--- title: The Falikov-Kimball Model - Introduction excerpt: The Falikov-Kimball is about the simplest possible testbed we could have for the many electron problem. layout: none image: --- <!DOCTYPE html> <html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang=""> <head> <meta charset="utf-8" /> <meta name="generator" content="pandoc" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="description" content="The Falikov-Kimball is about the simplest possible testbed we could have for the many electron problem." /> <title>The Falikov-Kimball Model - Introduction</title> <!-- <style> html { line-height: 1.5; font-family: Georgia, serif; font-size: 20px; color: #1a1a1a; background-color: #fdfdfd; } body { margin: 0 auto; max-width: 36em; padding-left: 50px; padding-right: 50px; padding-top: 50px; padding-bottom: 50px; hyphens: auto; overflow-wrap: break-word; text-rendering: optimizeLegibility; font-kerning: normal; } @media 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} div.csl-right-inline { margin-left:2em; padding-left:1em; } div.csl-indent { margin-left: 2em; } </style> --> <!-- <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js" type="text/javascript"></script> --> <!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script> --> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <!--[if lt IE 9]> <script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script> <![endif]--> <link rel="stylesheet" href="/assets/css/styles.css"> <script src="/assets/js/index.js"></script> </head> <body> {% include header.html %} <main> <nav id="TOC" role="doc-toc"> <ul> <li><a href="#contributions" id="toc-contributions">Contributions</a></li> <li><a href="#introduction" id="toc-introduction">Introduction</a> <ul> <li><a href="#localisation" id="toc-localisation">Localisation</a> <ul> <li><a href="#the-falikov-kimball-model" id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a></li> </ul></li> <li><a href="#falikov-kimball-and-hubbard-models" id="toc-falikov-kimball-and-hubbard-models">Falikov Kimball and Hubbard models</a> <ul> <li><a href="#hubbard-model" id="toc-hubbard-model">Hubbard model</a></li> <li><a href="#falikov-kimball-model" id="toc-falikov-kimball-model">Falikov-Kimball model</a></li> <li><a href="#thermodynamics-of-the-fk-model" id="toc-thermodynamics-of-the-fk-model">Thermodynamics of the FK model</a></li> <li><a href="#thermodynamics" id="toc-thermodynamics">Thermodynamics</a></li> <li><a href="#markov-chain-monte-carlo" id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li> </ul></li> <li><a href="#localisation-1" id="toc-localisation-1">Localisation</a> <ul> <li><a href="#thermalisation" id="toc-thermalisation">Thermalisation</a></li> <li><a href="#anderson-localisation" id="toc-anderson-localisation">Anderson Localisation</a></li> <li><a href="#many-body-localisation" id="toc-many-body-localisation">Many Body Localisation</a></li> <li><a href="#disorder-free-localisation" id="toc-disorder-free-localisation">Disorder Free localisation</a></li> <li><a href="#diagnostics-of-localisation" id="toc-diagnostics-of-localisation">Diagnostics of Localisation</a></li> </ul></li> <li><a href="#numerical-methods" id="toc-numerical-methods">Numerical Methods</a> <ul> <li><a href="#markov-chain-monte-carlo-1" id="toc-markov-chain-monte-carlo-1">Markov Chain Monte Carlo}</a></li> <li><a href="#applying-mcmc-to-the-fk-model" id="toc-applying-mcmc-to-the-fk-model">Applying MCMC to the FK model}</a></li> </ul></li> <li><a href="#markov-chain-monte-carlo-in-practice" id="toc-markov-chain-monte-carlo-in-practice">Markov Chain Monte-Carlo in Practice}</a> <ul> <li><a href="#quick-intro-to-mcmc" id="toc-quick-intro-to-mcmc">Quick Intro to MCMC}</a></li> <li><a href="#convergence-time" id="toc-convergence-time">Convergence Time}</a></li> <li><a href="#auto-correlation-time" id="toc-auto-correlation-time">Auto-correlation Time}</a></li> <li><a href="#the-metropolis-hastings-algorithm" id="toc-the-metropolis-hastings-algorithm">The Metropolis-Hastings Algorithm}</a></li> <li><a href="#choosing-the-proposal-distribution" id="toc-choosing-the-proposal-distribution">Choosing the proposal distribution}</a></li> <li><a href="#two-step-trick" id="toc-two-step-trick">Two Step Trick</a></li> </ul></li> </ul></li> <li><a href="#introduction-1" id="toc-introduction-1">Introduction</a></li> <li><a href="#the-long-ranged-falikov-kimball-model" id="toc-the-long-ranged-falikov-kimball-model">The Long-Ranged Falikov-Kimball Model</a></li> <li><a href="#the-phase-diagram" id="toc-the-phase-diagram">The Phase Diagram</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="#localisation-properties" id="toc-localisation-properties">Localisation Properties</a></li> <li><a href="#discussion-conclusion" id="toc-discussion-conclusion">Discussion & Conclusion</a></li> <li><a href="#acknowledgments" id="toc-acknowledgments">Acknowledgments</a></li> <li><a href="#detailed-balance" id="toc-detailed-balance"><span id="app:balance" label="app:balance"></span> DETAILED BALANCE</a></li> <li><a href="#uncorrelated-disorder-model" id="toc-uncorrelated-disorder-model"><span id="app:disorder_model" label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li> </ul> </nav> <h1 id="contributions">Contributions</h1> <p>This material is this chapter expands on work presented in</p> <p><span class="citation" data-cites="hodsonOnedimensionalLongRangeFalikovKimball2021"><sup><a href="#ref-hodsonOnedimensionalLongRangeFalikovKimball2021" role="doc-biblioref">1</a></sup></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>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. The three of us brought the project to fruition.</p> <h1 id="introduction">Introduction</h1> <h2 id="localisation">Localisation</h2> <p>The discovery of localisation in quantum systems surprising at the time given the seeming ubiquity of extended Bloch states. Later, when thermalisation in quantum systems gained interest, localisation phenomena again stood out as counterexamples to the eigenstate thermalisation hypothesis<span class="citation" data-cites="abaninRecentProgressManybody2017 srednickiChaosQuantumThermalization1994"><sup><a href="#ref-abaninRecentProgressManybody2017" role="doc-biblioref">2</a>,<a href="#ref-srednickiChaosQuantumThermalization1994" role="doc-biblioref">3</a></sup></span>, allowing quantum systems to avoid to retain memory of their initial conditions in the face of thermal noise.</p> <p>The simplest and first discovered kind is Anderson localisation, first studied in 1958<span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"><sup><a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">4</a></sup></span> in the context of non-interacting fermions subject to a static or quenched disorder potential <span class="math inline">\(V_j\)</span> drawn uniformly from the interval <span class="math inline">\([-W,W]\)</span></p> <p><span class="math display">\[ H = -t\sum_{\langle jk \rangle} c^\daggerger_j c_k + \sum_j V_j c_j^\daggerger c_j \]</span></p> <p>this model exhibits exponentially localised eigenfunctions <span class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\)</span> which cannot contribute to transport processes. Initially it was thought that in one dimensional disordered models, all states would be localised, however it was later shown that in the presence of correlated disorder, bands of extended states can exist<span class="citation" data-cites="izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"><sup><a href="#ref-izrailevLocalizationMobilityEdge1999" role="doc-biblioref">5</a>–<a href="#ref-izrailevAnomalousLocalizationLowDimensional2012" role="doc-biblioref">7</a></sup></span>.</p> <p>Later localisation was found in interacting many-body systems with quenched disorder:</p> <p><span class="math display">\[ H = -t\sum_{\langle jk \rangle} c^\daggerger_j c_k + \sum_j V_j c_j^\daggerger c_j + U\sum_{jk} n_j n_k \]</span></p> <p>where the number operators <span class="math inline">\(n_j = c^\dagger_j c_j\)</span>. Here, in contrast to the Anderson model, localisation phenomena can be proven robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL)<span class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016"><sup><a href="#ref-imbrieManyBodyLocalizationQuantum2016" role="doc-biblioref">8</a></sup></span>.</p> <p>Both MBL and Anderson localisation depend crucially on the presence of quenched disorder. This has led to ongoing interest in the possibility of disorder-free localisation, in which the disorder necessary to generate localisation is generated entirely from the dynamics of the model. This contracts with typical models of disordered systems in which disorder is explicielty introduced into the Hamilton or the initial state.</p> <p>The concept of disorder-free localisation was first proposed in the context of Helium mixtures<span class="citation" data-cites="kagan1984localization"><sup><a href="#ref-kagan1984localization" role="doc-biblioref">9</a></sup></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"><sup><a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">10</a>,<a href="#ref-schiulazDynamicsManybodyLocalized2015" role="doc-biblioref">11</a></sup></span> instead find that the models thermalise exponentially slowly in system size, which Ref.<span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"><sup><a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">10</a></sup></span> dubs Quasi-MBL.</p> <p>True disorder-free localisation does occur in exactly solveable models with extensively many conserved quantities<span class="citation" data-cites="smithDisorderFreeLocalization2017"><sup><a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">12</a></sup></span>. As conserved quantites have no time dynamics this can be thought of as taking the separation of timescales to the infinite limit.</p> <h3 id="the-falikov-kimball-model">The Falikov Kimball Model</h3> <p>In the Falikov Kimball (FK) model spinless fermions <span class="math inline">\(c_{i\uparrow}\)</span> are coupled via a repulsive on-site interaction to a classical degree of freedom <span class="math inline">\(n_{i\downarrow}\)</span>.</p> <p><span class="math display">\[\begin{aligned} H &= -t \sum_{<ij>} c^\daggerger_{i\uparrow}c_{j\uparrow} + U \sum_{i} (n_{i \uparrow} - 1/2)( n_{i\downarrow} - 1/2) \\ & - \mu \sum_i \left( n_{i \uparrow} + n_{i \downarrow} \right) + \sum_{ij} V_{ij} (n_{i\downarrow} - 1/2)(n_{j\downarrow} - 1/2) \end{aligned}\]</span> <strong>replace with hamiltonian from the paper</strong></p> <p>This notation emphasises that this can also be thought of as an asymmetric Hubbard model in which the spin down electrons cannot hop and are subject to an additional long range potential. However, to avoid the confusion of talking about two distinct species of spinless electrons we will use a different interpretation and refer to the classical degrees of freedom as the “ionic sector” and the quantum degrees of freedom as the “electronic sector”. The final term that induces interactions between the classical particles has been added by us to stabilise the formation of an ordered phase in 1D. The classical variables commute with the Hamiltonian <span class="math inline">\([H, n_{i\downarrow}] = 0\)</span> so like the lattice gauge model in Ref<span class="citation" data-cites="smithDisorderFreeLocalization2017"><sup><a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">12</a></sup></span>} the FK model has extensively many conserved quantities which can act as an effective disorder potential for the electronic sector.</p> <p>Due to Pauli exclusion, the maximum filling occurs when one of each species occupies each lattice site such that <span class="math inline">\(\sum_i (n_{i\downarrow} + n_{i\uparrow} )/ N = 2\)</span>. Here we focus on the half filled case which also displays particle-hole symmetry (see later).</p> <h2 id="falikov-kimball-and-hubbard-models">Falikov Kimball and Hubbard models</h2> <p>We will first introduce the standard Hubbard and Falikov-Kimball (FK) models then look at some of their properties. We’ll then cover why the Falikov-Kimball model represents an interesting system in which to study disorder free localisation.</p> <h3 id="hubbard-model">Hubbard model</h3> <p>The Hubbard model gives a very simple setting in which to study interacting, itinerant electrons. It is a tight binding model of spin half electrons with finite bandwidth <span class="math inline">\(t\)</span> and a repulsive on-site interaction <span class="math inline">\(U > 0\)</span>.</p> <p><span class="math display">\[ H = -\sum_{<ij>,\sigma} t_{\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U \sum_{i} (n_{i \uparrow} - 1/2)( n_{i\downarrow} - 1/2) - \mu \sum_i \left( n_{i \uparrow} + n_{i \downarrow} \right) \]</span></p> <p>in standard notation. The standard Hubbard model corresponds to the case <span class="math inline">\(t_{\uparrow} = t_{\downarrow}\)</span>. Here we have used the particle-hole symmetric version of the interaction term, which is more often given as <span class="math inline">\(n_{i \uparrow} n_{i\downarrow}\)</span>. The difference just amounts to a redefinition of the chemical potential.</p> <p>Hubbard originally used the model at half filling <span class="math inline">\(\mu = 0\)</span> to explain the Mott metal-insulator (MI) transition, however it has seen applications to high-temperature superconductivity and become target for cold-atom optical trap experiments.<span class="citation" data-cites="HubbardModelHalf2013"><sup><a href="#ref-HubbardModelHalf2013" role="doc-biblioref">13</a></sup></span>, greiner_quantum_2002, jordens_mott_2008}. While simple, only a few analytic results exist, namely the Bethe ansatz<span class="citation" data-cites="liebAbsenceMottTransition1968"><sup><a href="#ref-liebAbsenceMottTransition1968" role="doc-biblioref">14</a></sup></span>} which proves the absence of even a zero temperature phase transition in the 1D model and Nagaoka’s theorem<span class="citation" data-cites="nagaokaFerromagnetismNarrowAlmost1966"><sup><a href="#ref-nagaokaFerromagnetismNarrowAlmost1966" role="doc-biblioref">15</a></sup></span>} which proves that the three dimensional model has a ferromagnetic ground state in the vicinity of half filling.</p> <h3 id="falikov-kimball-model">Falikov-Kimball model</h3> <p>The Falikov-Kimball model corresponds to the case <span class="math inline">\(t_{\downarrow} = 0\)</span>. It can be interpreted as two coupled spinless electron bands with infinite mass ratio. An itinerant light species with creation operator <span class="math inline">\(c^\dagger_{i\uparrow}\)</span> coupled to an infinitely heavy, immobile species with density operator <span class="math inline">\(n_{i\downarrow}\)</span>. These are often called c and f electrons or electrons and ions. The model was first introduced by Hubbard in 1963 as a model of interacting localised and de-localised electron bands and gained its name from Falikov and Kimball’s use of it to study the MI transition in rare-earth materials<span class="citation" data-cites="hubbardj.ElectronCorrelationsNarrow1963"><sup><a href="#ref-hubbardj.ElectronCorrelationsNarrow1963" role="doc-biblioref">16</a></sup></span>, falicov_simple_1969}.</p> <p>Here we will use refer to the light spinless species as <code>electrons' with creation operator $c^\dagger_{i}$ and the heavy species as</code>ions’ with density operator <span class="math inline">\(n_i\)</span>. When the the density operator of the electrons is needed I’ll always use <span class="math inline">\(c^\dagger_{i}c_{i}\)</span>. We also set <span class="math inline">\(t = 1\)</span>.</p> <p><span class="math display">\[ H_{\mathrm{FK}} = -\sum_{<ij>} c^\dagger_{i}c_{j} + U \sum_{i} (c^\dagger_{i}c_{i} - 1/2)( n_i - 1/2) - \mu \sum_i \left(c^\dagger_{i}c_{i} + n_{i}\right) \]</span> % ### Particle-Hole Symmetry The Hubbard and FK models on a bipartite lattice have particle-hole (PH) symmetry <span class="math inline">\(P^\dagger H P = - H\)</span>, accordingly they have symmetric energy spectra. The associated symmetry operator <span class="math inline">\(P\)</span> exchanges creation and annihilation operators along with a sign change between the two sublattices.</p> <p><span class="math display">\[ d^\dagger_{i\sigma} = (-1)^i c_{i\sigma}\]</span> % The entirely filled state <span class="math inline">\(\ket{\Omega} = \sum_{j\rho} c^\dagger_{j\rho} \ket{0}\)</span> becomes the new vacuum state: <span class="math display">\[d_{i\sigma} \ket{\Omega} = (-1)^i c^\dagger_{i\sigma} \sum_{j\rho} c^\dagger_{j\rho} \ket{0} = 0\]</span> % The number operator <span class="math inline">\(n_{i\sigma} = 0,1\)</span> counts holes rather than particles: <span class="math display">\[ d^\dagger_{i\sigma} d_{i \sigma} = c_{i\sigma} c^\dagger_{i\sigma} = 1 - c^\dagger_{i\sigma} c_{i\sigma}\]</span> % With the last equality following from the fermionic commutation relations. In the case of nearest neighbour hopping on a bipartite lattice this transformation also leaves the hopping term unchanged: <span class="math display">\[ d^\dagger_{i\sigma} d_{j \sigma} = (-1)^{i+j} c_{i\sigma} c^\dagger_{j\sigma} = c^\dagger_{i\sigma} c_{j\sigma} \]</span> % Since when <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span> label sites on separate sublattices, <span class="math inline">\((-1)^{i-j} = -1\)</span> and this is absorbed into rearranging the operators via their anti-commutator.</p> <p>Defining the particle density <span class="math inline">\(\rho\)</span> as the number of fermions per site: <span class="math display">\[ \rho = \frac{1}{N} \sum_i \left( n_{i \uparrow} + n_{i \downarrow} \right) \]</span> % The PH symmetry maps the Hamiltonian to itself with the sign of the chemical potential reversed and the density is inverted about half filling: <span class="math display">\[ \text{PH} : H(t, U, \mu) \rightarrow H(t, U, -\mu) \]</span> <span class="math display">\[ \rho \rightarrow 2 - \rho \]</span> % The Hamiltonian is symmetric under PH at <span class="math inline">\(\mu = 0\)</span> and so must all the observables, hence half filling <span class="math inline">\(\rho = 1\)</span> occurs here. This symmetry and known observable acts as a useful test for the numerical calculations.</p> <h3 id="thermodynamics-of-the-fk-model">Thermodynamics of the FK model</h3> \begin{figure} <p>} \end{figure}</p> <p>At half filling and 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 charge density wave state in which the ions occupy one of the two sublattices A and B<span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"><sup><a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">17</a></sup></span>}. The order parameter is the square of the staggered magnetisation: <span class="math display">\[ M = \sum_{i \in A} n_i - \sum_{i \in B} n_i \]</span> % In the disordered phase Ref.<span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"><sup><a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">4</a></sup></span>} identifies an interplay between Anderson localisation at weak interaction and a Mott insulator phase in the strongly interacting regime.</p> <p>In the one dimensional FK model, however, Peierls’ argument<span class="citation" data-cites="peierlsIsingModelFerromagnetism1936"><sup><a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">18</a></sup></span>, kennedyItinerantElectronModel1986} and the Bethe ansatz<span class="citation" data-cites="liebAbsenceMottTransition1968"><sup><a href="#ref-liebAbsenceMottTransition1968" role="doc-biblioref">14</a></sup></span>} make it clear that there is no ordered CDW phase. Peierls’ argument is that one should 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 the order parameter. In the Ising model this would be having the spins pointing up in one part of the model and down in the other, for a CDW phase it means having the ions occupy the A sublattice in one part and the B sublattice in the other.</p> <p>Short range interactions will produce a constant energy penalty for such a domain wall that does not scale with system size while in 1D there are <span class="math inline">\(L\)</span> such states so the domain wall is associated with entropy <span class="math inline">\(S \propto \ln L\)</span> which dominates in the thermodynamic limit. The slow logarithmic scaling suggests we should be wary of finite size scaling effects.</p> <p>One dimensional systems are more amenable to numerical and experimental study so we add long range staggered interactions to bring back the ordered phase:</p> <p><span class="math display">\[ H_{\textrm{int}} = 4J \sum_{ij} \frac{(-1)^{|i-j|}}{ |i - j|^{\alpha} } (n_i - 1/2) (n_j - 1/2) = J \sum_{ij} |i - j|^{-\alpha} \tau_i \tau_j\]</span> % at half-filling the modified Hamiltonian is then: <span class="math display">\[ H_{\mathrm{FK}}^* &= -\sum_{<ij>} c^\dagger_{i}c_{j} + U \sum_{i} (c^\dagger_{i}c_{i} - 1/2)( n_i - 1/2) \\ &+ 4J \sum_{ij} \frac{(-1)^{|i-j|}}{ |i - j|^{\alpha} } (n_i - 1/2) (n_j - 1/2) \\ &= -\sum_{<ij>} c^\dagger_{i}c_{j} + 2U \sum_{i} (-1)^i (c^\dagger_{i}c_{i} - 1/2)\tau_i + J \sum_{ij} |i - j|^{-\alpha} \tau_i \tau_j \\ \]</span> % The form of this interaction comes from interpreting the f-electrons as a classical Ising chain using a staggered mapping <span class="math inline">\(\tau_i = (-1)^i (2n_i^ f - 1)\)</span> so that ferromagnetic order in the <span class="math inline">\(\tau_i\)</span> variables corresponds to a CDW state in the <span class="math inline">\(n_i^f\)</span> variables. It also preserves the particle hole symmetry because for the ions the PH transformation corresponds to <span class="math inline">\(n_i \rightarrow 1 - n_i\)</span>. When <span class="math inline">\(U = 0\)</span> the model decouples into a long ranged Ising model and free fermions.</p> <p>Our extension to the FK model could now be though of as spinless fermions coupled to a long range Ising (LRI) model. The LRI model has been extensively studied and its behaviour may be bear relation to the behaviour of our modified FK model.</p> <p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(\abs{i-j}) \tau_i \tau_j = J \sum_{i\neq j} |i - j|^{-\alpha} \tau_i \tau_j\]</span> % Rigorous renormalisation group arguments show that the LRI model has an ordered phase in 1D for $1 < < 2 $<span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"><sup><a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">19</a></sup></span>}. Peierls’ argument can be extended<span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"><sup><a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">20</a></sup></span>} to provide intuition for why this is the case. Again considering the energy difference between the ordered state <span class="math inline">\(\ket{\ldots\uparrow\uparrow\uparrow\uparrow\ldots}\)</span> and a domain wall state <span class="math inline">\(\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}\)</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> % 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. 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 in the thermodynamic limit then the free energy is analytic<span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968"><sup><a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">21</a></sup></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"><sup><a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">19</a></sup></span>}.</p> <p>With a power law form for <span class="math inline">\(J(n)\)</span>, there are three cases to consider:</p> <ol type="1"> <li>$ = 0$ For infinite range interactions the Ising model is exactly solveable and mean field theory is exact<span class="citation" data-cites="lipkinValidityManybodyApproximation1965"><sup><a href="#ref-lipkinValidityManybodyApproximation1965" role="doc-biblioref">22</a></sup></span>}.</li> <li>$ $ For slowly decaying interactions <span class="math inline">\(\sum_n J(n)\)</span> does not converge so the Hamiltonian is non-extensive, a case which won’t be further considered here.</li> <li>$ 1 < < 2 $ A phase transition to an ordered state at a finite temperature.</li> <li>$ = 2 $ The energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transition<span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"><sup><a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">20</a></sup></span>}.</li> <li>$ 2 < $ For quickly decaying interactions, domain walls have a finite energy penalty, hence Peirels’ argument holds and there is no phase transition.</li> </ol> <h3 id="thermodynamics">Thermodynamics</h3> <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"><sup><a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">17</a></sup></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"><sup><a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">18</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">23</a></sup></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> <h3 id="markov-chain-monte-carlo">Markov Chain Monte Carlo</h3> <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"><sup><a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">17</a></sup></span>}. Using a binder-cumulant method<span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"><sup><a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">24</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">25</a></sup></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. We show preliminary results that the in-gap states induced at finite temperature are localised while the states in the unperturbed bands remain extended, evidence for a mobility edge.</p> <div id="fig:binder" class="fignos"> <figure> <img src="/assets/thesis/figure_code/fk_chapter/binder.png" style="width:100.0%" alt="Figure 1: Hello I am the figure caption!" /> <figcaption aria-hidden="true"><span>Figure 1:</span> Hello I am the figure caption!</figcaption> </figure> </div> <p>Macro definitions in this cell <span class="math display">\[ \newcommand{\expval}[1]{\langle #1 \rangle} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|} \newcommand{\op}[2]{|#1\rangle \langle#2|} \]</span></p> <p><span class="math display">\[ \expval{O}, \op{\alpha}{\beta}, \ket{\psi} \]</span></p> <h2 id="localisation-1">Localisation</h2> <h3 id="thermalisation">Thermalisation</h3> <p>Isolated classical systems generally thermalise if they are large enough. Since classical dynamics is the limit of some underlying quantum dynamics, it seems reasonable to suggest that isolated quantum systems also thermalise in some related sense. However it is not immediately obvious what thermalisation should mean in a quantum setting where the presence of unitary time dynamics implies full information about a system’s initial state is always preserved.</p> <p>A potential solution lies in the eigenstate thermalisation hypothesis. It states that if a system thermalises, then we can isolate small patches of a larger system, trace everyhing else out, and get a thermal density matrix.</p> <p>Following Ref.<span class="citation" data-cites="abaninRecentProgressManybody2017"><sup><a href="#ref-abaninRecentProgressManybody2017" role="doc-biblioref">2</a></sup></span>, consider the time evolution of a local operator <span class="math inline">\(\hat{O}\)</span> <span class="math display">\[ \expval{\hat{O}}{\psi(t)} = \sum_{\alpha \beta} C^*_\alpha C_\beta e^{i(E_\alpha - E_\beta)} O_{\alpha \beta}\]</span></p> <p>Where <span class="math inline">\(C_\alpha\)</span> are determined by the initial state and <span class="math inline">\(O_{\alpha \beta} = \expval{\alpha | \hat{O} | \beta}\)</span> are the matrix elements of <span class="math inline">\(\hat{O}\)</span> with respect to the energy eigenstates. Srednicki<span class="citation" data-cites="srednickiChaosQuantumThermalization1994"><sup><a href="#ref-srednickiChaosQuantumThermalization1994" role="doc-biblioref">3</a></sup></span>} introduced the ansatz that for local operators:</p> <p><span class="math display">\[O_{\alpha \beta} = O(E)\delta_{\alpha\beta} + e^{-S(E)/2} f(E,\omega) R_{\alpha\beta}\]</span></p> <p>with <span class="math inline">\(E = (E_\alpha + E_\beta)\)</span>, <span class="math inline">\(\omega = (E_\alpha - E_\beta)\)</span> and <span class="math inline">\(R_{\alpha\beta}\)</span> are sampled from some distribution with zero mean and unit variance. The first term asserts that the diagonal elements are given by the thermal expectation value <span class="math inline">\(O(E) = Tr[e^{-\beta \hat{H}} \hat{O}]/\mathcal{Z}\)</span> with <span class="math inline">\(\beta\)</span> an effective temperature defined by equating the energy to the expectation of the Hamiltonian at that temperature <span class="math inline">\(E = Tr[H e^{-\beta \hat{H}}/\mathcal{Z}]\)</span>.</p> <p>The second term deals with thermodynamic fluctuations scaled by the entropy <span class="math inline">\(S(E) = -Tr(\rho \log \rho)\)</span> where <span class="math inline">\(\rho = e^{-\beta \hat{H}}\)</span> and <span class="math inline">\(\mathcal{Z} = Tr[e^{-\beta \hat{H}}]\)</span>.</p> <p>With this ansatz the long time average of the observable becomes equal to the thermal expectations with fluctuations suppressed by the entropic term <span class="math inline">\(e^{-S(E)}\)</span> and the rapidly varying phase factors <span class="math inline">\(e^{i(E_\alpha - E_\beta)}\)</span>. This statement of the ETH has verified for the quantum hard sphere model<span class="citation" data-cites="srednickiChaosQuantumThermalization1994"><sup><a href="#ref-srednickiChaosQuantumThermalization1994" role="doc-biblioref">3</a></sup></span> and numerically for other models<span class="citation" data-cites="khatamiFluctuationDissipationTheoremIsolated2013 dalessioQuantumChaosEigenstate2016"><sup><a href="#ref-khatamiFluctuationDissipationTheoremIsolated2013" role="doc-biblioref">26</a>,<a href="#ref-dalessioQuantumChaosEigenstate2016" role="doc-biblioref">27</a></sup></span>.</p> <p>An alternate view on ETH is the statement that in thermalising systems individual eigenstates look thermal when viewed locally. Take a eigenstate <span class="math inline">\(|\alpha\rangle\)</span> with energy <span class="math inline">\(E_\alpha\)</span> and as before define an effective temperature with <span class="math inline">\(E_\alpha = Tr[H e^{-\beta \hat{H}}/\mathcal{Z}]\)</span>. This statement of the ETH says that if we partition the system into subsystems A and B with a limit taken as B becomes very large, B will act as a heat bath for A. Specifically the reduced density matrix <span class="math inline">\(\rho_A = Tr_B \op{\alpha}{\alpha}\)</span> is equal to the thermal density matrix:</p> <p><span class="math display">\[\rho_A = Tr_B |\alpha\rangle \langle \alpha| = \mathcal{Z}^{-1} Tr_B [e^{-\beta \hat{H}}] \]</span></p> <p>Intuitively, for thermalisation to happen, the degrees of freedom must be sufficiently well coupled that energy transport occurs. This condition is broken by systems with localised states so a lack of thermalisation is often used as a diagnostic tool for localisation.</p> <h3 id="anderson-localisation">Anderson Localisation</h3> <p>Localisation was first studied by Anderson in 1958<span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"><sup><a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">4</a></sup></span> in the context of non-interacting fermions subject to a static or quenched disorder potential <span class="math inline">\(V_j\)</span> drawn uniformly from the interval <span class="math inline">\([-W,W]\)</span>:</p> <p><span class="math display">\[ H = -t\sum_{\expval{jk}} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j \]</span></p> <p>At sufficiently strong disorder the Anderson model exhibits exponentially localised eigenfunctions <span class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\)</span> which cannot contribute to diffusive transport processes. Except in 1D where any disorder strength is sufficient. Intuitively this happens because hopping processes between nearby sites become off-resonant, hindering the hybridisation that would normally lead to extended Bloch states<span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"><sup><a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">28</a></sup></span>.</p> <p>In one and two dimensions, all the states in the Anderson model are localised. In three dimensions there are mobility edges. Mobility edges are critical energies in the spectrum which separate delocalised states in a band from localised states which form a band tail<span class="citation" data-cites="abaninRecentProgressManybody2017"><sup><a href="#ref-abaninRecentProgressManybody2017" role="doc-biblioref">2</a></sup></span>}. An argument due to Lifshitz shows that the density of state of the band tail should decay exponentially and localised and extended stats cannot co-exist at the same energy as they would hybridise into extended states<span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"><sup><a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">28</a></sup></span>}.</p> <p>It was thought that mobility edges could not exist in 1D because all the states localised in the presence of any amount of disorder. This is true for uncorrelated potentials<span class="citation" data-cites="goldshteinPurePointSpectrum1977"><sup><a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">29</a></sup></span>}. However, it was shown that if the disorder potential <span class="math inline">\(V_j\)</span> contains spatial correlations mobility edges do exist in 1D<span class="citation" data-cites="izrailevLocalizationMobilityEdge1999"><sup><a href="#ref-izrailevLocalizationMobilityEdge1999" role="doc-biblioref">5</a></sup></span>, izrailevAnomalousLocalizationLowDimensional2012}. Ref.<span class="citation" data-cites="croyAndersonLocalization1D2011"><sup><a href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">6</a></sup></span>} extends this work to look at power law decay of the correlations: <span class="math display">\[ C(l) = \expval{V_i V_{i+l}} \propto l^{-\alpha} \]</span> % Figure <span class="math inline">\(\ref{fig:anderson_dos}\)</span> shows numerical calculations of the Localisation length (see later) and density of states for the power law correlated Anderson model. At the unperturbed band edges <span class="math inline">\(\abs{E} = 2\)</span>, the states transition from extended to localised. The behaviour close to the edge takes a universal scaling form with exponents dependant on <span class="math inline">\(\alpha\)</span>.</p> <h3 id="many-body-localisation">Many Body Localisation</h3> <p>A related phenomena known as many body localisation (MBL) was found in interacting systems with quenched disorder. A simple example comes from adding density-density interactions to the Anderson model:</p> <p><span class="math display">\[ H = -t\sum_{\expval{jk}} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U\sum_{jk} n_j n_k \]</span> % with <span class="math inline">\(n_j = c^\dagger_j c_j\)</span> Here, in contrast to the Anderson model, localisation phenomena can be proven robust to weak perturbations of the Hamiltonian<span class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016"><sup><a href="#ref-imbrieManyBodyLocalizationQuantum2016" role="doc-biblioref">8</a></sup></span>}.</p> <p>MBL is defined by the emergence of an extensive number of quasi-local operators called local integrals of motions (LIOMs) or l-bits. Following Ref.<span class="citation" data-cites="abaninRecentProgressManybody2017"><sup><a href="#ref-abaninRecentProgressManybody2017" role="doc-biblioref">2</a></sup></span>}, using a spin system with variables <span class="math inline">\(\sigma^z_i\)</span>, any operator can be written in the general form:</p> <p><span class="math display">\[ \tau^z_i = \sigma^z_i + \sum_{\alpha\beta kl} f_{kl}^{\alpha\beta} \sigma^\alpha_{i+k} \sigma_z\beta_{i+k} + ...\]</span> % what defines a MBL system is that there exist extensively many <span class="math inline">\(\tau^z_i\)</span> for which the coefficients decay exponentially with distance <span class="math inline">\(f_{kl}^{\alpha\beta} \propto e^{-\max(\abs{l},\abs{k}) / \xi}\)</span>. These LIOMs commute with the Hamiltonian and each other <span class="math inline">\([\hat{H}, \tau^z_i] = [\tau^z_i, \tau^z_j] = 0\)</span>. It is this extensive number of conserved local charges that leads to the localisation properties of MBL. It also has implications for the way entanglement grows over time in MBL systems.</p> <p>Since the Hamiltonian commutes with all the LIOMs and they are a complete operator basis, the Hamiltonian can be written as:</p> <p><span class="math display">\[\hat{H} = \sum_{i} h_i \tau^z_i + \sum_{ij} J_{ij} \tau^z_i \tau^z_j + \sum_{ijk} J_{ij} \tau^z_i \tau^z_j \tau^z_k+ ...\]</span> % Where again the couplings decay exponentially, albeit with a different length scale <span class="math inline">\(\Bar{\xi}\)</span>. From this form we see that distant l-bits can only become entangled on a timescale of:</p> <p><span class="math display">\[ t_{\mathrm{ent}}(r) \propto \frac{\hbar}{J_0} e^{r/\Bar{\xi}} \]</span> % and hence quantum correlations and entanglement propagates logarithmically in MBL systems<span class="citation" data-cites="imbrieDiagonalizationManyBodyLocalization2016"><sup><a href="#ref-imbrieDiagonalizationManyBodyLocalization2016" role="doc-biblioref">30</a></sup></span>}.</p> <h3 id="disorder-free-localisation">Disorder Free localisation</h3> <p>Both Anderson localisation and MBL depend on the presence of quenched disorder. Recently the idea of disorder-free localisation has gained traction, asking whether the disorder necessary to generate localisation can be generated entirely from the dynamics of a model itself.</p> <p>The idea was first proposed in the context of Helium mixtures<span class="citation" data-cites="kagan1984localization"><sup><a href="#ref-kagan1984localization" role="doc-biblioref">9</a></sup></span>} and then extended to heavy-light mixtures in which multiple species with large mass ratios interact, the idea being 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"><sup><a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">10</a></sup></span>,schiulazDynamicsManybodyLocalized2015} instead find that the models thermalise exponentially slowly in system size, which Ref.<span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"><sup><a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">10</a></sup></span>} dubs Quasi-MBL. A. Smith, J. Knolle et al instead looked at models containing an extensive number of conserved quantities and demonstrated true disorder free localisation<span class="citation" data-cites="smithDisorderFreeLocalization2017"><sup><a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">12</a></sup></span>}.</p> <h3 id="diagnostics-of-localisation">Diagnostics of Localisation</h3> <h4 id="inverse-participation-ratio">Inverse Participation Ratio</h4> <p>The inverse participation ratio is defined for a normalised wave function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i \abs{\psi_i}^2 = 1\)</span> as its fourth moment<span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"><sup><a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">28</a></sup></span>}: <span class="math display">\[ P^{-1} = \sum_i \abs{\psi_i}^4 \]</span> % It acts as a measure of the portion of space occupied by the wave function. For localised states it will be independent of system size while for plane wave states in d dimensions $P = L^d $. States may also be intermediate between localised and extended, described by their fractal dimensionality <span class="math inline">\(d > d* > 0\)</span>: <span class="math display">\[ P(L) \goeslike L^{d*} \]</span> % For extended states <span class="math inline">\(d* = 0\)</span> while for localised ones <span class="math inline">\(d* = 0\)</span>. In this work we take use an energy resolved IPR<span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"><sup><a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">4</a></sup></span>: <span class="math display">\[ DOS(\omega) = \sum_n \delta(\omega - \epsilon_n) IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_{n,i}}^4 \]</span> Where <span class="math inline">\(\psi_{n,i}\)</span> is the wavefunction corresponding to the energy <span class="math inline">\(\epsilon_n\)</span> at the ith site. In practice we bin the energies and IPRs into a fine energy grid and use Lorentzian smoothing if necessary.</p> <h4 id="transfer-matrix-approach">Transfer Matrix Approach</h4> <p>The transfer matrix method (TMM) can be used to calculate the localisation length of the eigenstates of a system. Following Refs.<span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"><sup><a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">28</a></sup></span>, smithDynamicalLocalizationMathbbZ2018}, for bi-linear, 1D Hamiltonians the method represents the action of <span class="math inline">\(H\)</span> on a state <span class="math inline">\(\ket{\psi} = \sum_i \psi_i \ket{i}\)</span> with energy E by a matrix equation: <span class="math display">\[ H &= - \sum_i (c^\dagger_i c_{i+1} + c^\dagger_{i+1} c_{i}) - \sum_i h_i c^\dagger_i c_i \\ E\ket{\psi} &= H \ket{\psi} \\ \label{eq:tmm_difference} E \psi_i &= -(\psi_{i+1} + \psi_{i-1}) - h_i \psi_i \]</span> % Where Eq. <span class="math inline">\(\ref{eq:tmm_difference}\)</span> can be represented by a matrix equation: <span class="math display">\[ \begin{pmatrix} \psi_{i+1}\\ \psi_{i} \end{pmatrix} = \begin{pmatrix} -(E + h_i) & -1\\ 1 & 0 \end{pmatrix} \begin{pmatrix} \psi_{i}\\ \psi_{i-1} \end{pmatrix} = T_i \begin{pmatrix} \psi_{i}\\ \psi_{i-1} \end{pmatrix} \]</span> % Defining a product along the chain: <span class="math display">\[Q_L = \prod_{i=0}^L T_i\]</span> % Oseledec’s theorem proves that there exists a limiting matrix <span class="math inline">\(\Gamma\)</span>: <span class="math display">\[ \Gamma = \lim_{L \to \infty} (Q_L Q_L^\dagger)^{\frac{1}{2L}} \]</span> % with eigenvalues <span class="math inline">\(\exp(\gamma_i)\)</span> where <span class="math inline">\(\gamma_i\)</span> are the Lyapunov exponents of <span class="math inline">\(Q_L\)</span>. The smallest Lyapunov exponent is the inverse of the localisation length of the state. In practice one takes <span class="math inline">\(Q_L\)</span> with L equal to the system size, finds the smallest eigenvalue q and estimates the localisation length by: <span class="math display">\[ \lambda = \frac{L}{\ln{q}} \]</span> % As noted by<span class="citation" data-cites="smithDynamicalLocalizationMathbbZ2018"><sup><a href="#ref-smithDynamicalLocalizationMathbbZ2018" role="doc-biblioref">31</a></sup></span> this method can be numerically unstable because the matrix elements diverge or vanish exponentially. To get around this, the authors break the matrix multiplication into chunks and the logarithms of the eigenvalues of each are stored separately.</p> <h2 id="numerical-methods">Numerical Methods</h2> <p>In this section we will define the Markov Chain Monte Carlo (MCMC) method in general then detail its application to the FK model. We will then cover methods applicable to the measurements obtained from MCMC. These include calculation of the density of states and energy resolved inverse participation ratio as well as phase transition diagnostics such as the Binder cumulant.</p> <h3 id="markov-chain-monte-carlo-1">Markov Chain Monte Carlo}</h3> <p>Markov Chain Monte Carlo (MCMC) is a technique for evaluating thermal expectation values <span class="math inline">\(\expval{O}\)</span> with respect to some physical system defined by a set of states <span class="math inline">\(\{x: x \in S\}\)</span> and a free energy <span class="math inline">\(F(x)\)</span><span class="citation" data-cites="krauthIntroductionMonteCarlo1998"><sup><a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">32</a></sup></span>}. The thermal expectation value is defined via a Boltzmann weighted sum over the entire states: <span class="math display">\[ \tex{O} &= \frac{1}{\Z} \sum_{x \in S} O(x) P(x) \\ P(x) &= \frac{1}{\Z} e^{-\beta F(x)} \\ \Z &= \sum_{x \in S} e^{-\beta F(x)} \]</span></p> <p>When the state space is too large to evaluate this sum directly, MCMC defines a stochastic algorithm which generates a random walk <span class="math inline">\(\{x_0\ldots x_i\ldots x_N\}\)</span> whose distribution <span class="math inline">\(p(x_i)\)</span> approaches a target distribution <span class="math inline">\(P(x)\)</span> in the large N limit.</p> <p><span class="math display">\[\lim_{i\to\infty} p(x_i) = P(x)\]</span></p> <p>In this case the target distribution will be the thermal one <span class="math inline">\(P(x) \rightarrow \Z^{-1} e^{-\beta F(x)}\)</span>. The major benefit of the method being that as long as one can express the desired <span class="math inline">\(P(x)\)</span> up to a multiplicative constant, MCMC can be applied. Since <span class="math inline">\(e^{-\beta F(x)}\)</span> is relatively easy to evaluate, MCMC provides a useful method for finite temperature physics.</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 display">\[ \tex{O} = \sum_{i = 0}^{N} O(x_i) + \mathcal{O}(\frac{1}{\sqrt{N}}) \]</span> The the samples in the random walk are correlated so the samples effectively contain less information than <span class="math inline">\(N\)</span> independent samples would. As a consequence the variance is larger than the <span class="math inline">\(\tex{O^2} - \tex{O}^2\)</span> form it would have if the estimates were uncorrelated. Methods of estimating the true variance of <span class="math inline">\(\tex{O}\)</span> and decided how many steps are needed will be considered later.</p> <p>Markov chains are defined by a transition function $(x_{i+1} x_i) $ giving the probability that a chain in state <span class="math inline">\(x_i\)</span> at time <span class="math inline">\(i\)</span> will transition to a state <span class="math inline">\(x_{i+1}\)</span>. Since we must transition somewhere at each step, this comes with the normalisation condition that <span class="math inline">\(\sum\limits_x \T(x' \rightarrow x) = 1\)</span>.</p> <p>If we define an ensemble of Markov chains and consider the distributions we get a sequence of distributions <span class="math inline">\(\{p_0(x), p_1(x), p_2(x)\ldots\}\)</span> with <span class="math display">\[p_{i+1}(x) &= \sum_{x' \in S} p_i(x') \T(x' \rightarrow x)\]</span> <span class="math inline">\(p_o(x)\)</span> might be a delta function on one particular starting state which would then be smoothed out by the transition function repeatedly.</p> <p>As we’d like to draw samples from the target distribution <span class="math inline">\(P(x)\)</span> the trick is to choose $(x_{i+1} x_i) $ such that :</p> <p><span class="math display">\[ P(x) &= \sum_{x'} P(x') \T(x' \rightarrow x) \]</span> In other words the MCMC dynamics defined by <span class="math inline">\(\T\)</span> must be constructed to have the target distribution as their only fixed point. This condition is called the global balance condition. Along with some more technical considerations such as ergodcity which won’t be considered here, global balance suffices to ensure that a MCMC method is correct.</p> <p>A sufficient but not necessary condition for global balance to hold is detailed balance:</p> <p><span class="math display">\[ P(x) \T(x \rightarrow x') = P(x') \T(x' \rightarrow x) \]</span> % In practice most algorithms are constructed to satisfy detailed balance though there are arguments that relaxing the condition can lead to faster algorithms<span class="citation" data-cites="kapferSamplingPolytopeHarddisk2013"><sup><a href="#ref-kapferSamplingPolytopeHarddisk2013" role="doc-biblioref">33</a></sup></span>}.</p> <p>The goal of MCMC is then to choose <span class="math inline">\(\T\)</span> so that it has the desired thermal distribution <span class="math inline">\(P(x)\)</span> as its fixed point and that it converges quickly onto it. This boils down to requiring that the matrix representation of <span class="math inline">\(T_{ij} = \T(x_i \to x_j)\)</span> has an eigenvector equal to <span class="math inline">\(P_i = P(x_i)\)</span> with eigenvalue 1 and all other eigenvalues with magnitude less than one. The convergence time depends on the magnitude of the second largest eigenvalue.</p> <p>In order to actually choose new states according to <span class="math inline">\(\T\)</span> one chooses states from a proposal distribution <span class="math inline">\(q(x_i \to x')\)</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">\(q(x'\to x_i)\)</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">\(x'\)</span> is then accepted or rejected with an acceptance probability <span class="math inline">\(\A(x'\to x_{i+1})\)</span>, if the proposal is rejected then <span class="math inline">\(x_{i+1} = x_{i}\)</span>. Now <span class="math inline">\(\T(x\to x') = q(x\to x')\A(x \to x')\)</span>.</p> <p>The Metropolis-Hasting algorithm is a slight extension of the original Metropolis algorithm that allows for non-symmetric proposal distributions $q(xx’) q(x’x) $. It can be derived starting from detailed balance<span class="citation" data-cites="krauthIntroductionMonteCarlo1998"><sup><a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">32</a></sup></span>}: <span class="math display">\[ P(x)\T(x \to x') &= P(x')\T(x' \to x) \\ P(x)q(x \to x')\A(x \to x') &= P(x')q(x' \to x)\A(x' \to x) \\ \label{eq:db2} \frac{\A(x \to x')}{\A(x' \to x)} &= \frac{P(x')q(x' \to x)}{P(x)q(x \to x')} = f(x, x')\\ \]</span> % The Metropolis-Hastings algorithm is the choice: <span class="math display">\[\label{eq:mh} \A(x \to x') = \min\left(1, f(x,x')\right)\]</span> % Noting that <span class="math inline">\(f(x,x') = 1/f(x',x)\)</span>, Eq. <span class="math inline">\(\ref{eq:mh}\)</span> can be seen to satisfy Eq. <span class="math inline">\(\ref{eq:db2}\)</span> by considering the two cases <span class="math inline">\(f(x,x') > 1\)</span> and <span class="math inline">\(f(x,x') < 1\)</span>.</p> <p>By choosing the proposal distribution such that <span class="math inline">\(f(x,x')\)</span> is as close as possible to one, the rate of rejections can be reduced and the algorithm sped up.</p> % <p>%Thinning, burn in, multiple runs %Simulated annealing and Parallel Tempering</p> <h3 id="applying-mcmc-to-the-fk-model">Applying MCMC to the FK model}</h3> <p>MCMC can be applied to sample over the classical degrees of freedom of the model. We take the full Hamiltonian and split it into a classical and a quantum part: <span class="math display">\[ H_{\mathrm{FK}} &= -\sum_{<ij>} c^\dagger_{i}c_{j} + U \sum_{i} (c^\dagger_{i}c_{i} - 1/2)( n_i - 1/2) \\ &+ \sum_{ij} J_{ij} (n_i - 1/2) (n_j - 1/2) - \mu \sum_i (c^\dagger_{i}c_{i} + n_i)\\ H_q &= -\sum_{<ij>} c^\dagger_{i}c_{j} + \sum_{i} \left(U(n_i - 1/2) - \mu\right) c^\dagger_{i}c_{i}\\ H_c &= \sum_i \mu n_i - \frac{U}{2}(n_i - 1/2) + \sum_{ij}J_{ij}(n_i - 1/2)(n_j - 1/2) \]</span> % There are <span class="math inline">\(2^N\)</span> possible ion configurations <span class="math inline">\(\{ n_i \}\)</span>, we define <span class="math inline">\(n^k_i\)</span> to be the occupation of the ith site of the kth configuration. The quantum part of the free energy can then be defined through the quantum partition function <span class="math inline">\(\Z^k\)</span> associated with each ionic state <span class="math inline">\(n^k_i\)</span>: <span class="math display">\[ F^k &= -1/\beta \ln{\Z^k} \\ \]</span> % Such that the overall partition function is: <span class="math display">\[ \Z &= \sum_k e^{- \beta H^k} Z^k \\ &= \sum_k e^{-\beta (H^k + F^k)} \\ \]</span> % Because fermions are limited to occupation numbers of 0 or 1 <span class="math inline">\(Z^k\)</span> simplifies nicely. If <span class="math inline">\(m^j_i = \{0,1\}\)</span> is defined as the occupation of the level with energy <span class="math inline">\(\epsilon^k_i\)</span> then the partition function is a sum over all the occupation states labelled by j: <span class="math display">\[ Z^k &= \Tr e^{-\beta F^k} = \sum_j e^{-\beta \sum_i m^j_i \epsilon^k_i}\\ &= \sum_j \prod_i e^{- \beta m^j_i \epsilon^k_i}= \prod_i \sum_j e^{- \beta m^j_i \epsilon^k_i}\\ &= \prod_i (1 + e^{- \beta \epsilon^k_i})\\ F^k &= -1/\beta \sum_k \ln{(1 + e^{- \beta \epsilon^k_i})} \]</span> % Observables can then be calculated from the partition function, for examples the occupation numbers:</p> <p><span class="math display">\[ \tex{N} &= \frac{1}{\beta} \frac{1}{Z} \frac{\partial Z}{\partial \mu} = - \frac{\partial F}{\partial \mu}\\ &= \frac{1}{\beta} \frac{1}{Z} \frac{\partial}{\partial \mu} \sum_k e^{-\beta (H^k + F^k)}\\ &= 1/Z \sum_k (N^k_{\mathrm{ion}} + N^k_{\mathrm{electron}}) e^{-\beta (H^k + F^k)}\\ \]</span> % with the definitions:</p> <p><span class="math display">\[ N^k_{\mathrm{ion}} &= - \frac{\partial H^k}{\partial \mu} = \sum_i n^k_i\\ N^k_{\mathrm{electron}} &= - \frac{\partial F^k}{\partial \mu} = \sum_i \left(1 + e^{\beta \epsilon^k_i}\right)^{-1}\\ \]</span> % The MCMC algorithm consists of performing a random walk over the states <span class="math inline">\(\{ n^k_i \}\)</span>. In the simplest case the proposal distribution corresponds to flipping a random site from occupied to unoccupied or vice versa, since this proposal is symmetric the acceptance function becomes: <span class="math display">\[ P(k) &= \Z^{-1} e^{-\beta(H^k + F^k)} \\ \A(k \to k') &= \min\left(1, \frac{P(k')}{P(k)}\right) = \min\left(1, e^{\beta(H^{k'} + F^{k'})-\beta(H^k + F^k)}\right) \]</span> % At each step <span class="math inline">\(F^k\)</span> is calculated by diagonalising the tri-diagonal matrix representation of <span class="math inline">\(H_q\)</span> with open boundary conditions. Observables are simply averages over the their value at each step of the random walk. The full spectrum and eigenbasis is too large to save to disk so usually running averages of key observables are taken as the walk progresses.</p> <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"><sup><a href="#ref-robertsWeakConvergenceOptimal1997" role="doc-biblioref">34</a></sup></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> <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"><sup><a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">32</a></sup></span>}: <span class="math display">\[ q(k \to k') &= \min\left(1, e^{\beta (H^{k'} - H^k)}\right) \\ \A(k \to k') &= \min\left(1, e^{\beta(F^{k'}- F^k)}\right) \]</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"><sup><a href="#ref-huangAcceleratedMonteCarlo2017" role="doc-biblioref">35</a></sup></span>} does this with restricted Boltzmann machines whose form is very similar to a classical spin model.</p> <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">\[ J'(x) &= \abs{\frac{L}{\pi}\sin \frac{\pi x}{L}}^{-\alpha} \\ J(x) &= \frac{J_0 J'(x)}{\sum_y J'(y)} \]</span> % 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> <p>The Binder cumulant is defined as: <span class="math display">\[U_B = 1 - \frac{\tex{\mu_4}}{3\tex{\mu_2}^2}\]</span> % where <span class="math display">\[\mu_n = \tex{(m - \tex{m})^n}\]</span> % are the central moments of the order parameter m: <span class="math display">\[m = \sum_i (-1)^i (2n_i - 1) / N\]</span> % The Binder cumulant evaluated against temperature can be used as a diagnostic for the existence of a phase transition. If multiple such curves are plotted for different system sizes, a crossing indicates the location of a critical point<span class="citation" data-cites="binderFiniteSizeScaling1981"><sup><a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">24</a></sup></span>, musialMonteCarloSimulations2002}.</p> <h2 id="markov-chain-monte-carlo-in-practice">Markov Chain Monte-Carlo in Practice}</h2> <h3 id="quick-intro-to-mcmc">Quick Intro to MCMC}</h3> <p>The main paper relies on extensively to evaluate thermal expectation values within the model by walking over states of the classical spin system <span class="math inline">\(S_i\)</span>. For a classical system, the thermal expectation value of some operator <span class="math inline">\(O\)</span> is defined by a Boltzmann weighted sum over the classical state space: <span class="math display">\[ \tex{O} &= \frac{1}{\Z} \sum_{\s \in S} O(x) P(x) \\ P(x) &= \frac{1}{\Z} e^{-\beta F(x)} \\ \Z &= \sum_{\s \in S} e^{-\beta F(x)} \]</span> While for a quantum system these sums are replaced by equivalent traces. The obvious approach to evaluate these sums numerically would be to directly loop over all the classical states in the system and perform the sum. But we all know know why this isn’t feasible: the state space is too large! Indeed even if we could do it, it would still be computationally wasteful since at low temperatures the sums are dominated by low energy excitations about the ground states of the system. Even worse, in our case we must fully solve the fermionic system via exact diagonalisation for each classical state in the sum, a very expensive operation!~.}</p> <p> 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> <p>%MCMC from an ensemble point of view In implementation can be boiled down to choosing a transition function $(_{t} _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">\(\T\)</span>. This defines a set of random walks <span class="math inline">\(\{\s_0\ldots \s_i\ldots \s_N\}\)</span>. Fig.~<span class="math inline">\(\ref{fig:single}\)</span> 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>Where the function here produces a state with probability determined by the and the transition function <span class="math inline">\(\T\)</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">\(\T\)</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">\(P(\s; \beta) = \Z^{-1} e^{-\beta F(\s)}\)</span>. This turns out to be quite easy to achieve using the Metropolis-Hasting algorithm.</p> <h3 id="convergence-time">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">\(\T\)</span> as the matrix with entries <span class="math inline">\(T_{ij} = \T(\s_j \rightarrow \s_i)\)</span> we can write the update rule for the ensemble probability as: <span class="math display">\[\vec{p}_{t+1} = \T \vec{p}_t \implies \vec{p}_{t} = \T^t \vec{p}_0\]</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 Assuming <span class="math inline">\(\T\)</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> <h3 id="auto-correlation-time">Auto-correlation Time}</h3> <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.~<span class="math inline">\(\ref{fig:raw}\)</span> 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.~<span class="math inline">\(\ref{fig:m_autocorr}\)</span> 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.~ 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>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> <h3 id="the-metropolis-hastings-algorithm">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">\(\A(\s \to \s')\)</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">\(\A(\s \to \s')\)</span>, if the proposal is rejected then <span class="math inline">\(\s_{i+1} = \s_{i}\)</span>. Hence:</p> <p><span class="math display">\[\T(x\to x') = q(x\to x')\A(x \to x')\]</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">\[ \A(x \to x') = \min\left(1, e^{-\beta\;\Delta F}\right)\]</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> <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> <h3 id="choosing-the-proposal-distribution">Choosing the proposal distribution}</h3> <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> <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> <h3 id="two-step-trick">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="cb1"><pre class="sourceCode python"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a></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> proposal(current_state)</span> <span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a></span> <span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a> df_classical <span class="op">=</span> classical_free_energy_change(current_state, new_state, parameters)</span> <span id="cb1-8"><a href="#cb1-8" 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="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a> f_quantum <span class="op">=</span> quantum_free_energy(current_state, new_state, parameters)</span> <span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a> </span> <span id="cb1-11"><a href="#cb1-11" 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="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span> <span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a> </span> <span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a> states[i] <span class="op">=</span> current_state</span> <span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a> </span></code></pre></div> <p>lets cite Figure<a href="#fig:binder">1</a></p> <p>lets cite to person<span class="citation" data-cites="trebstKitaevMaterials2022"><sup><a href="#ref-trebstKitaevMaterials2022" role="doc-biblioref">36</a></sup></span>. and then multple<span class="citation" data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"><sup><a href="#ref-trebstKitaevMaterials2022" role="doc-biblioref">36</a>,<a href="#ref-banerjeeProximateKitaevQuantum2016" role="doc-biblioref">37</a></sup></span>. what is we surround it by spaces?<span class="citation" data-cites="trebstKitaevMaterials2022"><sup><a href="#ref-trebstKitaevMaterials2022" role="doc-biblioref">36</a></sup></span></p> <div id="fig:phase_diagram" class="fignos"> <figure> <img src="pdf_figs/phase_diagram.svg" alt="Figure 2: Phase diagrams of the long-range 1D FK model. (a) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating (MI) phase by a critical temperature T_c, linear in J. (b) 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. (c) The order parameters, \tex{m^2}(solid) and 1 - f (dashed) describing the onset of the CDW phase of the long-range 1D FK model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}\abs{\sum_i (-1)^i \; \expval{c^\dag_{i}c_{i}}} . (d) The crossing of the Binder cumulant, B = \tex{m^4} / \tex{m^2}^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 (a) and (b). 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 varied." /> <figcaption aria-hidden="true"><span>Figure 2:</span> Phase diagrams of the long-range 1D <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model. (a) The TJ plane at <span class="math inline">\(U = 5\)</span>: the <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> ordered phase is separated from a disordered Mott insulating (MI) phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (b) The TU plane at <span class="math inline">\(J = 5\)</span>: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at <span class="math inline">\(E=0\)</span> in the single particle energy spectrum. <span class="math inline">\(U_c\)</span> is independent of temperature. At <span class="math inline">\(U = 0\)</span> the fermions are decoupled from the spins forming a simple Fermi gas. (c) The order parameters, <span class="math inline">\(\tex{m^2}\)</span>(solid) and <span class="math inline">\(1 - f\)</span> (dashed) describing the onset of the <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> phase of the long-range 1D <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model at low temperature with staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}\abs{\sum_i (-1)^i \; \expval{c^\dag_{i}c_{i}}}\)</span> . (d) The crossing of the Binder cumulant, <span class="math inline">\(B = \tex{m^4} / \tex{m^2}^2\)</span>, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in (a) and (b). All plots use system sizes <span class="math inline">\(N = [10,20,30,50,70,110,160,250]\)</span> and lines are coloured from <span class="math inline">\(N = 10\)</span> in dark blue to <span class="math inline">\(N = 250\)</span> in yellow. The parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span> except where explicitly varied.</figcaption> </figure> </div> <h1 id="introduction-1">Introduction</h1> <p>The <span data-acronym-label="FK" data-acronym-form="singular+long">FK</span> model is one of the simplest models of the correlated electron problem. It captures the essence of the interaction between itinerant and localized electrons, equivalent to a model of hopping fermions coupled to a classical Ising field. It was originally introduced to explain the metal-insulator transition in f-electron systems but in its long history it has been interpreted variously as a model of electrons and ions, binary alloys or of crystal formation <span class="citation" data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicovSimpleModelSemiconductorMetal1969 gruberFalicovKimballModelReview1996 gruberFalicovKimballModel2006"><sup><a href="#ref-hubbardj.ElectronCorrelationsNarrow1963" role="doc-biblioref">16</a>,<a href="#ref-falicovSimpleModelSemiconductorMetal1969" role="doc-biblioref">38</a>–<a href="#ref-gruberFalicovKimballModel2006" role="doc-biblioref">40</a></sup></span>. Despite its simplicity, the <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model has a rich phase diagram in <span class="math inline">\(D \geq 2\)</span> dimensions. For example, it shows an interaction-induced gap opening even at high temperatures, similar to the corresponding Hubbard Model <span class="citation" data-cites="brandtThermodynamicsCorrelationFunctions1989"><sup><a href="#ref-brandtThermodynamicsCorrelationFunctions1989" role="doc-biblioref">41</a></sup></span>. In 1D, the ground state phenomenology as a function of filling can be rich <span class="citation" data-cites="gruberGroundStatesSpinless1990"><sup><a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">42</a></sup></span> but the system is disordered for all <span class="math inline">\(T > 0\)</span> <span class="citation" data-cites="kennedyItinerantElectronModel1986"><sup><a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">23</a></sup></span>. Moreover, the model has been a test-bed for many-body methods, interest took off when an exact DMFT solution in the infinite dimensional case was found <span class="citation" data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"><sup><a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">43</a>–<a href="#ref-herrmannNonequilibriumDynamicalCluster2016" role="doc-biblioref">46</a></sup></span>.</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 <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> algorithm <span class="citation" data-cites="devriesGapsDensitiesStates1993 devriesSimplifiedHubbardModel1993 antipovInteractionTunedAndersonMott2016 debskiPossibilityDetectionFinite2016 herrmannSpreadingCorrelationsFalicovKimball2018 maskaThermodynamicsTwodimensionalFalicovKimball2006"><sup><a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">17</a>,<a href="#ref-devriesGapsDensitiesStates1993" role="doc-biblioref">47</a>–<a href="#ref-herrmannSpreadingCorrelationsFalicovKimball2018" role="doc-biblioref">51</a></sup></span>. The <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> 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 <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model to recent interest in disorder-free localisation <span class="citation" data-cites="smithDisorderFreeLocalization2017 smithDynamicalLocalizationMathbbZ2018 brenesManyBodyLocalizationDynamics2018"><sup><a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">12</a>,<a href="#ref-smithDynamicalLocalizationMathbbZ2018" role="doc-biblioref">31</a>,<a href="#ref-brenesManyBodyLocalizationDynamics2018" role="doc-biblioref">52</a></sup></span>.</p> <p>Dimensionality is crucial for the physics of both localisation and <span data-acronym-label="FTPT" data-acronym-form="plural+short">FTPTs</span>. 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 delocalization <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"><sup><a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">53</a>–<a href="#ref-dunlapAbsenceLocalizationRandomdimer1990" role="doc-biblioref">55</a></sup></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"><sup><a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">4</a>,<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">28</a>,<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">29</a>,<a href="#ref-abrahamsScalingTheoryLocalization1979" role="doc-biblioref">56</a></sup></span>. However, the absence of an <span data-acronym-label="FTPT" data-acronym-form="singular+short">FTPT</span> in the short ranged <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> 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"><sup><a href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">57</a>–<a href="#ref-yosidaMagneticPropertiesCuMn1957" role="doc-biblioref">60</a></sup></span> decays as <span class="math inline">\(r^{-1}\)</span> in 1D <span class="citation" data-cites="rusinCalculationRKKYRange2017a"><sup><a href="#ref-rusinCalculationRKKYRange2017a" role="doc-biblioref">61</a></sup></span>. This could in principle induce the necessary long-range interactions for the classical Ising background <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"><sup><a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">18</a>,<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">20</a></sup></span>. However, Kennedy and Lieb established rigorously that at half-filling a <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> phase only exists at <span class="math inline">\(T = 0\)</span> for the 1D <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model <span class="citation" data-cites="kennedyItinerantElectronModel1986"><sup><a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">23</a></sup></span>.</p> <p>Here, we construct a generalised one-dimensional <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model with long-range interactions which induces the otherwise forbidden <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> 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 <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"><sup><a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">49</a></sup></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 <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> 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>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> <h1 id="the-long-ranged-falikov-kimball-model">The Long-Ranged Falikov-Kimball Model</h1> <p>We interpret the <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model as a model of spinless fermions, <span class="math inline">\(c^\dag_{i}\)</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">\(J_{ij} = 4\kappa J (-1)^{\abs{i-j}} \abs{i-j}^{-\alpha}\)</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"><sup><a href="#ref-gruberFalicovKimballModelReview1996" role="doc-biblioref">39</a></sup></span>. <span class="math display">\[\begin{aligned} H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dag_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{i} (c^\dag_{i}c_{i+1} + \textit{h.c.)}\\ & + \sum_{i, j}^{N} J_{ij} S_i S_j \nonumber \label{eq:HFK}\end{aligned}\]</span></p> <p>In two or more dimensions, the <span class="math inline">\(J\!=0\!\)</span> <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model has a <span data-acronym-label="FTPT" data-acronym-form="singular+short">FTPT</span> to the <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> 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">\(f = 2 N^{-1}\abs{\sum_i (-1)^i \; \expval{c^\dag_{i}c_{i}}}\)</span> <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 maskaThermodynamicsTwodimensionalFalicovKimball2006"><sup><a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">17</a>,<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">49</a></sup></span>. This only exists at zero temperature in the short ranged 1D model <span class="citation" data-cites="kennedyItinerantElectronModel1986"><sup><a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">23</a></sup></span>. To study the <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> 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 <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> 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 <span data-acronym-label="LRI" data-acronym-form="singular+short">LRI</span> 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 <span data-acronym-label="LRI" data-acronym-form="singular+short">LRI</span> model, a power law decay of <span class="math inline">\(\alpha < 2\)</span> is required for a <span data-acronym-label="FTPT" data-acronym-form="singular+short">FTPT</span> 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"><sup><a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">20</a>,<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">21</a>,<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">62</a></sup></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">\(\abs{i - j}^{-\alpha} \rightarrow \abs{f(i - j)}^{-\alpha}\)</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"><sup><a href="#ref-fukuiOrderNClusterMonte2009" role="doc-biblioref">63</a></sup></span>. We only consider even system sizes given that odd system sizes are not commensurate with a <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> state.</p> <h1 id="the-phase-diagram">The Phase Diagram</h1> <p>Figs. [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>a] and [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>b] show the phase diagram for constant <span class="math inline">\(U=5\)</span> and constant <span class="math inline">\(J=5\)</span>, respectively. We determined the transition temperatures from the crossings of the Binder cumulants <span class="math inline">\(B_4 = \tex{m^4}/\tex{m^2}^2\)</span> <span class="citation" data-cites="binderFiniteSizeScaling1981"><sup><a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">24</a></sup></span>. For a representative set of parameters, Fig. [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>c] shows the order parameter <span class="math inline">\(\tex{m}^2\)</span>. Fig. [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>d] shows the Binder cumulants, both as functions of system size and temperature. The crossings confirm that the system has a <span data-acronym-label="FTPT" data-acronym-form="singular+short">FTPT</span> and that the ordered phase is not a finite size effect.</p> <p>The CDW transition temperature is largely independent from the strength of the interaction <span class="math inline">\(U\)</span>. This demonstrates that the phase transition is driven by the long-range term <span class="math inline">\(J\)</span> with little effect from the coupling to the fermions <span class="math inline">\(U\)</span>. The physics of the spin sector in our long-range <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model mimics that of the <span data-acronym-label="LRI" data-acronym-form="singular+short">LRI</span> model and is not significantly altered by the presence of the fermions, which shows that the long range tail expected from a basic fermion mediated RKKY interaction between the Ising spins is absent.</p> <p>Our main interest concerns the additional structure of the fermionic sector in the high temperature phase. Following Ref. <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"><sup><a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">49</a></sup></span>, we can distinguish between the Mott and Anderson insulating phases. The former is characterised by a gapped <span data-acronym-label="DOS" data-acronym-form="singular+short">DOS</span> in the absence of a <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span>. Thus, the opening of a gap for large <span class="math inline">\(U\)</span> is distinct from the gap-opening induced by the translational symmetry breaking in the CDW state below <span class="math inline">\(T_c\)</span>, see also Fig. [<a href="#fig:band_opening" data-reference-type="ref" data-reference="fig:band_opening">3</a>a]. The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates.</p> <h1 id="markov-chain-monte-carlo-and-emergent-disorder">Markov Chain Monte Carlo and Emergent Disorder</h1> <p>The results for the phase diagram were obtained with a classical <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> method which we discuss in the following. It allows us to solve our long-range <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> 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>. <span class="math display">\[\begin{aligned} H_s& = - \frac{U}{2}S_i + \sum_{i, j}^{N} J_{ij} S_i S_j \\ H_c& = \sum_i U S_i c^\dag_{i}c_{i} -t(c^\dag_{i}c_{i+1} + c^\dag_{i+1}c_{i}) \end{aligned}\]</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">\[\begin{aligned} \Z = \Tr e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s} \Tr_c e^{-\beta H_c} .\end{aligned}\]</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">\[\begin{aligned} \Z = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}\end{aligned}\]</span></p> <p><span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> 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">\(p(\vec{S}) = \Z^{-1} e^{-\beta E}\)</span> <span class="citation" data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"><sup><a href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">64</a>–<a href="#ref-wolffMonteCarloErrors2004" role="doc-biblioref">66</a></sup></span>. Hence, any observable can be estimated as a mean over the states visited by the walk. <span class="math display">\[\begin{aligned} \label{eq:thermal_expectation} \tex{O}& = \sum_{\vec{S}} p(\vec{S}) \tex{O}_{\vec{S}} = \sum_{i = 0}^{M} \tex{O}_{\vec{S}_i} + \mathcal{O}(\tfrac{1}{\sqrt{M}})\\ \tex{O}_{\vec{S}}& = \sum_{\nu} n_F(\epsilon_{\nu}) \expval{O}{\nu}\end{aligned}\]</span> 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 <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> 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"><sup><a href="#ref-kellyReversibilityStochasticNetworks1981" role="doc-biblioref">67</a></sup></span>. Here, we incorporate a modification to the standard Metropolis-Hastings algorithm <span class="citation" data-cites="hastingsMonteCarloSampling1970"><sup><a href="#ref-hastingsMonteCarloSampling1970" role="doc-biblioref">68</a></sup></span> gleaned from Krauth <span class="citation" data-cites="krauthIntroductionMonteCarlo1998"><sup><a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">32</a></sup></span>. Let us first recall the standard algorithm which decomposes the transition probability into <span class="math inline">\(\T(a \to b) = \p(a \to b)\A(a \to b)\)</span>. Here, <span class="math inline">\(\p\)</span> is the proposal distribution that we can directly sample from while <span class="math inline">\(\A\)</span> is the acceptance probability. The standard Metropolis-Hastings choice is <span class="math display">\[\A(a \to b) = \min\left(1, \frac{\p(b\to a)}{\p(a\to b)} e^{-\beta \Delta E}\right)\;,\]</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">\(\A(a \to b)\)</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">\(\A(a \to b)\)</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">\(\A\)</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"><sup><a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">69</a></sup></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">\[\A(a \to b) = \min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]</span></p> <p>See Appendix <a href="#app:balance" data-reference-type="ref" data-reference="app:balance">[app:balance]</a> for a proof that this satisfies the detailed balance condition.</p> <p>For the model parameters used in Fig. <a href="#fig:indiv_IPR" data-reference-type="ref" data-reference="fig:indiv_IPR">2</a>, we find 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 <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> 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 <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> 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 <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model <span class="citation" data-cites="huangAcceleratedMonteCarlo2017"><sup><a href="#ref-huangAcceleratedMonteCarlo2017" role="doc-biblioref">35</a></sup></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> <h1 id="localisation-properties">Localisation Properties</h1> <p>The <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> 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"><sup><a href="#ref-abrahamsScalingTheoryLocalization1979" role="doc-biblioref">56</a></sup></span>. An Anderson localised state centered around <span class="math inline">\(r_0\)</span> has magnitude that drops exponentially over some localisation length <span class="math inline">\(\xi\)</span> i.e <span class="math inline">\(|\psi(r)|^2 \sim \exp{-\abs{r - r_0}/\xi}\)</span>. Calculating <span class="math inline">\(\xi\)</span> directly is numerically demanding. Therefore, we determine if a given state is localised via the energy-resolved <span data-acronym-label="IPR" data-acronym-form="singular+short">IPR</span> and the <span data-acronym-label="DOS" data-acronym-form="singular+short">DOS</span> defined as <span class="math display">\[\begin{aligned} \mathrm{DOS}(\vec{S}, \omega)& = N^{-1} \sum_{i} \delta(\epsilon_i - \omega)\\ \mathrm{IPR}(\vec{S}, \omega)& = \; N^{-1} \mathrm{DOS}(\vec{S}, \omega)^{-1} \sum_{i,j} \delta(\epsilon_i - \omega)\;\psi^{4}_{i,j}\end{aligned}\]</span> where <span class="math inline">\(\epsilon_i\)</span> and <span class="math inline">\(\psi_{i,j}\)</span> are the <span class="math inline">\(i\)</span>th energy level and <span class="math inline">\(j\)</span>th element of the corresponding eigenfunction, both dependent on the background spin configuration <span class="math inline">\(\vec{S}\)</span>.</p> <p>The scaling of the IPR with system size <span class="math display">\[\mathrm{IPR} \propto N^{-\tau}\]</span> depends on the localisation properties of states at that energy. For delocalised states, e.g. Bloch waves, <span class="math inline">\(\tau\)</span> is the physical dimension. For fully localised states <span class="math inline">\(\tau\)</span> goes to zero in the thermodynamic limit. However, for special types of disorder such as binary disorder, the localisation lengths can be large comparable to the system size at hand, which can make it difficult to extract the correct scaling. An additional complication arises from the fact that the scaling exponent may display intermediate behaviours for correlated disorder and in the vicinity of a localisation-delocalisation transition <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008a"><sup><a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">28</a>,<a href="#ref-eversAndersonTransitions2008a" role="doc-biblioref">70</a></sup></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> <div id="fig:indiv_IPR" class="fignos"> <figure> <img src="pdf_figs/indiv_IPR.svg" alt="Figure 3: Energy resolved DOS(\omega) and \tau (the scaling exponent of IPR(\omega) against system size N). The left column shows the Anderson phase U = 2 at high T = 2.5 and the CDW phase at low T = 1.5 temperature. IPRs are evaluated for one of the in-gap states \omega_0/U = 0.057 and the center of the band \omega_1 U = 0.81. The right column shows instead the Mott and CDW phases at U = 5 with \omega_0/U = 0.24 and \omega_1/U = 0.571. For all the plots J = 5,\;\alpha = 1.25 and the fits for \tau use system sizes greater than 60. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalisation of the states." /> <figcaption aria-hidden="true"><span>Figure 3:</span> Energy resolved DOS(<span class="math inline">\(\omega\)</span>) and <span class="math inline">\(\tau\)</span> (the scaling exponent of IPR(<span class="math inline">\(\omega\)</span>) against system size <span class="math inline">\(N\)</span>). The left column shows the Anderson phase <span class="math inline">\(U = 2\)</span> at high <span class="math inline">\(T = 2.5\)</span> and the CDW phase at low <span class="math inline">\(T = 1.5\)</span> temperature. IPRs are evaluated for one of the in-gap states <span class="math inline">\(\omega_0/U = 0.057\)</span> and the center of the band <span class="math inline">\(\omega_1\)</span> <span class="math inline">\(U = 0.81\)</span>. The right column shows instead the Mott and CDW phases at <span class="math inline">\(U = 5\)</span> with <span class="math inline">\(\omega_0/U = 0.24\)</span> and <span class="math inline">\(\omega_1/U = 0.571\)</span>. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span> and the fits for <span class="math inline">\(\tau\)</span> use system sizes greater than 60. The measured <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\((0.06\pm0.01, 0.02\pm0.01\)</span> (b) <span class="math inline">\(0.04\pm0.02, 0.00\pm0.01\)</span> (c) <span class="math inline">\(0.05\pm0.03, 0.30\pm0.03\)</span> (d) <span class="math inline">\(0.06\pm0.04, 0.15\pm0.05\)</span> We show later that the apparent scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalisation of the states.</figcaption> </figure> </div> <div id="fig:band_opening" class="fignos"> <figure> <img src="pdf_figs/gap_openingboth.svg" alt="Figure 4: The DOS (a and c) and scaling exponent \tau (b and d) as a function of energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at U=2 while (c) and (d) show the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown a white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. U = 5,\;J = 5,\;\alpha = 1.25" /> <figcaption aria-hidden="true"><span>Figure 4:</span> The <span data-acronym-label="DOS" data-acronym-form="singular+long">DOS</span> (a and c) and scaling exponent <span class="math inline">\(\tau\)</span> (b and d) as a function of energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at <span class="math inline">\(U=2\)</span> while (c) and (d) show the CDW to gapped Mott phase transition at <span class="math inline">\(U=5\)</span>. Regions where the DOS is close to zero are shown a white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span></figcaption> </figure> </div> <div id="fig:indiv_IPR_disorder" class="fignos"> <figure> <img src="pdf_figs/indiv_IPR_disorder.svg" alt="Figure 5: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation 49, 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"><span>Figure 5:</span> A comparison of the full <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 < \rho < 1\)</span> matched to the largest corresponding FK model. As in Fig <a href="#fig:indiv_IPR" data-reference-type="ref" data-reference="fig:indiv_IPR">2</a>, the Energy resolved DOS(<span class="math inline">\(\omega\)</span>) and <span class="math inline">\(\tau\)</span> are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"><sup><a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">49</a></sup></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> </div> <p>Fig. <a href="#fig:indiv_IPR" data-reference-type="ref" data-reference="fig:indiv_IPR">2</a> shows the <span data-acronym-label="DOS" data-acronym-form="singular+short">DOS</span> and <span class="math inline">\(\tau\)</span>, the scaling exponent of the IPR with system size, for a representative set of parameters covering all three phases. The DOS is symmetric about <span class="math inline">\(0\)</span> because of the particle hole symmetry of the model. At high temperatures, all of the eigenstates are localised in both the Mott and Anderson phases (with <span class="math inline">\(\tau \leq 0.07\)</span> for our system sizes). We also checked that the states are localised by direct inspection. Note that there are in-gap states for instance at <span class="math inline">\(\omega_0\)</span>, below the upper band which are localized and smoothly connected across the phase transition.</p> <p>In the CDW phases at <span class="math inline">\(U=2\)</span> and <span class="math inline">\(U=5\)</span>, we find for the states within the gapped CDW bands, e.g. at <span class="math inline">\(\omega_1\)</span>, scaling exponents <span class="math inline">\(\tau = 0.30\pm0.03\)</span> and <span class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This surprising finding suggests that the CDW bands are partially delocalised with multi-fractal behaviour of the wavefunctions <span class="citation" data-cites="eversAndersonTransitions2008a"><sup><a href="#ref-eversAndersonTransitions2008a" role="doc-biblioref">70</a></sup></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"><sup><a href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">6</a>,<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">29</a></sup></span>. However, we later show by comparison to an uncorrelated Anderson model that these nonzero exponents are a finite size effect and the states are localised with a finite <span class="math inline">\(\xi\)</span> similar to the system size. As a result, the IPR does not scale correctly until the system size has grown much larger than <span class="math inline">\(\xi\)</span>. Fig. [<a href="#fig:indiv_IPR_disorder" data-reference-type="ref" data-reference="fig:indiv_IPR_disorder">4</a>] shows that the scaling of the IPR in the CDW phase does flatten out eventually.</p> <p>Next, we use the <span data-acronym-label="DOS" data-acronym-form="singular+short">DOS</span> and the scaling exponent <span class="math inline">\(\tau\)</span> to explore the localisation properties over the energy-temperature plane in Fig. <a href="#fig:band_opening" data-reference-type="ref" data-reference="fig:band_opening">3</a>. Gapped areas are shown in white, which highlights the distinction between the gapped Mott phase and the ungapped Anderson phase. In-gap states appear just below the critical point, smoothly filling the bandgap in the Anderson phase and forming islands in the Mott phase. As in the finite <span class="citation" data-cites="zondaGaplessRegimeCharge2019"><sup><a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">71</a></sup></span> and infinite dimensional <span class="citation" data-cites="hassanSpectralPropertiesChargedensitywave2007"><sup><a href="#ref-hassanSpectralPropertiesChargedensitywave2007" role="doc-biblioref">72</a></sup></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"><sup><a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">71</a></sup></span>.</p> <p>In order to understand the localization properties we can compare the behaviour of our model with that of a simpler Anderson disorder model (DM) in which the spins are replaced by a <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> background with uncorrelated binary defect potentials, see Appendix <a href="#app:disorder_model" data-reference-type="ref" data-reference="app:disorder_model">[app:disorder_model]</a>. Fig. [<a href="#fig:indiv_IPR_disorder" data-reference-type="ref" data-reference="fig:indiv_IPR_disorder">4</a>] compares the FK model to the disorder model at different system sizes, matching the defect densities of the disorder model to the FK model at <span class="math inline">\(N = 270\)</span> above and below the CDW transition. We find very good, even quantitative, agreement between the FK and disorder models, which suggests that correlations in the spin sector do not play a significant role. As we can sample directly from the disorder model, rather than through MCMC, the samples are uncorrelated. Hence we can evaluate much larger system sizes with the disorder model which enables us to pin down the correct localisation effects. In particular, what appear to be delocalized states for small system sizes eventually turn out to be states with large localization length. The localization length diverges towards the ordered zero temperature CDW state. Overall, we see that the interplay of interactions, here manifest as a peculiar binary potential, and localization can be very intricate and the added advantage of our 1D model is that we can explore very large system sizes for a complete understanding.</p> <h1 id="discussion-conclusion">Discussion & Conclusion</h1> <p>The <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model is one of the simplest non-trivial models of interacting fermions. We studied its thermodynamic and localisation properties brought down in dimensionality to 1D by adding a novel long-ranged coupling designed to stabilise the <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> phase present in dimension two and above. Our hybrid <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> approach elucidates a disorder-free localization mechanism within our translationally invariant system. Further, we demonstrate a significant speedup over the naive method. We show that our long-range <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> in 1D retains much of the rich phase diagram of its higher dimensional cousins. Careful scaling analysis indicates that all the single particle eigenstates eventually localise at nonzero temperature albeit only for very large system sizes of several thousand.</p> <p>Our work raises a number of interesting questions for future research. A straightforward but numerically challenging problem is to pin down the model’s behaviour closer to the critical point where correlations in the spin sector would become significant. Would this modify the localisation behaviour? Similar to other soluble models of disorder-free localisation, we expect intriguing out-of equilibrium physics, for example slow entanglement dynamics akin to more generic interacting systems <span class="citation" data-cites="hartLogarithmicEntanglementGrowth2020"><sup><a href="#ref-hartLogarithmicEntanglementGrowth2020" role="doc-biblioref">73</a></sup></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"><sup><a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">42</a></sup></span> such as the devil’s staircase <span class="citation" data-cites="michelettiCompleteDevilTextquotesingles1997"><sup><a href="#ref-michelettiCompleteDevilTextquotesingles1997" role="doc-biblioref">74</a></sup></span> could be stabilised at finite temperature. In a broader context, we envisage that long-range interactions can also be used to gain a deeper understanding of the temperature evolution of topological phases. One example would be a long-ranged <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> 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> <h1 id="acknowledgments">Acknowledgments</h1> <p>We wish to acknowledge the support of Alexander Belcik who was involved with the initial stages of the project. We thank Angus MacKinnon for helpful discussions, Sophie Nadel for input when preparing the figures and acknowledge support from the Imperial-TUM flagship partnership. This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) <a href="https://gtr.ukri.org/project/145404DD-ABAD-4CFB-A2D8-152A6AFCCEB7#/tabOverview">Project No. 2120140</a>.</p> <h1 id="detailed-balance"><span id="app:balance" label="app:balance"></span> DETAILED BALANCE</h1> <p>Given a <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> 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"><sup><a href="#ref-wolffMonteCarloErrors2004" role="doc-biblioref">66</a></sup></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="cb2" data-language="Python"><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></span> <span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a> c_dE <span class="op">=</span> classical_energy_change(</span> <span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a> current_state,</span> <span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a> new_state)</span> <span id="cb2-9"><a href="#cb2-9" 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="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a> q_dF <span class="op">=</span> quantum_free_energy_change(</span> <span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a> current_state,</span> <span id="cb2-12"><a href="#cb2-12" aria-hidden="true" tabindex="-1"></a> new_state)</span> <span id="cb2-13"><a href="#cb2-13" 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="cb2-14"><a href="#cb2-14" aria-hidden="true" tabindex="-1"></a> current_state <span class="op">=</span> new_state</span> <span id="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a></span> <span id="cb2-16"><a href="#cb2-16" 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> <h1 id="uncorrelated-disorder-model"><span id="app:disorder_model" label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</h1> <p>The disorder model referred to in the main text is defined by replacing the spin degree of freedom in the FK model <span class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder potential <span class="math inline">\(d_i = \pm \tfrac{1}{2}\)</span> controlled by a defect density <span class="math inline">\(\rho\)</span> such that <span class="math inline">\(d_i = -\tfrac{1}{2}\)</span> with probability <span class="math inline">\(\rho/2\)</span> and <span class="math inline">\(d_i = \tfrac{1}{2}\)</span> otherwise. <span class="math inline">\(\rho/2\)</span> is used rather than <span class="math inline">\(\rho\)</span> so that the disorder potential takes on the zero temperature CDW ground state at <span class="math inline">\(\rho = 0\)</span> and becomes a random choice over spin states at <span class="math inline">\(\rho = 1\)</span> i.e the infinite temperature limit. <span class="math display">\[\begin{aligned} H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} - \tfrac{1}{2}) \\ & -\;t \sum_{i} c^\dag_{i}c_{i+1} + c^\dag_{i+1}c_{i} \nonumber\end{aligned}\]</span></p> <div class="sourceCode" id="cb3"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div> <p></ij></ij></ij></ij></ij></ij></ij></p> <div id="refs" class="references csl-bib-body" data-line-spacing="2" 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">Hodson, T., Wilsher, J. & Knolle, J. 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