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---
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title: Background - Disorder & Localisation
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<title>Background - Disorder & Localisation</title>
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<ul>
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<li><a href="#bg-disorder-and-localisation"
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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"
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id="toc-localisation-anderson-many-body-and-disorder-free">Localisation:
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Anderson, Many Body and Disorder-Free</a></li>
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<li><a href="#disorder-and-spin-liquids"
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id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
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<li><a href="#amorphous-magnetism"
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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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</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<!-- Table of Contents -->
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<ul>
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<li><a href="#bg-disorder-and-localisation"
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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"
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id="toc-localisation-anderson-many-body-and-disorder-free">Localisation:
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Anderson, Many Body and Disorder-Free</a></li>
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<li><a href="#disorder-and-spin-liquids"
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id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
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<li><a href="#amorphous-magnetism"
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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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</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<!-- Main Page Body -->
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<div id="page-header">
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<p>2 Background</p>
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<hr />
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</div>
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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"
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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
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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
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time given the seeming ubiquity of extended Bloch states. Later, when
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thermalisation in quantum systems gained interest, localisation
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phenomena again stood out as counterexamples to the eigenstate
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thermalisation hypothesis <span class="citation"
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data-cites="abaninRecentProgressManybody2017 srednickiChaosQuantumThermalization1994"> [<a
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href="#ref-abaninRecentProgressManybody2017"
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role="doc-biblioref">1</a>,<a
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href="#ref-srednickiChaosQuantumThermalization1994"
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role="doc-biblioref">2</a>]</span>, allowing quantum systems to avoid to
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retain memory of their initial conditions in the face of thermal
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noise.</p>
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<p>The simplest and first discovered kind is Anderson localisation,
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first studied in 1958 <span class="citation"
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data-cites="andersonAbsenceDiffusionCertain1958"> [<a
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href="#ref-andersonAbsenceDiffusionCertain1958"
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role="doc-biblioref">3</a>]</span> in the context of non-interacting
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fermions subject to a static or quenched disorder potential <span
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class="math inline">\(V_j\)</span> drawn uniformly from the interval
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<span class="math inline">\([-W,W]\)</span></p>
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<p><span class="math display">\[
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H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger
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c_j
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\]</span></p>
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<p>this model exhibits exponentially localised eigenfunctions <span
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class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\)</span> which
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cannot contribute to transport processes. Initially it was thought that
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in one dimensional disordered models, all states would be localised,
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however it was later shown that in the presence of correlated disorder,
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bands of extended states can exist <span class="citation"
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data-cites="izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a
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href="#ref-izrailevLocalizationMobilityEdge1999"
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role="doc-biblioref">4</a>–<a
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href="#ref-izrailevAnomalousLocalizationLowDimensional2012"
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role="doc-biblioref">6</a>]</span>.</p>
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<p>Later localisation was found in interacting many-body systems with
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quenched disorder:</p>
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<p><span class="math display">\[
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H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger
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c_j + U\sum_{jk} n_j n_k
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\]</span></p>
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<p>where the number operators <span class="math inline">\(n_j =
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c^\dagger_j c_j\)</span>. Here, in contrast to the Anderson model,
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localisation phenomena can be proven robust to weak perturbations of the
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Hamiltonian. This is called many-body localisation (MBL) <span
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class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016"> [<a
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href="#ref-imbrieManyBodyLocalizationQuantum2016"
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role="doc-biblioref">7</a>]</span>.</p>
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<p>Both MBL and Anderson localisation depend crucially on the presence
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of quenched disorder. This has led to ongoing interest in the
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possibility of disorder-free localisation, in which the disorder
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necessary to generate localisation is generated entirely from the
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dynamics of the model. This contracts with typical models of disordered
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systems in which disorder is explicitly introduced into the Hamilton or
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the initial state.</p>
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<p>The concept of disorder-free localisation was first proposed in the
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context of Helium mixtures <span class="citation"
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data-cites="kagan1984localization"> [<a
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href="#ref-kagan1984localization" role="doc-biblioref">8</a>]</span> and
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then extended to heavy-light mixtures in which multiple species with
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large mass ratios interact. The idea is that the heavier particles act
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as an effective disorder potential for the lighter ones, inducing
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localisation. Two such models <span class="citation"
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data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015"> [<a
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href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
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role="doc-biblioref">9</a>,<a
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href="#ref-schiulazDynamicsManybodyLocalized2015"
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role="doc-biblioref">10</a>]</span> instead find that the models
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thermalise exponentially slowly in system size, which Ref. <span
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class="citation"
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data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"> [<a
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href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
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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
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with extensively many conserved quantities <span class="citation"
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data-cites="smithDisorderFreeLocalization2017"> [<a
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href="#ref-smithDisorderFreeLocalization2017"
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role="doc-biblioref">11</a>]</span>. As conserved quantities have no
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time dynamics this can be thought of as taking the separation of
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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
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href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3 The Long
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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"
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role="doc-biblioentry">
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<div class="csl-left-margin">[1] </div><div class="csl-right-inline">D.
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A. Abanin and Z. Papić, <em><a
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href="https://doi.org/10.1002/andp.201700169">Recent Progress in
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Many-Body Localization</a></em>, ANNALEN DER PHYSIK
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<strong>529</strong>, 1700169 (2017).</div>
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</div>
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<div id="ref-srednickiChaosQuantumThermalization1994" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[2] </div><div class="csl-right-inline">M.
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Srednicki, <em><a href="https://doi.org/10.1103/PhysRevE.50.888">Chaos
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and Quantum Thermalization</a></em>, Phys. Rev. E <strong>50</strong>,
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888 (1994).</div>
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</div>
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<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[3] </div><div class="csl-right-inline">P.
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W. Anderson, <em><a
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href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in
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Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492
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(1958).</div>
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</div>
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<div id="ref-izrailevLocalizationMobilityEdge1999" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[4] </div><div class="csl-right-inline">F.
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M. Izrailev and A. A. Krokhin, <em><a
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href="https://doi.org/10.1103/PhysRevLett.82.4062">Localization and the
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Mobility Edge in One-Dimensional Potentials with Correlated
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Disorder</a></em>, Phys. Rev. Lett. <strong>82</strong>, 4062
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(1999).</div>
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</div>
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<div id="ref-croyAndersonLocalization1D2011" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[5] </div><div class="csl-right-inline">A.
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Croy, P. Cain, and M. Schreiber, <em><a
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href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization
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in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B
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<strong>82</strong>, 107 (2011).</div>
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</div>
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<div id="ref-izrailevAnomalousLocalizationLowDimensional2012"
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class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[6] </div><div class="csl-right-inline">F.
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M. Izrailev, A. A. Krokhin, and N. M. Makarov, <em><a
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href="https://doi.org/10.1016/j.physrep.2011.11.002">Anomalous
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Localization in Low-Dimensional Systems with Correlated
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Disorder</a></em>, Physics Reports <strong>512</strong>, 125
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(2012).</div>
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</div>
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<div id="ref-imbrieManyBodyLocalizationQuantum2016" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[7] </div><div class="csl-right-inline">J.
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Z. Imbrie, <em><a href="https://doi.org/10.1007/s10955-016-1508-x">On
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Many-Body Localization for Quantum Spin Chains</a></em>, J Stat Phys
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<strong>163</strong>, 998 (2016).</div>
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</div>
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<div id="ref-kagan1984localization" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[8] </div><div class="csl-right-inline">Y.
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Kagan and L. Maksimov, <em>Localization in a System of Interacting
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Particles Diffusing in a Regular Crystal</em>, Zhurnal Eksperimental’noi
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i Teoreticheskoi Fiziki <strong>87</strong>, 348 (1984).</div>
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</div>
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<div id="ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016"
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class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[9] </div><div class="csl-right-inline">N.
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Y. Yao, C. R. Laumann, J. I. Cirac, M. D. Lukin, and J. E. Moore,
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<em><a
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href="https://doi.org/10.1103/PhysRevLett.117.240601">Quasi-Many-Body
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Localization in Translation-Invariant Systems</a></em>, Phys. Rev. Lett.
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<strong>117</strong>, 240601 (2016).</div>
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</div>
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<div id="ref-schiulazDynamicsManybodyLocalized2015" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[10] </div><div class="csl-right-inline">M.
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Schiulaz, A. Silva, and M. Müller, <em><a
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href="https://doi.org/10.1103/PhysRevB.91.184202">Dynamics in Many-Body
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Localized Quantum Systems Without Disorder</a></em>, Phys. Rev. B
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<strong>91</strong>, 184202 (2015).</div>
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</div>
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<div id="ref-smithDisorderFreeLocalization2017" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[11] </div><div class="csl-right-inline">A.
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Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, <em><a
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href="https://doi.org/10.1103/PhysRevLett.118.266601">Disorder-Free
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Localization</a></em>, Phys. Rev. Lett. <strong>118</strong>, 266601
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(2017).</div>
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</div>
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</div>
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</section>
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