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title: Background - Disorder & Localisation
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<title>Background - Disorder & Localisation</title>
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<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
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<ul>
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<li><a href="#localisation-anderson-many-body-and-disorder-free" id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: Anderson, Many Body and Disorder-Free</a></li>
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<li><a href="#disorder-and-spin-liquids" id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
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<li><a href="#amorphous-magnetism" id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
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<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
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<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
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<li><a href="#localisation-anderson-many-body-and-disorder-free" id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: Anderson, Many Body and Disorder-Free</a></li>
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<li><a href="#disorder-and-spin-liquids" id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
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<li><a href="#amorphous-magnetism" id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
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<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
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<p>2 Background</p>
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<hr />
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<section id="bg-disorder-and-localisation" class="level1">
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<h1>Disorder and Localisation</h1>
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<section id="localisation-anderson-many-body-and-disorder-free" class="level2">
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<h2>Localisation: Anderson, Many Body and Disorder-Free</h2>
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</section>
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<section id="disorder-and-spin-liquids" class="level2">
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<h2>Disorder and Spin liquids</h2>
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</section>
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<section id="amorphous-magnetism" class="level2">
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<h2>Amorphous Magnetism</h2>
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<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
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</section>
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<section id="localisation" class="level2">
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<h2>Localisation</h2>
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<p>The discovery of localisation in quantum systems surprising at the time given the seeming ubiquity of extended Bloch states. Later, when thermalisation in quantum systems gained interest, localisation phenomena again stood out as counterexamples to the eigenstate thermalisation hypothesis <span class="citation" data-cites="abaninRecentProgressManybody2017 srednickiChaosQuantumThermalization1994"> [<a href="#ref-abaninRecentProgressManybody2017" role="doc-biblioref">1</a>,<a href="#ref-srednickiChaosQuantumThermalization1994" role="doc-biblioref">2</a>]</span>, allowing quantum systems to avoid to retain memory of their initial conditions in the face of thermal noise.</p>
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<p>The simplest and first discovered kind is Anderson localisation, first studied in 1958 <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">3</a>]</span> in the context of non-interacting fermions subject to a static or quenched disorder potential <span class="math inline">\(V_j\)</span> drawn uniformly from the interval <span class="math inline">\([-W,W]\)</span></p>
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<p><span class="math display">\[
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H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j
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\]</span></p>
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<p>this model exhibits exponentially localised eigenfunctions <span class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\)</span> which cannot contribute to transport processes. Initially it was thought that in one dimensional disordered models, all states would be localised, however it was later shown that in the presence of correlated disorder, bands of extended states can exist <span class="citation" data-cites="izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a href="#ref-izrailevLocalizationMobilityEdge1999" role="doc-biblioref">4</a>–<a href="#ref-izrailevAnomalousLocalizationLowDimensional2012" role="doc-biblioref">6</a>]</span>.</p>
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<p>Later localisation was found in interacting many-body systems with quenched disorder:</p>
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<p><span class="math display">\[
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H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U\sum_{jk} n_j n_k
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\]</span></p>
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<p>where the number operators <span class="math inline">\(n_j = c^\dagger_j c_j\)</span>. Here, in contrast to the Anderson model, localisation phenomena can be proven robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL) <span class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016"> [<a href="#ref-imbrieManyBodyLocalizationQuantum2016" role="doc-biblioref">7</a>]</span>.</p>
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<p>Both MBL and Anderson localisation depend crucially on the presence of quenched disorder. This has led to ongoing interest in the possibility of disorder-free localisation, in which the disorder necessary to generate localisation is generated entirely from the dynamics of the model. This contracts with typical models of disordered systems in which disorder is explicitly introduced into the Hamilton or the initial state.</p>
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<p>The concept of disorder-free localisation was first proposed in the context of Helium mixtures <span class="citation" data-cites="kagan1984localization"> [<a href="#ref-kagan1984localization" role="doc-biblioref">8</a>]</span> and then extended to heavy-light mixtures in which multiple species with large mass ratios interact. The idea is that the heavier particles act as an effective disorder potential for the lighter ones, inducing localisation. Two such models <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">9</a>,<a href="#ref-schiulazDynamicsManybodyLocalized2015" role="doc-biblioref">10</a>]</span> instead find that the models thermalise exponentially slowly in system size, which Ref. <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">9</a>]</span> dubs Quasi-MBL.</p>
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<p>True disorder-free localisation does occur in exactly solvable models with extensively many conserved quantities <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">11</a>]</span>. As conserved quantities have no time dynamics this can be thought of as taking the separation of timescales to the infinite limit.</p>
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<p>-link to the FK model</p>
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<p>-link to the Kitaev Model</p>
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<p>-link to the physics of amorphous systems</p>
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<p>Next Chapter: <a href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falikov-Kimball Model</a></p>
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</section>
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</section>
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<section id="bibliography" class="level1 unnumbered">
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<h1 class="unnumbered">Bibliography</h1>
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<div id="refs" class="references csl-bib-body" role="doc-bibliography">
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<div id="ref-abaninRecentProgressManybody2017" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[1] </div><div class="csl-right-inline">D. A. Abanin and Z. Papić, <em><a href="https://doi.org/10.1002/andp.201700169">Recent Progress in Many-Body Localization</a></em>, ANNALEN DER PHYSIK <strong>529</strong>, 1700169 (2017).</div>
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</div>
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<div id="ref-srednickiChaosQuantumThermalization1994" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[2] </div><div class="csl-right-inline">M. Srednicki, <em><a href="https://doi.org/10.1103/PhysRevE.50.888">Chaos and Quantum Thermalization</a></em>, Phys. Rev. E <strong>50</strong>, 888 (1994).</div>
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</div>
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<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[3] </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>
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</div>
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<div id="ref-izrailevLocalizationMobilityEdge1999" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[4] </div><div class="csl-right-inline">F. M. Izrailev and A. A. Krokhin, <em><a href="https://doi.org/10.1103/PhysRevLett.82.4062">Localization and the Mobility Edge in One-Dimensional Potentials with Correlated Disorder</a></em>, Phys. Rev. Lett. <strong>82</strong>, 4062 (1999).</div>
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</div>
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<div id="ref-croyAndersonLocalization1D2011" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[5] </div><div class="csl-right-inline">A. Croy, P. Cain, and M. Schreiber, <em><a href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B <strong>82</strong>, 107 (2011).</div>
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</div>
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<div id="ref-izrailevAnomalousLocalizationLowDimensional2012" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[6] </div><div class="csl-right-inline">F. M. Izrailev, A. A. Krokhin, and N. M. Makarov, <em><a href="https://doi.org/10.1016/j.physrep.2011.11.002">Anomalous Localization in Low-Dimensional Systems with Correlated Disorder</a></em>, Physics Reports <strong>512</strong>, 125 (2012).</div>
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</div>
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<div id="ref-imbrieManyBodyLocalizationQuantum2016" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[7] </div><div class="csl-right-inline">J. Z. Imbrie, <em><a href="https://doi.org/10.1007/s10955-016-1508-x">On Many-Body Localization for Quantum Spin Chains</a></em>, J Stat Phys <strong>163</strong>, 998 (2016).</div>
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</div>
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<div id="ref-kagan1984localization" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[8] </div><div class="csl-right-inline">Y. Kagan and L. Maksimov, <em>Localization in a System of Interacting Particles Diffusing in a Regular Crystal</em>, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki <strong>87</strong>, 348 (1984).</div>
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</div>
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<div id="ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[9] </div><div class="csl-right-inline">N. Y. Yao, C. R. Laumann, J. I. Cirac, M. D. Lukin, and J. E. Moore, <em><a href="https://doi.org/10.1103/PhysRevLett.117.240601">Quasi-Many-Body Localization in Translation-Invariant Systems</a></em>, Phys. Rev. Lett. <strong>117</strong>, 240601 (2016).</div>
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</div>
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<div id="ref-schiulazDynamicsManybodyLocalized2015" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[10] </div><div class="csl-right-inline">M. Schiulaz, A. Silva, and M. Müller, <em><a href="https://doi.org/10.1103/PhysRevB.91.184202">Dynamics in Many-Body Localized Quantum Systems Without Disorder</a></em>, Phys. Rev. B <strong>91</strong>, 184202 (2015).</div>
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</div>
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<div id="ref-smithDisorderFreeLocalization2017" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[11] </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>
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</div>
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