mirror of
https://github.com/TomHodson/tomhodson.github.com.git
synced 2025-06-26 10:01:18 +02:00
add localisation stuff
This commit is contained in:
parent
b1749e8607
commit
5a856d2182
@ -29,8 +29,7 @@ image:
|
||||
<ul>
|
||||
<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
|
||||
<ul>
|
||||
<li><a href="#disorder-and-spin-liquids" id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
|
||||
<li><a href="#amorphous-magnetism" id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
|
||||
<li><a href="#diagnosing-localisation-in-practice" id="toc-diagnosing-localisation-in-practice">Diagnosing Localisation in practice</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -47,8 +46,7 @@ image:
|
||||
<ul>
|
||||
<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
|
||||
<ul>
|
||||
<li><a href="#disorder-and-spin-liquids" id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
|
||||
<li><a href="#amorphous-magnetism" id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
|
||||
<li><a href="#diagnosing-localisation-in-practice" id="toc-diagnosing-localisation-in-practice">Diagnosing Localisation in practice</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -62,11 +60,6 @@ image:
|
||||
</div>
|
||||
<section id="bg-disorder-and-localisation" class="level1">
|
||||
<h1>Disorder and Localisation</h1>
|
||||
<ul>
|
||||
<li><p>disorder starts with the very simple anderson model</p></li>
|
||||
<li><p>Quenched vs Annealed disorder</p></li>
|
||||
<li></li>
|
||||
</ul>
|
||||
<p>Disorder is a fact of life for the condensed matter physicist. No sample will ever be completely free of contamination or of structural defects. The classical Drude theory of electron conductivity envisages electrons as scattering off impurities. Hence we would expect the electrical conductivity to be proportional to the mean free path <span class="citation" data-cites="lagendijkFiftyYearsAnderson2009"> [<a href="#ref-lagendijkFiftyYearsAnderson2009" role="doc-biblioref">1</a>]</span>, decreasing smoothly as the number of defects increases. However, Anderson showed in 1958 <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">2</a>]</span> that at some critical level of disorder <strong>all</strong> single particle eigenstates localise. What would later be known as Anderson localisation is characterised by exponentially localised eigenfunctions <span class="math inline">\(\psi(x) \sim e^{-x/\lambda}\)</span> which cannot contribute to transport processes. The localisation length <span class="math inline">\(\lambda\)</span> is the typical scale of localised state and can be extracted with transmission matrix methods <span class="citation" data-cites="pendrySymmetryTransportWaves1994"> [<a href="#ref-pendrySymmetryTransportWaves1994" role="doc-biblioref">3</a>]</span>. Anderson localisation provided a different kind of insulator to that of the band insulator.</p>
|
||||
<p>The Anderson model is about the simplest model of disorder one could imagine <span id="eq:bg-anderson-model"><span class="math display">\[
|
||||
H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j
|
||||
@ -83,17 +76,31 @@ H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U
|
||||
<p>In Chapter 3 we will consider a generalised FK model in one dimension and how the disorder generated near a one dimensional thermodynamic phase transition interacts with localisation physics.</p>
|
||||
<p>So far we have considered disorder as a static or dynamic field coupled to a model defined on a translation invariant lattice. Another kind of disordered system that worthy of study are amorphous systems.</p>
|
||||
<p>Amorphous systems have disordered bond connectivity, so called <em>topological disorder</em>. As discussed in the introduction these include amorphous semiconductors such as amorphous Germanium and Silicon <span class="citation" data-cites="Yonezawa1983 zallen2008physics Weaire1971 betteridge1973possible"> [<a href="#ref-Yonezawa1983" role="doc-biblioref">19</a>–<a href="#ref-betteridge1973possible" role="doc-biblioref">22</a>]</span>. While materials do not have long range lattice structure they can enforce local constraints such as the approximate coordination number <span class="math inline">\(z = 4\)</span> of silicon.</p>
|
||||
<p>Topological disorder can be qualitatively different from other disordered systems. Disordered graphs are constrained by fixed coordination number and the Euler equation. The Harris <span class="citation" data-cites="harrisEffectRandomDefects1974"> [<a href="#ref-harrisEffectRandomDefects1974" role="doc-biblioref">23</a>]</span> and the Imry-Mar <span class="citation" data-cites="imryRandomFieldInstabilityOrdered1975"> [<a href="#ref-imryRandomFieldInstabilityOrdered1975" role="doc-biblioref">24</a>]</span> criteria are key results on the effect of disorder on thermodynamic phase transitions. The Harris criterion signals when disorder will affect the universal of a thermodynamic critical point. It states that for a critical point in a <span class="math inline">\(d\)</span>-dimensional system with correlation length scaling exponent, disorder will be relevant if <span class="math inline">\(\nu\)</span> if <span class="math inline">\(d\nu < 2\)</span>. The Imry-Ma criterion simply forbids the formation of long range ordered states in <span class="math inline">\(d \leq 2\)</span> dimensions in the presence of disorder. The latter criteria is violated in the presence of correlated disorder <span class="citation" data-cites="changlaniChargeDensityWaves2016"> [<a href="#ref-changlaniChargeDensityWaves2016" role="doc-biblioref">25</a>]</span> and both are modified for topological disorder. In chapter 4 we will put the Kitaev model onto two dimensional Voronoi lattices. These lattices are have fixed coordination number <span class="math inline">\(z=3\)</span> and must satisfy the Euler equation for the plane, this leads to strong anticorrelations which mean that topological disorder is effectively weaker than standard disorder here <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">26</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">27</a>]</span>]. This does not apply to the three dimensional Voronoi lattices where the Euler equation is a weaker constraint.</p>
|
||||
<p>Lastly it is worth exploring how quantum spin liquids and disorder interact. The KH model has been studied subject to both bond and site disorder <strong>cite</strong>. In some instances it seems that disorder can even promote the formation of a QSL ground state <span class="citation" data-cites="wenDisorderedRouteCoulomb2017"> [<a href="#ref-wenDisorderedRouteCoulomb2017" role="doc-biblioref">28</a>]</span>.</p>
|
||||
<ul>
|
||||
<li>localisation length and IPR scaling</li>
|
||||
<li>multifractality</li>
|
||||
</ul>
|
||||
<section id="disorder-and-spin-liquids" class="level2">
|
||||
<h2>Disorder and Spin liquids</h2>
|
||||
</section>
|
||||
<section id="amorphous-magnetism" class="level2">
|
||||
<h2>Amorphous Magnetism</h2>
|
||||
<p>Topological disorder can be qualitatively different from other disordered systems. Disordered graphs are constrained by fixed coordination number and the Euler equation. The Harris <span class="citation" data-cites="harrisEffectRandomDefects1974"> [<a href="#ref-harrisEffectRandomDefects1974" role="doc-biblioref">23</a>]</span> and the Imry-Mar <span class="citation" data-cites="imryRandomFieldInstabilityOrdered1975"> [<a href="#ref-imryRandomFieldInstabilityOrdered1975" role="doc-biblioref">24</a>]</span> criteria are key results on the effect of disorder on thermodynamic phase transitions. The Harris criterion signals when disorder will affect the universal of a thermodynamic critical point. It states that for a critical point in a <span class="math inline">\(d\)</span>-dimensional system with correlation length scaling exponent, disorder will be relevant if <span class="math inline">\(\nu\)</span> if <span class="math inline">\(d\nu < 2\)</span>. The Imry-Ma criterion simply forbids the formation of long range ordered states in <span class="math inline">\(d \leq 2\)</span> dimensions in the presence of disorder. The latter criteria is violated in the presence of correlated disorder <span class="citation" data-cites="changlaniChargeDensityWaves2016"> [<a href="#ref-changlaniChargeDensityWaves2016" role="doc-biblioref">25</a>]</span> and both are modified for topological disorder. In chapter 4 we will put the Kitaev model onto two dimensional Voronoi lattices. These lattices are have fixed coordination number <span class="math inline">\(z=3\)</span> and must satisfy the Euler equation for the plane, this leads to strong anti-correlations which mean that topological disorder is effectively weaker than standard disorder here <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">26</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">27</a>]</span>]. This does not apply to the three dimensional Voronoi lattices where the Euler equation is a weaker constraint.</p>
|
||||
<p>Lastly it is worth exploring how quantum spin liquids and disorder interact. The KH model has been studied subject to both flux <span class="citation" data-cites="Nasu_Thermal_2015"> [<a href="#ref-Nasu_Thermal_2015" role="doc-biblioref">28</a>]</span> and bond <span class="citation" data-cites="knolle_dynamics_2016"> [<a href="#ref-knolle_dynamics_2016" role="doc-biblioref">29</a>]</span> disorder. In some instances it seems that disorder can even promote the formation of a QSL ground state <span class="citation" data-cites="wenDisorderedRouteCoulomb2017"> [<a href="#ref-wenDisorderedRouteCoulomb2017" role="doc-biblioref">30</a>]</span>. I will look at how adding lattice disorder to the mix affects the picture. It has also been shown that the KH model exhibits disorder free localisation after a quantum quench <span class="citation" data-cites="zhuSubdiffusiveDynamicsCritical2021"> [<a href="#ref-zhuSubdiffusiveDynamicsCritical2021" role="doc-biblioref">31</a>]</span>.</p>
|
||||
<section id="diagnosing-localisation-in-practice" class="level2">
|
||||
<h2>Diagnosing Localisation in practice</h2>
|
||||
<figure>
|
||||
<img src="/assets/thesis/background_chapter/localisation_radius_vs_length.svg" id="fig:localisation_radius_vs_length" data-short-caption="Localisation length vs diameter" style="width:100.0%" alt="Figure 1: A localised state \psi in an potential well that has formed from random fluctuations in the disorder potential V(x). The localisation length \lambda governs how quickly the state decays away from the well while the diameter R of the state is controlled by the size of the well. Reproduced from [6]." />
|
||||
<figcaption aria-hidden="true">Figure 1: A localised state <span class="math inline">\(\psi\)</span> in an potential well that has formed from random fluctuations in the disorder potential <span class="math inline">\(V(x)\)</span>. The localisation length <span class="math inline">\(\lambda\)</span> governs how quickly the state decays away from the well while the diameter <span class="math inline">\(R\)</span> of the state is controlled by the size of the well. Reproduced from <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span>.</figcaption>
|
||||
</figure>
|
||||
<p>Looking at practical tools for diagnosing localisation, there are a few standard methods <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span>.</p>
|
||||
<p>The most direct method would be to fit a function of the form <span class="math inline">\(\psi(x) = f(x) e^{-|x-x_0|/\lambda}\)</span> to each single particle wavefunction to extract the localisation length <span class="math inline">\(\lambda\)</span>. This method is little used in practice since it requires storing and processing full wavefunctions which quickly becomes expensive for large systems.</p>
|
||||
<p>For low dimensional systems with quenched disorder, transmission matrix methods can be used to directly extract the localisation length. These work by turning the time independent Schrödinger equation <span class="math inline">\(\hat{H}|\psi\rangle = E|\psi\rangle\)</span> into a matrix equation linking the amplitude of <span class="math inline">\(\psi\)</span> on each <span class="math inline">\(d-1\)</span> dimensional slice of the system to the next and looking at average properties of this transmission matrix. This method is less useful for systems like the FK model where the disorder as a whole must be sampled from the thermodynamic ensemble. It is also problematic for the Kitaev Model on an amorphous lattice as the slicing procedure is complex to define in the absence of a regular lattice.</p>
|
||||
<p>A more versatile method is based on the inverse participation ratio. The inverse participation ratio is defined for a normalised wave function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i |\psi_i|^2 = 1\)</span> as its fourth moment <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span>:</p>
|
||||
<p><span class="math display">\[
|
||||
P^{-1} = \sum_i |\psi_i|^4
|
||||
\]</span></p>
|
||||
<p>The name derive from the fact that this operator acts as a measure of the volume where the wavefunction is significantly different from zero. They can alternatively be thougt of as providing a measure of the average diameter <span class="math inline">\(R\)</span> from <span class="math inline">\(R = P^{1/d}\)</span>, see fig. <a href="#fig:localisation_radius_vs_length">1</a> for the distinction between <span class="math inline">\(R\)</span> and <span class="math inline">\(\lambda\)</span>.</p>
|
||||
<p>For localised states, the <em>inverse</em> participation ratio <span class="math inline">\(P^{-1}\)</span> is independent of system size while for plane wave states in <span class="math inline">\(d\)</span> dimensions <span class="math inline">\(P^{-1} = L^{-d}\)</span>. States may also be intermediate between localised and extended, described by their fractal dimensionality <span class="math inline">\(d > d* > 0\)</span>:</p>
|
||||
<p><span class="math display">\[
|
||||
P(L)^{-1} \sim L^{-d*}
|
||||
\]</span></p>
|
||||
<p>For finite size systems, these relations only hold once the system size <span class="math inline">\(L\)</span> is much greater than the localisation length. When the localisation length is comparable to the system size the states contribute to transport. This is called weak localisation <span class="citation" data-cites="altshulerMagnetoresistanceHallEffect1980 dattaElectronicTransportMesoscopic1995"> [<a href="#ref-altshulerMagnetoresistanceHallEffect1980" role="doc-biblioref">32</a>,<a href="#ref-dattaElectronicTransportMesoscopic1995" role="doc-biblioref">33</a>]</span>.</p>
|
||||
<p>For extended states <span class="math inline">\(d* = 0\)</span> while for localised ones <span class="math inline">\(d* = 0\)</span>. In both chapters I will use an energy resolved IPR <span class="math display">\[
|
||||
DOS(\omega) = \sum_n \delta(\omega - \epsilon_n)\\
|
||||
IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) |\psi_{n,i}|^4
|
||||
\]</span> Where <span class="math inline">\(\psi_{n,i}\)</span> is the wavefunction corresponding to the energy <span class="math inline">\(\epsilon_n\)</span> at the ith site. In practice I bin the energies and IPRs into a fine energy grid and use the mean within each bin.</p>
|
||||
<p>Next Chapter: <a href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falikov-Kimball Model</a></p>
|
||||
</section>
|
||||
</section>
|
||||
@ -181,8 +188,23 @@ H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U
|
||||
<div id="ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[27] </div><div class="csl-right-inline">M. Schrauth, J. Portela, and F. Goth, <em><a href="https://doi.org/10.1103/PhysRevLett.121.100601">Violation of the Harris-Barghathi-Vojta Criterion</a></em>, Physical Review Letters <strong>121</strong>, (2018).</div>
|
||||
</div>
|
||||
<div id="ref-Nasu_Thermal_2015" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">J. Nasu, M. Udagawa, and Y. Motome, <em><a href="https://doi.org/10.1103/PhysRevB.92.115122">Thermal Fractionalization of Quantum Spins in a Kitaev Model: Temperature-Linear Specific Heat and Coherent Transport of Majorana Fermions</a></em>, Phys. Rev. B <strong>92</strong>, 115122 (2015).</div>
|
||||
</div>
|
||||
<div id="ref-knolle_dynamics_2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">J. Knolle, Dynamics of a Quantum Spin Liquid, Max Planck Institute for the Physics of Complex Systems, Dresden, 2016.</div>
|
||||
</div>
|
||||
<div id="ref-wenDisorderedRouteCoulomb2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">J.-J. Wen et al., <em><a href="https://doi.org/10.1103/PhysRevLett.118.107206">Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in ${\Mathrm{Pr}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$</a></em>, Phys. Rev. Lett. <strong>118</strong>, 107206 (2017).</div>
|
||||
<div class="csl-left-margin">[30] </div><div class="csl-right-inline">J.-J. Wen et al., <em><a href="https://doi.org/10.1103/PhysRevLett.118.107206">Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in ${\Mathrm{Pr}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$</a></em>, Phys. Rev. Lett. <strong>118</strong>, 107206 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-zhuSubdiffusiveDynamicsCritical2021" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">G.-Y. Zhu and M. Heyl, <em><a href="https://doi.org/10.1103/PhysRevResearch.3.L032069">Subdiffusive Dynamics and Critical Quantum Correlations in a Disorder-Free Localized Kitaev Honeycomb Model Out of Equilibrium</a></em>, Phys. Rev. Research <strong>3</strong>, L032069 (2021).</div>
|
||||
</div>
|
||||
<div id="ref-altshulerMagnetoresistanceHallEffect1980" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee, <em><a href="https://doi.org/10.1103/PhysRevB.22.5142">Magnetoresistance and Hall Effect in a Disordered Two-Dimensional Electron Gas</a></em>, Phys. Rev. B <strong>22</strong>, 5142 (1980).</div>
|
||||
</div>
|
||||
<div id="ref-dattaElectronicTransportMesoscopic1995" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[33] </div><div class="csl-right-inline">S. Datta, <em><a href="https://doi.org/10.1017/CBO9780511805776">Electronic Transport in Mesoscopic Systems</a></em> (Cambridge University Press, Cambridge, 1995).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -35,10 +35,6 @@ image:
|
||||
<li><a href="#two-step-trick" id="toc-two-step-trick">Two Step Trick</a></li>
|
||||
<li><a href="#scaling" id="toc-scaling">Scaling</a></li>
|
||||
<li><a href="#binder-cumulants" id="toc-binder-cumulants">Binder Cumulants</a></li>
|
||||
<li><a href="#diagnostics-of-localisation" id="toc-diagnostics-of-localisation">Diagnostics of Localisation</a>
|
||||
<ul>
|
||||
<li><a href="#inverse-participation-ratio" id="toc-inverse-participation-ratio">Inverse Participation Ratio</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#convergence-time" id="toc-convergence-time">Convergence Time</a></li>
|
||||
<li><a href="#auto-correlation-time" id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
</ul></li>
|
||||
@ -63,10 +59,6 @@ image:
|
||||
<li><a href="#two-step-trick" id="toc-two-step-trick">Two Step Trick</a></li>
|
||||
<li><a href="#scaling" id="toc-scaling">Scaling</a></li>
|
||||
<li><a href="#binder-cumulants" id="toc-binder-cumulants">Binder Cumulants</a></li>
|
||||
<li><a href="#diagnostics-of-localisation" id="toc-diagnostics-of-localisation">Diagnostics of Localisation</a>
|
||||
<ul>
|
||||
<li><a href="#inverse-participation-ratio" id="toc-inverse-participation-ratio">Inverse Participation Ratio</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#convergence-time" id="toc-convergence-time">Convergence Time</a></li>
|
||||
<li><a href="#auto-correlation-time" id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
</ul></li>
|
||||
@ -136,27 +128,20 @@ H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{
|
||||
<p>In order to reduce the effects of the boundary conditions and the finite size of the system we redefine and normalise the coupling matrix to have 0 derivative at its furthest extent rather than cutting off abruptly.</p>
|
||||
<p><span class="math display">\[
|
||||
\begin{aligned}
|
||||
J'(x) &= \abs{\frac{L}{\pi}\sin \frac{\pi x}{L}}^{-\alpha} \\
|
||||
J'(x) &= \frac{L}{\pi}\left|\;\sin \frac{\pi x}{L}\;\right|^{-\alpha} \\
|
||||
J(x) &= \frac{J_0 J'(x)}{\sum_y J'(y)}
|
||||
\end{aligned}\]</span> % The scaling ensures that, in the ordered phase, the overall potential felt by each site due to the rest of the system is independent of system size.</p>
|
||||
\end{aligned}
|
||||
\]</span></p>
|
||||
<p>The scaling ensures that, in the ordered phase, the overall potential felt by each site due to the rest of the system is independent of system size.</p>
|
||||
</section>
|
||||
<section id="binder-cumulants" class="level2">
|
||||
<h2>Binder Cumulants</h2>
|
||||
<p>The Binder cumulant is defined as: <span class="math display">\[U_B = 1 - \frac{\tex{\mu_4}}{3\tex{\mu_2}^2}\]</span> % where <span class="math display">\[\mu_n = \tex{(m - \tex{m})^n}\]</span> % are the central moments of the order parameter m: <span class="math display">\[m = \sum_i (-1)^i (2n_i - 1) / N\]</span> % The Binder cumulant evaluated against temperature can be used as a diagnostic for the existence of a phase transition. If multiple such curves are plotted for different system sizes, a crossing indicates the location of a critical point <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">9</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">10</a>]</span>.</p>
|
||||
</section>
|
||||
<section id="diagnostics-of-localisation" class="level2">
|
||||
<h2>Diagnostics of Localisation</h2>
|
||||
<section id="inverse-participation-ratio" class="level3">
|
||||
<h3>Inverse Participation Ratio</h3>
|
||||
<p>The inverse participation ratio is defined for a normalised wave function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i \abs{\psi_i}^2 = 1\)</span> as its fourth moment <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">11</a>]</span>: <span class="math display">\[
|
||||
P^{-1} = \sum_i \abs{\psi_i}^4
|
||||
\]</span> % It acts as a measure of the portion of space occupied by the wave function. For localised states it will be independent of system size while for plane wave states in d dimensions $P = L^d $. States may also be intermediate between localised and extended, described by their fractal dimensionality <span class="math inline">\(d > d* > 0\)</span>: <span class="math display">\[
|
||||
P(L) \goeslike L^{d*}
|
||||
\]</span> % For extended states <span class="math inline">\(d* = 0\)</span> while for localised ones <span class="math inline">\(d* = 0\)</span>. In this work we take use an energy resolved IPR <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">12</a>]</span>: <span class="math display">\[
|
||||
DOS(\omega) = \sum_n \delta(\omega - \epsilon_n)
|
||||
IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_{n,i}}^4
|
||||
\]</span> Where <span class="math inline">\(\psi_{n,i}\)</span> is the wavefunction corresponding to the energy <span class="math inline">\(\epsilon_n\)</span> at the ith site. In practice we bin the energies and IPRs into a fine energy grid and use Lorentzian smoothing if necessary.</p>
|
||||
<!-- ![An MCMC walk starting from the staggered charge density wave ground state for a system with $N = 100$ sites and 10,000 MCMC steps. In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation $m = N^{-1} \sum_i (-1)^i \; S_i$ order parameter is plotted below. At this temperature the thermal average of m is zero, while the initial state has m = 1. We see that it takes about 1000 steps for the system to converge, after which it moves about the m = 0 average with a finite auto-correlation time. $t = 1, \alpha = 1.25, T = 3, J = U = 5$ [\[fig:raw\]]{#fig:raw label="fig:raw"}](figs/lsr/raw_steps_single_flip.pdf){#fig:raw width="\\columnwidth"} -->
|
||||
<p>The Binder cumulant is defined as:</p>
|
||||
<p><span class="math display">\[
|
||||
U_B = 1 - \frac{\langle\mu_4\rangle}{3\langle\mu_2\rangle^2}
|
||||
\]</span></p>
|
||||
<p>where <span class="math inline">\(\mu_n = \langle(m - \langle m\rangle)^n\rangle\)</span> are the central moments of the order parameter <span class="math inline">\(m = \sum_i (-1)^i (2n_i - 1) / N\)</span>. The Binder cumulant evaluated against temperature is a diagnostic for the existence of a phase transition. If multiple such curves are plotted for different system sizes, a crossing indicates the location of a critical point while the lines do not cross for systems that don’t have a phase transition in the thermodynamic limit <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">9</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">10</a>]</span>.</p>
|
||||
<p><img src="/assets/thesis/fk_chapter/binder_cumulants/binder_cumulants.svg" id="fig:binder_cumulants" data-short-caption="Binder Cumulants" style="width:100.0%" alt="(Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. ¿fig:phase_diagram? . All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly mentioned." /> <!-- ![An MCMC walk starting from the staggered charge density wave ground state for a system with $N = 100$ sites and 10,000 MCMC steps. In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation $m = N^{-1} \sum_i (-1)^i \; S_i$ order parameter is plotted below. At this temperature the thermal average of m is zero, while the initial state has m = 1. We see that it takes about 1000 steps for the system to converge, after which it moves about the m = 0 average with a finite auto-correlation time. $t = 1, \alpha = 1.25, T = 3, J = U = 5$ [\[fig:raw\]]{#fig:raw label="fig:raw"}](figs/lsr/raw_steps_single_flip.pdf){#fig:raw width="\\columnwidth"} --></p>
|
||||
<p><span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> sidesteps these issues by defining a random walk that focuses on the states with the greatest Boltzmann weight. At low temperatures this means we need only visit a few low energy states to make good estimates while at high temperatures the weights become uniform so a small number of samples distributed across the state space suffice. However we will see that the method is not without difficulties of its own.</p>
|
||||
<!-- ![Two MCMC chains starting from the same initial state for a system with $N = 90$ sites and 1000 MCMC steps. In this simulation the MCMC step is defined differently: an attempt is made to flip n spins, where n is drawn from Uniform(1,N). This is repeated $N^2/100$ times for each step. This trades off computation time for storage space, as it makes the samples less correlated, giving smaller statistical error for a given number of stored samples. These simulations therefore have the potential to necessitate $N^2/100$ matrix diagonalisations for every MCMC sample, though this can be cut down with caching and other tricks. $t = 1, \alpha = 1.25, T = 2.2, J = U = 5$ [\[fig:single\]]{#fig:single label="fig:single"}](figs/lsr/single.pdf){#fig:single width="\\columnwidth"} -->
|
||||
<p>In implementation <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> can be boiled down to choosing a transition function <span class="math inline">\(\mathcal{T}(\s_{t} \rightarrow \s_t+1)\)</span> where <span class="math inline">\(\s\)</span> are vectors representing classical spin configurations. We start in some initial state <span class="math inline">\(\s_0\)</span> and then repeatedly jump to new states according to the probabilities given by <span class="math inline">\(\mathcal{T}\)</span>. This defines a set of random walks <span class="math inline">\(\{\s_0\ldots \s_i\ldots \s_N\}\)</span>. Fig. <a href="#fig:single" data-reference-type="ref" data-reference="fig:single">2</a> shows this in practice: we have a (rather small) ensemble of <span class="math inline">\(M = 2\)</span> walkers starting at the same point in state space and then spreading outwards by flipping spins along the way.</p>
|
||||
@ -166,7 +151,6 @@ IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_
|
||||
<p>Where the <code>sample_T</code> function here produces a state with probability determined by the <code>current_state</code> and the transition function <span class="math inline">\(\mathcal{T}\)</span>.</p>
|
||||
<p>If we ran many such walkers in parallel we could then approximate the distribution <span class="math inline">\(p_t(\s; \s_0)\)</span> which tells us where the walkers are likely to be after they’ve evolved for <span class="math inline">\(t\)</span> steps from an initial state <span class="math inline">\(\s_0\)</span>. We need to carefully choose <span class="math inline">\(\mathcal{T}\)</span> such that after a large number of steps <span class="math inline">\(k\)</span> (the convergence time) the probability <span class="math inline">\(p_t(\s;\s_0)\)</span> approaches the thermal distribution <span class="math inline">\(P(\s; \beta) = \mathcal{Z}^{-1} e^{-\beta F(\s)}\)</span>. This turns out to be quite easy to achieve using the Metropolis-Hasting algorithm.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="convergence-time" class="level2">
|
||||
<h2>Convergence Time</h2>
|
||||
<p>Considering <span class="math inline">\(p(\s)\)</span> as a vector <span class="math inline">\(\vec{p}\)</span> whose jth entry is the probability of the jth state <span class="math inline">\(p_j = p(\s_j)\)</span>, and writing <span class="math inline">\(\mathcal{T}\)</span> as the matrix with entries <span class="math inline">\(T_{ij} = \mathcal{T}(\s_j \rightarrow \s_i)\)</span> we can write the update rule for the ensemble probability as: <span class="math display">\[\vec{p}_{t+1} = \mathcal{T} \vec{p}_t \implies \vec{p}_{t} = \mathcal{T}^t \vec{p}_0\]</span> where <span class="math inline">\(\vec{p}_0\)</span> is vector which is one on the starting state and zero everywhere else. Since all states must transition to somewhere with probability one: <span class="math inline">\(\sum_i T_{ij} = 1\)</span>.</p>
|
||||
@ -214,12 +198,6 @@ IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_
|
||||
<div id="ref-musialMonteCarloSimulations2002" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">G. Musiał, L. Dȩbski, and G. Kamieniarz, <em><a href="https://doi.org/10.1103/PhysRevB.66.012407">Monte Carlo Simulations of Ising-Like Phase Transitions in the Three-Dimensional Ashkin-Teller Model</a></em>, Phys. Rev. B <strong>66</strong>, 012407 (2002).</div>
|
||||
</div>
|
||||
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </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-andersonAbsenceDiffusionCertain1958" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[12] </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>
|
||||
</section>
|
||||
|
||||
|
||||
@ -67,23 +67,19 @@ image:
|
||||
<section id="lrfk-results-phase-diagram" class="level2">
|
||||
<h2>Phase Diagram</h2>
|
||||
<p>Using the MCMC methods described in the previous section I will now discuss the results of extensive MCMC simulations of the model, starting with the phase diagram in the fermion spin coupling <span class="math inline">\(U\)</span>, the strength of the long range spin-spin coupling <span class="math inline">\(J\)</span> and the temperature <span class="math inline">\(T\)</span>.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/binder_cumulants/binder_cumulants.svg" id="fig:binder_cumulants" data-short-caption="Binder Cumulants" style="width:100.0%" alt="Figure 1: (Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. 2 . All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly mentioned." />
|
||||
<figcaption aria-hidden="true">Figure 1: (Left) The order parameters, <span class="math inline">\(\langle m^2 \rangle\)</span>(solid) and <span class="math inline">\(1 - f\)</span> (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle\)</span> . (Right) The crossing of the Binder cumulant, <span class="math inline">\(B = \langle m^4 \rangle / \langle m^2 \rangle^2\)</span>, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. <a href="#fig:phase_diagram">2</a> . All plots use system sizes <span class="math inline">\(N = [10,20,30,50,70,110,160,250]\)</span> and lines are coloured from <span class="math inline">\(N = 10\)</span> in dark blue to <span class="math inline">\(N = 250\)</span> in yellow. The parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span> except where explicitly mentioned.</figcaption>
|
||||
</figure>
|
||||
<p>Fig fig. <a href="#fig:phase_diagram">2</a> shows the phase diagram for constant <span class="math inline">\(U=5\)</span> and constant <span class="math inline">\(J=5\)</span>, respectively. The transition temperatures were determined from the crossings of the Binder cumulants <span class="math inline">\(B_4 = \langle m^4 \rangle /\langle m^2 \rangle^2\)</span> <span class="citation" data-cites="binderFiniteSizeScaling1981"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">1</a>]</span>. For a representative set of parameters, fig. <a href="#fig:binder_cumulants">1</a> shows the order parameter <span class="math inline">\(\langle m \rangle^2\)</span> and the Binder cumulants, both as functions of system size and temperature. The crossings confirm that the system has a FTPT and that the ordered phase is not a finite size effect.</p>
|
||||
<p>Fig fig. <a href="#fig:phase_diagram">1</a> shows the phase diagram for constant <span class="math inline">\(U=5\)</span> and constant <span class="math inline">\(J=5\)</span>, respectively. The transition temperatures were determined from the crossings of the Binder cumulants <span class="math inline">\(B_4 = \langle m^4 \rangle /\langle m^2 \rangle^2\)</span> <span class="citation" data-cites="binderFiniteSizeScaling1981"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">1</a>]</span>. For a representative set of parameters, fig. <strong>¿fig:binder_cumulants?</strong> shows the order parameter <span class="math inline">\(\langle m \rangle^2\)</span> and the Binder cumulants, both as functions of system size and temperature. The crossings confirm that the system has a FTPT and that the ordered phase is not a finite size effect.</p>
|
||||
<p>The CDW transition temperature is largely independent from the strength of the interaction <span class="math inline">\(U\)</span>. This demonstrates that the phase transition is driven by the long-range term <span class="math inline">\(J\)</span> with little effect from the coupling to the fermions <span class="math inline">\(U\)</span>. The physics of the spin sector in the long-range FK model mimics that of the long range Ising (LRI) model and is not significantly altered by the presence of the fermions. In two dimensions the transition to the CDW phase is mediated by an RKYY-like interaction <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">2</a>]</span> but this is insufficient to stabilise long range order in one dimension. That the critical temperature <span class="math inline">\(T_c\)</span> does not depend on <span class="math inline">\(U\)</span> in our model further confirms this.</p>
|
||||
<p>The main order parameters for this model is the staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> that signals the onset of a charge density wave (CDW) phase at low temperature. However, my main interest concerns the additional structure of the fermionic sector in the high temperature phase. Following Ref. <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">3</a>]</span>, we can distinguish between the Mott and Anderson insulating phases. The Mott insulator is characterised by a gapped DOS in the absence of a CDW, instead the gap is driven entirely by interaction effect. Thus, the opening of a gap for large <span class="math inline">\(U\)</span> is distinct from the gap-opening induced by the translational symmetry breaking in the CDW state below <span class="math inline">\(T_c\)</span>, see also fig. <strong>¿fig:gap_opening?</strong>. The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates hence it also has insulating character.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg" id="fig:phase_diagram" data-short-caption="Long Range Falicov Kimball Model Phase Diagram" style="width:100.0%" alt="Figure 2: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature T_c, linear in J. (Right) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas." />
|
||||
<figcaption aria-hidden="true">Figure 2: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at <span class="math inline">\(U = 5\)</span>: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (Right) The TU plane at <span class="math inline">\(J = 5\)</span>: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at <span class="math inline">\(E=0\)</span> in the single particle energy spectrum. <span class="math inline">\(U_c\)</span> is independent of temperature. At <span class="math inline">\(U = 0\)</span> the fermions are decoupled from the spins forming a simple Fermi gas.</figcaption>
|
||||
<img src="/assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg" id="fig:phase_diagram" data-short-caption="Long Range Falicov Kimball Model Phase Diagram" style="width:100.0%" alt="Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature T_c, linear in J. (Right) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas." />
|
||||
<figcaption aria-hidden="true">Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at <span class="math inline">\(U = 5\)</span>: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (Right) The TU plane at <span class="math inline">\(J = 5\)</span>: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at <span class="math inline">\(E=0\)</span> in the single particle energy spectrum. <span class="math inline">\(U_c\)</span> is independent of temperature. At <span class="math inline">\(U = 0\)</span> the fermions are decoupled from the spins forming a simple Fermi gas.</figcaption>
|
||||
</figure>
|
||||
</section>
|
||||
<section id="localisation-properties" class="level2">
|
||||
<h2>Localisation Properties</h2>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/DOS/DOS.svg" id="fig:DOS" data-short-caption="Energy resolved DOS($\omega$) and $\tau$ (the scaling exponent of IPR($\omega$) against system size $N$)." style="width:100.0%" alt="Figure 3: Energy resolved DOS(\omega) against system size N in all three phases. The charge density wave phase is shown in both the high and low U regime for completeness. The top left panel shows the Anderson phase at U = 2 and high T = 2.5, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at U = 2 and low T = 1.5, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at U = 5 and high T = 2.5 the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at U = 5 and T = 1.5 is qualitatively similar to the lower left panel except that the gap scales with U. For all the plots J = 5,\;\alpha = 1.25." />
|
||||
<figcaption aria-hidden="true">Figure 3: Energy resolved DOS(<span class="math inline">\(\omega\)</span>) against system size <span class="math inline">\(N\)</span> in all three phases. The charge density wave phase is shown in both the high and low <span class="math inline">\(U\)</span> regime for completeness. The top left panel shows the Anderson phase at <span class="math inline">\(U = 2\)</span> and high <span class="math inline">\(T = 2.5\)</span>, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at <span class="math inline">\(U = 2\)</span> and low <span class="math inline">\(T = 1.5\)</span>, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at <span class="math inline">\(U = 5\)</span> and high <span class="math inline">\(T = 2.5\)</span> the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at <span class="math inline">\(U = 5\)</span> and <span class="math inline">\(T = 1.5\)</span> is qualitatively similar to the lower left panel except that the gap scales with <span class="math inline">\(U\)</span>. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span>.</figcaption>
|
||||
<img src="/assets/thesis/fk_chapter/DOS/DOS.svg" id="fig:DOS" data-short-caption="Energy resolved DOS($\omega$) and $\tau$ (the scaling exponent of IPR($\omega$) against system size $N$)." style="width:100.0%" alt="Figure 2: Energy resolved DOS(\omega) against system size N in all three phases. The charge density wave phase is shown in both the high and low U regime for completeness. The top left panel shows the Anderson phase at U = 2 and high T = 2.5, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at U = 2 and low T = 1.5, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at U = 5 and high T = 2.5 the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at U = 5 and T = 1.5 is qualitatively similar to the lower left panel except that the gap scales with U. For all the plots J = 5,\;\alpha = 1.25." />
|
||||
<figcaption aria-hidden="true">Figure 2: Energy resolved DOS(<span class="math inline">\(\omega\)</span>) against system size <span class="math inline">\(N\)</span> in all three phases. The charge density wave phase is shown in both the high and low <span class="math inline">\(U\)</span> regime for completeness. The top left panel shows the Anderson phase at <span class="math inline">\(U = 2\)</span> and high <span class="math inline">\(T = 2.5\)</span>, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at <span class="math inline">\(U = 2\)</span> and low <span class="math inline">\(T = 1.5\)</span>, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at <span class="math inline">\(U = 5\)</span> and high <span class="math inline">\(T = 2.5\)</span> the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at <span class="math inline">\(U = 5\)</span> and <span class="math inline">\(T = 1.5\)</span> is qualitatively similar to the lower left panel except that the gap scales with <span class="math inline">\(U\)</span>. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span>.</figcaption>
|
||||
</figure>
|
||||
<p>The MCMC formulation suggests viewing the spin configurations as a form of annealed binary disorder whose probability distribution is given by the Boltzmann weight <span class="math inline">\(e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]}\)</span>. This makes apparent the link to the study of disordered systems and Anderson localisation. While these systems are typically studied by defining the probability distribution for the quenched disorder potential externally, here we have a translation invariant system with disorder as a natural consequence of the Ising background field conserved under the dynamics.</p>
|
||||
<p>In the limits of zero and infinite temperature, our model becomes a simple tight-binding model for the fermions. At zero temperature, the spin background is in one of the two translation invariant AFM ground states with two gapped fermionic CDW bands at energies <span class="math display">\[E_{\pm} = \pm\sqrt{\frac{1}{4}U^2 + 2t^2(1 + \cos ka)^2}\;.\]</span></p>
|
||||
@ -91,32 +87,39 @@ image:
|
||||
\mathrm{DOS}(\vec{S}, \omega)& = N^{-1} \sum_{i} \delta(\epsilon_i - \omega)\\
|
||||
\mathrm{IPR}(\vec{S}, \omega)& = \; N^{-1} \mathrm{DOS}(\vec{S}, \omega)^{-1} \sum_{i,j} \delta(\epsilon_i - \omega)\;\psi^{4}_{i,j}\end{aligned}\]</span> where <span class="math inline">\(\epsilon_i\)</span> and <span class="math inline">\(\psi_{i,j}\)</span> are the <span class="math inline">\(i\)</span>th energy level and <span class="math inline">\(j\)</span>th element of the corresponding eigenfunction, both dependent on the background spin configuration <span class="math inline">\(\vec{S}\)</span>.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/DOS/IPR_scaling.svg" id="fig:IPR_scaling" data-short-caption="Energy resolved DOS($\omega$) and $\tau$ (the scaling exponent of IPR($\omega$) against system size $N$)." style="width:100.0%" alt="Figure 4: The IPR(\omega) scaling with N at fixed energy for each phase and for points both in the gap (\omega_0) and in a band (\omega_1). The slope of the line yields the scaling exponent \tau defined by \mathrm{IPR} \propto N^{-\tau}). \tau close to zero implies that the states at that energy are localised while \tau = -d corresponds to extended states where d is the system dimension. All but the bands of the charge density wave phase are approximately localised with \tau is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots J = 5,\;\alpha = 1.25. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states." />
|
||||
<figcaption aria-hidden="true">Figure 4: The IPR(<span class="math inline">\(\omega\)</span>) scaling with <span class="math inline">\(N\)</span> at fixed energy for each phase and for points both in the gap (<span class="math inline">\(\omega_0\)</span>) and in a band (<span class="math inline">\(\omega_1\)</span>). The slope of the line yields the scaling exponent <span class="math inline">\(\tau\)</span> defined by <span class="math inline">\(\mathrm{IPR} \propto N^{-\tau}\)</span>). <span class="math inline">\(\tau\)</span> close to zero implies that the states at that energy are localised while <span class="math inline">\(\tau = -d\)</span> corresponds to extended states where <span class="math inline">\(d\)</span> is the system dimension. All but the bands of the charge density wave phase are approximately localised with <span class="math inline">\(\tau\)</span> is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span>. The measured <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\((0.06\pm0.01, 0.02\pm0.01\)</span> (b) <span class="math inline">\(0.04\pm0.02, 0.00\pm0.01\)</span> (c) <span class="math inline">\(0.05\pm0.03, 0.30\pm0.03\)</span> (d) <span class="math inline">\(0.06\pm0.04, 0.15\pm0.05\)</span> We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states.</figcaption>
|
||||
<img src="/assets/thesis/fk_chapter/DOS/IPR_scaling.svg" id="fig:IPR_scaling" data-short-caption="Energy resolved DOS($\omega$) and $\tau$ (the scaling exponent of IPR($\omega$) against system size $N$)." style="width:100.0%" alt="Figure 3: The IPR(\omega) scaling with N at fixed energy for each phase and for points both in the gap (\omega_0) and in a band (\omega_1). The slope of the line yields the scaling exponent \tau defined by \mathrm{IPR} \propto N^{-\tau}). \tau close to zero implies that the states at that energy are localised while \tau = -d corresponds to extended states where d is the system dimension. All but the bands of the charge density wave phase are approximately localised with \tau is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots J = 5,\;\alpha = 1.25. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states." />
|
||||
<figcaption aria-hidden="true">Figure 3: The IPR(<span class="math inline">\(\omega\)</span>) scaling with <span class="math inline">\(N\)</span> at fixed energy for each phase and for points both in the gap (<span class="math inline">\(\omega_0\)</span>) and in a band (<span class="math inline">\(\omega_1\)</span>). The slope of the line yields the scaling exponent <span class="math inline">\(\tau\)</span> defined by <span class="math inline">\(\mathrm{IPR} \propto N^{-\tau}\)</span>). <span class="math inline">\(\tau\)</span> close to zero implies that the states at that energy are localised while <span class="math inline">\(\tau = -d\)</span> corresponds to extended states where <span class="math inline">\(d\)</span> is the system dimension. All but the bands of the charge density wave phase are approximately localised with <span class="math inline">\(\tau\)</span> is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span>. The measured <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\((0.06\pm0.01, 0.02\pm0.01\)</span> (b) <span class="math inline">\(0.04\pm0.02, 0.00\pm0.01\)</span> (c) <span class="math inline">\(0.05\pm0.03, 0.30\pm0.03\)</span> (d) <span class="math inline">\(0.06\pm0.04, 0.15\pm0.05\)</span> We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states.</figcaption>
|
||||
</figure>
|
||||
<p>The scaling of the IPR with system size</p>
|
||||
<p><span class="math display">\[\mathrm{IPR} \propto N^{-\tau}\]</span></p>
|
||||
<p>depends on the localisation properties of states at that energy. For delocalised states, e.g. Bloch waves, <span class="math inline">\(\tau\)</span> is the physical dimension. For fully localised states <span class="math inline">\(\tau\)</span> goes to zero in the thermodynamic limit. However, for special types of disorder such as binary disorder, the localisation lengths can be large comparable to the system size at hand, which can make it difficult to extract the correct scaling. An additional complication arises from the fact that the scaling exponent may display intermediate behaviours for correlated disorder and in the vicinity of a localisation-delocalisation transition <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">5</a>,<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">6</a>]</span>. The thermal defects of the CDW phase lead to a binary disorder potential with a finite correlation length, which in principle could result in delocalized eigenstates.</p>
|
||||
<p>The key question for our system is then: How is the <span class="math inline">\(T=0\)</span> CDW phase with fully delocalized fermionic states connected to the fully localized phase at high temperatures?</p>
|
||||
<p>For a representative set of parameters covering all three phases fig. <a href="#fig:DOS">3</a> shows the density of states as function of energy while fig. <a href="#fig:IPR_scaling">4</a> shows <span class="math inline">\(\tau\)</span>, the scaling exponent of the IPR with system size, The DOS is symmetric about <span class="math inline">\(0\)</span> because of the particle hole symmetry of the model. At high temperatures, all of the eigenstates are localised in both the Mott and Anderson phases (with <span class="math inline">\(\tau \leq 0.07\)</span> for our system sizes). We also checked that the states are localised by direct inspection. Note that there are in-gap states for instance at <span class="math inline">\(\omega_0\)</span>, below the upper band which are localized and smoothly connected across the phase transition.</p>
|
||||
<p>For a representative set of parameters covering all three phases fig. <a href="#fig:DOS">2</a> shows the density of states as function of energy while fig. <a href="#fig:IPR_scaling">3</a> shows <span class="math inline">\(\tau\)</span>, the scaling exponent of the IPR with system size, The DOS is symmetric about <span class="math inline">\(0\)</span> because of the particle hole symmetry of the model. At high temperatures, all of the eigenstates are localised in both the Mott and Anderson phases (with <span class="math inline">\(\tau \leq 0.07\)</span> for our system sizes). We also checked that the states are localised by direct inspection. Note that there are in-gap states for instance at <span class="math inline">\(\omega_0\)</span>, below the upper band which are localized and smoothly connected across the phase transition.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U5.svg" id="fig:gap_opening_U5" data-short-caption="DOS and Scaling Exponents for the transition from CDW to the Mott phase" style="width:100.0%" alt="Figure 5: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
|
||||
<figcaption aria-hidden="true">Figure 5: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW to gapped Mott phase transition at <span class="math inline">\(U=5\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
|
||||
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U5.svg" id="fig:gap_opening_U5" data-short-caption="DOS and Scaling Exponents for the transition from CDW to the Mott phase" style="width:100.0%" alt="Figure 4: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
|
||||
<figcaption aria-hidden="true">Figure 4: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW to gapped Mott phase transition at <span class="math inline">\(U=5\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
|
||||
</figure>
|
||||
<p>In the CDW phases at <span class="math inline">\(U=2\)</span> and <span class="math inline">\(U=5\)</span>, we find for the states within the gapped CDW bands, e.g. at <span class="math inline">\(\omega_1\)</span>, scaling exponents <span class="math inline">\(\tau = 0.30\pm0.03\)</span> and <span class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This surprising finding suggests that the CDW bands are partially delocalised with multi-fractal behaviour of the wavefunctions <span class="citation" data-cites="eversAndersonTransitions2008"> [<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">6</a>]</span>. This phenomenon would be unexpected in a 1D model as they generally do not support delocalisation in the presence of disorder except as the result of correlations in the emergent disorder potential <span class="citation" data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"> [<a href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">7</a>,<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">8</a>]</span>. However, we later show by comparison to an uncorrelated Anderson model that these nonzero exponents are a finite size effect and the states are localised with a finite <span class="math inline">\(\xi\)</span> similar to the system size, an example of weak localisation. As a result, the IPR does not scale correctly until the system size has grown much larger than <span class="math inline">\(\xi\)</span>. fig. <strong>¿fig:DM_IPR_scaling?</strong> shows that the scaling of the IPR in the CDW phase does flatten out eventually.</p>
|
||||
<p>In the CDW phases at <span class="math inline">\(U=2\)</span> and <span class="math inline">\(U=5\)</span>, we find for the states within the gapped CDW bands, e.g. at <span class="math inline">\(\omega_1\)</span>, scaling exponents <span class="math inline">\(\tau = 0.30\pm0.03\)</span> and <span class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This surprising finding suggests that the CDW bands are partially delocalised with multi-fractal behaviour of the wavefunctions <span class="citation" data-cites="eversAndersonTransitions2008"> [<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">6</a>]</span>. This phenomenon would be unexpected in a 1D model as they generally do not support delocalisation in the presence of disorder except as the result of correlations in the emergent disorder potential <span class="citation" data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"> [<a href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">7</a>,<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">8</a>]</span>. However, we later show by comparison to an uncorrelated Anderson model that these nonzero exponents are a finite size effect and the states are localised with a finite <span class="math inline">\(\xi\)</span> similar to the system size, an example of weak localisation. As a result, the IPR does not scale correctly until the system size has grown much larger than <span class="math inline">\(\xi\)</span>. fig. <a href="#fig:DM_IPR_scaling">7</a> shows that the scaling of the IPR in the CDW phase does flatten out eventually.</p>
|
||||
<p>Next, we use the DOS and the scaling exponent <span class="math inline">\(\tau\)</span> to explore the localisation properties over the energy-temperature plane in fig. <strong>¿fig:gap_opening?</strong>. Gapped areas are shown in white, which highlights the distinction between the gapped Mott phase and the ungapped Anderson phase. In-gap states appear just below the critical point, smoothly filling the bandgap in the Anderson phase and forming islands in the Mott phase. As in the finite <span class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">9</a>]</span> and infinite dimensional <span class="citation" data-cites="hassanSpectralPropertiesChargedensitywave2007"> [<a href="#ref-hassanSpectralPropertiesChargedensitywave2007" role="doc-biblioref">10</a>]</span> cases, the in-gap states merge and are pushed to lower energy for decreasing U as the <span class="math inline">\(T=0\)</span> CDW gap closes. Intuitively, the presence of in-gap states can be understood as a result of domain wall fluctuations away from the AFM ordered background. These domain walls act as local potentials for impurity-like bound states <span class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">9</a>]</span>.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U2.svg" id="fig:gap_opening_U2" data-short-caption="DOS and Scaling Exponents for the transition from CDW to the Anderson Phase" style="width:100.0%" alt="Figure 6: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at U=2. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
|
||||
<figcaption aria-hidden="true">Figure 6: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at <span class="math inline">\(U=2\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
|
||||
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U2.svg" id="fig:gap_opening_U2" data-short-caption="DOS and Scaling Exponents for the transition from CDW to the Anderson Phase" style="width:100.0%" alt="Figure 5: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at U=2. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
|
||||
<figcaption aria-hidden="true">Figure 5: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at <span class="math inline">\(U=2\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
|
||||
</figure>
|
||||
<p>In order to understand the localization properties we can compare the behaviour of our model with that of a simpler Anderson disorder model (DM) in which the spins are replaced by a CDW background with uncorrelated binary defect potentials. This is defined by replacing the spin degree of freedom in the FK model <span class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder potential <span class="math inline">\(d_i = \pm \tfrac{1}{2}\)</span> controlled by a defect density <span class="math inline">\(\rho\)</span> such that <span class="math inline">\(d_i = -\tfrac{1}{2}\)</span> with probability <span class="math inline">\(\rho/2\)</span> and <span class="math inline">\(d_i = \tfrac{1}{2}\)</span> otherwise. <span class="math inline">\(\rho/2\)</span> is used rather than <span class="math inline">\(\rho\)</span> so that the disorder potential takes on the zero temperature CDW ground state at <span class="math inline">\(\rho = 0\)</span> and becomes a random choice over spin states at <span class="math inline">\(\rho = 1\)</span> i.e the infinite temperature limit.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) \\
|
||||
& -\;t \sum_{i} c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i}
|
||||
\end{aligned}\]</span></p>
|
||||
<p>fig. <strong>¿fig:DM_DOS?</strong> and fig. <strong>¿fig:DM_IPR_scaling?</strong> compare the FK model to the disorder model at different system sizes, matching the defect densities of the disorder model to the FK model at <span class="math inline">\(N = 270\)</span> above and below the CDW transition. We find very good, even quantitative, agreement between the FK and disorder models, which suggests that correlations in the spin sector do not play a significant role.</p>
|
||||
<p>fig. <a href="#fig:DM_DOS">6</a> and fig. <a href="#fig:DM_IPR_scaling">7</a> compare the FK model to the disorder model at different system sizes, matching the defect densities of the disorder model to the FK model at <span class="math inline">\(N = 270\)</span> above and below the CDW transition. We find very good, even quantitative, agreement between the FK and disorder models, which suggests that correlations in the spin sector do not play a significant role.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/disorder_model/DM_DOS.svg" id="fig:DM_DOS" data-short-caption="A comparison of the full FK model to a simple binary disorder model." style="width:100.0%" alt="Figure 6: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 2, the Energy resolved DOS(\omega) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters." />
|
||||
<figcaption aria-hidden="true">Figure 6: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 < \rho < 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:DOS">2</a>, the Energy resolved DOS(<span class="math inline">\(\omega\)</span>) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters.</figcaption>
|
||||
</figure>
|
||||
<p>As we can sample directly from the disorder model, rather than through MCMC, the samples are uncorrelated. Hence we can evaluate much larger system sizes with the disorder model which enables us to pin down the correct localisation effects. In particular, what appear to be delocalized states for small system sizes eventually turn out to be states with large localization length. The localization length diverges towards the ordered zero temperature CDW state. The interplay of interactions, which here produce as peculiar binary potential, and localization can be very intricate and the added advantage of a 1D model is that we can explore very large system sizes.</p>
|
||||
<p><img src="/assets/thesis/fk_chapter/disorder_model/DM_DOS.svg" id="fig:DM_DOS" data-short-caption="A comparison of the full FK model to a simple binary disorder model." style="width:100.0%" alt="A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 3, the Energy resolved DOS(\omega) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters." /> <img src="/assets/thesis/fk_chapter/disorder_model/DM_IPR_scaling.svg" id="fig:DM_IPR_scaling" data-short-caption="A comparison of the full FK model to a simple binary disorder model." style="width:100.0%" alt="A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 4 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation [3], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N > 400" /></p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/disorder_model/DM_IPR_scaling.svg" id="fig:DM_IPR_scaling" data-short-caption="A comparison of the full FK model to a simple binary disorder model." style="width:100.0%" alt="Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 3 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation [3], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N > 400" />
|
||||
<figcaption aria-hidden="true">Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 < \rho < 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:IPR_scaling">3</a> <span class="math inline">\(\tau(\omega)\)</span> the scaling of IPR(<span class="math inline">\(\omega\)</span>) with system size, , is shown both in gap (<span class="math inline">\(\omega_0\)</span>) and in the band (<span class="math inline">\(\omega_1\)</span>). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">3</a>]</span>, hence all the states are indeed localised as one would expect in 1D. The disorder model <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span class="math inline">\(0.01\pm0.04, -0.01\pm0.04\)</span> (c) <span class="math inline">\(0.05\pm0.06, 0.04\pm0.06\)</span> (d) <span class="math inline">\(-0.03\pm0.06, 0.01\pm0.06\)</span>. The lines are fit on system sizes <span class="math inline">\(N > 400\)</span></figcaption>
|
||||
</figure>
|
||||
</section>
|
||||
</section>
|
||||
<section id="fk-conclusion" class="level1">
|
||||
|
||||
2109
assets/thesis/background_chapter/localisation_radius_vs_length.svg
Normal file
2109
assets/thesis/background_chapter/localisation_radius_vs_length.svg
Normal file
File diff suppressed because one or more lines are too long
|
After Width: | Height: | Size: 158 KiB |
Loading…
x
Reference in New Issue
Block a user