mirror of
https://github.com/TomHodson/tomhodson.github.com.git
synced 2025-06-26 10:01:18 +02:00
762 lines
39 KiB
HTML
762 lines
39 KiB
HTML
---
|
||
title: Background - The Falikov Kimball Model
|
||
excerpt:
|
||
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" />
|
||
<title>Background - The Falikov Kimball Model</title>
|
||
|
||
|
||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||
<script src="/assets/js/thesis_scrollspy.js"></script>
|
||
|
||
<link rel="stylesheet" href="/assets/css/styles.css">
|
||
<script src="/assets/js/index.js"></script>
|
||
</head>
|
||
<body>
|
||
|
||
<!--Capture the table of contents from pandoc as a jekyll variable -->
|
||
{% capture tableOfContents %}
|
||
<br>
|
||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||
<ul>
|
||
<li><a href="#the-falikov-kimball-model"
|
||
id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
|
||
<ul>
|
||
<li><a href="#the-model" id="toc-the-model">The Model</a></li>
|
||
<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
|
||
Diagrams</a></li>
|
||
<li><a href="#long-ranged-ising-model"
|
||
id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
|
||
</ul></li>
|
||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||
</ul>
|
||
</nav>
|
||
{% endcapture %}
|
||
|
||
<!-- Give the table of contents to header as a variable so it can be put into the sidebar-->
|
||
{% include header.html extra=tableOfContents %}
|
||
|
||
<main>
|
||
|
||
<!-- Table of Contents -->
|
||
<!-- <nav id="TOC" role="doc-toc">
|
||
<ul>
|
||
<li><a href="#the-falikov-kimball-model"
|
||
id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
|
||
<ul>
|
||
<li><a href="#the-model" id="toc-the-model">The Model</a></li>
|
||
<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
|
||
Diagrams</a></li>
|
||
<li><a href="#long-ranged-ising-model"
|
||
id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
|
||
</ul></li>
|
||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||
</ul>
|
||
</nav>
|
||
-->
|
||
|
||
<!-- Main Page Body -->
|
||
<div id="page-header">
|
||
<p>2 Background</p>
|
||
<hr />
|
||
</div>
|
||
<section id="the-falikov-kimball-model" class="level1">
|
||
<h1>The Falikov Kimball Model</h1>
|
||
<section id="the-model" class="level2">
|
||
<h2>The Model</h2>
|
||
<p>The Falikov-Kimball (FK) model is one of the simplest models of the
|
||
correlated electron problem. It captures the essence of the interaction
|
||
between itinerant and localized 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>
|
||
<p><span class="math display">\[\begin{aligned}
|
||
H_{\mathrm{FK}} = & \;U \sum_{i} (d^\dagger_{i}d_{i} -
|
||
\tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle
|
||
i,j\rangle} c^\dagger_{i}c_{j}.\\
|
||
\end{aligned}\]</span></p>
|
||
<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 relabel the up and down spin electron
|
||
states and removed the hopping term for one species, 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>
|
||
<p>Like other exactly solvable models <span class="citation"
|
||
data-cites="smithDisorderFreeLocalization2017"> [<a
|
||
href="#ref-smithDisorderFreeLocalization2017"
|
||
role="doc-biblioref">6</a>]</span> and the Kitaev Model, the FK model
|
||
possesses extensively many conserved degrees of freedom <span
|
||
class="math inline">\([d^\dagger_{i}d_{i}, H] = 0\)</span>. The Hilbert
|
||
space therefore breaks up into a set of sectors in which these operators
|
||
take a definite value. Crucially, this reduces the interaction term
|
||
<span class="math inline">\((d^\dagger_{i}d_{i} -
|
||
\tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\)</span> from being
|
||
quartic in fermion operators to quadratic. This is what makes the FK
|
||
model exactly solvable, in contrast to the Hubbard model.</p>
|
||
<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 from this point we will only
|
||
consider the half-filled point.</p>
|
||
<p>At half-filling and on bipartite lattices, FK the model is
|
||
particle-hole (PH) 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> and this 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 even 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>. See appendix <a
|
||
href="../6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">A.1</a>
|
||
for a full derivation of the PH symmetry.</p>
|
||
<div id="fig:simple_DOS" class="fignos">
|
||
<figure>
|
||
<img src="/assets/thesis/background_chapter/simple_DOS.svg"
|
||
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 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." />
|
||
<figcaption aria-hidden="true"><span>Figure 1:</span> 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 one dimension. (a) With not 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.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<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">\(S_i = 1 -
|
||
2d^\dagger_id_i = \pm\tfrac{1}{2}\)</span> which I will refer to as the
|
||
spins.</p>
|
||
<p><span class="math display">\[\begin{aligned}
|
||
H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
|
||
\tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
|
||
\end{aligned}\]</span></p>
|
||
<p>The FK model can be solved exactly with dynamic mean field theory in
|
||
the infinite dimensional <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>.</p>
|
||
</section>
|
||
<section id="phase-diagrams" class="level2">
|
||
<h2>Phase Diagrams</h2>
|
||
<div id="fig:fk_phase_diagram" class="fignos">
|
||
<figure>
|
||
<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg"
|
||
data-short-caption="Fermi-Hubbard and Falikov-Kimball Temperatue-Interaction Phase Diagrams"
|
||
style="width:100.0%"
|
||
alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]" />
|
||
<figcaption aria-hidden="true"><span>Figure 2:</span> Schematic Phase
|
||
diagram of the Falikov-Kimball model in dimensions greater than two. At
|
||
low temperature the classical fermions (spins) settle into an ordered
|
||
charge density wave state (antiferromagnetic state). The schematic
|
||
diagram for the Hubbard model is the same. Reproduced from <span
|
||
class="citation"
|
||
data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014"> [<a
|
||
href="#ref-antipovCriticalExponentsStrongly2014"
|
||
role="doc-biblioref">10</a>,<a
|
||
href="#ref-antipovInteractionTunedAndersonMott2016"
|
||
role="doc-biblioref">14</a>]</span></figcaption>
|
||
</figure>
|
||
</div>
|
||
<p>In dimensions greater than one, the FK model exhibits a phase
|
||
transition at some <span class="math inline">\(U\)</span> dependent
|
||
critical temperature <span class="math inline">\(T_c(U)\)</span> to a
|
||
low temperature ordered phase <span class="citation"
|
||
data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
|
||
href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
|
||
role="doc-biblioref">15</a>]</span>. In terms of the immobile electrons
|
||
this corresponds to them occupying only one of the two sublattices A and
|
||
B this is known as a charge density wave (CDW) phase. In terms of spins
|
||
this is an AFM phase.</p>
|
||
<p>In the disordered region above <span
|
||
class="math inline">\(T_c(U)\)</span> there are two insulating phases.
|
||
For weak interactions <span class="math inline">\(U << t\)</span>,
|
||
thermal fluctuations in the spins act as an effective disorder potential
|
||
for the fermions, causing them to localise and giving rise to an
|
||
Anderson insulating state <span class="citation"
|
||
data-cites="andersonAbsenceDiffusionCertain1958"> [<a
|
||
href="#ref-andersonAbsenceDiffusionCertain1958"
|
||
role="doc-biblioref">16</a>]</span> which we will discuss more in
|
||
section <a
|
||
href="../2_Background/2.3_Disorder.html#bg-disorder-and-localisation">2.3</a>.
|
||
For strong interactions <span class="math inline">\(U >>
|
||
t\)</span>, the spins are not ordered but nevertheless their interaction
|
||
with the electrons opens a gap, leading a Mott insulator analogous to
|
||
that of the Hubbard model <span class="citation"
|
||
data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
|
||
href="#ref-brandtThermodynamicsCorrelationFunctions1989"
|
||
role="doc-biblioref">17</a>]</span>.</p>
|
||
<p>By contrast, in the one dimensional FK model there is no
|
||
finite-temperature phase transition (FTPT) to an ordered CDW phase <span
|
||
class="citation" data-cites="liebAbsenceMottTransition1968"> [<a
|
||
href="#ref-liebAbsenceMottTransition1968"
|
||
role="doc-biblioref">18</a>]</span>. Indeed dimensionality is crucial
|
||
for the physics of both localisation and FTPTs. In one dimension,
|
||
disorder generally dominates: even the weakest disorder exponentially
|
||
localises <em>all</em> single particle eigenstates. Only longer-range
|
||
correlations of the disorder potential can potentially induce
|
||
localisation-delocalisation transitions in one dimension <span
|
||
class="citation"
|
||
data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a
|
||
href="#ref-aubryAnalyticityBreakingAnderson1980"
|
||
role="doc-biblioref">19</a>–<a
|
||
href="#ref-dunlapAbsenceLocalizationRandomdimer1990"
|
||
role="doc-biblioref">21</a>]</span>. Thermodynamically, short-range
|
||
interactions cannot overcome thermal defects in one dimension which
|
||
prevents ordered phases at non-zero temperature <span class="citation"
|
||
data-cites="goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a
|
||
href="#ref-goldshteinPurePointSpectrum1977"
|
||
role="doc-biblioref">22</a>–<a
|
||
href="#ref-kramerLocalizationTheoryExperiment1993"
|
||
role="doc-biblioref">24</a>]</span>.</p>
|
||
<p>However, the absence of an FTPT in the short ranged FK chain is far
|
||
from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY)
|
||
interaction mediated by the fermions <span class="citation"
|
||
data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a
|
||
href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">25</a>–<a
|
||
href="#ref-yosidaMagneticPropertiesCuMn1957"
|
||
role="doc-biblioref">28</a>]</span> decays as <span
|
||
class="math inline">\(r^{-1}\)</span> in one dimension <span
|
||
class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a
|
||
href="#ref-rusinCalculationRKKYRange2017"
|
||
role="doc-biblioref">29</a>]</span>. This could in principle induce the
|
||
necessary long-range interactions for the classical Ising background to
|
||
order at low temperatures <span class="citation"
|
||
data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a
|
||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||
role="doc-biblioref">30</a>,<a
|
||
href="#ref-peierlsIsingModelFerromagnetism1936"
|
||
role="doc-biblioref">31</a>]</span>. However, Kennedy and Lieb
|
||
established rigorously that at half-filling a CDW phase only exists at
|
||
<span class="math inline">\(T = 0\)</span> for the one dimensional FK
|
||
model <span class="citation"
|
||
data-cites="kennedyItinerantElectronModel1986"> [<a
|
||
href="#ref-kennedyItinerantElectronModel1986"
|
||
role="doc-biblioref">32</a>]</span>.</p>
|
||
<p>Based on this primacy of dimensionality, we will go digging into the
|
||
one dimensional case. In chapter <a
|
||
href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#fk-model">3</a>
|
||
we will construct a generalised one-dimensional FK model with long-range
|
||
interactions which induces the otherwise forbidden CDW phase at non-zero
|
||
temperature. To do this we will draw on theory of the Long Range Ising
|
||
Model which is the subject of the next section.</p>
|
||
</section>
|
||
<section id="long-ranged-ising-model" class="level2">
|
||
<h2>Long Ranged Ising model</h2>
|
||
<p>The suppression of phase transitions is a common phenomena in one
|
||
dimensional systems and the Ising model serves as a great illustration
|
||
of this. In terms of classical spins <span class="math inline">\(S_i =
|
||
\pm \frac{1}{2}\)</span> the standard Ising model reads</p>
|
||
<p><span class="math display">\[H_{\mathrm{I}} = \sum_{\langle ij
|
||
\rangle} S_i S_j\]</span></p>
|
||
<p>Like the FK model, the Ising model shows an FTPT to an ordered state
|
||
only in two dimensions and above. This can be understood via Peierls’
|
||
argument <span class="citation"
|
||
data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a
|
||
href="#ref-peierlsIsingModelFerromagnetism1936"
|
||
role="doc-biblioref">31</a>,<a
|
||
href="#ref-kennedyItinerantElectronModel1986"
|
||
role="doc-biblioref">32</a>]</span> to be a consequence of the low
|
||
energy penalty for domain walls in one dimensional systems.</p>
|
||
<p>Following Peierls’ argument, consider the difference in free energy
|
||
<span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span>
|
||
between an ordered state and a state with single domain wall in a
|
||
discrete order parameter. If this value is negative it implies that the
|
||
ordered state is unstable with respect to domain wall defects, and they
|
||
will thus proliferate, destroying the ordered phase. If we consider the
|
||
scaling of the two terms with system size <span
|
||
class="math inline">\(L\)</span> we see that short range interactions
|
||
produce a constant energy penalty <span class="math inline">\(\Delta
|
||
E\)</span> for a domain wall. In contrast, the number of such single
|
||
domain wall states scales linearly with system size so the entropy is
|
||
<span class="math inline">\(\propto \ln L\)</span>. Thus the entropic
|
||
contribution dominates (eventually) in the thermodynamic limit and no
|
||
finite temperature order is possible. In two dimensions and above, the
|
||
energy penalty of a domain wall scales like <span
|
||
class="math inline">\(L^{d-1}\)</span> which is why they can support
|
||
ordered phases. This argument does not quite apply to the FK model
|
||
because of the aforementioned RKKY interaction. Instead this argument
|
||
will give us insight into how to recover an ordered phase in the one
|
||
dimensional FK model.</p>
|
||
<p>In contrast the long range Ising (LRI) model <span
|
||
class="math inline">\(H_{\mathrm{LRI}}\)</span> can have an FTPT in one
|
||
dimension.</p>
|
||
<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|)
|
||
S_i S_j = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\]</span></p>
|
||
<p>Renormalisation group analyses show that the LRI model has an ordered
|
||
phase in 1D for <span class="math inline">\(1 < \alpha <
|
||
2\)</span> <span class="citation"
|
||
data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
|
||
href="#ref-dysonExistencePhasetransitionOnedimensional1969"
|
||
role="doc-biblioref">33</a>]</span>. Peierls’ argument can be
|
||
extended <span class="citation"
|
||
data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
|
||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||
role="doc-biblioref">30</a>]</span> to long range interactions to
|
||
provide intuition for why this is the case. Again considering the energy
|
||
difference between the ordered state <span
|
||
class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span>
|
||
and a domain wall state <span
|
||
class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\)</span>.
|
||
In the case of the LRI model, careful counting shows that this energy
|
||
penalty is <span class="math display">\[\Delta E \propto
|
||
\sum_{n=1}^{\infty} n J(n)\]</span></p>
|
||
<p>because each interaction between spins separated across the domain by
|
||
a bond length <span class="math inline">\(n\)</span> can be drawn
|
||
between <span class="math inline">\(n\)</span> equivalent pairs of
|
||
sites. The behaviour then depends crucially on the sum scales with
|
||
system size. Ruelle proved rigorously for a very general class of 1D
|
||
systems, that if <span class="math inline">\(\Delta E\)</span> or its
|
||
many-body generalisation converges to a constant in the thermodynamic
|
||
limit then the free energy is analytic <span class="citation"
|
||
data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a
|
||
href="#ref-ruelleStatisticalMechanicsOnedimensional1968"
|
||
role="doc-biblioref">34</a>]</span>. This rules out a finite order phase
|
||
transition, though not one of the Kosterlitz-Thouless type. Dyson also
|
||
proves this though with a slightly different condition on <span
|
||
class="math inline">\(J(n)\)</span> <span class="citation"
|
||
data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
|
||
href="#ref-dysonExistencePhasetransitionOnedimensional1969"
|
||
role="doc-biblioref">33</a>]</span>.</p>
|
||
<p>With a power law form for <span class="math inline">\(J(n)\)</span>,
|
||
there are a few cases to consider:</p>
|
||
<p>For <span class="math inline">\(\alpha = 0\)</span> i.e infinite
|
||
range interactions, the Ising model is exactly solvable and mean field
|
||
theory is exact <span class="citation"
|
||
data-cites="lipkinValidityManybodyApproximation1965"> [<a
|
||
href="#ref-lipkinValidityManybodyApproximation1965"
|
||
role="doc-biblioref">35</a>]</span>. This limit is the same as the
|
||
infinite dimensional limit.</p>
|
||
<p>For <span class="math inline">\(\alpha \leq 1\)</span> we have very
|
||
slowly decaying interactions. <span class="math inline">\(\Delta
|
||
E\)</span> does not converge as a function of system size so the
|
||
Hamiltonian is non-extensive, a topic not without some considerable
|
||
controversy <span class="citation"
|
||
data-cites="grossNonextensiveHamiltonianSystems2002 lutskoQuestioningValidityNonextensive2011 wangCommentNonextensiveHamiltonian2003"> [<a
|
||
href="#ref-grossNonextensiveHamiltonianSystems2002"
|
||
role="doc-biblioref">36</a>–<a
|
||
href="#ref-wangCommentNonextensiveHamiltonian2003"
|
||
role="doc-biblioref">38</a>]</span> that we will not consider further
|
||
here.</p>
|
||
<p>For <span class="math inline">\(1 < \alpha < 2\)</span>, we get
|
||
a phase transition to an ordered state at a finite temperature, this is
|
||
what we want!</p>
|
||
<p>For <span class="math inline">\(\alpha = 2\)</span>, the energy of
|
||
domain walls diverges logarithmically, and this turns out to be a
|
||
Kostelitz-Thouless transition <span class="citation"
|
||
data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
|
||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||
role="doc-biblioref">30</a>]</span>.</p>
|
||
<p>Finally, for <span class="math inline">\(2 < \alpha\)</span> we
|
||
have very quickly decaying interactions and domain walls again have a
|
||
finite energy penalty, hence Peirels’ argument holds and there is no
|
||
phase transition.</p>
|
||
<p>One final complexity is that for <span
|
||
class="math inline">\(\tfrac{3}{2} < \alpha < 2\)</span>
|
||
renormalisation group methods show that the critical point has
|
||
non-universal critical exponents that depend on <span
|
||
class="math inline">\(\alpha\)</span> <span class="citation"
|
||
data-cites="fisherCriticalExponentsLongRange1972"> [<a
|
||
href="#ref-fisherCriticalExponentsLongRange1972"
|
||
role="doc-biblioref">39</a>]</span>. To avoid this potential confounding
|
||
factors we will park ourselves at <span class="math inline">\(\alpha =
|
||
1.25\)</span> when we apply these ideas to the FK model.</p>
|
||
<p>Were we to extend this to arbitrary dimension <span
|
||
class="math inline">\(d\)</span> we would find that thermodynamics
|
||
properties generally both <span class="math inline">\(d\)</span> and
|
||
<span class="math inline">\(\alpha\)</span>, long range interactions can
|
||
modify the ‘effective dimension’ of thermodynamic systems <span
|
||
class="citation"
|
||
data-cites="angeliniRelationsShortrangeLongrange2014"> [<a
|
||
href="#ref-angeliniRelationsShortrangeLongrange2014"
|
||
role="doc-biblioref">40</a>]</span>.</p>
|
||
<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"
|
||
style="width:100.0%"
|
||
alt="Figure 3: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied." />
|
||
<figcaption aria-hidden="true"><span>Figure 3:</span> The thermodynamic
|
||
behaviour of the long range Ising model <span
|
||
class="math inline">\(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i -
|
||
j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span
|
||
class="math inline">\(\alpha\)</span> is varied.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<p>Next Section: <a
|
||
href="../2_Background/2.2_HKM_Model.html#the-kitaev-honeycomb-model">The
|
||
Kitaev Honeycomb Model</a></p>
|
||
</section>
|
||
</section>
|
||
<section id="bibliography" class="level1 unnumbered">
|
||
<h1 class="unnumbered">Bibliography</h1>
|
||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||
<div id="ref-hubbardj.ElectronCorrelationsNarrow1963" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[1] </div><div
|
||
class="csl-right-inline">Hubbard, J., <em><a
|
||
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>
|
||
<div id="ref-gruberFalicovKimballModelReview1996" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">C.
|
||
Gruber and N. Macris, <em>The Falicov-Kimball Model: A Review of Exact
|
||
Results and Extensions</em>, Helvetica Physica Acta <strong>69</strong>,
|
||
(1996).</div>
|
||
</div>
|
||
<div id="ref-gruberFalicovKimballModel2006" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">C.
|
||
Gruber and D. Ueltschi, <em>The Falicov-Kimball model</em>, in
|
||
<em>Encyclopedia of mathematical physics. 1, 1,</em> (Elsevier,
|
||
Amsterdam, 2006).</div>
|
||
</div>
|
||
<div id="ref-devriesSimplifiedHubbardModel1993" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">P.
|
||
de Vries, K. Michielsen, and H. De Raedt, <em><a
|
||
href="https://doi.org/10.1007/BF01308754">The Simplified Hubbard Model
|
||
in One and Two Dimensions</a></em>, Z. Physik B - Condensed Matter
|
||
<strong>92</strong>, 353 (1993).</div>
|
||
</div>
|
||
<div id="ref-smithDisorderFreeLocalization2017" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">A.
|
||
Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, <em><a
|
||
href="https://doi.org/10.1103/PhysRevLett.118.266601">Disorder-Free
|
||
Localization</a></em>, Phys. Rev. Lett. <strong>118</strong>, 266601
|
||
(2017).</div>
|
||
</div>
|
||
<div id="ref-jedrzejewskiFalicovKimballModels2001" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">J.
|
||
Jedrzejewski and R. Lemanski, <em><a
|
||
href="http://adsabs.harvard.edu/abs/2001AcPPB..32.3243J">Falicov--Kimball
|
||
Models of Collective Phenomena in Solids (A Concise Guide)</a></em>,
|
||
Acta Physica Polonica B <strong>32</strong>, 3243 (2001).</div>
|
||
</div>
|
||
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">C.
|
||
Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a
|
||
href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the
|
||
Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B
|
||
<strong>41</strong>, 2198 (1990).</div>
|
||
</div>
|
||
<div id="ref-gruberFalicovKimballModel2005" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">C.
|
||
Gruber and D. Ueltschi, <em><a
|
||
href="http://arxiv.org/abs/math-ph/0502041">The Falicov-Kimball
|
||
Model</a></em>, arXiv:math-Ph/0502041 (2005).</div>
|
||
</div>
|
||
<div id="ref-antipovCriticalExponentsStrongly2014" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">A.
|
||
E. Antipov, E. Gull, and S. Kirchner, <em><a
|
||
href="https://doi.org/10.1103/PhysRevLett.112.226401">Critical Exponents
|
||
of Strongly Correlated Fermion Systems from Diagrammatic Multiscale
|
||
Methods</a></em>, Phys. Rev. Lett. <strong>112</strong>, 226401
|
||
(2014).</div>
|
||
</div>
|
||
<div id="ref-ribicNonlocalCorrelationsSpectral2016" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">T.
|
||
Ribic, G. Rohringer, and K. Held, <em><a
|
||
href="https://doi.org/10.1103/PhysRevB.93.195105">Nonlocal Correlations
|
||
and Spectral Properties of the Falicov-Kimball Model</a></em>, Phys.
|
||
Rev. B <strong>93</strong>, 195105 (2016).</div>
|
||
</div>
|
||
<div id="ref-freericksExactDynamicalMeanfield2003" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">J.
|
||
K. Freericks and V. Zlatić, <em><a
|
||
href="https://doi.org/10.1103/RevModPhys.75.1333">Exact Dynamical
|
||
Mean-Field Theory of the Falicov-Kimball Model</a></em>, Rev. Mod. Phys.
|
||
<strong>75</strong>, 1333 (2003).</div>
|
||
</div>
|
||
<div id="ref-herrmannNonequilibriumDynamicalCluster2016"
|
||
class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">A.
|
||
J. Herrmann, N. Tsuji, M. Eckstein, and P. Werner, <em><a
|
||
href="https://doi.org/10.1103/PhysRevB.94.245114">Nonequilibrium
|
||
Dynamical Cluster Approximation Study of the Falicov-Kimball
|
||
Model</a></em>, Phys. Rev. B <strong>94</strong>, 245114 (2016).</div>
|
||
</div>
|
||
<div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">A.
|
||
E. Antipov, Y. Javanmard, P. Ribeiro, and S. Kirchner, <em><a
|
||
href="https://doi.org/10.1103/PhysRevLett.117.146601">Interaction-Tuned
|
||
Anderson Versus Mott Localization</a></em>, Phys. Rev. Lett.
|
||
<strong>117</strong>, 146601 (2016).</div>
|
||
</div>
|
||
<div id="ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
|
||
class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">M.
|
||
M. Maśka and K. Czajka, <em><a
|
||
href="https://doi.org/10.1103/PhysRevB.74.035109">Thermodynamics of the
|
||
Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo
|
||
Study</a></em>, Phys. Rev. B <strong>74</strong>, 035109 (2006).</div>
|
||
</div>
|
||
<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">P.
|
||
W. Anderson, <em><a
|
||
href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in
|
||
Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492
|
||
(1958).</div>
|
||
</div>
|
||
<div id="ref-brandtThermodynamicsCorrelationFunctions1989"
|
||
class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">U.
|
||
Brandt and C. Mielsch, <em><a
|
||
href="https://doi.org/10.1007/BF01321824">Thermodynamics and Correlation
|
||
Functions of the Falicov-Kimball Model in Large Dimensions</a></em>, Z.
|
||
Physik B - Condensed Matter <strong>75</strong>, 365 (1989).</div>
|
||
</div>
|
||
<div id="ref-liebAbsenceMottTransition1968" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">E.
|
||
H. Lieb and F. Y. Wu, <em><a
|
||
href="https://doi.org/10.1103/PhysRevLett.20.1445">Absence of Mott
|
||
Transition in an Exact Solution of the Short-Range, One-Band Model in
|
||
One Dimension</a></em>, Phys. Rev. Lett. <strong>20</strong>, 1445
|
||
(1968).</div>
|
||
</div>
|
||
<div id="ref-aubryAnalyticityBreakingAnderson1980" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">S.
|
||
Aubry and G. André, <em>Analyticity Breaking and Anderson Localization
|
||
in Incommensurate Lattices</em>, Proceedings, VIII International
|
||
Colloquium on Group-Theoretical Methods in Physics <strong>3</strong>,
|
||
18 (1980).</div>
|
||
</div>
|
||
<div id="ref-dassarmaLocalizationMobilityEdges1990" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[20] </div><div class="csl-right-inline">S.
|
||
Das Sarma, S. He, and X. C. Xie, <em><a
|
||
href="https://doi.org/10.1103/PhysRevB.41.5544">Localization, Mobility
|
||
Edges, and Metal-Insulator Transition in a Class of One-Dimensional
|
||
Slowly Varying Deterministic Potentials</a></em>, Phys. Rev. B
|
||
<strong>41</strong>, 5544 (1990).</div>
|
||
</div>
|
||
<div id="ref-dunlapAbsenceLocalizationRandomdimer1990" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[21] </div><div class="csl-right-inline">D.
|
||
H. Dunlap, H.-L. Wu, and P. W. Phillips, <em><a
|
||
href="https://doi.org/10.1103/PhysRevLett.65.88">Absence of Localization
|
||
in a Random-Dimer Model</a></em>, Phys. Rev. Lett. <strong>65</strong>,
|
||
88 (1990).</div>
|
||
</div>
|
||
<div id="ref-goldshteinPurePointSpectrum1977" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[22] </div><div class="csl-right-inline">I.
|
||
Ya. Gol’dshtein, S. A. Molchanov, and L. A. Pastur, <em><a
|
||
href="https://doi.org/10.1007/BF01135526">A Pure Point Spectrum of the
|
||
Stochastic One-Dimensional Schrödinger Operator</a></em>, Funct Anal Its
|
||
Appl <strong>11</strong>, 1 (1977).</div>
|
||
</div>
|
||
<div id="ref-abrahamsScalingTheoryLocalization1979" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[23] </div><div class="csl-right-inline">E.
|
||
Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan,
|
||
<em><a href="https://doi.org/10.1103/PhysRevLett.42.673">Scaling Theory
|
||
of Localization: Absence of Quantum Diffusion in Two
|
||
Dimensions</a></em>, Phys. Rev. Lett. <strong>42</strong>, 673
|
||
(1979).</div>
|
||
</div>
|
||
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[24] </div><div class="csl-right-inline">B.
|
||
Kramer and A. MacKinnon, <em><a
|
||
href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory
|
||
and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469
|
||
(1993).</div>
|
||
</div>
|
||
<div id="ref-kasuyaTheoryMetallicFerro1956" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[25] </div><div class="csl-right-inline">T.
|
||
Kasuya, <em><a href="https://doi.org/10.1143/PTP.16.45">A Theory of
|
||
Metallic Ferro- and Antiferromagnetism on Zener’s Model</a></em>, Prog
|
||
Theor Phys <strong>16</strong>, 45 (1956).</div>
|
||
</div>
|
||
<div id="ref-rudermanIndirectExchangeCoupling1954" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[26] </div><div class="csl-right-inline">M.
|
||
A. Ruderman and C. Kittel, <em><a
|
||
href="https://doi.org/10.1103/PhysRev.96.99">Indirect Exchange Coupling
|
||
of Nuclear Magnetic Moments by Conduction Electrons</a></em>, Phys. Rev.
|
||
<strong>96</strong>, 99 (1954).</div>
|
||
</div>
|
||
<div id="ref-vanvleckNoteInteractionsSpins1962" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[27] </div><div class="csl-right-inline">J.
|
||
H. Van Vleck, <em><a
|
||
href="https://doi.org/10.1103/RevModPhys.34.681">Note on the
|
||
Interactions Between the Spins of Magnetic Ions or Nuclei in
|
||
Metals</a></em>, Rev. Mod. Phys. <strong>34</strong>, 681 (1962).</div>
|
||
</div>
|
||
<div id="ref-yosidaMagneticPropertiesCuMn1957" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">K.
|
||
Yosida, <em><a href="https://doi.org/10.1103/PhysRev.106.893">Magnetic
|
||
Properties of Cu-Mn Alloys</a></em>, Phys. Rev. <strong>106</strong>,
|
||
893 (1957).</div>
|
||
</div>
|
||
<div id="ref-rusinCalculationRKKYRange2017" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">T.
|
||
M. Rusin and W. Zawadzki, <em><a
|
||
href="https://doi.org/10.1016/j.jmmm.2017.06.007">On Calculation of RKKY
|
||
Range Function in One Dimension</a></em>, Journal of Magnetism and
|
||
Magnetic Materials <strong>441</strong>, 387 (2017).</div>
|
||
</div>
|
||
<div id="ref-thoulessLongRangeOrderOneDimensional1969" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[30] </div><div class="csl-right-inline">D.
|
||
J. Thouless, <em><a
|
||
href="https://doi.org/10.1103/PhysRev.187.732">Long-Range Order in
|
||
One-Dimensional Ising Systems</a></em>, Phys. Rev. <strong>187</strong>,
|
||
732 (1969).</div>
|
||
</div>
|
||
<div id="ref-peierlsIsingModelFerromagnetism1936" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">R.
|
||
Peierls, <em><a href="https://doi.org/10.1017/S0305004100019174">On
|
||
Ising’s Model of Ferromagnetism</a></em>, Mathematical Proceedings of
|
||
the Cambridge Philosophical Society <strong>32</strong>, 477
|
||
(1936).</div>
|
||
</div>
|
||
<div id="ref-kennedyItinerantElectronModel1986" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">T.
|
||
Kennedy and E. H. Lieb, <em><a
|
||
href="https://doi.org/10.1016/0378-4371(86)90188-3">An Itinerant
|
||
Electron Model with Crystalline or Magnetic Long Range Order</a></em>,
|
||
Physica A: Statistical Mechanics and Its Applications
|
||
<strong>138</strong>, 320 (1986).</div>
|
||
</div>
|
||
<div id="ref-dysonExistencePhasetransitionOnedimensional1969"
|
||
class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">[33] </div><div class="csl-right-inline">F.
|
||
J. Dyson, <em><a href="https://doi.org/10.1007/BF01645907">Existence of
|
||
a Phase-Transition in a One-Dimensional Ising Ferromagnet</a></em>,
|
||
Commun.Math. Phys. <strong>12</strong>, 91 (1969).</div>
|
||
</div>
|
||
<div id="ref-ruelleStatisticalMechanicsOnedimensional1968"
|
||
class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">[34] </div><div class="csl-right-inline">D.
|
||
Ruelle, <em><a
|
||
href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical
|
||
Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys.
|
||
<strong>9</strong>, 267 (1968).</div>
|
||
</div>
|
||
<div id="ref-lipkinValidityManybodyApproximation1965" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[35] </div><div class="csl-right-inline">H.
|
||
J. Lipkin, N. Meshkov, and A. J. Glick, <em><a
|
||
href="https://doi.org/10.1016/0029-5582(65)90862-X">Validity of
|
||
Many-Body Approximation Methods for a Solvable Model. (I). Exact
|
||
Solutions and Perturbation Theory</a></em>, Nuclear Physics
|
||
<strong>62</strong>, 188 (1965).</div>
|
||
</div>
|
||
<div id="ref-grossNonextensiveHamiltonianSystems2002" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[36] </div><div class="csl-right-inline">D.
|
||
H. E. Gross, <em><a
|
||
href="https://doi.org/10.1016/S0378-4371(01)00646-X">Non-Extensive
|
||
Hamiltonian Systems Follow Boltzmann’s Principle Not Tsallis Statistics.
|
||
-- Phase Transitions, Second Law of Thermodynamics</a></em>, Physica A:
|
||
Statistical Mechanics and Its Applications <strong>305</strong>, 99
|
||
(2002).</div>
|
||
</div>
|
||
<div id="ref-lutskoQuestioningValidityNonextensive2011"
|
||
class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">[37] </div><div class="csl-right-inline">J.
|
||
F. Lutsko and J. P. Boon, <em><a
|
||
href="https://doi.org/10.1209/0295-5075/95/20006">Questioning the
|
||
Validity of Non-Extensive Thermodynamics for Classical Hamiltonian
|
||
Systems</a></em>, EPL <strong>95</strong>, 20006 (2011).</div>
|
||
</div>
|
||
<div id="ref-wangCommentNonextensiveHamiltonian2003" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[38] </div><div class="csl-right-inline">Q.
|
||
A. Wang, <em><a
|
||
href="https://hal.archives-ouvertes.fr/hal-00009462">Comment on
|
||
<span>“Nonextensive Hamiltonian Systems Follow Boltzmann’s Principle Not
|
||
Tsallis Statistics-Phase Transition, Second Law of
|
||
Thermodynamics”</span> by Gross</a></em>, (2003).</div>
|
||
</div>
|
||
<div id="ref-fisherCriticalExponentsLongRange1972" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[39] </div><div class="csl-right-inline">M.
|
||
E. Fisher, S. Ma, and B. G. Nickel, <em><a
|
||
href="https://doi.org/10.1103/PhysRevLett.29.917">Critical Exponents for
|
||
Long-Range Interactions</a></em>, Phys. Rev. Lett. <strong>29</strong>,
|
||
917 (1972).</div>
|
||
</div>
|
||
<div id="ref-angeliniRelationsShortrangeLongrange2014" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">[40] </div><div class="csl-right-inline">M.
|
||
C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a
|
||
href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between
|
||
Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E
|
||
<strong>89</strong>, 062120 (2014).</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
|
||
|
||
</main>
|
||
</body>
|
||
</html>
|