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---
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title: 2.1_FK_Model
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<title>2.1_FK_Model</title>
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<ul>
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<li><a href="#the-falikov-kimball-model"
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id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
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<ul>
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<li><a href="#the-model" id="toc-the-model">The Model</a></li>
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<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
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Diagrams</a></li>
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<li><a href="#long-ranged-ising-model"
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id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
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</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<ul>
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<li><a href="#the-falikov-kimball-model"
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id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
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<ul>
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<li><a href="#the-model" id="toc-the-model">The Model</a></li>
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<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
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Diagrams</a></li>
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<li><a href="#long-ranged-ising-model"
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id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
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</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<section id="the-falikov-kimball-model" class="level1">
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<h1>The Falikov Kimball Model</h1>
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<section id="the-model" class="level2">
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<h2>The Model</h2>
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<p>The Falikov-Kimball (FK) model is one of the simplest models of the
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correlated electron problem. It captures the essence of the interaction
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between itinerant and localized electrons. It was originally introduced
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to explain the metal-insulator transition in f-electron systems but in
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its long history it has been interpreted variously as a model of
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electrons and ions, binary alloys or of crystal formation <span
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class="citation"
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data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicovSimpleModelSemiconductorMetal1969 gruberFalicovKimballModelReview1996 gruberFalicovKimballModel2006"> [<a
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href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
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role="doc-biblioref">1</a>–<a href="#ref-gruberFalicovKimballModel2006"
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role="doc-biblioref">4</a>]</span>. In terms of immobile fermions <span
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class="math inline">\(d_i\)</span> and light fermions <span
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class="math inline">\(c_i\)</span> and with chemical potential fixed at
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half-filling, the model reads</p>
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<p><span class="math display">\[\begin{aligned}
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H_{\mathrm{FK}} = & \;U \sum_{i} (d^\dagger_{i}d_{i} -
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\tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle
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i,j\rangle} c^\dagger_{i}c_{j}.\\
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\end{aligned}\]</span></p>
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<p>The connection to the Hubbard model is that we have relabel the up
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and down spin electron states and removed the hopping term for one
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species, the equivalent of taking the limit of infinite mass ratio <span
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class="citation" data-cites="devriesSimplifiedHubbardModel1993"> [<a
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href="#ref-devriesSimplifiedHubbardModel1993"
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role="doc-biblioref">5</a>]</span>.</p>
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<p>Like other exactly solvable models <span class="citation"
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data-cites="smithDisorderFreeLocalization2017"> [<a
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href="#ref-smithDisorderFreeLocalization2017"
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role="doc-biblioref">6</a>]</span> and the Kitaev Model, the FK model
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possesses extensively many conserved degrees of freedom <span
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class="math inline">\([d^\dagger_{i}d_{i}, H] = 0\)</span>. The Hilbert
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space therefore breaks up into a set of sectors in which these operators
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take a definite value. Crucially, this reduces the interaction term
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<span class="math inline">\((d^\dagger_{i}d_{i} -
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\tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\)</span> from being
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quartic in fermion operators to quadratic. This is what makes the FK
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model exactly solvable, in contrast to the Hubbard model.</p>
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<p>Due to Pauli exclusion, maximum filling occurs when each lattice site
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is fully occupied, <span class="math inline">\(\langle n_c + n_d \rangle
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= 2\)</span>. Here we will focus on the half filled case <span
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class="math inline">\(\langle n_c + n_d \rangle = 1\)</span>. Doping the
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model away from the half-filled point leads to rich physics including
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superconductivity <span class="citation"
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data-cites="jedrzejewskiFalicovKimballModels2001"> [<a
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href="#ref-jedrzejewskiFalicovKimballModels2001"
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role="doc-biblioref">7</a>]</span>.</p>
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<p>At half-filling and on bipartite lattices, FK the model is
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particle-hole symmetric. That is, the Hamiltonian anticommutes with the
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particle hole operator <span
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class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a
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consequence the energy spectrum is symmetric about <span
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class="math inline">\(E = 0\)</span> and this is the Fermi energy. The
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particle hole operator corresponds to the substitution <span
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class="math inline">\(c^\dagger_i \rightarrow \epsilon_i c_i,
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d^\dagger_i \rightarrow d_i\)</span> where <span
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class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and
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<span class="math inline">\(-1\)</span> for the even sublattice <span
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class="citation" data-cites="gruberFalicovKimballModel2005"> [<a
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href="#ref-gruberFalicovKimballModel2005"
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role="doc-biblioref">8</a>]</span>. The absence of a hopping term for
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the heavy electrons means they do not need the factor of <span
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class="math inline">\(\epsilon_i\)</span>.</p>
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<div id="fig:simple_DOS" class="fignos">
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<figure>
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<img src="/assets/thesis/background_chapter/simple_DOS.svg"
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data-short-caption="Cubic Lattice dispersion with disorder"
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style="width:100.0%"
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alt="Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in one dimension. (a) With not external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder." />
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<figcaption aria-hidden="true"><span>Figure 1:</span> The dispersion
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(upper row) and density of states (lower row) obtained from a cubic
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lattice model <span class="math inline">\(H = \sum_{i} V_i
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c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle}
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c^\dagger_{i}c_{j}\)</span> in one dimension. (a) With not external
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potential. (b) With a static charge density wave background <span
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class="math inline">\(V_i = (-1)^i\)</span> (c) A static charge density
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wave background with 2% binary disorder.</figcaption>
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</figure>
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</div>
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<p>We will later add a long range interaction between the localised
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electrons so we will replace the immobile fermions with a classical
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Ising field <span class="math inline">\(S_i = 1 - 2d^\dagger_id_i =
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\pm\tfrac{1}{2}\)</span>.</p>
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<p><span class="math display">\[\begin{aligned}
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H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
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\tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
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\end{aligned}\]</span></p>
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<p>The FK model can be solved exaclty with dynamic mean field theory in
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the infinite dimensional <span class="citation"
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data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a
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href="#ref-antipovCriticalExponentsStrongly2014"
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role="doc-biblioref">9</a>–<a
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href="#ref-herrmannNonequilibriumDynamicalCluster2016"
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role="doc-biblioref">12</a>]</span>.</p>
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<ul>
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<li>displays disorder free localisation</li>
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</ul>
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</section>
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<section id="phase-diagrams" class="level2">
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<h2>Phase Diagrams</h2>
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<div id="fig:fk_phase_diagram" class="fignos">
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<figure>
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<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg"
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data-short-caption="Fermi-Hubbard and Falikov-Kimball Temperatue-Interaction Phase Diagrams"
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style="width:100.0%"
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alt="Figure 2: Schematic Phase diagrams of the Fermi-Hubbard (left) and Falikov-Kimball model (right) showing temperature (T) and repulsive interaction strength (U). Hubbard model diagram adapted from [13], Falikov-Kimball model from [14,15]" />
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<figcaption aria-hidden="true"><span>Figure 2:</span> Schematic Phase
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diagrams of the Fermi-Hubbard (left) and Falikov-Kimball model (right)
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showing temperature (T) and repulsive interaction strength (U). Hubbard
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model diagram adapted from <span class="citation"
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data-cites="micnasSuperconductivityNarrowbandSystems1990"> [<a
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href="#ref-micnasSuperconductivityNarrowbandSystems1990"
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role="doc-biblioref">13</a>]</span>, Falikov-Kimball model from <span
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class="citation"
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data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014a"> [<a
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href="#ref-antipovInteractionTunedAndersonMott2016"
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role="doc-biblioref">14</a>,<a
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href="#ref-antipovCriticalExponentsStrongly2014a"
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role="doc-biblioref">15</a>]</span></figcaption>
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</figure>
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</div>
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<ul>
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<li>rich phase diagram in 2d Despite its simplicity, the FK model has a
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rich phase diagram in <span class="math inline">\(D \geq 2\)</span>
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dimensions. For example, it shows an interaction-induced gap opening
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even at high temperatures, similar to the corresponding Hubbard
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Model <span class="citation"
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data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
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href="#ref-brandtThermodynamicsCorrelationFunctions1989"
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role="doc-biblioref">16</a>]</span>.</li>
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</ul>
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<p>At half filling and in dimensions greater than one, the FK model
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exhibits a phase transition at some <span
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class="math inline">\(U\)</span> dependent critical temperature <span
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class="math inline">\(T_c(U)\)</span> to a low temperature charge
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density wave state in which the spins order antiferromagnetically. This
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corresponds to the heavy electrons occupying one of the two sublattices
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A and B <span class="citation"
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data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
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href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
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role="doc-biblioref">17</a>]</span>. In the disordered region above
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<span class="math inline">\(T_c(U)\)</span> there is a transition
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between an Anderson insulator phase at weak interaction and a Mott
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insulator phase in the strongly interacting regime <span
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class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a
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href="#ref-andersonAbsenceDiffusionCertain1958"
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role="doc-biblioref">18</a>]</span>.</p>
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<ul>
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<li>superconductivity when doped</li>
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</ul>
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<p>In 1D, the ground state phenomenology as the model is doped away from
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the half-filled state can be rich <span class="citation"
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data-cites="gruberGroundStatesSpinless1990"> [<a
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href="#ref-gruberGroundStatesSpinless1990"
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role="doc-biblioref">19</a>]</span> but the system is disordered for all
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<span class="math inline">\(T > 0\)</span> <span class="citation"
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data-cites="kennedyItinerantElectronModel1986"> [<a
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href="#ref-kennedyItinerantElectronModel1986"
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role="doc-biblioref">20</a>]</span>.</p>
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<p>In the one dimensional FK model there is no ordered CDW phase <span
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class="citation" data-cites="liebAbsenceMottTransition1968"> [<a
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href="#ref-liebAbsenceMottTransition1968"
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role="doc-biblioref">21</a>]</span>. The supression of phase transition
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is a common phenomena in one dimensional systems. It can be understood
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via Peierls’ argument <span class="citation"
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data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a
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href="#ref-kennedyItinerantElectronModel1986"
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role="doc-biblioref">20</a>,<a
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href="#ref-peierlsIsingModelFerromagnetism1936"
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role="doc-biblioref">22</a>]</span> to be a consequence of the low
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energy penalty for domain walls in one dimensional systems.</p>
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<p>Following Peierls’ argument, consider the difference in free energy
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<span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span>
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between an ordered state and a state with single domain wall in a
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discrete order parameter. Short range interactions produce a constant
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energy penalty for such a domain wall that does not scale with system
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size. In contrast, the number of such single domain wall states scales
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linearly so the entropy is <span class="math inline">\(\propto \ln
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L\)</span>. Thus the entropic contribution dominates (eventually) in the
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thermodynamic limit and no finite temperature order is possible. In two
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dimensions and above, the energy penalty of a domain wall scales like
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<span class="math inline">\(L^{d-1}\)</span> so they can support ordered
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phases.</p>
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</section>
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<section id="long-ranged-ising-model" class="level2">
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<h2>Long Ranged Ising model</h2>
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<p>Our extension to the FK model could now be though of as spinless
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fermions coupled to a long range Ising (LRI) model. The LRI model has
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been extensively studied and its behaviour may be bear relation to the
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behaviour of our modified FK model.</p>
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<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|)
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\tau_i \tau_j = J \sum_{i\neq j} |i - j|^{-\alpha} \tau_i
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\tau_j\]</span></p>
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<p>Renormalisation group analyses show that the LRI model has an ordered
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phase in 1D for $1 < < 2 $ <span class="citation"
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data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
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||
href="#ref-dysonExistencePhasetransitionOnedimensional1969"
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role="doc-biblioref">23</a>]</span>. Peierls’ argument can be
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extended <span class="citation"
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data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
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href="#ref-thoulessLongRangeOrderOneDimensional1969"
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role="doc-biblioref">24</a>]</span> to long range interactions to
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provide intuition for why this is the case. Again considering the energy
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difference between the ordered state <span
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class="math inline">\(\ket{\ldots\uparrow\uparrow\uparrow\uparrow\ldots}\)</span>
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and a domain wall state <span
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class="math inline">\(\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}\)</span>.
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In the case of the LRI model, careful counting shows that this energy
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penalty is: <span class="math display">\[\Delta E \propto
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\sum_{n=1}^{\infty} n J(n)\]</span></p>
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<p>because each interaction between spins separated across the domain by
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a bond length <span class="math inline">\(n\)</span> can be drawn
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between <span class="math inline">\(n\)</span> equivalent pairs of
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sites. Ruelle proved rigorously for a very general class of 1D systems,
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||
that if <span class="math inline">\(\Delta E\)</span> or its many-body
|
||
generalisation converges in the thermodynamic limit then the free energy
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||
is analytic <span class="citation"
|
||
data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a
|
||
href="#ref-ruelleStatisticalMechanicsOnedimensional1968"
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||
role="doc-biblioref">25</a>]</span>. This rules out a finite order phase
|
||
transition, though not one of the Kosterlitz-Thouless type. Dyson also
|
||
proves this though with a slightly different condition on <span
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||
class="math inline">\(J(n)\)</span> <span class="citation"
|
||
data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
|
||
href="#ref-dysonExistencePhasetransitionOnedimensional1969"
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||
role="doc-biblioref">23</a>]</span>.</p>
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||
<p>With a power law form for <span class="math inline">\(J(n)\)</span>,
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||
there are three cases to consider:</p>
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||
<ol type="1">
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||
<li>$ = 0$ For infinite range interactions the Ising model is exactly
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||
solveable and mean field theory is exact <span class="citation"
|
||
data-cites="lipkinValidityManybodyApproximation1965"> [<a
|
||
href="#ref-lipkinValidityManybodyApproximation1965"
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||
role="doc-biblioref">26</a>]</span>.</li>
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||
<li>$ $ For slowly decaying interactions <span
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||
class="math inline">\(\sum_n J(n)\)</span> does not converge so the
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||
Hamiltonian is non-extensive, a case which won’t be further considered
|
||
here.</li>
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||
<li>$ 1 < < 2 $ A phase transition to an ordered state at a finite
|
||
temperature.</li>
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||
<li>$ = 2 $ The energy of domain walls diverges logarithmically, and
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||
this turns out to be a Kostelitz-Thouless transition <span
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||
class="citation"
|
||
data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
|
||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||
role="doc-biblioref">24</a>]</span>.</li>
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||
<li>$ 2 < $ For quickly decaying interactions, domain walls have a
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||
finite energy penalty, hence Peirels’ argument holds and there is no
|
||
phase transition.</li>
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||
</ol>
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||
<div id="fig:alpha_diagram" class="fignos">
|
||
<figure>
|
||
<img src="/assets/thesis/background_chapter/alpha_diagram.svg"
|
||
data-short-caption="Long Range Ising Model Behaviour"
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||
style="width:100.0%" alt="Figure 3: " />
|
||
<figcaption aria-hidden="true"><span>Figure 3:</span> </figcaption>
|
||
</figure>
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</div>
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<div class="sourceCode" id="cb1"><pre
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||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||
</section>
|
||
</section>
|
||
<section id="bibliography" class="level1 unnumbered">
|
||
<h1 class="unnumbered">Bibliography</h1>
|
||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||
<div id="ref-hubbardj.ElectronCorrelationsNarrow1963" class="csl-entry"
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||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[1] </div><div
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||
class="csl-right-inline">Hubbard, J., <em><a
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||
href="https://doi.org/10.1098/rspa.1963.0204">Electron Correlations in
|
||
Narrow Energy Bands</a></em>, Proceedings of the Royal Society of
|
||
London. Series A. Mathematical and Physical Sciences
|
||
<strong>276</strong>, 238 (1963).</div>
|
||
</div>
|
||
<div id="ref-falicovSimpleModelSemiconductorMetal1969" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">L.
|
||
M. Falicov and J. C. Kimball, <em><a
|
||
href="https://doi.org/10.1103/PhysRevLett.22.997">Simple Model for
|
||
Semiconductor-Metal Transitions: Sm${\mathrm{B}}_{6}$ and
|
||
Transition-Metal Oxides</a></em>, Phys. Rev. Lett. <strong>22</strong>,
|
||
997 (1969).</div>
|
||
</div>
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</main>
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</body>
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</html>
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