csl-right-inline">G.-Y. Zhu and M. Heyl, <em><a href="https://doi.org/10.1103/PhysRevResearch.3.L032069">Subdiffusive Dynamics and Critical Quantum Correlations in a Disorder-Free Localized Kitaev Honeycomb Model Out of Equilibrium</a></em>, Phys. Rev. Research <strong>3</strong>, L032069 (2021).</div>
|
||||
</div>
|
||||
<div id="ref-eversAndersonTransitions2008" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[34] </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>
|
||||
</div>
|
||||
<div id="ref-altshulerMagnetoresistanceHallEffect1980" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee, <em><a href="https://doi.org/10.1103/PhysRevB.22.5142">Magnetoresistance and Hall Effect in a Disordered Two-Dimensional Electron Gas</a></em>, Phys. Rev. B <strong>22</strong>, 5142 (1980).</div>
|
||||
<div class="csl-left-margin">[35] </div><div class="csl-right-inline">B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee, <em><a href="https://doi.org/10.1103/PhysRevB.22.5142">Magnetoresistance and Hall Effect in a Disordered Two-Dimensional Electron Gas</a></em>, Phys. Rev. B <strong>22</strong>, 5142 (1980).</div>
|
||||
</div>
|
||||
<div id="ref-dattaElectronicTransportMesoscopic1995" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[33] </div><div class="csl-right-inline">S. Datta, <em><a href="https://doi.org/10.1017/CBO9780511805776">Electronic Transport in Mesoscopic Systems</a></em> (Cambridge University Press, Cambridge, 1995).</div>
|
||||
<div class="csl-left-margin">[36] </div><div class="csl-right-inline">S. Datta, <em><a href="https://doi.org/10.1017/CBO9780511805776">Electronic Transport in Mesoscopic Systems</a></em> (Cambridge University Press, Cambridge, 1995).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
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<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 Falicov-Kimball Model</a></li>
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<li><a href="#sec:lrfk-model" id="toc-sec:lrfk-model">The Model</a></li>
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<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 Falicov-Kimball Model</a></li>
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<p>3 The Long Range Falicov-Kimball Model</p>
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<section id="chap:3-the-long-range-falicov-kimball-model" class="level1">
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<h1>3 The Long Range Falicov-Kimball Model</h1>
|
||||
<p><strong>Contributions</strong></p>
|
||||
<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>
|
||||
<p>The code is available online <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">2</a>]</span>.</p>
|
||||
<p>Johannes had the initial idea to use a long range Ising term to stabilise order in a one dimension Falikov-Kimball model. Josef developed a proof of concept during a summer project at Imperial along with Alexander Belcik. I wrote the simulation code and performed all the analysis presented here.</p>
|
||||
<p><strong>Chapter Summary</strong></p>
|
||||
<p>The paper is organised as follows. First, I will introduce the long range Falicov-Kimball (LRFK) model and motivate its definition. Second, I will present the <a href="../3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html#sec:lrfk-methods">methods</a> used to solve it numerically, including Markov chain Monte Carlo and finite size scaling. I will then present and interpret the <a href="../3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html#sec:lrfk-results">results</a> obtained.</p>
|
||||
</section>
|
||||
<section id="sec:lrfk-model" class="level1">
|
||||
<h1>The Model</h1>
|
||||
<p>Dimensionality is crucial for the physics of both localisation and phase transitions. We have already seen that the one dimensional standard FK model cannot support an ordered phase at finite temperatures and therefore has no finite temperature phase transition (FTPT).</p>
|
||||
<p>On bipartite lattices in dimensions 2 and above the FK model exhibits a finite temperature phase transition to an ordered charge density wave (CDW) phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">3</a>]</span>. In this phase, the spins order anti-ferromagnetically, breaking the <span class="math inline">\(\mathbb{Z}_2\)</span> symmetry. In 1D, however, Periel’s argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">4</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">5</a>]</span> states that domain walls only introduce a constant energy penalty into the free energy while bringing a entropic contribution logarithmic in system size. Hence the 1D model does not have a finite temperature phase transition. However 1D systems are much easier to study numerically and admit simpler realisations experimentally. We therefore introduce a long-range coupling between the ions in order to stabilise a CDW phase in 1D.</p>
|
||||
<p>We interpret the FK model as a model of spinless fermions, <span class="math inline">\(c^\dagger_{i}\)</span>, hopping on a 1D lattice against a classical Ising spin background, <span class="math inline">\(S_i \in {\pm \frac{1}{2}}\)</span>. The fermions couple to the spins via an onsite interaction with strength <span class="math inline">\(U\)</span> which we supplement by a long-range interaction, <span class="math display">\[
|
||||
J_{ij} = 4\kappa J\; (-1)^{|i-j|} |i-j|^{-\alpha},
|
||||
\]</span></p>
|
||||
<p>between the spins. The additional coupling is very similar to that of the long range Ising model, it stabilises the Antiferromagnetic (AFM) order of the Ising spins which promotes the finite temperature CDW phase of the fermionic sector.</p>
|
||||
<p>The hopping strength of the electrons, <span class="math inline">\(t = 1\)</span>, sets the overall energy scale and we concentrate throughout on the particle-hole symmetric point at zero chemical potential and half filling <span class="citation" data-cites="gruberFalicovKimballModelReview1996"> [<a href="#ref-gruberFalicovKimballModelReview1996" role="doc-biblioref">6</a>]</span>.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{i} (c^\dagger_{i}c_{i+1} + \textit{h.c.)}\\
|
||||
& + \sum_{i, j}^{N} J_{ij} S_i S_j
|
||||
\label{eq:HFK}\end{aligned}\]</span></p>
|
||||
<p>Without proper normalisation, the long range coupling would render the critical temperature strongly system size dependent for small system sizes. Within a mean field approximation the critical temperature scales with the effective coupling to all the neighbours of each site, which for a system with <span class="math inline">\(N\)</span> sites is <span class="math inline">\(\sum_{i=1}^{N} i^{-\alpha}\)</span>. Hence, the normalisation <span class="math inline">\(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\)</span>, renders the critical temperature independent of system size in the mean field approximation. This greatly improves the finite size behaviour of the model.</p>
|
||||
<p>Taking the limit <span class="math inline">\(U = 0\)</span> decouples the spins from the fermions, which gives a spin sector governed by a classical LRI model. Note, the transformation of the spins <span class="math inline">\(S_i \to (-1)^{i} S_i\)</span> maps the AFM model to the FM one. As discussed in the background section, Peierls’ classic argument can be extended to long range couplings to show that, for the 1D LRI model, a power law decay of <span class="math inline">\(\alpha < 2\)</span> is required for a FTPT as the energy of defect domain then scales with the system size and can overcome the entropic contribution. A renormalisation group analysis supports this finding and shows that the critical exponents are only universal for <span class="math inline">\(\alpha \leq 3/2\)</span> <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968 thoulessLongRangeOrderOneDimensional1969 angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">7</a>–<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">9</a>]</span>. In the following, we choose <span class="math inline">\(\alpha = 5/4\)</span> to avoid the additional complexity of non-universal critical points.</p>
|
||||
<p>Next Section: <a href="../3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html">Methods</a></p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-hodsonOnedimensionalLongrangeFalikovKimball2021" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">T. Hodson, J. Willsher, and J. Knolle, <em><a href="https://doi.org/10.1103/PhysRevB.104.045116">One-Dimensional Long-Range Falikov-Kimball Model: Thermal Phase Transition and Disorder-Free Localization</a></em>, Phys. Rev. B <strong>104</strong>, 045116 (2021).</div>
|
||||
</div>
|
||||
<div id="ref-hodsonMCMCFKModel2021" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">T. Hodson, <em><a href="https://doi.org/10.5281/zenodo.4593904">Markov Chain Monte Carlo for the Kitaev Model</a></em>, (2021).</div>
|
||||
</div>
|
||||
<div id="ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">M. M. Maśka and K. Czajka, <em><a href="https://doi.org/10.1103/PhysRevB.74.035109">Thermodynamics of the Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo Study</a></em>, Phys. Rev. B <strong>74</strong>, 035109 (2006).</div>
|
||||
</div>
|
||||
<div id="ref-peierlsIsingModelFerromagnetism1936" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">R. Peierls, <em><a href="https://doi.org/10.1017/S0305004100019174">On Ising’s Model of Ferromagnetism</a></em>, Mathematical Proceedings of the Cambridge Philosophical Society <strong>32</strong>, 477 (1936).</div>
|
||||
</div>
|
||||
<div id="ref-kennedyItinerantElectronModel1986" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">T. Kennedy and E. H. Lieb, <em><a href="https://doi.org/10.1016/0378-4371(86)90188-3">An Itinerant Electron Model with Crystalline or Magnetic Long Range Order</a></em>, Physica A: Statistical Mechanics and Its Applications <strong>138</strong>, 320 (1986).</div>
|
||||
</div>
|
||||
<div id="ref-gruberFalicovKimballModelReview1996" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">C. Gruber and N. Macris, <em>The Falicov-Kimball Model: A Review of Exact Results and Extensions</em>, Helvetica Physica Acta <strong>69</strong>, (1996).</div>
|
||||
</div>
|
||||
<div id="ref-ruelleStatisticalMechanicsOnedimensional1968" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">D. Ruelle, <em><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys. <strong>9</strong>, 267 (1968).</div>
|
||||
</div>
|
||||
<div id="ref-thoulessLongRangeOrderOneDimensional1969" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">D. J. Thouless, <em><a href="https://doi.org/10.1103/PhysRev.187.732">Long-Range Order in One-Dimensional Ising Systems</a></em>, Phys. Rev. <strong>187</strong>, 732 (1969).</div>
|
||||
</div>
|
||||
<div id="ref-angeliniRelationsShortrangeLongrange2014" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E <strong>89</strong>, 062120 (2014).</div>
|
||||
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|
||||
<li><a href="#thermodynamics-of-the-lrfk-model" id="toc-thermodynamics-of-the-lrfk-model">Thermodynamics of the LRFK Model</a></li>
|
||||
<li><a href="#markov-chain-monte-carlo-and-emergent-disorder" id="toc-markov-chain-monte-carlo-and-emergent-disorder">Markov Chain Monte Carlo and Emergent Disorder</a></li>
|
||||
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|
||||
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|
||||
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<p>3 The Long Range Falicov-Kimball Model</p>
|
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|
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|
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<section id="sec:lrfk-methods" class="level1">
|
||||
<h1>Methods</h1>
|
||||
<p>To evaluate thermodynamic averages I perform classical Markov Chain Monte Carlo random walks over the space of spin configurations of the LRFK model, at each step diagonalising the effective electronic Hamiltonian <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">1</a>]</span>. Using a binder-cumulant method <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">2</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">3</a>]</span>, I demonstrate the model has a finite temperature phase transition when the interaction is sufficiently long ranged. In this section I will discuss the thermodynamics of the model and how they are amenable to an exact Markov Chain Monte Carlo method.</p>
|
||||
<section id="thermodynamics-of-the-lrfk-model" class="level2">
|
||||
<h2>Thermodynamics of the LRFK Model</h2>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/lsr/pdf_figs/raw_steps_single_flip.svg" id="fig:raw_steps_single_flip" data-short-caption="Comparison of different proposal distributions" style="width:100.0%" alt="Figure 1: Two MCMC walks starting from the CDW state for a system with N = 100 sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation m = N^{-1} \sum_i (-1)^i \; S_i order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5" />
|
||||
<figcaption aria-hidden="true">Figure 1: Two MCMC walks starting from the CDW state for a system with <span class="math inline">\(N = 100\)</span> sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. <span class="math inline">\(t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5\)</span></figcaption>
|
||||
</figure>
|
||||
<p>The classical Markov Chain Monte Carlo (MCMC) method which we discuss in the following allows us to solve our long-range FK model efficiently, yielding unbiased estimates of thermal expectation values.</p>
|
||||
<p>Since the spin configurations are classical, the LRFK Hamiltonian can be split into a classical spin part <span class="math inline">\(H_s\)</span> and an operator valued part <span class="math inline">\(H_c\)</span>.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
H_s& = - \frac{U}{2}S_i + \sum_{i, j}^{N} J_{ij} S_i S_j \\
|
||||
H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i}) \end{aligned}\]</span></p>
|
||||
<p>The partition function can then be written as a sum over spin configurations, <span class="math inline">\(\vec{S} = (S_0, S_1...S_{N-1})\)</span>:</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \mathrm{Tr} e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s} \mathrm{Tr}_c e^{-\beta H_c} .\end{aligned}\]</span></p>
|
||||
<p>The contribution of <span class="math inline">\(H_c\)</span> to the grand canonical partition function can be obtained by performing the sum over eigenstate occupation numbers giving <span class="math inline">\(-\beta F_c[\vec{S}] = \sum_k \ln{(1 + e^{- \beta \epsilon_k})}\)</span> where <span class="math inline">\({\epsilon_k[\vec{S}]}\)</span> are the eigenvalues of the matrix representation of <span class="math inline">\(H_c\)</span> determined through exact diagonalisation. This gives a partition function containing a classical energy which corresponds to the long-range interaction of the spins, and a free energy which corresponds to the quantum subsystem.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}\end{aligned}\]</span></p>
|
||||
</section>
|
||||
<section id="markov-chain-monte-carlo-and-emergent-disorder" class="level2">
|
||||
<h2>Markov Chain Monte Carlo and Emergent Disorder</h2>
|
||||
<p>Classical MCMC defines a weighted random walk over the spin states <span class="math inline">\((\vec{S}_0, \vec{S}_1, \vec{S}_2, ...)\)</span>, such that the likelihood of visiting a particular state converges to its Boltzmann probability <span class="math inline">\(p(\vec{S}) = \mathcal{Z}^{-1} e^{-\beta E}\)</span>. Hence, any observable can be estimated as a mean over the states visited by the walk <span class="citation" data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"> [<a href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">4</a>–<a href="#ref-wolffMonteCarloErrors2004" role="doc-biblioref">6</a>]</span>,</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\label{eq:thermal_expectation}
|
||||
\langle O \rangle & = \sum_{\vec{S}} p(\vec{S}) \langle O \rangle\\
|
||||
& = \sum_{i = 0}^{M} \langle O\rangle \pm \mathcal{O}(M^{-\tfrac{1}{2}})
|
||||
\end{aligned}\]</span></p>
|
||||
<p>where the former sum runs over the entire state space while the later runs over all the state visited by a particular MCMC run.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\langle O \rangle_{\vec{S}}& = \sum_{\nu} n_F(\epsilon_{\nu}) \langle O \rangle{\nu}
|
||||
\end{aligned}\]</span></p>
|
||||
<p>Where <span class="math inline">\(\nu\)</span> runs over the eigenstates of <span class="math inline">\(H_c\)</span> for a particular spin configuration and <span class="math inline">\(n_F(\epsilon) = \left(e^{-\beta\epsilon} + 1\right)^{-1}\)</span> is the Fermi function.</p>
|
||||
<p>The choice of the transition function for MCMC is under-determined as one only needs to satisfy a set of balance conditions for which there are many solutions <span class="citation" data-cites="kellyReversibilityStochasticNetworks1981"> [<a href="#ref-kellyReversibilityStochasticNetworks1981" role="doc-biblioref">7</a>]</span>. Here, we incorporate a modification to the standard Metropolis-Hastings algorithm <span class="citation" data-cites="hastingsMonteCarloSampling1970"> [<a href="#ref-hastingsMonteCarloSampling1970" role="doc-biblioref">8</a>]</span> gleaned from Krauth <span class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">9</a>]</span>.</p>
|
||||
<p>The standard algorithm decomposes the transition probability into <span class="math inline">\(\mathcal{T}(a \to b) = p(a \to b)\mathcal{A}(a \to b)\)</span>. Here, <span class="math inline">\(p\)</span> is the proposal distribution that we can directly sample from while <span class="math inline">\(\mathcal{A}\)</span> is the acceptance probability. The standard Metropolis-Hastings choice is</p>
|
||||
<p><span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, \frac{p(b\to a)}{p(a\to b)} e^{-\beta \Delta E}\right)\;,\]</span></p>
|
||||
<p>with <span class="math inline">\(\Delta E = E_b - E_a\)</span>. The walk then proceeds by sampling a state <span class="math inline">\(b\)</span> from <span class="math inline">\(p\)</span> and moving to <span class="math inline">\(b\)</span> with probability <span class="math inline">\(\mathcal{A}(a \to b)\)</span>. The latter operation is typically implemented by performing a transition if a uniform random sample from the unit interval is less than <span class="math inline">\(\mathcal{A}(a \to b)\)</span> and otherwise repeating the current state as the next step in the random walk. The proposal distribution is often symmetric so does not appear in <span class="math inline">\(\mathcal{A}\)</span>. Here, we flip a small number of sites in <span class="math inline">\(b\)</span> at random to generate proposals, which is a symmetric proposal.</p>
|
||||
<p>In our computations <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">10</a>]</span> the modification to this algorithm that we employ is based on the observation that the free energy of the FK system is composed of a classical part which is much quicker to compute than the quantum part. Hence, we can obtain a computational speed up by first considering the value of the classical energy difference <span class="math inline">\(\Delta H_s\)</span> and rejecting the transition if the former is too high. We only compute the quantum energy difference <span class="math inline">\(\Delta F_c\)</span> if the transition is accepted. We then perform a second rejection sampling step based upon it. This corresponds to two nested comparisons with the majority of the work only occurring if the first test passes. This modified scheme has the acceptance function <span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]</span></p>
|
||||
<p>For the model parameters used, we find that with our new scheme the matrix diagonalisation is skipped around 30% of the time at <span class="math inline">\(T = 2.5\)</span> and up to 80% at <span class="math inline">\(T = 1.5\)</span>. We observe that for <span class="math inline">\(N = 50\)</span>, the matrix diagonalisation, if it occurs, occupies around 60% of the total computation time for a single step. This rises to 90% at N = 300 and further increases for larger N. We therefore get the greatest speedup for large system sizes at low temperature where many prospective transitions are rejected at the classical stage and the matrix computation takes up the greatest fraction of the total computation time. The upshot is that we find a speedup of up to a factor of 10 at the cost of very little extra algorithmic complexity.</p>
|
||||
<p>Our two-step method should be distinguished from the more common method for speeding up MCMC which is to add asymmetry to the proposal distribution to make it as similar as possible to <span class="math inline">\(\min\left(1, e^{-\beta \Delta E}\right)\)</span>. This reduces the number of rejected states, which brings the algorithm closer in efficiency to a direct sampling method. However it comes at the expense of requiring a way to directly sample from this complex distribution, a problem which MCMC was employed to solve in the first place. For example, recent work trains restricted Boltzmann machines (RBMs) to generate samples for the proposal distribution of the FK model <span class="citation" data-cites="huangAcceleratedMonteCarlo2017"> [<a href="#ref-huangAcceleratedMonteCarlo2017" role="doc-biblioref">11</a>]</span>. The RBMs are chosen as a parametrisation of the proposal distribution that can be efficiently sampled from while offering sufficient flexibility that they can be adjusted to match the target distribution. Our proposed method is considerably simpler and does not require training while still reaping some of the benefits of reduced computation.</p>
|
||||
</section>
|
||||
<section id="scaling" class="level2">
|
||||
<h2>Scaling</h2>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/binder_cumulants/binder_cumulants.svg" id="fig:binder_cumulants" data-short-caption="Binder Cumulants" style="width:100.0%" alt="Figure 2: (Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. ¿fig:phase-diagram-lrfk?. All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly mentioned." />
|
||||
<figcaption aria-hidden="true">Figure 2: (Left) The order parameters, <span class="math inline">\(\langle m^2 \rangle\)</span>(solid) and <span class="math inline">\(1 - f\)</span> (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle\)</span> . (Right) The crossing of the Binder cumulant, <span class="math inline">\(B = \langle m^4 \rangle / \langle m^2 \rangle^2\)</span>, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. <strong>¿fig:phase-diagram-lrfk?</strong>. All plots use system sizes <span class="math inline">\(N = [10,20,30,50,70,110,160,250]\)</span> and lines are coloured from <span class="math inline">\(N = 10\)</span> in dark blue to <span class="math inline">\(N = 250\)</span> in yellow. The parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span> except where explicitly mentioned.</figcaption>
|
||||
</figure>
|
||||
<p>To improve the scaling of finite size effects, we make the replacement <span class="math inline">\(|i - j|^{-\alpha} \rightarrow |f(i - j)|^{-\alpha}\)</span>, in both <span class="math inline">\(J_{ij}\)</span> and <span class="math inline">\(\kappa\)</span>, where <span class="math inline">\(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\)</span>. <span class="math inline">\(f\)</span> is smooth across the circular boundary and its effect effect diminished for larger systems <span class="citation" data-cites="fukuiOrderNClusterMonte2009"> [<a href="#ref-fukuiOrderNClusterMonte2009" role="doc-biblioref">12</a>]</span>. We only consider even system sizes given that odd system sizes are not commensurate with a CDW state.</p>
|
||||
<p>To identify critical points we use the the Binder cumulant <span class="math inline">\(U_B\)</span> defined by</p>
|
||||
<p><span class="math display">\[
|
||||
U_B = 1 - \frac{\langle\mu_4\rangle}{3\langle\mu_2\rangle^2}
|
||||
\]</span></p>
|
||||
<p>where <span class="math inline">\(\mu_n = \langle(m - \langle m\rangle)^n\rangle\)</span> are the central moments of the order parameter <span class="math inline">\(m = \sum_i (-1)^i (2n_i - 1) / N\)</span>. The Binder cumulant evaluated against temperature is a diagnostic for the existence of a phase transition. If multiple such curves are plotted for different system sizes, a crossing indicates the location of a critical point while the lines do not cross for systems that don’t have a phase transition in the thermodynamic limit <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">2</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">3</a>]</span>.</p>
|
||||
<p>Next Section: <a href="../3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html">Results</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">
|
||||
<div id="ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">M. M. Maśka and K. Czajka, <em><a href="https://doi.org/10.1103/PhysRevB.74.035109">Thermodynamics of the Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo Study</a></em>, Phys. Rev. B <strong>74</strong>, 035109 (2006).</div>
|
||||
</div>
|
||||
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </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>
|
||||
</div>
|
||||
<div id="ref-musialMonteCarloSimulations2002" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">G. Musiał, L. Dȩbski, and G. Kamieniarz, <em><a href="https://doi.org/10.1103/PhysRevB.66.012407">Monte Carlo Simulations of Ising-Like Phase Transitions in the Three-Dimensional Ashkin-Teller Model</a></em>, Phys. Rev. B <strong>66</strong>, 012407 (2002).</div>
|
||||
</div>
|
||||
<div id="ref-binderGuidePracticalWork1988" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">K. Binder and D. W. Heermann, <em><a href="https://doi.org/10.1007/978-3-662-08854-8_3">Guide to Practical Work with the Monte Carlo Method</a></em>, in <em>Monte Carlo Simulation in Statistical Physics: An Introduction</em>, edited by K. Binder and D. W. Heermann (Springer Berlin Heidelberg, Berlin, Heidelberg, 1988), pp. 68–112.</div>
|
||||
</div>
|
||||
<div id="ref-kerteszAdvancesComputerSimulation1998" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">J. Kertesz and I. Kondor, editors, <em><a href="https://doi.org/10.1007/BFb0105456">Advances in Computer Simulation: Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July 1996</a></em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
|
||||
</div>
|
||||
<div id="ref-wolffMonteCarloErrors2004" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">U. Wolff, <em><a href="https://doi.org/10.1016/S0010-4655(03)00467-3">Monte Carlo Errors with Less Errors</a></em>, Computer Physics Communications <strong>156</strong>, 143 (2004).</div>
|
||||
</div>
|
||||
<div id="ref-kellyReversibilityStochasticNetworks1981" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">F. P. Kelly, <em><a href="https://doi.org/10.2307/2287860">Reversibility and Stochastic Networks / F.P. Kelly</a></em>, SERBIULA (Sistema Librum 2.0) <strong>76</strong>, (1981).</div>
|
||||
</div>
|
||||
<div id="ref-hastingsMonteCarloSampling1970" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">W. K. Hastings, <em><a href="https://doi.org/10.1093/biomet/57.1.97">Monte Carlo Sampling Methods Using Markov Chains and Their Applications</a></em>, Biometrika <strong>57</strong>, 97 (1970).</div>
|
||||
</div>
|
||||
<div id="ref-krauthIntroductionMonteCarlo1998" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">W. Krauth, <em><a href="https://doi.org/10.1007/BFb0105456">Introduction To Monte Carlo Algorithms</a></em>, in <em>Advances in Computer Simulation: Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July 1996</em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
|
||||
</div>
|
||||
<div id="ref-hodsonMCMCFKModel2021" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">T. Hodson, <em><a href="https://doi.org/10.5281/zenodo.4593904">Markov Chain Monte Carlo for the Kitaev Model</a></em>, (2021).</div>
|
||||
</div>
|
||||
<div id="ref-huangAcceleratedMonteCarlo2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">L. Huang and L. Wang, <em><a href="https://doi.org/10.1103/PhysRevB.95.035105">Accelerated Monte Carlo Simulations with Restricted Boltzmann Machines</a></em>, Phys. Rev. B <strong>95</strong>, 035105 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-fukuiOrderNClusterMonte2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">K. Fukui and S. Todo, <em><a href="https://doi.org/10.1016/j.jcp.2008.12.022">Order-N Cluster Monte Carlo Method for Spin Systems with Long-Range Interactions</a></em>, Journal of Computational Physics <strong>228</strong>, 2629 (2009).</div>
|
||||
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|
||||
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|
||||
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||||
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<li><a href="#lrfk-results-phase-diagram" id="toc-lrfk-results-phase-diagram">Phase Diagram</a></li>
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<li><a href="#fk-conclusion" id="toc-fk-conclusion">Discussion and Conclusion</a></li>
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<p>3 The Long Range Falicov-Kimball Model</p>
|
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|
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|
||||
<section id="sec:lrfk-results" class="level1">
|
||||
<h1>Results</h1>
|
||||
<p>Looking at the results of our MCMC simulations, we find a rich phase diagram with a CDW FTPT and interaction-tuned Anderson versus Mott localized phases similar to the 2D FK model <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>. We explore the localization properties of the fermionic sector and find that the localisation lengths vary dramatically across the phases and for different energies. Although moderate system sizes indicate the coexistence of localized and delocalized states within the CDW phase, we find quantitatively similar behaviour in a model of uncorrelated binary disorder on a CDW background. For large system sizes, i.e. for our 1D disorder model we can treat linear sizes of several thousand sites, we find that all states are eventually localized with a localization length which diverges towards zero temperature.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg" id="fig:phase-diagram-lrfk" data-short-caption="Long Range Falicov Kimball Model Phase Diagram" style="width:100.0%" alt="Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature T_c, linear in J. (Right) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas." />
|
||||
<figcaption aria-hidden="true">Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at <span class="math inline">\(U = 5\)</span>: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (Right) The TU plane at <span class="math inline">\(J = 5\)</span>: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at <span class="math inline">\(E=0\)</span> in the single particle energy spectrum. <span class="math inline">\(U_c\)</span> is independent of temperature. At <span class="math inline">\(U = 0\)</span> the fermions are decoupled from the spins forming a simple Fermi gas.</figcaption>
|
||||
</figure>
|
||||
<section id="lrfk-results-phase-diagram" class="level2">
|
||||
<h2>Phase Diagram</h2>
|
||||
<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>
|
||||
<p>Fig fig. <a href="#fig:phase-diagram-lrfk">1</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">2</a>]</span>.</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">3</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">1</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>. The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates hence it also has insulating character.</p>
|
||||
</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($\omega$) in the difference phases." style="width:100.0%" alt="Figure 2: Energy resolved DOS(\omega) against system size N in all three phases. The charge density wave phase is shown in both the high and low U regime for completeness. The top left panel shows the Anderson phase at U = 2 and high T = 2.5, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at U = 2 and low T = 1.5, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at U = 5 and high T = 2.5 the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at U = 5 and T = 1.5 is qualitatively similar to the lower left panel except that the gap scales with U. For all the plots J = 5,\;\alpha = 1.25." />
|
||||
<figcaption aria-hidden="true">Figure 2: Energy resolved DOS(<span class="math inline">\(\omega\)</span>) against system size <span class="math inline">\(N\)</span> in all three phases. The charge density wave phase is shown in both the high and low <span class="math inline">\(U\)</span> regime for completeness. The top left panel shows the Anderson phase at <span class="math inline">\(U = 2\)</span> and high <span class="math inline">\(T = 2.5\)</span>, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at <span class="math inline">\(U = 2\)</span> and low <span class="math inline">\(T = 1.5\)</span>, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at <span class="math inline">\(U = 5\)</span> and high <span class="math inline">\(T = 2.5\)</span> the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at <span class="math inline">\(U = 5\)</span> and <span class="math inline">\(T = 1.5\)</span> is qualitatively similar to the lower left panel except that the gap scales with <span class="math inline">\(U\)</span>. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span>.</figcaption>
|
||||
</figure>
|
||||
<p>The MCMC formulation suggests viewing the spin configurations as a form of annealed binary disorder whose probability distribution is given by the Boltzmann weight <span class="math inline">\(e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]}\)</span>. This makes apparent the link to the study of disordered systems and Anderson localisation. While these systems are typically studied by defining the probability distribution for the quenched disorder potential externally, here we have a translation invariant system with disorder as a natural consequence of the Ising background field conserved under the dynamics.</p>
|
||||
<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">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>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/DOS/IPR_scaling.svg" id="fig:IPR_scaling" data-short-caption="Scaling of IPR($\omega$) against system size $N$." style="width:100.0%" alt="Figure 3: The IPR(\omega) scaling with N at fixed energy for each phase and for points both in the gap (\omega_0) and in a band (\omega_1). The slope of the line yields the scaling exponent \tau defined by \mathrm{IPR} \propto N^{-\tau}). \tau close to zero implies that the states at that energy are localised while \tau = -d corresponds to extended states where d is the system dimension. All but the bands of the charge density wave phase are approximately localised with \tau is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots J = 5,\;\alpha = 1.25. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states." />
|
||||
<figcaption aria-hidden="true">Figure 3: The IPR(<span class="math inline">\(\omega\)</span>) scaling with <span class="math inline">\(N\)</span> at fixed energy for each phase and for points both in the gap (<span class="math inline">\(\omega_0\)</span>) and in a band (<span class="math inline">\(\omega_1\)</span>). The slope of the line yields the scaling exponent <span class="math inline">\(\tau\)</span> defined by <span class="math inline">\(\mathrm{IPR} \propto N^{-\tau}\)</span>). <span class="math inline">\(\tau\)</span> close to zero implies that the states at that energy are localised while <span class="math inline">\(\tau = -d\)</span> corresponds to extended states where <span class="math inline">\(d\)</span> is the system dimension. All but the bands of the charge density wave phase are approximately localised with <span class="math inline">\(\tau\)</span> is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span>. The measured <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\((0.06\pm0.01, 0.02\pm0.01\)</span> (b) <span class="math inline">\(0.04\pm0.02, 0.00\pm0.01\)</span> (c) <span class="math inline">\(0.05\pm0.03, 0.30\pm0.03\)</span> (d) <span class="math inline">\(0.06\pm0.04, 0.15\pm0.05\)</span> We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states.</figcaption>
|
||||
</figure>
|
||||
<p>The scaling of the IPR with system size</p>
|
||||
<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">2</a> shows the density of states as function of energy while fig. <a href="#fig:IPR_scaling">3</a> shows <span class="math inline">\(\tau\)</span>, the scaling exponent of the IPR with system size, The DOS is symmetric about <span class="math inline">\(0\)</span> because of the particle hole symmetry of the model. At high temperatures, all of the eigenstates are localised in both the Mott and Anderson phases (with <span class="math inline">\(\tau \leq 0.07\)</span> for our system sizes). We also checked that the states are localised by direct inspection. Note that there are in-gap states for instance at <span class="math inline">\(\omega_0\)</span>, below the upper band which are localized and smoothly connected across the phase transition.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U5.svg" id="fig:gap_opening_U5" data-short-caption="The transition from CDW to the Mott phase" style="width:100.0%" alt="Figure 4: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
|
||||
<figcaption aria-hidden="true">Figure 4: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW to gapped Mott phase transition at <span class="math inline">\(U=5\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
|
||||
</figure>
|
||||
<p>In the CDW phases at <span class="math inline">\(U=2\)</span> and <span class="math inline">\(U=5\)</span>, we find for the states within the gapped CDW bands, e.g. at <span class="math inline">\(\omega_1\)</span>, scaling exponents <span class="math inline">\(\tau = 0.30\pm0.03\)</span> and <span class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This surprising finding suggests that the CDW bands are partially delocalised with multi-fractal behaviour of the wavefunctions <span class="citation" data-cites="eversAndersonTransitions2008"> [<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">6</a>]</span>. This phenomenon would be unexpected in a 1D model as they generally do not support delocalisation in the presence of disorder except as the result of correlations in the emergent disorder potential <span class="citation" data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"> [<a href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">7</a>,<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">8</a>]</span>. However, we later show by comparison to an uncorrelated Anderson model that these nonzero exponents are a finite size effect and the states are localised with a finite <span class="math inline">\(\xi\)</span> similar to the system size, an example of weak localisation. As a result, the IPR does not scale correctly until the system size has grown much larger than <span class="math inline">\(\xi\)</span>. fig. <a href="#fig:DM_IPR_scaling">7</a> shows that the scaling of the IPR in the CDW phase does flatten out eventually.</p>
|
||||
<p>Next, we use the DOS and the scaling exponent <span class="math inline">\(\tau\)</span> to explore the localisation properties over the energy-temperature plane in fig. <a href="#fig:gap_opening_U2">5</a> and fig. <a href="#fig:gap_opening_U5">4</a>. Gapped areas are shown in white, which highlights the distinction between the gapped Mott phase and the ungapped Anderson phase. In-gap states appear just below the critical point, smoothly filling the bandgap in the Anderson phase and forming islands in the Mott phase. As in the finite <span class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">9</a>]</span> and infinite dimensional <span class="citation" data-cites="hassanSpectralPropertiesChargedensitywave2007"> [<a href="#ref-hassanSpectralPropertiesChargedensitywave2007" role="doc-biblioref">10</a>]</span> cases, the in-gap states merge and are pushed to lower energy for decreasing U as the <span class="math inline">\(T=0\)</span> CDW gap closes. Intuitively, the presence of in-gap states can be understood as a result of domain wall fluctuations away from the AFM ordered background. These domain walls act as local potentials for impurity-like bound states <span class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">9</a>]</span>.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U2.svg" id="fig:gap_opening_U2" data-short-caption="The transition from CDW to the Anderson Phase" style="width:100.0%" alt="Figure 5: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at U=2. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
|
||||
<figcaption aria-hidden="true">Figure 5: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at <span class="math inline">\(U=2\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
|
||||
</figure>
|
||||
<p>In order to understand the localization properties we can compare the behaviour of our model with that of a simpler Anderson disorder model (DM) in which the spins are replaced by a CDW background with uncorrelated binary defect potentials. This is defined by replacing the spin degree of freedom in the FK model <span class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder potential <span class="math inline">\(d_i = \pm \tfrac{1}{2}\)</span> controlled by a defect density <span class="math inline">\(\rho\)</span> such that <span class="math inline">\(d_i = -\tfrac{1}{2}\)</span> with probability <span class="math inline">\(\rho/2\)</span> and <span class="math inline">\(d_i = \tfrac{1}{2}\)</span> otherwise. <span class="math inline">\(\rho/2\)</span> is used rather than <span class="math inline">\(\rho\)</span> so that the disorder potential takes on the zero temperature CDW ground state at <span class="math inline">\(\rho = 0\)</span> and becomes a random choice over spin states at <span class="math inline">\(\rho = 1\)</span> i.e the infinite temperature limit.</p>
|
||||
<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. <a href="#fig:DM_DOS">6</a> and fig. <a href="#fig:DM_IPR_scaling">7</a> compare the FK model to the disorder model at different system sizes, matching the defect densities of the disorder model to the FK model at <span class="math inline">\(N = 270\)</span> above and below the CDW transition. We find very good, even quantitative, agreement between the FK and disorder models, which suggests that correlations in the spin sector do not play a significant role.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/disorder_model/DM_DOS.svg" id="fig:DM_DOS" data-short-caption="FK model compared to binary disorder model: DOS" style="width:100.0%" alt="Figure 6: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 2, the Energy resolved DOS(\omega) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters." />
|
||||
<figcaption aria-hidden="true">Figure 6: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 < \rho < 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:DOS">2</a>, the Energy resolved DOS(<span class="math inline">\(\omega\)</span>) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters.</figcaption>
|
||||
</figure>
|
||||
<p>As we can sample directly from the disorder model, rather than through MCMC, the samples are uncorrelated. Hence we can evaluate much larger system sizes with the disorder model which enables us to pin down the correct localisation effects. In particular, what appear to be delocalized states for small system sizes eventually turn out to be states with large localization length. The localization length diverges towards the ordered zero temperature CDW state. The interplay of interactions, which here produce as peculiar binary potential, and localization can be very intricate and the added advantage of a 1D model is that we can explore very large system sizes.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/disorder_model/DM_IPR_scaling.svg" id="fig:DM_IPR_scaling" data-short-caption="FK model compared to binary disorder model: IPR Scaling" style="width:100.0%" alt="Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 3 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation [1], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N > 400" />
|
||||
<figcaption aria-hidden="true">Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 < \rho < 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:IPR_scaling">3</a> <span class="math inline">\(\tau(\omega)\)</span> the scaling of IPR(<span class="math inline">\(\omega\)</span>) with system size, , is shown both in gap (<span class="math inline">\(\omega_0\)</span>) and in the band (<span class="math inline">\(\omega_1\)</span>). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>, hence all the states are indeed localised as one would expect in 1D. The disorder model <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span class="math inline">\(0.01\pm0.04, -0.01\pm0.04\)</span> (c) <span class="math inline">\(0.05\pm0.06, 0.04\pm0.06\)</span> (d) <span class="math inline">\(-0.03\pm0.06, 0.01\pm0.06\)</span>. The lines are fit on system sizes <span class="math inline">\(N > 400\)</span></figcaption>
|
||||
</figure>
|
||||
</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 one dimension by adding a novel long-ranged coupling designed to stabilise the CDW phase present in dimension two and above.</p>
|
||||
<p>Our MCMC approach emphasises the presence of a disorder-free localization mechanism within our translationally invariant system. Further, it gives a significant speed up over the naive method. We show that our LRFK model 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 non-zero 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">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> as well as superconductor like states <span class="citation" data-cites="caiVisualizingEvolutionMott2016"> [<a href="#ref-caiVisualizingEvolutionMott2016" role="doc-biblioref">14</a>]</span> could be stabilised at finite temperature.</p>
|
||||
<p>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>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">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </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>
|
||||
</div>
|
||||
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </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>
|
||||
</div>
|
||||
<div id="ref-rusinCalculationRKKYRange2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">T. M. Rusin and W. Zawadzki, <em><a href="https://doi.org/10.1016/j.jmmm.2017.06.007">On Calculation of RKKY Range Function in One Dimension</a></em>, Journal of Magnetism and Magnetic Materials <strong>441</strong>, 387 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-abrahamsScalingTheoryLocalization1979" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, <em><a href="https://doi.org/10.1103/PhysRevLett.42.673">Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions</a></em>, Phys. Rev. Lett. <strong>42</strong>, 673 (1979).</div>
|
||||
</div>
|
||||
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
</div>
|
||||
<div id="ref-eversAndersonTransitions2008" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
</div>
|
||||
<div id="ref-croyAndersonLocalization1D2011" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">A. Croy, P. Cain, and M. Schreiber, <em><a href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B <strong>82</strong>, 107 (2011).</div>
|
||||
</div>
|
||||
<div id="ref-goldshteinPurePointSpectrum1977" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">I. Ya. Gol’dshtein, S. A. Molchanov, and L. A. Pastur, <em><a href="https://doi.org/10.1007/BF01135526">A Pure Point Spectrum of the Stochastic One-Dimensional Schrödinger Operator</a></em>, Funct Anal Its Appl <strong>11</strong>, 1 (1977).</div>
|
||||
</div>
|
||||
<div id="ref-zondaGaplessRegimeCharge2019" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">M. Žonda, J. Okamoto, and M. Thoss, <em><a href="https://doi.org/10.1103/PhysRevB.100.075124">Gapless Regime in the Charge Density Wave Phase of the Finite Dimensional Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>100</strong>, 075124 (2019).</div>
|
||||
</div>
|
||||
<div id="ref-hassanSpectralPropertiesChargedensitywave2007" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">S. R. Hassan and H. R. Krishnamurthy, <em><a href="https://doi.org/10.1103/PhysRevB.76.205109">Spectral Properties in the Charge-Density-Wave Phase of the Half-Filled Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>76</strong>, 205109 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-hartLogarithmicEntanglementGrowth2020" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">O. Hart, S. Gopalakrishnan, and C. Castelnovo, <em><a href="http://arxiv.org/abs/2009.00618">Logarithmic Entanglement Growth from Disorder-Free Localisation in the Two-Leg Compass Ladder</a></em>, arXiv:2009.00618 [Cond-Mat] (2020).</div>
|
||||
</div>
|
||||
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">C. Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>41</strong>, 2198 (1990).</div>
|
||||
</div>
|
||||
<div id="ref-michelettiCompleteDevilTextquotesingles1997" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">C. Micheletti, A. B. Harris, and J. M. Yeomans, <em><a href="https://doi.org/10.1088/0305-4470/30/21/002">A Complete Devil\textquotesingles Staircase in the Falicov - Kimball Model</a></em>, J. Phys. A: Math. Gen. <strong>30</strong>, L711 (1997).</div>
|
||||
</div>
|
||||
<div id="ref-caiVisualizingEvolutionMott2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">P. Cai, W. Ruan, Y. Peng, C. Ye, X. Li, Z. Hao, X. Zhou, D.-H. Lee, and Y. Wang, <em><a href="https://doi.org/10.1038/nphys3840">Visualizing the Evolution from the Mott Insulator to a Charge-Ordered Insulator in Lightly Doped Cuprates</a></em>, Nature Phys <strong>12</strong>, 11 (2016).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
</main>
|
||||
</body>
|
||||
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|
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|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<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="#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="#fk-model" id="toc-fk-model">The Model</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
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|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
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||||
<ul>
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||||
<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="#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="#fk-model" id="toc-fk-model">The Model</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -54,31 +54,32 @@ image:
|
||||
<p>3 The Long Range Falikov-Kimball Model</p>
|
||||
<hr />
|
||||
</div>
|
||||
<section id="chap:3-the-long-range-falicov-kimball-model" class="level1">
|
||||
<section id="chap:3-the-long-range-falikov-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>
|
||||
<p>the code for which is available is available at <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">2</a>]</span>.</p>
|
||||
<p>The code for which is available is available at <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">2</a>]</span>.</p>
|
||||
<p><strong>Contributions</strong></p>
|
||||
<p>Johannes had the initial idea to use a long range Ising term to stablise order in a one dimension Falikov-Kimball model. Josef developed a proof of concept during a summer project at Imperial along with Alexander Belcik. I wrote the simulation code and performed all the analysis presented here.</p>
|
||||
<p><strong>Chapter Summary</strong></p>
|
||||
<p>The paper is organised as follows. First, we introduce the model and present its phase diagram. Second, we present the methods used to solve it numerically. Last, we investigate the model’s localisation properties and conclude.</p>
|
||||
<p>The paper is organised as follows. First, I will introduce the long range Falicov-Kimball (LRFK) model and motivate its definition. Second, I will present the methods used to solve it numerically, including Markov chain Monte Carlo and finite size scaling. I will then present and interpret the results obtained.</p>
|
||||
<p>Dimensionality is crucial for the physics of both localisation and phase transitions. We have already seen that the one dimensional standard FK model cannot support an ordered phase at finite temperatures. Here, we construct a generalised one-dimensional FK model with long-range interactions. The long-range interactions are able to induce the otherwise forbidden CDW phase at non-zero temperature.</p>
|
||||
</section>
|
||||
<section id="fk-model" class="level1">
|
||||
<h1>The Model</h1>
|
||||
<p>Dimensionality is crucial for the physics of both localisation and FTPTs. In 1D, 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 delocalisation <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">3</a>–<a href="#ref-dunlapAbsenceLocalizationRandomdimer1990" role="doc-biblioref">5</a>]</span>. Thermodynamically, short-range interactions cannot overcome thermal defects in 1D which prevents ordered phases at nonzero temperature <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958 goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">6</a>–<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">9</a>]</span>. 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">10</a>–<a href="#ref-yosidaMagneticPropertiesCuMn1957" role="doc-biblioref">13</a>]</span> decays as <span class="math inline">\(r^{-1}\)</span> in 1D <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">14</a>]</span>. This could in principle induce the necessary long-range interactions for the classical Ising background <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">15</a>,<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">16</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 1D FK model <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">17</a>]</span>.</p>
|
||||
<p>Here, we construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. We find a rich phase diagram with a CDW FTPT and interaction-tuned Anderson versus Mott localized phases similar to the 2D FK model <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">18</a>]</span>. We explore the localization properties of the fermionic sector and find that the localisation lengths vary dramatically across the phases and for different energies. Although moderate system sizes indicate the coexistence of localized and delocalized states within the CDW phase, we find quantitatively similar behaviour in a model of uncorrelated binary disorder on a CDW background. For large system sizes, i.e. for our 1D disorder model we can treat linear sizes of several thousand sites, we find that all states are eventually localized with a localization length which diverges towards zero temperature.</p>
|
||||
<p>On bipartite lattices in dimensions 2 and above the FK model exhibits a finite temperature phase transition to an ordered charge density wave (CDW) phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">19</a>]</span>. In this phase, the ions are confined to one of the two sublattices, breaking the <span class="math inline">\(\mathbb{Z}_2\)</span> symmetry.</p>
|
||||
<p>In 1D, however, Periel’s argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">16</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">17</a>]</span> states that domain walls only introduce a constant energy penalty into the free energy while bringing a entropic contribution logarithmic in system size. Hence the 1D model does not have a finite temperature phase transition. However 1D systems are much easier to study numerically and admit simpler realisations experimentally. We therefore introduce a long range coupling between the ions in order to stabilise a CDW phase in 1D. This leads to a disordered system that is gaped by the CDW background but with correlated fluctuations leading to a disorder-free correlation induced mobility edge in one dimension.</p>
|
||||
<p>The presence of the classical field makes the model amenable to an exact numerical treatment at finite temperature via a sign problem free MCMC algorithm <span class="citation" data-cites="devriesGapsDensitiesStates1993 devriesSimplifiedHubbardModel1993 antipovInteractionTunedAndersonMott2016 debskiPossibilityDetectionFinite2016 herrmannSpreadingCorrelationsFalicovKimball2018 maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">18</a>–<a href="#ref-herrmannSpreadingCorrelationsFalicovKimball2018" role="doc-biblioref">23</a>]</span>. The MCMC treatment motivates a view of the classical background field as a disorder potential, which suggests an intimate link to localisation physics. Indeed, thermal fluctuations of the classical sector act as disorder potentials drawn from a thermal distribution and the emergence of disorder in a translationally invariant Hamiltonian links the FK model to recent interest in disorder-free localisation <span class="citation" data-cites="smithDisorderFreeLocalization2017 smithDynamicalLocalizationMathbbZ2018 brenesManyBodyLocalizationDynamics2018"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">24</a>–<a href="#ref-brenesManyBodyLocalizationDynamics2018" role="doc-biblioref">26</a>]</span>.</p>
|
||||
<p>To evaluate thermodynamic averages we perform a classical Markov Chain Monte Carlo random walk over the space of ionic configurations, at each step diagonalising the effective electronic Hamiltonian <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">19</a>]</span>. Using a binder-cumulant method <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">27</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">28</a>]</span>, we demonstrate the model has a finite temperature phase transition when the interaction is sufficiently long ranged. We then estimate the density of states and the inverse participation ratio as a function of energy to diagnose localisation properties.</p>
|
||||
<p>We interpret the FK model as a model of spinless fermions, <span class="math inline">\(c^\dagger_{i}\)</span>, hopping on a 1D lattice against a classical Ising spin background, <span class="math inline">\(S_i \in {\pm \frac{1}{2}}\)</span>. The fermions couple to the spins via an onsite interaction with strength <span class="math inline">\(U\)</span> which we supplement by a long-range interaction, <span class="math inline">\(J_{ij} = 4\kappa J (-1)^{|i-j|} |i-j|^{-\alpha}\)</span>, between the spins. The normalisation, <span class="math inline">\(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\)</span>, renders the 0th order mean field critical temperature independent of system size. The hopping strength of the electrons, <span class="math inline">\(t = 1\)</span>, sets the overall energy scale and we concentrate throughout on the particle-hole symmetric point at zero chemical potential and half filling <span class="citation" data-cites="gruberFalicovKimballModelReview1996"> [<a href="#ref-gruberFalicovKimballModelReview1996" role="doc-biblioref">29</a>]</span>. <span class="math display">\[\begin{aligned}
|
||||
<p>On bipartite lattices in dimensions 2 and above the FK model exhibits a finite temperature phase transition to an ordered charge density wave (CDW) phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">3</a>]</span>. In this phase, the spins order anti-ferromagnetically, breaking the <span class="math inline">\(\mathbb{Z}_2\)</span> symmetry. In 1D, however, Periel’s argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">4</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">5</a>]</span> states that domain walls only introduce a constant energy penalty into the free energy while bringing a entropic contribution logarithmic in system size. Hence the 1D model does not have a finite temperature phase transition. However 1D systems are much easier to study numerically and admit simpler realisations experimentally. We therefore introduce a long-range coupling between the ions in order to stabilise a CDW phase in 1D.</p>
|
||||
<p>We interpret the FK model as a model of spinless fermions, <span class="math inline">\(c^\dagger_{i}\)</span>, hopping on a 1D lattice against a classical Ising spin background, <span class="math inline">\(S_i \in {\pm \frac{1}{2}}\)</span>. The fermions couple to the spins via an onsite interaction with strength <span class="math inline">\(U\)</span> which we supplement by a long-range interaction, <span class="math display">\[
|
||||
J_{ij} = 4\kappa J\; (-1)^{|i-j|} |i-j|^{-\alpha},
|
||||
\]</span></p>
|
||||
<p>between the spins. The additional coupling is very similar to that of the long range Ising model, it stabilises the Antiferromagnetic (AFM) order of the Ising spins which promotes the finite temperature CDW phase of the fermionic sector.</p>
|
||||
<p>The hopping strength of the electrons, <span class="math inline">\(t = 1\)</span>, sets the overall energy scale and we concentrate throughout on the particle-hole symmetric point at zero chemical potential and half filling <span class="citation" data-cites="gruberFalicovKimballModelReview1996"> [<a href="#ref-gruberFalicovKimballModelReview1996" role="doc-biblioref">6</a>]</span>.</p>
|
||||
<p>$$$$</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{i} (c^\dagger_{i}c_{i+1} + \textit{h.c.)}\\
|
||||
& + \sum_{i, j}^{N} J_{ij} S_i S_j \nonumber
|
||||
& + \sum_{i, j}^{N} J_{ij} S_i S_j
|
||||
\label{eq:HFK}\end{aligned}\]</span></p>
|
||||
<p>In two or more dimensions, the <span class="math inline">\(J\!=0\!\)</span> FK model has a FTPT to the CDW phase with non-zero staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}|\sum_i (-1)^i \; \expval{c^\dagger_{i}c_{i}}|\)</span> <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">18</a>,<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">19</a>]</span>. This only exists at zero temperature in the short ranged 1D model <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">17</a>]</span>. To study the CDW phase at finite temperature in 1D, we add an additional coupling that is both long-ranged and staggered by a factor <span class="math inline">\((-1)^{|i-j|}\)</span>. The additional coupling stabilises the Antiferromagnetic (AFM) order of the Ising spins which promotes the finite temperature CDW phase of the fermionic sector.</p>
|
||||
<p>Taking the limit <span class="math inline">\(U = 0\)</span> decouples the spins from the fermions, which gives a spin sector governed by a classical LRI model. Note, the transformation of the spins <span class="math inline">\(S_i \to (-1)^{i} S_i\)</span> maps the AFM model to the FM one. We recall that Peierls’ classic argument can be extended to show that, for the 1D LRI model, a power law decay of <span class="math inline">\(\alpha < 2\)</span> is required for a FTPT as the energy of defect domain then scales with the system size and can overcome the entropic contribution. A renormalisation group analysis supports this finding and shows that the critical exponents are only universal for <span class="math inline">\(\alpha \leq 3/2\)</span> <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968 thoulessLongRangeOrderOneDimensional1969 angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">15</a>,<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">30</a>,<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">31</a>]</span>. In the following, we choose <span class="math inline">\(\alpha = 5/4\)</span> to avoid this additional complexity.</p>
|
||||
<p>To improve the scaling of finite size effects, we make the replacement <span class="math inline">\(|i - j|^{-\alpha} \rightarrow |f(i - j)|^{-\alpha}\)</span>, in both <span class="math inline">\(J_{ij}\)</span> and <span class="math inline">\(\kappa\)</span>, where <span class="math inline">\(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\)</span>, which is smooth across the circular boundary <span class="citation" data-cites="fukuiOrderNClusterMonte2009"> [<a href="#ref-fukuiOrderNClusterMonte2009" role="doc-biblioref">32</a>]</span>. We only consider even system sizes given that odd system sizes are not commensurate with a CDW state.</p>
|
||||
<p>Without proper normalisation, the long range coupling would render the critical temperature strongly system size dependent for small system sizes. Within a mean field approximation the critical temperature scales with the effective coupling to all the neighbours of each site, which for a system with <span class="math inline">\(N\)</span> sites is <span class="math inline">\(\sum_{i=1}^{N} i^{-\alpha}\)</span>. Hence, the normalisation <span class="math inline">\(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\)</span>, renders the critical temperature independent of system size in the mean field approximation. This greatly improves the finite size behaviour of the model.</p>
|
||||
<p>Taking the limit <span class="math inline">\(U = 0\)</span> decouples the spins from the fermions, which gives a spin sector governed by a classical LRI model. Note, the transformation of the spins <span class="math inline">\(S_i \to (-1)^{i} S_i\)</span> maps the AFM model to the FM one. As discussed in the background section, Peierls’ classic argument can be extended to long range couplings to show that, for the 1D LRI model, a power law decay of <span class="math inline">\(\alpha < 2\)</span> is required for a FTPT as the energy of defect domain then scales with the system size and can overcome the entropic contribution. A renormalisation group analysis supports this finding and shows that the critical exponents are only universal for <span class="math inline">\(\alpha \leq 3/2\)</span> <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968 thoulessLongRangeOrderOneDimensional1969 angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">7</a>–<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">9</a>]</span>. In the following, we choose <span class="math inline">\(\alpha = 5/4\)</span> to avoid the additional complexity of non-universal critical points.</p>
|
||||
<p>Next Section: <a href="../3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html">Methods</a></p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
@ -90,95 +91,26 @@ H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\
|
||||
<div id="ref-hodsonMCMCFKModel2021" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">T. Hodson, <em><a href="https://doi.org/10.5281/zenodo.4593904">Markov Chain Monte Carlo for the Kitaev Model</a></em>, (2021).</div>
|
||||
</div>
|
||||
<div id="ref-aubryAnalyticityBreakingAnderson1980" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </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>
|
||||
</div>
|
||||
<div id="ref-dassarmaLocalizationMobilityEdges1990" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </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>
|
||||
</div>
|
||||
<div id="ref-dunlapAbsenceLocalizationRandomdimer1990" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">D. H. Dunlap, H.-L. Wu, and P. W. Phillips, <em><a href="https://doi.org/10.1103/PhysRevLett.65.88">Absence of Localization in a Random-Dimer Model</a></em>, Phys. Rev. Lett. <strong>65</strong>, 88 (1990).</div>
|
||||
</div>
|
||||
<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">P. W. Anderson, <em><a href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492 (1958).</div>
|
||||
</div>
|
||||
<div id="ref-goldshteinPurePointSpectrum1977" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">I. Ya. Gol’dshtein, S. A. Molchanov, and L. A. Pastur, <em><a href="https://doi.org/10.1007/BF01135526">A Pure Point Spectrum of the Stochastic One-Dimensional Schrödinger Operator</a></em>, Funct Anal Its Appl <strong>11</strong>, 1 (1977).</div>
|
||||
</div>
|
||||
<div id="ref-abrahamsScalingTheoryLocalization1979" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, <em><a href="https://doi.org/10.1103/PhysRevLett.42.673">Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions</a></em>, Phys. Rev. Lett. <strong>42</strong>, 673 (1979).</div>
|
||||
</div>
|
||||
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </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>
|
||||
</div>
|
||||
<div id="ref-kasuyaTheoryMetallicFerro1956" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">T. Kasuya, <em><a href="https://doi.org/10.1143/PTP.16.45">A Theory of Metallic Ferro- and Antiferromagnetism on Zener’s Model</a></em>, Prog Theor Phys <strong>16</strong>, 45 (1956).</div>
|
||||
</div>
|
||||
<div id="ref-rudermanIndirectExchangeCoupling1954" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">M. A. Ruderman and C. Kittel, <em><a href="https://doi.org/10.1103/PhysRev.96.99">Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons</a></em>, Phys. Rev. <strong>96</strong>, 99 (1954).</div>
|
||||
</div>
|
||||
<div id="ref-vanvleckNoteInteractionsSpins1962" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">J. H. Van Vleck, <em><a href="https://doi.org/10.1103/RevModPhys.34.681">Note on the Interactions Between the Spins of Magnetic Ions or Nuclei in Metals</a></em>, Rev. Mod. Phys. <strong>34</strong>, 681 (1962).</div>
|
||||
</div>
|
||||
<div id="ref-yosidaMagneticPropertiesCuMn1957" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">K. Yosida, <em><a href="https://doi.org/10.1103/PhysRev.106.893">Magnetic Properties of Cu-Mn Alloys</a></em>, Phys. Rev. <strong>106</strong>, 893 (1957).</div>
|
||||
</div>
|
||||
<div id="ref-rusinCalculationRKKYRange2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">T. M. Rusin and W. Zawadzki, <em><a href="https://doi.org/10.1016/j.jmmm.2017.06.007">On Calculation of RKKY Range Function in One Dimension</a></em>, Journal of Magnetism and Magnetic Materials <strong>441</strong>, 387 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-thoulessLongRangeOrderOneDimensional1969" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">D. J. Thouless, <em><a href="https://doi.org/10.1103/PhysRev.187.732">Long-Range Order in One-Dimensional Ising Systems</a></em>, Phys. Rev. <strong>187</strong>, 732 (1969).</div>
|
||||
<div id="ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">M. M. Maśka and K. Czajka, <em><a href="https://doi.org/10.1103/PhysRevB.74.035109">Thermodynamics of the Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo Study</a></em>, Phys. Rev. B <strong>74</strong>, 035109 (2006).</div>
|
||||
</div>
|
||||
<div id="ref-peierlsIsingModelFerromagnetism1936" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">R. Peierls, <em><a href="https://doi.org/10.1017/S0305004100019174">On Ising’s Model of Ferromagnetism</a></em>, Mathematical Proceedings of the Cambridge Philosophical Society <strong>32</strong>, 477 (1936).</div>
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">R. Peierls, <em><a href="https://doi.org/10.1017/S0305004100019174">On Ising’s Model of Ferromagnetism</a></em>, Mathematical Proceedings of the Cambridge Philosophical Society <strong>32</strong>, 477 (1936).</div>
|
||||
</div>
|
||||
<div id="ref-kennedyItinerantElectronModel1986" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">T. Kennedy and E. H. Lieb, <em><a href="https://doi.org/10.1016/0378-4371(86)90188-3">An Itinerant Electron Model with Crystalline or Magnetic Long Range Order</a></em>, Physica A: Statistical Mechanics and Its Applications <strong>138</strong>, 320 (1986).</div>
|
||||
</div>
|
||||
<div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
</div>
|
||||
<div id="ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">M. M. Maśka and K. Czajka, <em><a href="https://doi.org/10.1103/PhysRevB.74.035109">Thermodynamics of the Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo Study</a></em>, Phys. Rev. B <strong>74</strong>, 035109 (2006).</div>
|
||||
</div>
|
||||
<div id="ref-devriesGapsDensitiesStates1993" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[20] </div><div class="csl-right-inline">P. de Vries, K. Michielsen, and H. De Raedt, <em><a href="https://doi.org/10.1103/PhysRevLett.70.2463">Gaps in Densities of States of Two Hubbard-Like Models</a></em>, Phys. Rev. Lett. <strong>70</strong>, 2463 (1993).</div>
|
||||
</div>
|
||||
<div id="ref-devriesSimplifiedHubbardModel1993" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[21] </div><div class="csl-right-inline">P. de Vries, K. Michielsen, and H. De Raedt, <em><a href="https://doi.org/10.1007/BF01308754">The Simplified Hubbard Model in One and Two Dimensions</a></em>, Z. Physik B - Condensed Matter <strong>92</strong>, 353 (1993).</div>
|
||||
</div>
|
||||
<div id="ref-debskiPossibilityDetectionFinite2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[22] </div><div class="csl-right-inline">L. Debski, <em><a href="https://doi.org/10.1080/01411594.2015.1080257">The Possibility of Detection of Finite Temperature Stripe Ordering in 2d Spinless Falicov–Kimball Model</a></em>, Phase Transitions <strong>89</strong>, 249 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-herrmannSpreadingCorrelationsFalicovKimball2018" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[23] </div><div class="csl-right-inline">A. J. Herrmann, A. E. Antipov, and P. Werner, <em><a href="https://doi.org/10.1103/PhysRevB.97.165107">Spreading of Correlations in the Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>97</strong>, 165107 (2018).</div>
|
||||
</div>
|
||||
<div id="ref-smithDisorderFreeLocalization2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[24] </div><div class="csl-right-inline">A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, <em><a href="https://doi.org/10.1103/PhysRevLett.118.266601">Disorder-Free Localization</a></em>, Phys. Rev. Lett. <strong>118</strong>, 266601 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-smithDynamicalLocalizationMathbbZ2018" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[25] </div><div class="csl-right-inline">A. Smith, J. Knolle, R. Moessner, and D. L. Kovrizhin, <em><a href="https://doi.org/10.1103/PhysRevB.97.245137">Dynamical Localization in $\Mathbb{Z}_2$ Lattice Gauge Theories</a></em>, Phys. Rev. B <strong>97</strong>, 245137 (2018).</div>
|
||||
</div>
|
||||
<div id="ref-brenesManyBodyLocalizationDynamics2018" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[26] </div><div class="csl-right-inline">M. Brenes, M. Dalmonte, M. Heyl, and A. Scardicchio, <em><a href="https://doi.org/10.1103/PhysRevLett.120.030601">Many-Body Localization Dynamics from Gauge Invariance</a></em>, Phys. Rev. Lett. <strong>120</strong>, 030601 (2018).</div>
|
||||
</div>
|
||||
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[27] </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>
|
||||
</div>
|
||||
<div id="ref-musialMonteCarloSimulations2002" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">G. Musiał, L. Dȩbski, and G. Kamieniarz, <em><a href="https://doi.org/10.1103/PhysRevB.66.012407">Monte Carlo Simulations of Ising-Like Phase Transitions in the Three-Dimensional Ashkin-Teller Model</a></em>, Phys. Rev. B <strong>66</strong>, 012407 (2002).</div>
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">T. Kennedy and E. H. Lieb, <em><a href="https://doi.org/10.1016/0378-4371(86)90188-3">An Itinerant Electron Model with Crystalline or Magnetic Long Range Order</a></em>, Physica A: Statistical Mechanics and Its Applications <strong>138</strong>, 320 (1986).</div>
|
||||
</div>
|
||||
<div id="ref-gruberFalicovKimballModelReview1996" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">C. Gruber and N. Macris, <em>The Falicov-Kimball Model: A Review of Exact Results and Extensions</em>, Helvetica Physica Acta <strong>69</strong>, (1996).</div>
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">C. Gruber and N. Macris, <em>The Falicov-Kimball Model: A Review of Exact Results and Extensions</em>, Helvetica Physica Acta <strong>69</strong>, (1996).</div>
|
||||
</div>
|
||||
<div id="ref-ruelleStatisticalMechanicsOnedimensional1968" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[30] </div><div class="csl-right-inline">D. Ruelle, <em><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys. <strong>9</strong>, 267 (1968).</div>
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">D. Ruelle, <em><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys. <strong>9</strong>, 267 (1968).</div>
|
||||
</div>
|
||||
<div id="ref-thoulessLongRangeOrderOneDimensional1969" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">D. J. Thouless, <em><a href="https://doi.org/10.1103/PhysRev.187.732">Long-Range Order in One-Dimensional Ising Systems</a></em>, Phys. Rev. <strong>187</strong>, 732 (1969).</div>
|
||||
</div>
|
||||
<div id="ref-angeliniRelationsShortrangeLongrange2014" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E <strong>89</strong>, 062120 (2014).</div>
|
||||
</div>
|
||||
<div id="ref-fukuiOrderNClusterMonte2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">K. Fukui and S. Todo, <em><a href="https://doi.org/10.1016/j.jcp.2008.12.022">Order-N Cluster Monte Carlo Method for Spin Systems with Long-Range Interactions</a></em>, Journal of Computational Physics <strong>228</strong>, 2629 (2009).</div>
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E <strong>89</strong>, 062120 (2014).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -31,12 +31,7 @@ image:
|
||||
<ul>
|
||||
<li><a href="#thermodynamics-of-the-lrfk-model" id="toc-thermodynamics-of-the-lrfk-model">Thermodynamics of the LRFK Model</a></li>
|
||||
<li><a href="#markov-chain-monte-carlo-and-emergent-disorder" id="toc-markov-chain-monte-carlo-and-emergent-disorder">Markov Chain Monte Carlo and Emergent Disorder</a></li>
|
||||
<li><a href="#application-to-the-fk-model" id="toc-application-to-the-fk-model">Application to the FK Model</a></li>
|
||||
<li><a href="#two-step-trick" id="toc-two-step-trick">Two Step Trick</a></li>
|
||||
<li><a href="#scaling" id="toc-scaling">Scaling</a></li>
|
||||
<li><a href="#binder-cumulants" id="toc-binder-cumulants">Binder Cumulants</a></li>
|
||||
<li><a href="#convergence-time" id="toc-convergence-time">Convergence Time</a></li>
|
||||
<li><a href="#auto-correlation-time" id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -55,12 +50,7 @@ image:
|
||||
<ul>
|
||||
<li><a href="#thermodynamics-of-the-lrfk-model" id="toc-thermodynamics-of-the-lrfk-model">Thermodynamics of the LRFK Model</a></li>
|
||||
<li><a href="#markov-chain-monte-carlo-and-emergent-disorder" id="toc-markov-chain-monte-carlo-and-emergent-disorder">Markov Chain Monte Carlo and Emergent Disorder</a></li>
|
||||
<li><a href="#application-to-the-fk-model" id="toc-application-to-the-fk-model">Application to the FK Model</a></li>
|
||||
<li><a href="#two-step-trick" id="toc-two-step-trick">Two Step Trick</a></li>
|
||||
<li><a href="#scaling" id="toc-scaling">Scaling</a></li>
|
||||
<li><a href="#binder-cumulants" id="toc-binder-cumulants">Binder Cumulants</a></li>
|
||||
<li><a href="#convergence-time" id="toc-convergence-time">Convergence Time</a></li>
|
||||
<li><a href="#auto-correlation-time" id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -74,129 +64,99 @@ image:
|
||||
</div>
|
||||
<section id="fk-methods" class="level1">
|
||||
<h1>Methods</h1>
|
||||
<p>To evaluate thermodynamic averages I perform classical Markov Chain Monte Carlo random walks over the space of spin configurations of the LRFK model, at each step diagonalising the effective electronic Hamiltonian <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">1</a>]</span>. Using a binder-cumulant method <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">2</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">3</a>]</span>, I demonstrate the model has a finite temperature phase transition when the interaction is sufficiently long ranged.</p>
|
||||
<section id="thermodynamics-of-the-lrfk-model" class="level2">
|
||||
<h2>Thermodynamics of the LRFK Model</h2>
|
||||
<p>The results for the phase diagram were obtained with a classical Markov Chain Monte Carlo (MCMC) method which we discuss in the following. It allows us to solve our long-range FK model efficiently, yielding unbiased estimates of thermal expectation values and linking it to disorder physics in a translationally invariant setting.</p>
|
||||
<p>Since the spin configurations are classical, the Hamiltonian can be split into a classical spin part <span class="math inline">\(H_s\)</span> and an operator valued part <span class="math inline">\(H_c\)</span>.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/lsr/pdf_figs/raw_steps_single_flip.svg" id="fig:raw_steps_single_flip" data-short-caption="Comparison of different proposal distributions" style="width:100.0%" alt="Figure 1: Two MCMC walks starting from the staggered charge density wave ground state for a system with N = 100 sites and 10,000 MCMC steps but at a temperature close to the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation m = N^{-1} \sum_i (-1)^i \; S_i order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. t = 1, \alpha = 1.25, T = 2.5,5, J = U = 5" />
|
||||
<figcaption aria-hidden="true">Figure 1: Two MCMC walks starting from the staggered charge density wave ground state for a system with <span class="math inline">\(N = 100\)</span> sites and 10,000 MCMC steps but at a temperature close to the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. <span class="math inline">\(t = 1, \alpha = 1.25, T = 2.5,5, J = U = 5\)</span></figcaption>
|
||||
</figure>
|
||||
<p>The classical Markov Chain Monte Carlo (MCMC) method which we discuss in the following allows us to solve our long-range FK model efficiently, yielding unbiased estimates of thermal expectation values.</p>
|
||||
<p>Since the spin configurations are classical, the LRFK Hamiltonian can be split into a classical spin part <span class="math inline">\(H_s\)</span> and an operator valued part <span class="math inline">\(H_c\)</span>.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
H_s& = - \frac{U}{2}S_i + \sum_{i, j}^{N} J_{ij} S_i S_j \\
|
||||
H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i}) \end{aligned}\]</span></p>
|
||||
<p>The partition function can then be written as a sum over spin configurations, <span class="math inline">\(\vec{S} = (S_0, S_1...S_{N-1})\)</span>:</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \mathrm{Tr} e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s} \mathrm{Tr}_c e^{-\beta H_c} .\end{aligned}\]</span></p>
|
||||
<p>The contribution of <span class="math inline">\(H_c\)</span> to the grand canonical partition function can be obtained by performing the sum over eigenstate occupation numbers giving <span class="math inline">\(-\beta F_c[\vec{S}] = \sum_k \ln{(1 + e^{- \beta \epsilon_k})}\)</span> where <span class="math inline">\({\epsilon_k[\vec{S}]}\)</span> are the eigenvalues of the matrix representation of <span class="math inline">\(H_c\)</span> determined through exact diagonalisation. This gives a partition function containing a classical energy which corresponds to the long-range interaction of the spins, and a free energy which corresponds to the quantum subsystem. <span class="math display">\[\begin{aligned}
|
||||
<p>The contribution of <span class="math inline">\(H_c\)</span> to the grand canonical partition function can be obtained by performing the sum over eigenstate occupation numbers giving <span class="math inline">\(-\beta F_c[\vec{S}] = \sum_k \ln{(1 + e^{- \beta \epsilon_k})}\)</span> where <span class="math inline">\({\epsilon_k[\vec{S}]}\)</span> are the eigenvalues of the matrix representation of <span class="math inline">\(H_c\)</span> determined through exact diagonalisation. This gives a partition function containing a classical energy which corresponds to the long-range interaction of the spins, and a free energy which corresponds to the quantum subsystem.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}\end{aligned}\]</span></p>
|
||||
</section>
|
||||
<section id="markov-chain-monte-carlo-and-emergent-disorder" class="level2">
|
||||
<h2>Markov Chain Monte Carlo and Emergent Disorder</h2>
|
||||
<p>Classical MCMC defines a weighted random walk over the spin states <span class="math inline">\((\vec{S}_0, \vec{S}_1, \vec{S}_2, ...)\)</span>, such that the likelihood of visiting a particular state converges to its Boltzmann probability <span class="math inline">\(p(\vec{S}) = \mathcal{Z}^{-1} e^{-\beta E}\)</span>. Hence, any observable can be estimated as a mean over the states visited by the walk <span class="citation" data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"> [<a href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">1</a>–<a href="#ref-wolffMonteCarloErrors2004" role="doc-biblioref">3</a>]</span>, <span class="math display">\[\begin{aligned}
|
||||
<p>Classical MCMC defines a weighted random walk over the spin states <span class="math inline">\((\vec{S}_0, \vec{S}_1, \vec{S}_2, ...)\)</span>, such that the likelihood of visiting a particular state converges to its Boltzmann probability <span class="math inline">\(p(\vec{S}) = \mathcal{Z}^{-1} e^{-\beta E}\)</span>. Hence, any observable can be estimated as a mean over the states visited by the walk <span class="citation" data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"> [<a href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">4</a>–<a href="#ref-wolffMonteCarloErrors2004" role="doc-biblioref">6</a>]</span>,</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\label{eq:thermal_expectation}
|
||||
\langle O \rangle & = \sum_{\vec{S}} p(\vec{S}) \langle O \rangle_{\vec{S}}\\
|
||||
& = \sum_{i = 0}^{M} \langle O\rangle_{\vec{S}_i} \pm \mathcal{O}(\tfrac{1}{\sqrt{M}})
|
||||
\end{aligned}\]</span> where the former sum runs over the entire state space while the later runs over all the state visited by a particular MCMC run.</p>
|
||||
\langle O \rangle & = \sum_{\vec{S}} p(\vec{S}) \langle O \rangle\\
|
||||
& = \sum_{i = 0}^{M} \langle O\rangle \pm \mathcal{O}(M^{-\tfrac{1}{2}})
|
||||
\end{aligned}\]</span></p>
|
||||
<p>where the former sum runs over the entire state space while the later runs over all the state visited by a particular MCMC run.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\langle O \rangle_{\vec{S}}& = \sum_{\nu} n_F(\epsilon_{\nu}) \langle O \rangle{\nu}
|
||||
\end{aligned}\]</span></p>
|
||||
<p>Where <span class="math inline">\(\nu\)</span> runs over the eigenstates of <span class="math inline">\(H_c\)</span> for a particular spin configuration and <span class="math inline">\(n_F(\epsilon) = \left(e^{-\beta\epsilon} + 1\right)^{-1}\)</span> is the Fermi function.</p>
|
||||
<p>The choice of the transition function for MCMC is under-determined as one only needs to satisfy a set of balance conditions for which there are many solutions <span class="citation" data-cites="kellyReversibilityStochasticNetworks1981"> [<a href="#ref-kellyReversibilityStochasticNetworks1981" role="doc-biblioref">4</a>]</span>. Here, we incorporate a modification to the standard Metropolis-Hastings algorithm <span class="citation" data-cites="hastingsMonteCarloSampling1970"> [<a href="#ref-hastingsMonteCarloSampling1970" role="doc-biblioref">5</a>]</span> gleaned from Krauth <span class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">6</a>]</span>. Let us first recall the standard algorithm which decomposes the transition probability into <span class="math inline">\(\mathcal{T}(a \to b) = p(a \to b)\mathcal{A}(a \to b)\)</span>. Here, <span class="math inline">\(p\)</span> is the proposal distribution that we can directly sample from while <span class="math inline">\(\mathcal{A}\)</span> is the acceptance probability. The standard Metropolis-Hastings choice is <span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, \frac{p(b\to a)}{p(a\to b)} e^{-\beta \Delta E}\right)\;,\]</span> with <span class="math inline">\(\Delta E = E_b - E_a\)</span>. The walk then proceeds by sampling a state <span class="math inline">\(b\)</span> from <span class="math inline">\(p\)</span> and moving to <span class="math inline">\(b\)</span> with probability <span class="math inline">\(\mathcal{A}(a \to b)\)</span>. The latter operation is typically implemented by performing a transition if a uniform random sample from the unit interval is less than <span class="math inline">\(\mathcal{A}(a \to b)\)</span> and otherwise repeating the current state as the next step in the random walk. The proposal distribution is often symmetric so does not appear in <span class="math inline">\(\mathcal{A}\)</span>. Here, we flip a small number of sites in <span class="math inline">\(b\)</span> at random to generate proposals, which is indeed symmetric.</p>
|
||||
<p>In our computations <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">7</a>]</span> we employ a modification of the algorithm which is based on the observation that the free energy of the <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> system is composed of a classical part which is much quicker to compute than the quantum part. Hence, we can obtain a computational speedup by first considering the value of the classical energy difference <span class="math inline">\(\Delta H_s\)</span> and rejecting the transition if the former is too high. We only compute the quantum energy difference <span class="math inline">\(\Delta F_c\)</span> if the transition is accepted. We then perform a second rejection sampling step based upon it. This corresponds to two nested comparisons with the majority of the work only occurring if the first test passes and has the acceptance function <span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]</span></p>
|
||||
<p>For the model parameters used in Fig. [1], we find that with our new scheme the matrix diagonalisation is skipped around 30% of the time at <span class="math inline">\(T = 2.5\)</span> and up to 80% at <span class="math inline">\(T = 1.5\)</span>. We observe that for <span class="math inline">\(N = 50\)</span>, the matrix diagonalisation, if it occurs, occupies around 60% of the total computation time for a single step. This rises to 90% at N = 300 and further increases for larger N. We therefore get the greatest speedup for large system sizes at low temperature where many prospective transitions are rejected at the classical stage and the matrix computation takes up the greatest fraction of the total computation time. The upshot is that we find a speedup of up to a factor of 10 at the cost of very little extra algorithmic complexity.</p>
|
||||
<p>Our two-step method should be distinguished from the more common method for speeding up MCMC which is to add asymmetry to the proposal distribution to make it as similar as possible to <span class="math inline">\(\min\left(1, e^{-\beta \Delta E}\right)\)</span>. This reduces the number of rejected states, which brings the algorithm closer in efficiency to a direct sampling method. However it comes at the expense of requiring a way to directly sample from this complex distribution, a problem which MCMC was employed to solve in the first place. For example, recent work trains restricted Boltzmann machines (RBMs) to generate samples for the proposal distribution of the FK model <span class="citation" data-cites="huangAcceleratedMonteCarlo2017"> [<a href="#ref-huangAcceleratedMonteCarlo2017" role="doc-biblioref">8</a>]</span>. The RBMs are chosen as a parametrisation of the proposal distribution that can be efficiently sampled from while offering sufficient flexibility that they can be adjusted to match the target distribution. Our proposed method is considerably simpler and does not require training while still reaping some of the benefits of reduced computation.</p>
|
||||
</section>
|
||||
<section id="application-to-the-fk-model" class="level2">
|
||||
<h2>Application to the FK Model</h2>
|
||||
<p>We will work in the grand canonical ensemble of fixed temperature, chemical potential and volume.</p>
|
||||
<p>Since the spin configurations are classical, the Hamiltonian can be split into a classical spin part <span class="math inline">\(H_s\)</span> and an operator valued part <span class="math inline">\(H_c\)</span>. <span class="math display">\[\begin{aligned}
|
||||
H_s& = - \frac{U}{2}S_i + \sum_{i, j}^{N} J_{ij} S_i S_j \\
|
||||
H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i})
|
||||
\end{aligned}\]</span> The partition function can then be written as a sum over spin configurations, <span class="math inline">\(\vec{S} = (S_0, S_1...S_{N-1})\)</span>: <span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \mathrm{Tr} e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s} \mathrm{Tr}_c e^{-\beta H_c} .
|
||||
\end{aligned}
|
||||
\]</span> The contribution of <span class="math inline">\(H_c\)</span> to the grand canonical partition function can be obtained by performing the sum over eigenstate occupation numbers giving <span class="math inline">\(-\beta F_c[\vec{S}] = \sum_k \ln{(1 + e^{- \beta \epsilon_k})}\)</span> where <span class="math inline">\({\epsilon_k[\vec{S}]}\)</span> are the eigenvalues of the matrix representation of <span class="math inline">\(H_c\)</span> determined through exact diagonalisation. This gives a partition function containing a classical energy which corresponds to the long-range interaction of the spins, and a free energy which corresponds to the quantum subsystem. <span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}
|
||||
\end{aligned}\]</span></p>
|
||||
</section>
|
||||
<section id="two-step-trick" class="level2">
|
||||
<h2>Two Step Trick</h2>
|
||||
<p>Here, we incorporate a modification to the standard Metropolis-Hastings algorithm <span class="citation" data-cites="hastingsMonteCarloSampling1970"> [<a href="#ref-hastingsMonteCarloSampling1970" role="doc-biblioref">5</a>]</span> gleaned from Krauth <span class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">6</a>]</span>.</p>
|
||||
<p>In our computations <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">7</a>]</span> we employ a modification of the algorithm which is based on the observation that the free energy of the FK system is composed of a classical part which is much quicker to compute than the quantum part. Hence, we can obtain a computational speedup by first considering the value of the classical energy difference <span class="math inline">\(\Delta H_s\)</span> and rejecting the transition if the former is too high. We only compute the quantum energy difference <span class="math inline">\(\Delta F_c\)</span> if the transition is accepted. We then perform a second rejection sampling step based upon it. This corresponds to two nested comparisons with the majority of the work only occurring if the first test passes and has the acceptance function <span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]</span></p>
|
||||
<p>For the model parameters <span class="math inline">\(U=2/5, T = 1.5 / 2.5, J = 5,\;\alpha = 1.25\)</span>, we find that with our new scheme the matrix diagonalisation is skipped around 30% of the time at <span class="math inline">\(T = 2.5\)</span> and up to 80% at <span class="math inline">\(T = 1.5\)</span>. We observe that for <span class="math inline">\(N = 50\)</span>, the matrix diagonalisation, if it occurs, occupies around 60% of the total computation time for a single step. This rises to 90% at N = 300 and further increases for larger N. We therefore get the greatest speedup for large system sizes at low temperature where many prospective transitions are rejected at the classical stage and the matrix computation takes up the greatest fraction of the total computation time. The upshot is that we find a speedup of up to a factor of 10 at the cost of very little extra algorithmic complexity.</p>
|
||||
<p>Our two-step method should be distinguished from the more common method for speeding up MCMC which is to add asymmetry to the proposal distribution to make it as similar as possible to <span class="math inline">\(\min\left(1, e^{-\beta \Delta E}\right)\)</span>. This reduces the number of rejected states, which brings the algorithm closer in efficiency to a direct sampling method. However it comes at the expense of requiring a way to directly sample from this complex distribution, a problem which MCMC was employed to solve in the first place. For example, recent work trains restricted Boltzmann machines (RBMs) to generate samples for the proposal distribution of the FK model <span class="citation" data-cites="huangAcceleratedMonteCarlo2017"> [<a href="#ref-huangAcceleratedMonteCarlo2017" role="doc-biblioref">8</a>]</span>. The RBMs are chosen as a parametrisation of the proposal distribution that can be efficiently sampled from while offering sufficient flexibility that they can be adjusted to match the target distribution. Our proposed method is considerably simpler and does not require training while still reaping some of the benefits of reduced computation.</p>
|
||||
<p>The choice of the transition function for MCMC is under-determined as one only needs to satisfy a set of balance conditions for which there are many solutions <span class="citation" data-cites="kellyReversibilityStochasticNetworks1981"> [<a href="#ref-kellyReversibilityStochasticNetworks1981" role="doc-biblioref">7</a>]</span>. Here, we incorporate a modification to the standard Metropolis-Hastings algorithm <span class="citation" data-cites="hastingsMonteCarloSampling1970"> [<a href="#ref-hastingsMonteCarloSampling1970" role="doc-biblioref">8</a>]</span> gleaned from Krauth <span class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">9</a>]</span>.</p>
|
||||
<p>The standard algorithm decomposes the transition probability into <span class="math inline">\(\mathcal{T}(a \to b) = p(a \to b)\mathcal{A}(a \to b)\)</span>. Here, <span class="math inline">\(p\)</span> is the proposal distribution that we can directly sample from while <span class="math inline">\(\mathcal{A}\)</span> is the acceptance probability. The standard Metropolis-Hastings choice is</p>
|
||||
<p><span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, \frac{p(b\to a)}{p(a\to b)} e^{-\beta \Delta E}\right)\;,\]</span></p>
|
||||
<p>with <span class="math inline">\(\Delta E = E_b - E_a\)</span>. The walk then proceeds by sampling a state <span class="math inline">\(b\)</span> from <span class="math inline">\(p\)</span> and moving to <span class="math inline">\(b\)</span> with probability <span class="math inline">\(\mathcal{A}(a \to b)\)</span>. The latter operation is typically implemented by performing a transition if a uniform random sample from the unit interval is less than <span class="math inline">\(\mathcal{A}(a \to b)\)</span> and otherwise repeating the current state as the next step in the random walk. The proposal distribution is often symmetric so does not appear in <span class="math inline">\(\mathcal{A}\)</span>. Here, we flip a small number of sites in <span class="math inline">\(b\)</span> at random to generate proposals, which is a symmetric proposal.</p>
|
||||
<p>In our computations <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">10</a>]</span> the modification to this algorithm that we employ is based on the observation that the free energy of the FK system is composed of a classical part which is much quicker to compute than the quantum part. Hence, we can obtain a computational speed up by first considering the value of the classical energy difference <span class="math inline">\(\Delta H_s\)</span> and rejecting the transition if the former is too high. We only compute the quantum energy difference <span class="math inline">\(\Delta F_c\)</span> if the transition is accepted. We then perform a second rejection sampling step based upon it. This corresponds to two nested comparisons with the majority of the work only occurring if the first test passes. This modified scheme has the acceptance function <span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]</span></p>
|
||||
<p>For the model parameters used, we find that with our new scheme the matrix diagonalisation is skipped around 30% of the time at <span class="math inline">\(T = 2.5\)</span> and up to 80% at <span class="math inline">\(T = 1.5\)</span>. We observe that for <span class="math inline">\(N = 50\)</span>, the matrix diagonalisation, if it occurs, occupies around 60% of the total computation time for a single step. This rises to 90% at N = 300 and further increases for larger N. We therefore get the greatest speedup for large system sizes at low temperature where many prospective transitions are rejected at the classical stage and the matrix computation takes up the greatest fraction of the total computation time. The upshot is that we find a speedup of up to a factor of 10 at the cost of very little extra algorithmic complexity.</p>
|
||||
<p>Our two-step method should be distinguished from the more common method for speeding up MCMC which is to add asymmetry to the proposal distribution to make it as similar as possible to <span class="math inline">\(\min\left(1, e^{-\beta \Delta E}\right)\)</span>. This reduces the number of rejected states, which brings the algorithm closer in efficiency to a direct sampling method. However it comes at the expense of requiring a way to directly sample from this complex distribution, a problem which MCMC was employed to solve in the first place. For example, recent work trains restricted Boltzmann machines (RBMs) to generate samples for the proposal distribution of the FK model <span class="citation" data-cites="huangAcceleratedMonteCarlo2017"> [<a href="#ref-huangAcceleratedMonteCarlo2017" role="doc-biblioref">11</a>]</span>. The RBMs are chosen as a parametrisation of the proposal distribution that can be efficiently sampled from while offering sufficient flexibility that they can be adjusted to match the target distribution. Our proposed method is considerably simpler and does not require training while still reaping some of the benefits of reduced computation.</p>
|
||||
</section>
|
||||
<section id="scaling" class="level2">
|
||||
<h2>Scaling</h2>
|
||||
<p>In order to reduce the effects of the boundary conditions and the finite size of the system we redefine and normalise the coupling matrix to have 0 derivative at its furthest extent rather than cutting off abruptly.</p>
|
||||
<p><span class="math display">\[
|
||||
\begin{aligned}
|
||||
J'(x) &= \frac{L}{\pi}\left|\;\sin \frac{\pi x}{L}\;\right|^{-\alpha} \\
|
||||
J(x) &= \frac{J_0 J'(x)}{\sum_y J'(y)}
|
||||
\end{aligned}
|
||||
\]</span></p>
|
||||
<p>The scaling ensures that, in the ordered phase, the overall potential felt by each site due to the rest of the system is independent of system size.</p>
|
||||
</section>
|
||||
<section id="binder-cumulants" class="level2">
|
||||
<h2>Binder Cumulants</h2>
|
||||
<p>The Binder cumulant is defined as:</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 2: (Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. ¿fig:phase_diagram? . All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly mentioned." />
|
||||
<figcaption aria-hidden="true">Figure 2: (Left) The order parameters, <span class="math inline">\(\langle m^2 \rangle\)</span>(solid) and <span class="math inline">\(1 - f\)</span> (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle\)</span> . (Right) The crossing of the Binder cumulant, <span class="math inline">\(B = \langle m^4 \rangle / \langle m^2 \rangle^2\)</span>, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. <strong>¿fig:phase_diagram?</strong> . All plots use system sizes <span class="math inline">\(N = [10,20,30,50,70,110,160,250]\)</span> and lines are coloured from <span class="math inline">\(N = 10\)</span> in dark blue to <span class="math inline">\(N = 250\)</span> in yellow. The parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span> except where explicitly mentioned.</figcaption>
|
||||
</figure>
|
||||
<p>To improve the scaling of finite size effects, we make the replacement <span class="math inline">\(|i - j|^{-\alpha} \rightarrow |f(i - j)|^{-\alpha}\)</span>, in both <span class="math inline">\(J_{ij}\)</span> and <span class="math inline">\(\kappa\)</span>, where <span class="math inline">\(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\)</span>. <span class="math inline">\(f\)</span> is smooth across the circular boundary and its effect effect diminished for larger systems/ <span class="citation" data-cites="fukuiOrderNClusterMonte2009"> [<a href="#ref-fukuiOrderNClusterMonte2009" role="doc-biblioref">12</a>]</span>. We only consider even system sizes given that odd system sizes are not commensurate with a CDW state.</p>
|
||||
<p>To identify critical points we use the the Binder cumulant <span class="math inline">\(U_B\)</span> defined by</p>
|
||||
<p><span class="math display">\[
|
||||
U_B = 1 - \frac{\langle\mu_4\rangle}{3\langle\mu_2\rangle^2}
|
||||
\]</span></p>
|
||||
<p>where <span class="math inline">\(\mu_n = \langle(m - \langle m\rangle)^n\rangle\)</span> are the central moments of the order parameter <span class="math inline">\(m = \sum_i (-1)^i (2n_i - 1) / N\)</span>. The Binder cumulant evaluated against temperature is a diagnostic for the existence of a phase transition. If multiple such curves are plotted for different system sizes, a crossing indicates the location of a critical point while the lines do not cross for systems that don’t have a phase transition in the thermodynamic limit <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">9</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">10</a>]</span>.</p>
|
||||
<p><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="(Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. ¿fig:phase_diagram? . 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." /> <!-- ![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>
|
||||
<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>
|
||||
<!-- ![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>
|
||||
<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>
|
||||
<section id="convergence-time" class="level2">
|
||||
<h2>Convergence Time</h2>
|
||||
<p>Considering <span class="math inline">\(p(\s)\)</span> as a vector <span class="math inline">\(\vec{p}\)</span> whose jth entry is the probability of the jth state <span class="math inline">\(p_j = p(\s_j)\)</span>, and writing <span class="math inline">\(\mathcal{T}\)</span> as the matrix with entries <span class="math inline">\(T_{ij} = \mathcal{T}(\s_j \rightarrow \s_i)\)</span> we can write the update rule for the ensemble probability as: <span class="math display">\[\vec{p}_{t+1} = \mathcal{T} \vec{p}_t \implies \vec{p}_{t} = \mathcal{T}^t \vec{p}_0\]</span> where <span class="math inline">\(\vec{p}_0\)</span> is vector which is one on the starting state and zero everywhere else. Since all states must transition to somewhere with probability one: <span class="math inline">\(\sum_i T_{ij} = 1\)</span>.</p>
|
||||
<p>Matrices that satisfy this are called stochastic matrices exactly because they model these kinds of Markov processes. It can be shown that they have real eigenvalues, and ordering them by magnitude, that <span class="math inline">\(\lambda_0 = 1\)</span> and <span class="math inline">\(0 < \lambda_{i\neq0} < 1\)</span>. Assuming <span class="math inline">\(\mathcal{T}\)</span> has been chosen correctly, its single eigenvector with eigenvalue 1 will be the thermal distribution [^3] so repeated application of the transition function eventually leads there, while memory of the initial conditions decays exponentially with a convergence time <span class="math inline">\(k\)</span> determined by <span class="math inline">\(\lambda_1\)</span>. In practice this means that one throws away the data from the beginning of the random walk in order reduce the dependence on the initial conditions and be close enough to the target distribution.</p>
|
||||
</section>
|
||||
<section id="auto-correlation-time" class="level2">
|
||||
<h2>Auto-correlation Time</h2>
|
||||
<!-- ![(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>
|
||||
<p>where <span class="math inline">\(\mu_n = \langle(m - \langle m\rangle)^n\rangle\)</span> are the central moments of the order parameter <span class="math inline">\(m = \sum_i (-1)^i (2n_i - 1) / N\)</span>. The Binder cumulant evaluated against temperature is a diagnostic for the existence of a phase transition. If multiple such curves are plotted for different system sizes, a crossing indicates the location of a critical point while the lines do not cross for systems that don’t have a phase transition in the thermodynamic limit <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">2</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">3</a>]</span>.</p>
|
||||
<p>Next Section: <a href="../3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html">Results</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">
|
||||
<div id="ref-binderGuidePracticalWork1988" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">K. Binder and D. W. Heermann, <em><a href="https://doi.org/10.1007/978-3-662-08854-8_3">Guide to Practical Work with the Monte Carlo Method</a></em>, in <em>Monte Carlo Simulation in Statistical Physics: An Introduction</em>, edited by K. Binder and D. W. Heermann (Springer Berlin Heidelberg, Berlin, Heidelberg, 1988), pp. 68–112.</div>
|
||||
</div>
|
||||
<div id="ref-kerteszAdvancesComputerSimulation1998" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">J. Kertesz and I. Kondor, editors, <em><a href="https://doi.org/10.1007/BFb0105456">Advances in Computer Simulation: Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July 1996</a></em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
|
||||
</div>
|
||||
<div id="ref-wolffMonteCarloErrors2004" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">U. Wolff, <em><a href="https://doi.org/10.1016/S0010-4655(03)00467-3">Monte Carlo Errors with Less Errors</a></em>, Computer Physics Communications <strong>156</strong>, 143 (2004).</div>
|
||||
</div>
|
||||
<div id="ref-kellyReversibilityStochasticNetworks1981" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">F. P. Kelly, <em><a href="https://doi.org/10.2307/2287860">Reversibility and Stochastic Networks / F.P. Kelly</a></em>, SERBIULA (Sistema Librum 2.0) <strong>76</strong>, (1981).</div>
|
||||
</div>
|
||||
<div id="ref-hastingsMonteCarloSampling1970" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">W. K. Hastings, <em><a href="https://doi.org/10.1093/biomet/57.1.97">Monte Carlo Sampling Methods Using Markov Chains and Their Applications</a></em>, Biometrika <strong>57</strong>, 97 (1970).</div>
|
||||
</div>
|
||||
<div id="ref-krauthIntroductionMonteCarlo1998" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">W. Krauth, <em><a href="https://doi.org/10.1007/BFb0105456">Introduction To Monte Carlo Algorithms</a></em>, in <em>Advances in Computer Simulation: Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July 1996</em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
|
||||
</div>
|
||||
<div id="ref-hodsonMCMCFKModel2021" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">T. Hodson, <em><a href="https://doi.org/10.5281/zenodo.4593904">Markov Chain Monte Carlo for the Kitaev Model</a></em>, (2021).</div>
|
||||
</div>
|
||||
<div id="ref-huangAcceleratedMonteCarlo2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">L. Huang and L. Wang, <em><a href="https://doi.org/10.1103/PhysRevB.95.035105">Accelerated Monte Carlo Simulations with Restricted Boltzmann Machines</a></em>, Phys. Rev. B <strong>95</strong>, 035105 (2017).</div>
|
||||
<div id="ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">M. M. Maśka and K. Czajka, <em><a href="https://doi.org/10.1103/PhysRevB.74.035109">Thermodynamics of the Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo Study</a></em>, Phys. Rev. B <strong>74</strong>, 035109 (2006).</div>
|
||||
</div>
|
||||
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </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>
|
||||
<div class="csl-left-margin">[2] </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>
|
||||
</div>
|
||||
<div id="ref-musialMonteCarloSimulations2002" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">G. Musiał, L. Dȩbski, and G. Kamieniarz, <em><a href="https://doi.org/10.1103/PhysRevB.66.012407">Monte Carlo Simulations of Ising-Like Phase Transitions in the Three-Dimensional Ashkin-Teller Model</a></em>, Phys. Rev. B <strong>66</strong>, 012407 (2002).</div>
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">G. Musiał, L. Dȩbski, and G. Kamieniarz, <em><a href="https://doi.org/10.1103/PhysRevB.66.012407">Monte Carlo Simulations of Ising-Like Phase Transitions in the Three-Dimensional Ashkin-Teller Model</a></em>, Phys. Rev. B <strong>66</strong>, 012407 (2002).</div>
|
||||
</div>
|
||||
<div id="ref-binderGuidePracticalWork1988" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">K. Binder and D. W. Heermann, <em><a href="https://doi.org/10.1007/978-3-662-08854-8_3">Guide to Practical Work with the Monte Carlo Method</a></em>, in <em>Monte Carlo Simulation in Statistical Physics: An Introduction</em>, edited by K. Binder and D. W. Heermann (Springer Berlin Heidelberg, Berlin, Heidelberg, 1988), pp. 68–112.</div>
|
||||
</div>
|
||||
<div id="ref-kerteszAdvancesComputerSimulation1998" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">J. Kertesz and I. Kondor, editors, <em><a href="https://doi.org/10.1007/BFb0105456">Advances in Computer Simulation: Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July 1996</a></em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
|
||||
</div>
|
||||
<div id="ref-wolffMonteCarloErrors2004" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">U. Wolff, <em><a href="https://doi.org/10.1016/S0010-4655(03)00467-3">Monte Carlo Errors with Less Errors</a></em>, Computer Physics Communications <strong>156</strong>, 143 (2004).</div>
|
||||
</div>
|
||||
<div id="ref-kellyReversibilityStochasticNetworks1981" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">F. P. Kelly, <em><a href="https://doi.org/10.2307/2287860">Reversibility and Stochastic Networks / F.P. Kelly</a></em>, SERBIULA (Sistema Librum 2.0) <strong>76</strong>, (1981).</div>
|
||||
</div>
|
||||
<div id="ref-hastingsMonteCarloSampling1970" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">W. K. Hastings, <em><a href="https://doi.org/10.1093/biomet/57.1.97">Monte Carlo Sampling Methods Using Markov Chains and Their Applications</a></em>, Biometrika <strong>57</strong>, 97 (1970).</div>
|
||||
</div>
|
||||
<div id="ref-krauthIntroductionMonteCarlo1998" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">W. Krauth, <em><a href="https://doi.org/10.1007/BFb0105456">Introduction To Monte Carlo Algorithms</a></em>, in <em>Advances in Computer Simulation: Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July 1996</em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
|
||||
</div>
|
||||
<div id="ref-hodsonMCMCFKModel2021" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">T. Hodson, <em><a href="https://doi.org/10.5281/zenodo.4593904">Markov Chain Monte Carlo for the Kitaev Model</a></em>, (2021).</div>
|
||||
</div>
|
||||
<div id="ref-huangAcceleratedMonteCarlo2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">L. Huang and L. Wang, <em><a href="https://doi.org/10.1103/PhysRevB.95.035105">Accelerated Monte Carlo Simulations with Restricted Boltzmann Machines</a></em>, Phys. Rev. B <strong>95</strong>, 035105 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-fukuiOrderNClusterMonte2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">K. Fukui and S. Todo, <em><a href="https://doi.org/10.1016/j.jcp.2008.12.022">Order-N Cluster Monte Carlo Method for Spin Systems with Long-Range Interactions</a></em>, Journal of Computational Physics <strong>228</strong>, 2629 (2009).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -64,16 +64,17 @@ image:
|
||||
</div>
|
||||
<section id="fk-results" class="level1">
|
||||
<h1>Results</h1>
|
||||
<section id="lrfk-results-phase-diagram" class="level2">
|
||||
<h2>Phase Diagram</h2>
|
||||
<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>
|
||||
<p>Fig fig. <a href="#fig:phase_diagram">1</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. <strong>¿fig:binder_cumulants?</strong> 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>
|
||||
<p>Looking at the results of our MCMC simulations, we find a rich phase diagram with a CDW FTPT and interaction-tuned Anderson versus Mott localized phases similar to the 2D FK model <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>. We explore the localization properties of the fermionic sector and find that the localisation lengths vary dramatically across the phases and for different energies. Although moderate system sizes indicate the coexistence of localized and delocalized states within the CDW phase, we find quantitatively similar behaviour in a model of uncorrelated binary disorder on a CDW background. For large system sizes, i.e. for our 1D disorder model we can treat linear sizes of several thousand sites, we find that all states are eventually localized with a localization length which diverges towards zero temperature.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg" id="fig:phase_diagram" data-short-caption="Long Range Falicov Kimball Model Phase Diagram" style="width:100.0%" alt="Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature T_c, linear in J. (Right) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas." />
|
||||
<figcaption aria-hidden="true">Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at <span class="math inline">\(U = 5\)</span>: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (Right) The TU plane at <span class="math inline">\(J = 5\)</span>: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at <span class="math inline">\(E=0\)</span> in the single particle energy spectrum. <span class="math inline">\(U_c\)</span> is independent of temperature. At <span class="math inline">\(U = 0\)</span> the fermions are decoupled from the spins forming a simple Fermi gas.</figcaption>
|
||||
</figure>
|
||||
<section id="lrfk-results-phase-diagram" class="level2">
|
||||
<h2>Phase Diagram</h2>
|
||||
<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>
|
||||
<p>Fig fig. <a href="#fig:phase_diagram">1</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">2</a>]</span>. For a representative set of parameters, fig. <strong>¿fig:binder_cumulants?</strong> 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">3</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">1</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>
|
||||
</section>
|
||||
<section id="localisation-properties" class="level2">
|
||||
<h2>Localisation Properties</h2>
|
||||
@ -117,8 +118,8 @@ H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dagger_{i}c_{i} - \tfra
|
||||
</figure>
|
||||
<p>As we can sample directly from the disorder model, rather than through MCMC, the samples are uncorrelated. Hence we can evaluate much larger system sizes with the disorder model which enables us to pin down the correct localisation effects. In particular, what appear to be delocalized states for small system sizes eventually turn out to be states with large localization length. The localization length diverges towards the ordered zero temperature CDW state. The interplay of interactions, which here produce as peculiar binary potential, and localization can be very intricate and the added advantage of a 1D model is that we can explore very large system sizes.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/disorder_model/DM_IPR_scaling.svg" id="fig:DM_IPR_scaling" data-short-caption="A comparison of the full FK model to a simple binary disorder model." style="width:100.0%" alt="Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 3 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation [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" />
|
||||
<figcaption aria-hidden="true">Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 < \rho < 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:IPR_scaling">3</a> <span class="math inline">\(\tau(\omega)\)</span> the scaling of IPR(<span class="math inline">\(\omega\)</span>) with system size, , is shown both in gap (<span class="math inline">\(\omega_0\)</span>) and in the band (<span class="math inline">\(\omega_1\)</span>). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">3</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>
|
||||
<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="Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 3 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation [1], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N > 400" />
|
||||
<figcaption aria-hidden="true">Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 < \rho < 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:IPR_scaling">3</a> <span class="math inline">\(\tau(\omega)\)</span> the scaling of IPR(<span class="math inline">\(\omega\)</span>) with system size, , is shown both in gap (<span class="math inline">\(\omega_0\)</span>) and in the band (<span class="math inline">\(\omega_1\)</span>). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>, hence all the states are indeed localised as one would expect in 1D. The disorder model <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span class="math inline">\(0.01\pm0.04, -0.01\pm0.04\)</span> (c) <span class="math inline">\(0.05\pm0.06, 0.04\pm0.06\)</span> (d) <span class="math inline">\(-0.03\pm0.06, 0.01\pm0.06\)</span>. The lines are fit on system sizes <span class="math inline">\(N > 400\)</span></figcaption>
|
||||
</figure>
|
||||
</section>
|
||||
</section>
|
||||
@ -132,14 +133,14 @@ H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dagger_{i}c_{i} - \tfra
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </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>
|
||||
</div>
|
||||
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" role="doc-biblioentry">
|
||||
<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>
|
||||
<div class="csl-left-margin">[2] </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>
|
||||
</div>
|
||||
<div id="ref-rusinCalculationRKKYRange2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">T. M. Rusin and W. Zawadzki, <em><a href="https://doi.org/10.1016/j.jmmm.2017.06.007">On Calculation of RKKY Range Function in One Dimension</a></em>, Journal of Magnetism and Magnetic Materials <strong>441</strong>, 387 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </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>
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">T. M. Rusin and W. Zawadzki, <em><a href="https://doi.org/10.1016/j.jmmm.2017.06.007">On Calculation of RKKY Range Function in One Dimension</a></em>, Journal of Magnetism and Magnetic Materials <strong>441</strong>, 387 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-abrahamsScalingTheoryLocalization1979" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, <em><a href="https://doi.org/10.1103/PhysRevLett.42.673">Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions</a></em>, Phys. Rev. Lett. <strong>42</strong>, 673 (1979).</div>
|
||||
|
||||
@ -34,26 +34,11 @@ image:
|
||||
<li><a href="#implementation-of-mcmc" id="toc-implementation-of-mcmc">Implementation of MCMC</a></li>
|
||||
<li><a href="#global-and-detailed-balance-equations" id="toc-global-and-detailed-balance-equations">Global and Detailed balance equations</a></li>
|
||||
<li><a href="#the-metropolis-hastings-algorithm" id="toc-the-metropolis-hastings-algorithm">The Metropolis-Hastings Algorithm</a></li>
|
||||
<li><a href="#implementation-of-the-mh-algorithm" id="toc-implementation-of-the-mh-algorithm">Implementation of the MH Algorithm</a></li>
|
||||
<li><a href="#the-metropolis-hasting-algorithm" id="toc-the-metropolis-hasting-algorithm">The Metropolis-Hasting Algorithm</a></li>
|
||||
<li><a href="#metropolis-hastings" id="toc-metropolis-hastings">Metropolis-Hastings</a>
|
||||
<li><a href="#app-mcmc-two-step-trick" id="toc-app-mcmc-two-step-trick">Two Step Trick</a>
|
||||
<ul>
|
||||
<li><a href="#dup1-the-metropolis-hastings-algorithm" id="toc-dup1-the-metropolis-hastings-algorithm">dup1 The Metropolis-Hastings Algorithm</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#app-mcmc-two-step-trick" id="toc-app-mcmc-two-step-trick">Two Step Trick</a></li>
|
||||
<li><a href="#detailed-balance-for-the-two-step-method" id="toc-detailed-balance-for-the-two-step-method">Detailed Balance for the two step method</a>
|
||||
<ul>
|
||||
<li><a href="#get-rid-of-me-two-step-trick" id="toc-get-rid-of-me-two-step-trick">get rid of me Two Step Trick</a></li>
|
||||
<li><a href="#app-mcmc-autocorrelation" id="toc-app-mcmc-autocorrelation">Auto-correlation Time</a></li>
|
||||
<li><a href="#tuning-the-proposal-distribution" id="toc-tuning-the-proposal-distribution">Tuning the proposal distribution</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#proposal-distributions" id="toc-proposal-distributions">Proposal Distributions</a></li>
|
||||
<li><a href="#choosing-the-proposal-distribution" id="toc-choosing-the-proposal-distribution">Choosing the proposal distribution</a></li>
|
||||
<li><a href="#perturbation-mcmc" id="toc-perturbation-mcmc">Perturbation MCMC</a>
|
||||
<ul>
|
||||
<li><a href="#app-mcmc-convergence" id="toc-app-mcmc-convergence">Convergence Time</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#detailed-balance" id="toc-detailed-balance">Detailed Balance</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -75,26 +60,11 @@ image:
|
||||
<li><a href="#implementation-of-mcmc" id="toc-implementation-of-mcmc">Implementation of MCMC</a></li>
|
||||
<li><a href="#global-and-detailed-balance-equations" id="toc-global-and-detailed-balance-equations">Global and Detailed balance equations</a></li>
|
||||
<li><a href="#the-metropolis-hastings-algorithm" id="toc-the-metropolis-hastings-algorithm">The Metropolis-Hastings Algorithm</a></li>
|
||||
<li><a href="#implementation-of-the-mh-algorithm" id="toc-implementation-of-the-mh-algorithm">Implementation of the MH Algorithm</a></li>
|
||||
<li><a href="#the-metropolis-hasting-algorithm" id="toc-the-metropolis-hasting-algorithm">The Metropolis-Hasting Algorithm</a></li>
|
||||
<li><a href="#metropolis-hastings" id="toc-metropolis-hastings">Metropolis-Hastings</a>
|
||||
<li><a href="#app-mcmc-two-step-trick" id="toc-app-mcmc-two-step-trick">Two Step Trick</a>
|
||||
<ul>
|
||||
<li><a href="#dup1-the-metropolis-hastings-algorithm" id="toc-dup1-the-metropolis-hastings-algorithm">dup1 The Metropolis-Hastings Algorithm</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#app-mcmc-two-step-trick" id="toc-app-mcmc-two-step-trick">Two Step Trick</a></li>
|
||||
<li><a href="#detailed-balance-for-the-two-step-method" id="toc-detailed-balance-for-the-two-step-method">Detailed Balance for the two step method</a>
|
||||
<ul>
|
||||
<li><a href="#get-rid-of-me-two-step-trick" id="toc-get-rid-of-me-two-step-trick">get rid of me Two Step Trick</a></li>
|
||||
<li><a href="#app-mcmc-autocorrelation" id="toc-app-mcmc-autocorrelation">Auto-correlation Time</a></li>
|
||||
<li><a href="#tuning-the-proposal-distribution" id="toc-tuning-the-proposal-distribution">Tuning the proposal distribution</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#proposal-distributions" id="toc-proposal-distributions">Proposal Distributions</a></li>
|
||||
<li><a href="#choosing-the-proposal-distribution" id="toc-choosing-the-proposal-distribution">Choosing the proposal distribution</a></li>
|
||||
<li><a href="#perturbation-mcmc" id="toc-perturbation-mcmc">Perturbation MCMC</a>
|
||||
<ul>
|
||||
<li><a href="#app-mcmc-convergence" id="toc-app-mcmc-convergence">Convergence Time</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#detailed-balance" id="toc-detailed-balance">Detailed Balance</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -131,13 +101,14 @@ p(S) &= \frac{1}{\mathcal{Z}} e^{-\beta H(S)} \\
|
||||
</section>
|
||||
<section id="implementation-of-mcmc" class="level2">
|
||||
<h2>Implementation of MCMC</h2>
|
||||
<p>In implementation MCMC can be boiled down to choosing a transition function $(S_{t} S_{t+1}) $ 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>.</p>
|
||||
<p>In implementation MCMC 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>.</p>
|
||||
<p>In pseudo-code one could write the MCMC simulation for a single walker as:</p>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> sample_T(current_state) </span>
|
||||
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span></code></pre></div>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co"># A skeleton implementation of MCMC</span></span>
|
||||
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> sample_T(current_state) </span>
|
||||
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span></code></pre></div>
|
||||
<p>Where the <code>sample_T</code> function samples directly from the transition function <span class="math inline">\(\mathcal{T}\)</span>.</p>
|
||||
<p>If we run many such walkers in parallel we can then approximate the distribution <span class="math inline">\(p_t(S; S)\)</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 the the probability <span class="math inline">\(p_t(S; S_0)\)</span> approaches the distribution of interest. In this case the thermal distribution <span class="math inline">\(P(S; \beta) = \mathcal{Z}^{-1} e^{-\beta F(S)}\)</span>.</p>
|
||||
</section>
|
||||
@ -189,128 +160,64 @@ P(S)q(S \to S')\mathcal{A}(S \to S') = P(S')q(S' \to S)\mathcal{
|
||||
<p>Noting that <span class="math inline">\(f(S,S') = 1/f(S',S)\)</span>, we can see that the MH algorithm satifies detailed balance by considering the two cases <span class="math inline">\(f(S,S') > 1\)</span> and <span class="math inline">\(f(S,S') < 1\)</span>.</p>
|
||||
<p>By choosing the proposal distribution such that <span class="math inline">\(f(S,S')\)</span> is as close as possible to one, the rate of rejections can be reduced and the algorithm sped up. This can be challenging though, as getting <span class="math inline">\(f(S,S')\)</span> close to 1 would imply that we can directly sample from a distribution very close to the target distribution. As MCMC is usually applied to problems for which there is virtually no hope of sampling directly from the target distribution, it’s rare that one can do so approximately.</p>
|
||||
<p>When the proposal distribution is symmetric as ours is, it cancels out in the expression for the acceptance function and the Metropolis-Hastings algorithm is simply the choice</p>
|
||||
<p><span class="math display">\[\mathcal{A}(S \to S') = \min\left(1, e^{-\beta\;\Delta F}\right)\]</span></p>
|
||||
<p><span class="math display">\[
|
||||
\mathcal{A}(S \to S') = \min\left(1, e^{-\beta\;\Delta F}\right)
|
||||
\]</span></p>
|
||||
<p>where <span class="math inline">\(F\)</span> is the overall free energy of the system, including both the quantum and classical sector.</p>
|
||||
</section>
|
||||
<section id="implementation-of-the-mh-algorithm" class="level2">
|
||||
<h2>Implementation of the MH Algorithm</h2>
|
||||
<p>To implement the acceptance function in practice we pick a random number in the unit interval and accept if it is less than <span class="math inline">\(e^{-\beta\;\Delta F}\)</span>:</p>
|
||||
<div class="sourceCode" id="cb2"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> proposal(current_state)</span>
|
||||
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a> df <span class="op">=</span> free_energy_change(current_state, new_state, parameters)</span>
|
||||
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>) <span class="op"><</span> exp(<span class="op">-</span>beta <span class="op">*</span> df):</span>
|
||||
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span>
|
||||
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a> </span>
|
||||
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span></code></pre></div>
|
||||
</section>
|
||||
<section id="the-metropolis-hasting-algorithm" class="level2">
|
||||
<h2>The Metropolis-Hasting Algorithm</h2>
|
||||
</section>
|
||||
<section id="metropolis-hastings" class="level2">
|
||||
<h2>Metropolis-Hastings</h2>
|
||||
<p>In order to actually choose new states according to <span class="math inline">\(\mathcal{T}\)</span> one chooses states from a proposal distribution <span class="math inline">\(q(S_i \to S')\)</span> that can be directly sampled from. For instance, this might mean flipping a single random spin in a spin chain, in which case <span class="math inline">\(q(x_i\to x_i)\)</span> is the uniform distribution on states reachable by one spin flip from <span class="math inline">\(x_i\)</span>. The proposal <span class="math inline">\(S'\)</span> is then accepted or rejected with an acceptance probability <span class="math inline">\(\mathcal{A}(x_i\to x_{i+1})\)</span>, if the proposal is rejected then <span class="math inline">\(x_{i+1} = x_{i}\)</span>. Now <span class="math inline">\(\mathcal{T}(S\to S') = q(S\to S')\mathcal{A}(S \to S')\)</span>.</p>
|
||||
<section id="dup1-the-metropolis-hastings-algorithm" class="level3">
|
||||
<h3>dup1 The Metropolis-Hastings Algorithm</h3>
|
||||
<p>MH breaks up the transition function into a proposal distribution <span class="math inline">\(q(S \to S')\)</span> and an acceptance function <span class="math inline">\(\mathcal{A}(S \to S')\)</span>. <span class="math inline">\(q\)</span> needs to be something that we can directly sample from, and in our case generally takes the form of flipping some number of spins in <span class="math inline">\(S\)</span>, i.e if we’re flipping a single random spin in the spin chain, <span class="math inline">\(q(S \to S')\)</span> is the uniform distribution on states reachable by one spin flip from <span class="math inline">\(S\)</span>. This also gives the nice symmetry property that <span class="math inline">\(q(S \to S') = q(S' \to S)\)</span>.</p>
|
||||
<p>The proposal <span class="math inline">\(S'\)</span> is then accepted or rejected with an acceptance probability <span class="math inline">\(\mathcal{A}(S \to S')\)</span>, if the proposal is rejected then <span class="math inline">\(S_{i+1} = S_{i}\)</span>. Hence:</p>
|
||||
<p><span class="math display">\[\mathcal{T}(S\to S') = q(S\to S')\mathcal{A}(S \to S')\]</span></p>
|
||||
<p>When the proposal distribution is symmetric as ours is, it cancels out in the expression for the acceptance function and the Metropolis-Hastings algorithm is simply the choice: <span class="math display">\[ \mathcal{A}(S \to S') = \min\left(1, e^{-\beta\;\Delta F}\right)\]</span> Where <span class="math inline">\(F\)</span> is the overall free energy of the system, including both the quantum and classical sector.</p>
|
||||
<p>To implement the acceptance function in practice we pick a random number in the unit interval and accept if it is less than <span class="math inline">\(e^{-\beta\;\Delta F}\)</span>:</p>
|
||||
<div class="sourceCode" id="cb3"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> proposal(current_state)</span>
|
||||
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a> df <span class="op">=</span> free_energy_change(current_state, new_state, parameters)</span>
|
||||
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>) <span class="op"><</span> exp(<span class="op">-</span>beta <span class="op">*</span> df):</span>
|
||||
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span>
|
||||
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a> </span>
|
||||
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span></code></pre></div>
|
||||
<p>This has the effect of always accepting proposed states that are lower in energy and sometimes accepting those that are higher in energy than the current state.</p>
|
||||
</section>
|
||||
<div class="sourceCode" id="cb2"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># An implementation of the standard MH algorithm</span></span>
|
||||
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> proposal(current_state)</span>
|
||||
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a> df <span class="op">=</span> free_energy_change(current_state, new_state, parameters)</span>
|
||||
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>) <span class="op"><</span> exp(<span class="op">-</span>beta <span class="op">*</span> df):</span>
|
||||
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span>
|
||||
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a> </span>
|
||||
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span></code></pre></div>
|
||||
</section>
|
||||
<section id="app-mcmc-two-step-trick" class="level2">
|
||||
<h2>Two Step Trick</h2>
|
||||
<p>Our method already relies heavily on the split between the classical and quantum sector to derive a sign problem free MCMC algorithm but it turns out that there is a further trick we can play with it. The free energy term is the sum of an easy to compute classical energy and a more expensive quantum free energy, we can split the acceptance function into two in such as way as to avoid having to compute the full exact diagonalisation some of the time:</p>
|
||||
<div class="sourceCode" id="cb4" data-language="Python"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> proposal(current_state)</span>
|
||||
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a> df_classical <span class="op">=</span> classical_free_energy_change(current_state, new_state, parameters)</span>
|
||||
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> exp(<span class="op">-</span>beta <span class="op">*</span> df_classical) <span class="op"><</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>):</span>
|
||||
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a> f_quantum <span class="op">=</span> quantum_free_energy(current_state, new_state, parameters)</span>
|
||||
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a> </span>
|
||||
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> exp(<span class="op">-</span> beta <span class="op">*</span> df_quantum) <span class="op"><</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>):</span>
|
||||
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span>
|
||||
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a> </span>
|
||||
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span>
|
||||
<span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a> </span></code></pre></div>
|
||||
</section>
|
||||
<section id="detailed-balance-for-the-two-step-method" class="level2">
|
||||
<h2>Detailed Balance for the two step method</h2>
|
||||
<p>Given a MCMC algorithm with target distribution <span class="math inline">\(\pi(a)\)</span> and transition function <span class="math inline">\(\mathcal{T}\)</span> the detailed balance condition is sufficient (along with some technical constraints <span class="citation" data-cites="wolffMonteCarloErrors2004"> [<a href="#ref-wolffMonteCarloErrors2004" role="doc-biblioref">5</a>]</span>) to guarantee that in the long time limit the algorithm produces samples from <span class="math inline">\(\pi\)</span>. <span class="math display">\[\pi(a)\mathcal{T}(a \to b) = \pi(b)\mathcal{T}(b \to a)\]</span></p>
|
||||
<p>In pseudo-code, our two step method corresponds to two nested comparisons with the majority of the work only occurring if the first test passes:</p>
|
||||
<div class="sourceCode" id="cb5"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> proposal(current_state)</span>
|
||||
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a> c_dE <span class="op">=</span> classical_energy_change(</span>
|
||||
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a> current_state,</span>
|
||||
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a> new_state)</span>
|
||||
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>) <span class="op"><</span> exp(<span class="op">-</span>beta <span class="op">*</span> c_dE):</span>
|
||||
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a> q_dF <span class="op">=</span> quantum_free_energy_change(</span>
|
||||
<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a> current_state,</span>
|
||||
<span id="cb5-12"><a href="#cb5-12" aria-hidden="true" tabindex="-1"></a> new_state)</span>
|
||||
<span id="cb5-13"><a href="#cb5-13" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>) <span class="op"><</span> exp(<span class="op">-</span> beta <span class="op">*</span> q_dF):</span>
|
||||
<span id="cb5-14"><a href="#cb5-14" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span>
|
||||
<span id="cb5-15"><a href="#cb5-15" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb5-16"><a href="#cb5-16" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span></code></pre></div>
|
||||
<p>Defining <span class="math inline">\(r_c = e^{-\beta H_c}\)</span> and <span class="math inline">\(r_q = e^{-\beta F_q}\)</span> our target distribution is <span class="math inline">\(\pi(a) = r_c r_q\)</span>. This method has <span class="math inline">\(\mathcal{T}(a\to b) = q(a\to b)\mathcal{A}(a \to b)\)</span> with symmetric <span class="math inline">\(p(a \to b) = \pi(b \to a)\)</span> and <span class="math inline">\(\mathcal{A} = \min\left(1, r_c\right) \min\left(1, r_q\right)\)</span></p>
|
||||
<div class="sourceCode" id="cb3"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Our two step MH implementation for models with classical and quantum energy terms</span></span>
|
||||
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> proposal(current_state)</span>
|
||||
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> df_classical <span class="op">=</span> classical_free_energy_change(current_state, new_state, parameters)</span>
|
||||
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> exp(<span class="op">-</span>beta <span class="op">*</span> df_classical) <span class="op"><</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>):</span>
|
||||
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a> f_quantum <span class="op">=</span> quantum_free_energy(current_state, new_state, parameters)</span>
|
||||
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a> </span>
|
||||
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> exp(<span class="op">-</span> beta <span class="op">*</span> df_quantum) <span class="op"><</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>):</span>
|
||||
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span>
|
||||
<span id="cb3-13"><a href="#cb3-13" aria-hidden="true" tabindex="-1"></a> </span>
|
||||
<span id="cb3-14"><a href="#cb3-14" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span>
|
||||
<span id="cb3-15"><a href="#cb3-15" aria-hidden="true" tabindex="-1"></a> </span></code></pre></div>
|
||||
<p>As discussed in the main text, for the model parameters used, we find that with our new scheme the matrix diagonalisation is skipped around 30% of the time at <span class="math inline">\(T = 2.5\)</span> and up to 80% at <span class="math inline">\(T = 1.5\)</span>. We observe that for <span class="math inline">\(N = 50\)</span>, the matrix diagonalisation, if it occurs, occupies around 60% of the total computation time for a single step. This rises to 90% at N = 300 and further increases for larger N. We therefore get the greatest speedup for large system sizes at low temperature where many prospective transitions are rejected at the classical stage and the matrix computation takes up the greatest fraction of the total computation time. The upshot is that we find a speedup of up to a factor of 10 at the cost of very little extra algorithmic complexity.</p>
|
||||
<p>This modified scheme has the acceptance function <span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]</span></p>
|
||||
<p>We can show that this satisfies the detailed balance equations as follows. Defining <span class="math inline">\(r_c = e^{-\beta H_c}\)</span> and <span class="math inline">\(r_q = e^{-\beta F_q}\)</span> our target distribution is <span class="math inline">\(\pi(a) = r_c r_q\)</span>. This method has <span class="math inline">\(\mathcal{T}(a\to b) = q(a\to b)\mathcal{A}(a \to b)\)</span> with symmetric <span class="math inline">\(p(a \to b) = \pi(b \to a)\)</span> and <span class="math inline">\(\mathcal{A} = \min\left(1, r_c\right) \min\left(1, r_q\right)\)</span></p>
|
||||
<p>Substituting this into the detailed balance equation gives: <span class="math display">\[\mathcal{T}(a \to b)/\mathcal{T}(b \to a) = \pi(b)/\pi(a) = r_c r_q\]</span></p>
|
||||
<p>Taking the LHS and substituting in our transition function: <span class="math display">\[\begin{aligned}
|
||||
\mathcal{T}(a \to b)/\mathcal{T}(b \to a) = \frac{\min\left(1, r_c\right) \min\left(1, r_q\right)}{ \min\left(1, 1/r_c\right) \min\left(1, 1/r_q\right)}\end{aligned}\]</span></p>
|
||||
<p>which simplifies to <span class="math inline">\(r_c r_q\)</span> as <span class="math inline">\(\min(1,r)/\min(1,1/r) = r\)</span> for <span class="math inline">\(r > 0\)</span>.</p>
|
||||
<section id="get-rid-of-me-two-step-trick" class="level3">
|
||||
<h3>get rid of me Two Step Trick</h3>
|
||||
<p>Our method already relies heavily on the split between the classical and quantum sector to derive a sign problem free MCMC algorithm but it turns out that there is a further trick we can play with it. The free energy term is the sum of an easy to compute classical energy and a more expensive quantum free energy, we can split the acceptance function into two in such as way as to avoid having to compute the full exact diagonalisation some of the time:</p>
|
||||
<div class="sourceCode" id="cb6"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> proposal(current_state)</span>
|
||||
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a> df_classical <span class="op">=</span> classical_free_energy_change(current_state, new_state, parameters)</span>
|
||||
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> exp(<span class="op">-</span>beta <span class="op">*</span> df_classical) <span class="op"><</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>):</span>
|
||||
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a> f_quantum <span class="op">=</span> quantum_free_energy(current_state, new_state, parameters)</span>
|
||||
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a> </span>
|
||||
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> exp(<span class="op">-</span> beta <span class="op">*</span> df_quantum) <span class="op"><</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>):</span>
|
||||
<span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span>
|
||||
<span id="cb6-13"><a href="#cb6-13" aria-hidden="true" tabindex="-1"></a> </span>
|
||||
<span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span>
|
||||
<span id="cb6-15"><a href="#cb6-15" aria-hidden="true" tabindex="-1"></a> </span></code></pre></div>
|
||||
</section>
|
||||
<section id="app-mcmc-autocorrelation" class="level3">
|
||||
<h3>Auto-correlation Time</h3>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/lsr/figs/m_autocorr.png" id="fig:m_autocorr" data-short-caption="no title" style="width:100.0%" alt="Figure 1: (Upper) 10 MCMC chains starting from the same initial state for a system with N = 150 sites and 3000 MCMC steps. At each MCMC step, n spins are flipped where n is drawn from Uniform(1,N) and this is repeated N^2/100 times. The simulations therefore have the potential to necessitate 10*N^2 matrix diagonalisations for each 100 MCMC steps. (Lower) The normalised auto-correlation (\langle m_i m_{i-j}\rangle - \langle m_i\rangle \langle m_i \rangle) / Var(m_i)) averaged over i. It can be seen that even with each MCMC step already being composed of many individual flip attempts, the auto-correlation is still non negligible and must be taken into account in the statistics. t = 1, \alpha = 1.25, T = 2.2, J = U = 5" />
|
||||
<img src="/assets/thesis/fk_chapter/lsr/figs/m_autocorr.png" id="fig:m_autocorr" data-short-caption="Autocorrelation in MCMC" style="width:100.0%" alt="Figure 1: (Upper) 10 MCMC chains starting from the same initial state for a system with N = 150 sites and 3000 MCMC steps. At each MCMC step, n spins are flipped where n is drawn from Uniform(1,N) and this is repeated N^2/100 times. The simulations therefore have the potential to necessitate 10*N^2 matrix diagonalisations for each 100 MCMC steps. (Lower) The normalised auto-correlation (\langle m_i m_{i-j}\rangle - \langle m_i\rangle \langle m_i \rangle) / Var(m_i)) averaged over i. It can be seen that even with each MCMC step already being composed of many individual flip attempts, the auto-correlation is still non negligible and must be taken into account in the statistics. t = 1, \alpha = 1.25, T = 2.2, J = U = 5" />
|
||||
<figcaption aria-hidden="true">Figure 1: (Upper) 10 MCMC chains starting from the same initial state for a system with <span class="math inline">\(N = 150\)</span> sites and 3000 MCMC steps. At each MCMC step, n spins are flipped where n is drawn from Uniform(1,N) and this is repeated <span class="math inline">\(N^2/100\)</span> times. The simulations therefore have the potential to necessitate <span class="math inline">\(10*N^2\)</span> matrix diagonalisations for each 100 MCMC steps. (Lower) The normalised auto-correlation <span class="math inline">\((\langle m_i m_{i-j}\rangle - \langle m_i\rangle \langle m_i \rangle) / Var(m_i))\)</span> averaged over <span class="math inline">\(i\)</span>. It can be seen that even with each MCMC step already being composed of many individual flip attempts, the auto-correlation is still non negligible and must be taken into account in the statistics. <span class="math inline">\(t = 1, \alpha = 1.25, T = 2.2, J = U = 5\)</span></figcaption>
|
||||
</figure>
|
||||
<p>At this stage one might think we’re done. We can indeed draw independent samples from <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. ?? 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. autocorrelation 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 or equivalently as the number of MCMC steps after which the samples are correlated below some cut-off, see <span class="citation" data-cites="krauthIntroductionMonteCarlo1996"> [<a href="#ref-krauthIntroductionMonteCarlo1996" role="doc-biblioref">9</a>]</span> for a more rigorous definition involving a sum over the auto-correlation function. 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>At this stage one might think we’re done. We can indeed draw independent samples from our target Boltzmann distribution by starting from some arbitrary initial state and doing <span class="math inline">\(k\)</span> steps to arrive at a sample. These are not, however, independent samples. In fig. <a href="#fig:m_autocorr">1</a> it is already clear that the samples of the order parameter <span class="math inline">\(m\)</span> have some auto-correlation because only a few spins are flipped each step. Even when the number of spins flipped per step is increased that it can be an important effect near the phase transition. Let’s define the auto-correlation time <span class="math inline">\(\tau(O)\)</span> informally as the number of MCMC samples of some observable O that are statistically equal to one independent sample or equivalently as the number of MCMC steps after which the samples are correlated below some cut-off, see <span class="citation" data-cites="krauthIntroductionMonteCarlo1996"> [<a href="#ref-krauthIntroductionMonteCarlo1996" role="doc-biblioref">9</a>]</span>. The auto-correlation time is generally shorter than the convergence time so it therefore makes sense from an efficiency standpoint to run a single walker for many MCMC steps rather than to run a huge ensemble for <span class="math inline">\(k\)</span> steps each.</p>
|
||||
<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">\[
|
||||
\tex{O} = \sum_{i = 0}^{N} O(S_i) + \mathcal{O}(\frac{1}{\sqrt{N}})
|
||||
\langle O \rangle = \sum_{i = 0}^{N} O(S_i) + \mathcal{O}(\frac{1}{\sqrt{N}})
|
||||
\]</span></p>
|
||||
<p>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>
|
||||
<p>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">\(\langle O^2 \rangle - \langle O \rangle ^2\)</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">\(\langle O \rangle\)</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 conceptually simple.</p>
|
||||
</section>
|
||||
<section id="tuning-the-proposal-distribution" class="level3">
|
||||
<h3>Tuning the proposal distribution</h3>
|
||||
<figure>
|
||||
<img src="../figure_code/fk_chapter/lsr/figs/autocorr_multiple_proposals.png" id="fig:autocorr_multiple_proposals" data-short-caption="no title" style="width:100.0%" alt="Figure 2: Simulations showing how the autocorrelation of the order parameter depends on the proposal distribution used at different temperatures, we see that at T = 1.5 < T_c a single spin flip is likely the best choice, while at the high temperature T = 2.5 > T_c flipping two sites or a mixture of flipping two and 1 sites is likely a better choice. $t = 1, = 1.25, J = U = 5 $" />
|
||||
<img src="/assets/thesis/fk_chapter/lsr/figs/autocorr_multiple_proposals.png" id="fig:autocorr_multiple_proposals" data-short-caption="Comparison of different proposal distributions" style="width:100.0%" alt="Figure 2: Simulations showing how the autocorrelation of the order parameter depends on the proposal distribution used at different temperatures, we see that at T = 1.5 < T_c a single spin flip is likely the best choice, while at the high temperature T = 2.5 > T_c flipping two sites or a mixture of flipping two and 1 sites is likely a better choice. $t = 1, = 1.25, J = U = 5 $" />
|
||||
<figcaption aria-hidden="true">Figure 2: Simulations showing how the autocorrelation of the order parameter depends on the proposal distribution used at different temperatures, we see that at <span class="math inline">\(T = 1.5 < T_c\)</span> a single spin flip is likely the best choice, while at the high temperature <span class="math inline">\(T = 2.5 > T_c\)</span> flipping two sites or a mixture of flipping two and 1 sites is likely a better choice. $t = 1, = 1.25, J = U = 5 $</figcaption>
|
||||
</figure>
|
||||
<p>Now we can discuss how to minimise the auto-correlations. The general principle is that one must balance the proposal distribution between two extremes. Choose overlay small steps, like flipping only a single spin and the acceptance rate will be high because <span class="math inline">\(\Delta F\)</span> will usually be small, but each state will be very similar to the previous and the auto-correlations will be high too, making sampling inefficient. On the other hand, overlay large steps, like randomising a large portion of the spins each step, will result in very frequent rejections, especially at low temperatures.</p>
|
||||
@ -321,71 +228,11 @@ P(S)q(S \to S')\mathcal{A}(S \to S') = P(S')q(S' \to S)\mathcal{
|
||||
<li>Choosing n from Uniform(1, N) and then flipping n sites for some fixed N.</li>
|
||||
<li>Attempting to tune the proposal distribution for each parameter regime.</li>
|
||||
</ol>
|
||||
<p>Fro Figure~<span class="math inline">\(\ref{fig:comparison}\)</span> we see that even at moderately high temperatures <span class="math inline">\(T > T_c\)</span> flipping one or two sites is the best choice. However for some simulations at very high temperature flipping more spins is warranted. Tuning the proposal distribution automatically seems like something that would not yield enough benefit for the additional complexity it would require.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="proposal-distributions" class="level2">
|
||||
<h2>Proposal Distributions</h2>
|
||||
<p>In a MCMC method a key property is the proportion of the time that proposals are accepted, the acceptance rate. If this rate is too low the random walk is trying to take overly large steps in energy space which problematic because it means very few new samples will be generated. If it is too high it implies the steps are too small, a problem because then the walk will take longer to explore the state space and the samples will be highly correlated. Ideal values for the acceptance rate can be calculated under certain assumptions <span class="citation" data-cites="robertsWeakConvergenceOptimal1997"> [<a href="#ref-robertsWeakConvergenceOptimal1997" role="doc-biblioref">10</a>]</span>. Here we monitor the acceptance rate and if it is too high we re-run the MCMC with a modified proposal distribution that has a chance to propose moves that flip multiple sites at a time.</p>
|
||||
<p>In addition we exploit the particle-hole symmetry of the problem by occasionally proposing a flip of the entire state. This works because near half-filling, flipping the occupations of all the sites will produce a state at or near the energy of the current one.</p>
|
||||
</section>
|
||||
<section id="choosing-the-proposal-distribution" class="level2">
|
||||
<h2>Choosing the proposal distribution</h2>
|
||||
<p><img src="figs/lsr/autocorr_multiple_proposals.png" title="fig:" id="fig:comparison" alt="Figure 3: t = 1, \alpha = 1.25, J = U = 5 [fig:comparison]" /> Simulations showing how the autocorrelation of the order parameter depends on the proposal distribution used at different temperatures, we see that at <span class="math inline">\(T = 1.5 < T_c\)</span> a single spin flip is likely the best choice, while at the high temperature <span class="math inline">\(T = 2.5 > T_c\)</span> flipping two sites or a mixture of flipping two and 1 sites is likely a better choice.</p>
|
||||
<p>Now we can discuss how to minimise the auto-correlations. The general principle is that one must balance the proposal distribution between two extremes. Choose overlay small steps, like flipping only a single spin and the acceptance rate will be high because <span class="math inline">\(\Delta F\)</span> will usually be small, but each state will be very similar to the previous and the auto-correlations will be high too, making sampling inefficient. On the other hand, overlay large steps, like randomising a large portion of the spins each step, will result in very frequent rejections, especially at low temperatures.</p>
|
||||
<p>I evaluated a few different proposal distributions for use with the FK model.</p>
|
||||
<ol type="1">
|
||||
<li><p>Flipping a single random site</p></li>
|
||||
<li><p>Flipping N random sites for some N</p></li>
|
||||
<li><p>Choosing n from Uniform(1, N) and then flipping n sites for some fixed N.</p></li>
|
||||
<li><p>Attempting to tune the proposal distribution for each parameter regime.</p></li>
|
||||
</ol>
|
||||
<p>Fro Figure <a href="#fig:comparison" data-reference-type="ref" data-reference="fig:comparison">4</a> we see that even at moderately high temperatures <span class="math inline">\(T > T_c\)</span> flipping one or two sites is the best choice. However for some simulations at very high temperature flipping more spins is warranted. Tuning the proposal distribution automatically seems like something that would not yield enough benefit for the additional complexity it would require.</p>
|
||||
</section>
|
||||
<section id="perturbation-mcmc" class="level2">
|
||||
<h2>Perturbation MCMC</h2>
|
||||
<p>The matrix diagonalisation is the most computationally expensive step of the process, a speed up can be obtained by modifying the proposal distribution to depend on the classical part of the energy, a trick gleaned from Ref. <span class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">6</a>]</span>: <span class="math display">\[
|
||||
\begin{aligned}
|
||||
q(k \to k') &= \min\left(1, e^{\beta (H^{k'} - H^k)}\right) \\
|
||||
\mathcal{A}(k \to k') &= \min\left(1, e^{\beta(F^{k'}- F^k)}\right)
|
||||
\end{aligned}\]</span> % This allows the method to reject some states without performing the diagonalisation at no cost to the accuracy of the MCMC method.</p>
|
||||
<p>An extension of this idea is to try to define a classical model with a similar free energy dependence on the classical state as the full quantum, Ref. <span class="citation" data-cites="huangAcceleratedMonteCarlo2017"> [<a href="#ref-huangAcceleratedMonteCarlo2017" role="doc-biblioref">11</a>]</span> does this with restricted Boltzmann machines whose form is very similar to a classical spin model.</p>
|
||||
<section id="app-mcmc-convergence" class="level3">
|
||||
<h3>Convergence Time</h3>
|
||||
<p>Considering <span class="math inline">\(p(S)\)</span> as a vector <span class="math inline">\(\vec{p}\)</span> whose jth entry is the probability of the jth state <span class="math inline">\(p_j = p(S_j)\)</span>, and writing <span class="math inline">\(\mathcal{T}\)</span> as the matrix with entries <span class="math inline">\(T_{ij} = \mathcal{T}(S_j \rightarrow S_i)\)</span> we can write the update rule for the ensemble probability as: <span class="math display">\[\vec{p}_{t+1} = \mathcal{T} \vec{p}_t \implies \vec{p}_{t} = \mathcal{T}^t \vec{p}_0\]</span> where <span class="math inline">\(\vec{p}_0\)</span> is vector which is one on the starting state and zero everywhere else. Since all states must transition to somewhere with probability one: <span class="math inline">\(\sum_i T_{ij} = 1\)</span>.</p>
|
||||
<p>Matrices that satisfy this are called stochastic matrices exactly because they model these kinds of Markov processes. It can be shown that they have real eigenvalues, and ordering them by magnitude, that <span class="math inline">\(\lambda_0 = 1\)</span> and <span class="math inline">\(0 < \lambda_{i\neq0} < 1\)</span>. %https://en.wikipedia.org/wiki/Stochastic_matrix</p>
|
||||
<p>Assuming <span class="math inline">\(\mathcal{T}\)</span> has been chosen correctly, its single eigenvector with eigenvalue 1 will be the thermal distribution so repeated application of the transition function eventually leads there, while memory of the initial conditions decays exponentially with a convergence time <span class="math inline">\(k\)</span> determined by <span class="math inline">\(\lambda_1\)</span>. In practice this means that one throws away the data from the beginning of the random walk in order reduce the dependence on the initial conditions and be close enough to the target distribution.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="detailed-balance" class="level2">
|
||||
<h2>Detailed Balance</h2>
|
||||
<p>Given a MCMC algorithm with target distribution <span class="math inline">\(\pi(a)\)</span> and transition function <span class="math inline">\(\T\)</span> the detailed balance condition is sufficient (along with some technical constraints <span class="citation" data-cites="wolffMonteCarloErrors2004"> [<a href="#ref-wolffMonteCarloErrors2004" role="doc-biblioref">5</a>]</span>) to guarantee that in the long time limit the algorithm produces samples from <span class="math inline">\(\pi\)</span>. <span class="math display">\[\pi(a)\T(a \to b) = \pi(b)\T(b \to a)\]</span></p>
|
||||
<p>In pseudo-code, our two step method corresponds to two nested comparisons with the majority of the work only occurring if the first test passes:</p>
|
||||
<div class="sourceCode" id="cb7"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>current_state <span class="op">=</span> initial_state</span>
|
||||
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N_steps):</span>
|
||||
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a> new_state <span class="op">=</span> proposal(current_state)</span>
|
||||
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a> c_dE <span class="op">=</span> classical_energy_change(</span>
|
||||
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a> current_state,</span>
|
||||
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a> new_state)</span>
|
||||
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>) <span class="op"><</span> exp(<span class="op">-</span>beta <span class="op">*</span> c_dE):</span>
|
||||
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a> q_dF <span class="op">=</span> quantum_free_energy_change(</span>
|
||||
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a> current_state,</span>
|
||||
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a> new_state)</span>
|
||||
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> uniform(<span class="dv">0</span>,<span class="dv">1</span>) <span class="op"><</span> exp(<span class="op">-</span> beta <span class="op">*</span> q_dF):</span>
|
||||
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span>
|
||||
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span></code></pre></div>
|
||||
<p>Defining <span class="math inline">\(r_c = e^{-\beta H_c}\)</span> and <span class="math inline">\(r_q = e^{-\beta F_q}\)</span> our target distribution is <span class="math inline">\(\pi(a) = r_c r_q\)</span>. This method has <span class="math inline">\(\T(a\to b) = q(a\to b)\A(a \to b)\)</span> with symmetric <span class="math inline">\(p(a \to b) = \p(b \to a)\)</span> and <span class="math inline">\(\A = \min\left(1, r_c\right) \min\left(1, r_q\right)\)</span></p>
|
||||
<p>Substituting this into the detailed balance equation gives: <span class="math display">\[\T(a \to b)/\T(b \to a) = \pi(b)/\pi(a) = r_c r_q\]</span></p>
|
||||
<p>Taking the LHS and substituting in our transition function: <span class="math display">\[\begin{aligned}
|
||||
\T(a \to b)/\T(b \to a) = \frac{\min\left(1, r_c\right) \min\left(1, r_q\right)}{ \min\left(1, 1/r_c\right) \min\left(1, 1/r_q\right)}\end{aligned}\]</span></p>
|
||||
<p>which simplifies to <span class="math inline">\(r_c r_q\)</span> as <span class="math inline">\(\min(1,r)/\min(1,1/r) = r\)</span> for <span class="math inline">\(r > 0\)</span>.</p>
|
||||
<div class="sourceCode" id="cb8"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p>Fro fig. <a href="#fig:autocorr_multiple_proposals">2</a> we see that even at moderately high temperatures <span class="math inline">\(T > T_c\)</span> flipping one or two sites is the best choice. However for some simulations at very high temperature flipping more spins is warranted.</p>
|
||||
<p>Next Section: <a href="../6_Appendices/A.3_Lattice_Generation.html">Lattice Generation</a></p>
|
||||
</section>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
@ -416,12 +263,6 @@ q(k \to k') &= \min\left(1, e^{\beta (H^{k'} - H^k)}\right) \\
|
||||
<div id="ref-krauthIntroductionMonteCarlo1996" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">W. Krauth, <em><a href="http://arxiv.org/abs/cond-mat/9612186">Introduction To Monte Carlo Algorithms</a></em>, arXiv:cond-Mat/9612186 (1996).</div>
|
||||
</div>
|
||||
<div id="ref-robertsWeakConvergenceOptimal1997" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">G. O. Roberts, A. Gelman, and W. R. Gilks, <em><a href="https://doi.org/10.1214/aoap/1034625254">Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms</a></em>, Ann. Appl. Probab. <strong>7</strong>, 110 (1997).</div>
|
||||
</div>
|
||||
<div id="ref-huangAcceleratedMonteCarlo2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">L. Huang and L. Wang, <em><a href="https://doi.org/10.1103/PhysRevB.95.035105">Accelerated Monte Carlo Simulations with Restricted Boltzmann Machines</a></em>, Phys. Rev. B <strong>95</strong>, 035105 (2017).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
@ -11,12 +11,12 @@
|
||||
<li><a href="./2_Background/2.2_HKM_Model.html#the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a></li>
|
||||
<li><a href="./2_Background/2.4_Disorder.html#disorder-and-localisation">Disorder and Localisation</a></li>
|
||||
</ul>
|
||||
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falikov-Kimball Model</a></li>
|
||||
<li><a href="./3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falicov-Kimball Model</a></li>
|
||||
<ul>
|
||||
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">The Model</a></li>
|
||||
<li><a href="./3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html#methods">Methods</a></li>
|
||||
<li><a href="./3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html#results">Results</a></li>
|
||||
<li><a href="./3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html#discussion-and-conclusion">Discussion and Conclusion</a></li>
|
||||
<li><a href="./3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">The Model</a></li>
|
||||
<li><a href="./3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html#methods">Methods</a></li>
|
||||
<li><a href="./3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html#results">Results</a></li>
|
||||
<li><a href="./3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html#discussion-and-conclusion">Discussion and Conclusion</a></li>
|
||||
</ul>
|
||||
<li><a href="./4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">4 The Amorphous Kitaev Model</a></li>
|
||||
<ul>
|
||||
@ -36,6 +36,9 @@
|
||||
<li><a href="./6_Appendices/A.1.2_Fermion_Free_Energy.html">Evaluation of the Fermion Free Energy</a></li>
|
||||
<li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
|
||||
<li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#a-skeleton-implementation-of-mcmc">A skeleton implementation of MCMC</a></li>
|
||||
<li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#an-implementation-of-the-standard-mh-algorithm">An implementation of the standard MH algorithm</a></li>
|
||||
<li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#our-two-step-mh-implementation-for-models-with-classical-and-quantum-energy-terms">Our two step MH implementation for models with classical and quantum energy terms</a></li>
|
||||
<li><a href="./6_Appendices/A.3_Lattice_Generation.html#lattice-generation">Lattice Generation</a></li>
|
||||
<li><a href="./6_Appendices/A.4_Lattice_Colouring.html#lattice-colouring">Lattice Colouring</a></li>
|
||||
<li><a href="./6_Appendices/A.5_The_Projector.html#the-projector">The Projector</a></li>
|
||||
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Reference in New Issue
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