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title: Background - The Falicov Kimball Model
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<ul>
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<li><a href="#chap:2-background" id="toc-chap:2-background">2 Background</a></li>
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<li><a href="#the-falicov-kimball-model" id="toc-the-falicov-kimball-model">The Falicov Kimball Model</a>
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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 Diagrams</a></li>
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<li><a href="#long-ranged-ising-model" id="toc-long-ranged-ising-model">Long-Ranged Ising model</a></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<li><a href="#chap:2-background" id="toc-chap:2-background">2 Background</a></li>
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<li><a href="#the-falicov-kimball-model" id="toc-the-falicov-kimball-model">The Falicov Kimball Model</a>
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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 Diagrams</a></li>
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<li><a href="#long-ranged-ising-model" id="toc-long-ranged-ising-model">Long-Ranged Ising model</a></li>
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<p>2 Background</p>
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<section id="chap:2-background" class="level1">
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<h1>2 Background</h1>
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</section>
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<section id="the-falicov-kimball-model" class="level1">
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<h1>The Falicov 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 Falicov-Kimball (FK) model is one of the simplest models of the correlated electron problem. It captures the essence of the interaction between itinerant and localised electrons. 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"> [<a href="#ref-hubbardj.ElectronCorrelationsNarrow1963" role="doc-biblioref">1</a>–<a href="#ref-gruberFalicovKimballModel2006" role="doc-biblioref">4</a>]</span>. In terms of immobile fermions <span class="math inline">\(d_i\)</span> and light fermions <span class="math inline">\(c_i\)</span> and with chemical potential fixed at half-filling, the model reads</p>
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<p><span class="math display">\[\begin{aligned}
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H_{\mathrm{FK}} = & \;U \sum_{i} (d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
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\end{aligned}\]</span></p>
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<p>Here we will only discuss the hypercubic lattices, i.e the chain, the square lattice, the cubic lattice and so on. The connection to the Hubbard model is that we have relabelled the up and down spin electron states and removed the hopping term for one spin state, the equivalent of taking the limit of infinite mass ratio <span class="citation" data-cites="devriesSimplifiedHubbardModel1993"> [<a href="#ref-devriesSimplifiedHubbardModel1993" role="doc-biblioref">5</a>]</span>.</p>
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<p>Like other exactly solvable models <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">6</a>]</span>, the FK model possesses extensively many conserved degrees of freedom <span class="math inline">\([d^\dagger_{i}d_{i}, H] = 0\)</span>. Similarly, the Kitaev model contains an extensive number of conserved fluxes. So in both models, the Hilbert space breaks up into a set of sectors in which these operators take a definite value. Crucially, this reduces the interaction terms in the model from being quartic in fermion operators to quadratic. This is what makes the two models exactly solvable, in contrast to the Hubbard model. For the FK model the interaction term <span class="math inline">\((d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\)</span> becomes quadratic when <span class="math inline">\(d^\dagger_{i}d_{i}\)</span> is replaced with on of its eigenvalues <span class="math inline">\(\{0,1\}\)</span>. The same thing happens in the Kitaev model, though after first applying a clever transformation which we will discuss later.</p>
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<p>Due to Pauli exclusion, maximum filling occurs when each lattice site is fully occupied, <span class="math inline">\(\langle n_c + n_d \rangle = 2\)</span>. Here we will focus on the half filled case <span class="math inline">\(\langle n_c + n_d \rangle = 1\)</span>. The ground state phenomenology as the model is doped away from the half-filled state can be rich <span class="citation" data-cites="jedrzejewskiFalicovKimballModels2001 gruberGroundStatesSpinless1990"> [<a href="#ref-jedrzejewskiFalicovKimballModels2001" role="doc-biblioref">7</a>,<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">8</a>]</span> but the half-filled point has symmetries that make it particularly interesting. From this point on we will only consider the half-filled point.</p>
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<p>At half-filling and on bipartite lattices, the FK the model is particle-hole symmetric. That is, the Hamiltonian anticommutes with the particle hole operator <span class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a consequence, the energy spectrum is symmetric about <span class="math inline">\(E = 0\)</span>, which is the Fermi energy. The particle hole operator corresponds to the substitution <span class="math inline">\(c^\dagger_i \rightarrow \epsilon_i c_i, d^\dagger_i \rightarrow d_i\)</span> where <span class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and <span class="math inline">\(-1\)</span> for the B sublattice <span class="citation" data-cites="gruberFalicovKimballModel2005"> [<a href="#ref-gruberFalicovKimballModel2005" role="doc-biblioref">9</a>]</span>. The absence of a hopping term for the heavy electrons means they do not need the factor of <span class="math inline">\(\epsilon_i\)</span> but they would need it in the corresponding Hubbard model. See appendix <a href="../6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">A.1</a> for a full derivation of the particle-hole symmetry.</p>
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<figure>
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<img src="/assets/thesis/background_chapter/simple_DOS.svg" id="fig-simple_DOS" data-short-caption="Cubic Lattice dispersion with disorder" style="width:100.0%" alt="Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in 1D. (a) With no 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. The top rows shows the analytic dispersion in orange compared with the integral of the DOS in dotted black." />
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<figcaption aria-hidden="true">Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model <span class="math inline">\(H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}\)</span> in 1D. (a) With no external potential. (b) With a static charge density wave background <span class="math inline">\(V_i = (-1)^i\)</span> (c) A static charge density wave background with 2% binary disorder. The top rows shows the analytic dispersion in orange compared with the integral of the DOS in dotted black.</figcaption>
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</figure>
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<p>We will later add a long-range interaction between the localised electrons at which point we will replace the immobile fermions with a classical Ising field <span class="math inline">\(S_i = 1 - 2d^\dagger_id_i = \pm1\)</span> which I will refer to as the spins.</p>
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<p><span class="math display">\[\begin{aligned}
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H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
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\end{aligned}\]</span></p>
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<p>The FK model can be solved exactly with dynamic mean field theory in the infinite dimensional limit <span class="citation" data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">10</a>–<a href="#ref-herrmannNonequilibriumDynamicalCluster2016" role="doc-biblioref">13</a>]</span> but has radically different behaviour in lower dimensional systems.</p>
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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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<figure>
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<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg" id="fig-fk_phase_diagram" data-short-caption="Falicov-Kimball Temperatue-Interaction Phase Diagrams" style="width:100.0%" alt="Figure 2: Schematic Phase diagram of the Falicov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]" />
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<figcaption aria-hidden="true">Figure 2: Schematic Phase diagram of the Falicov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">10</a>,<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">14</a>]</span></figcaption>
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</figure>
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<p>In dimensions greater than one, the FK model exhibits a phase transition at some <span class="math inline">\(U\)</span> dependent critical temperature <span class="math inline">\(T_c(U)\)</span> to a low temperature ordered phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">15</a>]</span>. In terms of the heavy electrons this corresponds to them occupying only one of the two sublattices A and B, known as a Charge Density Wave (CDW) phase. In terms of spins this is an antiferromagnetic phase.</p>
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<p>In the disordered region above <span class="math inline">\(T_c(U)\)</span> there are two insulating phases. For weak interactions <span class="math inline">\(U << t\)</span>, thermal fluctuations in the spins act as an effective disorder potential for the fermions, causing them to localise and giving rise to an Anderson insulating (AI) phase <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">16</a>]</span> which we will discuss more in section <a href="../2_Background/2.4_Disorder.html#bg-disorder-and-localisation">2.3</a>. For strong interactions <span class="math inline">\(U >> t\)</span>, the spins are not ordered but nevertheless their interaction with the electrons opens a gap, leading to a Mott insulator analogous to that of the Hubbard model <span class="citation" data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a href="#ref-brandtThermodynamicsCorrelationFunctions1989" role="doc-biblioref">17</a>]</span>. The presence of an interaction driven phase like the Mott insulator in an exactly solvable model is part of what makes the FK model such an interesting system.</p>
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<p>By contrast, in the 1D FK model there is no Finite-Temperature Phase Transition (FTPT) to an ordered CDW phase <span class="citation" data-cites="liebAbsenceMottTransition1968"> [<a href="#ref-liebAbsenceMottTransition1968" role="doc-biblioref">18</a>]</span>. Indeed dimensionality is crucial for the physics of both localisation and FTPTs. In 1D, disorder generally dominates: even the weakest disorder exponentially localises <em>all</em> single particle eigenstates. In the 1D FK model this means the whole spectrum is localised at all finite temperatures <span class="citation" data-cites="goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">19</a>–<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">21</a>]</span>. Though at low temperatures the localisation length may be so large that the states appear extended in finite sized systems <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">14</a>]</span>. Only longer-range correlations of the disorder potential can potentially induce localisation-delocalisation transitions in 1D <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">22</a>–<a href="#ref-dunlapAbsenceLocalizationRandomdimer1990" role="doc-biblioref">24</a>]</span></p>
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<p>The absence of finite temperature ordered phases in 1D systems is a general feature. It can be understood as a consequence of the fact that domain walls are energetically cheap in 1D. Thermodynamically, short-range interactions just cannot overcome the entropy of thermal defects in 1D. However, the addition of longer range interactions can overcome this <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">25</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">26</a>]</span>.</p>
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<p>However, the absence of an FTPT in the short ranged FK chain is far from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the fermions <span class="citation" data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">27</a>–<a href="#ref-yosidaMagneticPropertiesCuMn1957" role="doc-biblioref">30</a>]</span> decays as <span class="math inline">\(r^{-1}\)</span> in 1D <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">31</a>]</span>. This could in principle induce the necessary long-range interactions for the classical Ising background to order at low temperatures <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">25</a>,<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">32</a>]</span>. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at <span class="math inline">\(T = 0\)</span> for the 1D FK model <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">26</a>]</span>.</p>
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<p>The 1D FK model has been studied numerically, perturbatively in interaction strength <span class="math inline">\(U\)</span> and in the continuum limit <span class="citation" data-cites="bursillOneDimensionalContinuum1994"> [<a href="#ref-bursillOneDimensionalContinuum1994" role="doc-biblioref">33</a>]</span>. The main results are that for attractive <span class="math inline">\(U > U_c\)</span> the system forms electron spin bound state ‘atoms’ which repel on another <span class="citation" data-cites="gruberGroundStateEnergyLowTemperature1993"> [<a href="#ref-gruberGroundStateEnergyLowTemperature1993" role="doc-biblioref">34</a>]</span> and that the ground state phase diagram has a has a fractal structure as a function of electron filling, a devil’s staircase <span class="citation" data-cites="freericksTwostateOnedimensionalSpinless1990 michelettiCompleteDevilStaircase1997"> [<a href="#ref-freericksTwostateOnedimensionalSpinless1990" role="doc-biblioref">35</a>,<a href="#ref-michelettiCompleteDevilStaircase1997" role="doc-biblioref">36</a>]</span>.</p>
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<p>Based on this primacy of dimensionality, we will go digging into the 1D case. In chapter <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3</a> we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long-Range Ising (LRI) model which is the subject of the next section.</p>
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</section>
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<section id="long-ranged-ising-model" class="level2">
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<h2>Long-Ranged Ising model</h2>
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<p>The suppression of phase transitions is a common phenomena in 1D systems and the Ising model serves as the canonical illustration of this. In terms of classical spins <span class="math inline">\(S_i = \pm 1\)</span> the standard Ising model reads</p>
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<p><span class="math display">\[H_{\mathrm{I}} = \sum_{\langle ij \rangle} S_i S_j\]</span></p>
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<p>Like the FK model, the Ising model shows an FTPT to an ordered state only in 2D and above. This can be understood via Peierls’ argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">25</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">26</a>]</span> to be a consequence of the low energy penalty for domain walls in 1D systems.</p>
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<figure>
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<img src="/assets/thesis/intro_chapter/ising_model_domain_wall.svg" id="fig-ising_model_domain_wall" data-short-caption="Domain walls in the long-range Ising Model" style="width:100.0%" alt="Figure 3: Domain walls in the 1D Ising model cost finite energy because they affect only one interaction while in the Long-Range Ising (LRI) model it depends on how the interactions decay with distance." />
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<figcaption aria-hidden="true">Figure 3: Domain walls in the 1D Ising model cost finite energy because they affect only one interaction while in the Long-Range Ising (LRI) model it depends on how the interactions decay with distance.</figcaption>
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</figure>
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<p>Following Peierls’ argument, consider the difference in free energy <span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span> between an ordered state and a state with single domain wall as in fig. <a href="#fig:ising_model_domain_wall">3</a>. If this value is negative it implies that the ordered state is unstable with respect to domain wall defects, and they will thus proliferate, destroying the ordered phase. If we consider the scaling of the two terms with system size <span class="math inline">\(L\)</span> we see that short range interactions produce a constant energy penalty <span class="math inline">\(\Delta E\)</span> for a domain wall. In contrast, the number of such single domain wall states scales linearly with system size so the entropy is <span class="math inline">\(\propto \ln L\)</span>. Thus the entropic contribution dominates (eventually) in the thermodynamic limit and no finite temperature order is possible. In 2D and above, the energy penalty of a large domain wall scales like <span class="math inline">\(L^{d-1}\)</span> which is why they can support ordered phases. This argument does not quite apply to the FK model because of the aforementioned RKKY interaction. Instead this argument will give us insight into how to recover an ordered phase in the 1D FK model.</p>
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<p>In contrast the LRI model <span class="math inline">\(H_{\mathrm{LRI}}\)</span> can have an FTPT in 1D.</p>
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<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|) S_i S_j = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\]</span></p>
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<p>Renormalisation group analyses show that the LRI model has an ordered phase in 1D for <span class="math inline">\(1 < \alpha < 2\)</span> <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">37</a>]</span>. Peierls’ argument can be extended <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">32</a>]</span> to long-range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state <span class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span> and a domain wall state <span class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\)</span>. In the case of the LRI model, careful counting shows that this energy penalty is <span id="eq:bg-dw-penalty"><span class="math display">\[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\qquad{(1)}\]</span></span></p>
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<p>because each interaction between spins separated across the domain by a bond length <span class="math inline">\(n\)</span> can be drawn between <span class="math inline">\(n\)</span> equivalent pairs of sites. The behaviour then depends crucially on how eq. <a href="#eq:bg-dw-penalty">1</a> scales with system size. Ruelle proved rigorously for a very general class of 1D systems that if <span class="math inline">\(\Delta E\)</span> or its many-body generalisation converges to a constant in the thermodynamic limit then the free energy is analytic <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">38</a>]</span>. This rules out a finite order phase transition, though not one of the Kosterlitz-Thouless type. Dyson also proves this though with a slightly different condition on <span class="math inline">\(J(n)\)</span> <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">37</a>]</span>.</p>
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<p>With a power law form for <span class="math inline">\(J(n)\)</span>, there are a few cases to consider:</p>
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<p>For <span class="math inline">\(\alpha = 0\)</span> i.e infinite range interactions, the Ising model is exactly solvable and mean field theory is exact <span class="citation" data-cites="lipkinValidityManybodyApproximation1965"> [<a href="#ref-lipkinValidityManybodyApproximation1965" role="doc-biblioref">39</a>]</span>. This limit is the same as the infinite dimensional limit.</p>
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<p>For <span class="math inline">\(\alpha \leq 1\)</span> we have very slowly decaying interactions. <span class="math inline">\(\Delta E\)</span> does not converge as a function of system size so the Hamiltonian is non-extensive, a topic not without some considerable controversy <span class="citation" data-cites="grossNonextensiveHamiltonianSystems2002 lutskoQuestioningValidityNonextensive2011 wangCommentNonextensiveHamiltonian2003"> [<a href="#ref-grossNonextensiveHamiltonianSystems2002" role="doc-biblioref">40</a>–<a href="#ref-wangCommentNonextensiveHamiltonian2003" role="doc-biblioref">42</a>]</span> that we will not consider further here.</p>
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<p>For <span class="math inline">\(1 < \alpha < 2\)</span>, we get a phase transition to an ordered state at a finite temperature, this is what we want!</p>
|
||
<p>For <span class="math inline">\(\alpha = 2\)</span>, the energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transition <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">32</a>]</span>.</p>
|
||
<p>Finally, for <span class="math inline">\(2 < \alpha\)</span> we have very quickly decaying interactions and domain walls again have a finite energy penalty, hence Peirels’ argument holds and there is no phase transition.</p>
|
||
<p>One final complexity is that for <span class="math inline">\(\tfrac{3}{2} < \alpha < 2\)</span> renormalisation group methods show that the critical point has non-universal critical exponents that depend on <span class="math inline">\(\alpha\)</span> <span class="citation" data-cites="fisherCriticalExponentsLongRange1972"> [<a href="#ref-fisherCriticalExponentsLongRange1972" role="doc-biblioref">43</a>]</span>. To avoid this potential confounding factors we will park ourselves at <span class="math inline">\(\alpha = 1.25\)</span> when we apply these ideas to the FK model.</p>
|
||
<p>Were we to extend this to arbitrary dimension <span class="math inline">\(d\)</span> we would find that thermodynamics properties generally both <span class="math inline">\(d\)</span> and <span class="math inline">\(\alpha\)</span>, long-range interactions can modify the ‘effective dimension’ of thermodynamic systems <span class="citation" data-cites="angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">44</a>]</span>.</p>
|
||
<figure>
|
||
<img src="/assets/thesis/background_chapter/alpha_diagram.svg" id="fig-alpha_diagram" data-short-caption="Long-Range Ising Model Behaviour" style="width:100.0%" alt="Figure 4: The thermodynamic behaviour of the long-range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied. In my simulations I stick to a value of \alpha = \tfrac{5}{4} the complexity of non-universal critical exponents." />
|
||
<figcaption aria-hidden="true">Figure 4: The thermodynamic behaviour of the long-range Ising model <span class="math inline">\(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span class="math inline">\(\alpha\)</span> is varied. In my simulations I stick to a value of <span class="math inline">\(\alpha = \tfrac{5}{4}\)</span> the complexity of non-universal critical exponents.</figcaption>
|
||
</figure>
|
||
<p>Next Section: <a href="../2_Background/2.2_HKM_Model.html">The Kitaev Honeycomb Model</a></p>
|
||
</section>
|
||
</section>
|
||
<section id="bibliography" class="level1 unnumbered">
|
||
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<div class="csl-left-margin">[44] </div><div class="csl-right-inline">M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E <strong>89</strong>, 062120 (2014).</div>
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