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_thesis/1_Introduction/1_Intro.html
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@ -213,6 +213,7 @@ Insulators</a></li>
<li><a href="#quantum-spin-liquids" <li><a href="#quantum-spin-liquids"
id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li> id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
<li><a href="#outline" id="toc-outline">Outline</a></li> <li><a href="#outline" id="toc-outline">Outline</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
{% endcapture %} {% endcapture %}
@ -230,6 +231,7 @@ Insulators</a></li>
<li><a href="#quantum-spin-liquids" <li><a href="#quantum-spin-liquids"
id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li> id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
<li><a href="#outline" id="toc-outline">Outline</a></li> <li><a href="#outline" id="toc-outline">Outline</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
@ -266,9 +268,10 @@ that of the individual objects.</p>
data-short-caption="A murmuration of Starlings" style="width:100.0%" data-short-caption="A murmuration of Starlings" style="width:100.0%"
alt="Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0" /> alt="Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0" />
<figcaption aria-hidden="true"><span>Figure 1:</span> A murmuration of <figcaption aria-hidden="true"><span>Figure 1:</span> A murmuration of
starlings. Dorset, UK. Credit <a href="twitter.com/arripay">Tanya starlings. Dorset, UK. Credit <a
Hart</a>, “Studland Starlings”, 2017, <a href="https://twitter.com/arripay">Tanya Hart</a>, “Studland Starlings”,
href="creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 2017, <a
href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA
3.0</a></figcaption> 3.0</a></figcaption>
</figure> </figure>
</div> </div>
@ -356,7 +359,7 @@ methods.</p>
<figure> <figure>
<img src="/assets/thesis/intro_chapter/venn_diagram.svg" <img src="/assets/thesis/intro_chapter/venn_diagram.svg"
data-short-caption="Interacting Quantum Many Body Systems Venn Diagram" data-short-caption="Interacting Quantum Many Body Systems Venn Diagram"
style="width:57.0%" style="width:100.0%"
alt="Figure 2: Three key adjectives. Many Body, the fact of describing systems in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials." /> alt="Figure 2: Three key adjectives. Many Body, the fact of describing systems in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials." />
<figcaption aria-hidden="true"><span>Figure 2:</span> Three key <figcaption aria-hidden="true"><span>Figure 2:</span> Three key
adjectives. Many Body, the fact of describing systems in the limit of adjectives. Many Body, the fact of describing systems in the limit of
@ -531,14 +534,14 @@ href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">26</a>,<a
href="#ref-ribakGaplessExcitationsGround2017" href="#ref-ribakGaplessExcitationsGround2017"
role="doc-biblioref">27</a>]</span>, giving rise to what is known as a role="doc-biblioref">27</a>]</span>, giving rise to what is known as a
quantum spin liquid (QSL) state.</p> quantum spin liquid (QSL) state.</p>
<p>Landau theory characterises phases of matter as inextricably linked <p>Landau-Ginzburg-Wilson theory characterises phases of matter as
to the emergence of long range order via a spontaneously broken inextricably linked to the emergence of long range order via a
symmetry. The fractional quantum Hall (FQH) state, discovered in the spontaneously broken symmetry. The fractional quantum Hall (FQH) state,
1980s is an explicit example of an electronic system that falls outside discovered in the 1980s is an explicit example of an electronic system
of the Landau paradigm. FQH systems exhibit fractionalised excitations that falls outside of the Landau-Ginzburg-Wilson paradigm. FQH systems
linked to their ground state having long range entanglement and exhibit fractionalised excitations linked to their ground state having
non-trivial topological properties <span class="citation" long range entanglement and non-trivial topological properties <span
data-cites="broholmQuantumSpinLiquids2020"> [<a class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a
href="#ref-broholmQuantumSpinLiquids2020" href="#ref-broholmQuantumSpinLiquids2020"
role="doc-biblioref">40</a>]</span>. Quantum spin liquids are the role="doc-biblioref">40</a>]</span>. Quantum spin liquids are the
analogous phase of matter for spin systems. Remarkably the existence of analogous phase of matter for spin systems. Remarkably the existence of
@ -595,33 +598,40 @@ problem via a mapping to Majorana fermions which yields an extensive
number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes
tied to an emergent gauge field. The model is remarkable not only for tied to an emergent gauge field. The model is remarkable not only for
its QSL ground state, it supports a rich phase diagram hosting gapless, its QSL ground state, it supports a rich phase diagram hosting gapless,
Abelian and non-Abelian phases and a finite temperature phase transition Abelian and non-Abelian phases <span class="citation"
to a thermal metal state <span class="citation" data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a
href="#ref-knolleDynamicsFractionalizationQuantum2015"
role="doc-biblioref">51</a>]</span> and a finite temperature phase
transition to a thermal metal state <span class="citation"
data-cites="selfThermallyInducedMetallic2019"> [<a data-cites="selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019" href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">51</a>]</span>. It has also been proposed that it role="doc-biblioref">52</a>]</span>. It been proposed that its
could be used to support topological quantum computing <span non-Abelian excitations could be used to support robust topological
class="citation" quantum computing [<span class="citation"
data-cites="kitaev_fault-tolerant_2003"> [<a
href="#ref-kitaev_fault-tolerant_2003"
role="doc-biblioref">53</a>]</span>; <span class="citation"
data-cites="freedmanTopologicalQuantumComputation2003"> [<a data-cites="freedmanTopologicalQuantumComputation2003"> [<a
href="#ref-freedmanTopologicalQuantumComputation2003" href="#ref-freedmanTopologicalQuantumComputation2003"
role="doc-biblioref">52</a>]</span>.</p> role="doc-biblioref">54</a>]</span>;
nayakNonAbelianAnyonsTopological2008].</p>
<p>It is by now understood that the Kitaev model on any tri-coordinated <p>It is by now understood that the Kitaev model on any tri-coordinated
<span class="math inline">\(z=3\)</span> graph has conserved plaquette <span class="math inline">\(z=3\)</span> graph has conserved plaquette
operators and local symmetries <span class="citation" operators and local symmetries <span class="citation"
data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007" data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007"
role="doc-biblioref">53</a>,<a href="#ref-Baskaran2008" role="doc-biblioref">55</a>,<a href="#ref-Baskaran2008"
role="doc-biblioref">54</a>]</span> which allow a mapping onto effective role="doc-biblioref">56</a>]</span> which allow a mapping onto effective
free Majorana fermion problems in a background of static <span free Majorana fermion problems in a background of static <span
class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation" class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation"
data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a
href="#ref-Nussinov2009" role="doc-biblioref">55</a>–<a href="#ref-Nussinov2009" role="doc-biblioref">57</a>–<a
href="#ref-hermanns2015weyl" role="doc-biblioref">58</a>]</span>. href="#ref-hermanns2015weyl" role="doc-biblioref">60</a>]</span>.
However, depending on lattice symmetries, finding the ground state flux However, depending on lattice symmetries, finding the ground state flux
sector and understanding the QSL properties can still be sector and understanding the QSL properties can still be
challenging <span class="citation" challenging <span class="citation"
data-cites="eschmann2019thermodynamics Peri2020"> [<a data-cites="eschmann2019thermodynamics Peri2020"> [<a
href="#ref-eschmann2019thermodynamics" role="doc-biblioref">59</a>,<a href="#ref-eschmann2019thermodynamics" role="doc-biblioref">61</a>,<a
href="#ref-Peri2020" role="doc-biblioref">60</a>]</span>.</p> href="#ref-Peri2020" role="doc-biblioref">62</a>]</span>.</p>
<p><strong>paragraph about amorphous lattices</strong></p> <p><strong>paragraph about amorphous lattices</strong></p>
<p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin <p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin
liquid by extending the Kitaev honeycomb model to random lattices with liquid by extending the Kitaev honeycomb model to random lattices with
@ -640,6 +650,7 @@ localisation.</p>
<p>In Chapter 3 I introduce the Long Range Falikov-Kimball Model in <p>In Chapter 3 I introduce the Long Range Falikov-Kimball Model in
greater detail. I will present results that. Chapter 4 focusses on the greater detail. I will present results that. Chapter 4 focusses on the
Amorphous Kitaev Model.</p> Amorphous Kitaev Model.</p>
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-king2012murmurations" class="csl-entry" <div id="ref-king2012murmurations" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
@ -1032,32 +1043,48 @@ Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons
in an Exactly Solved Model and Beyond</a></em>, Annals of Physics in an Exactly Solved Model and Beyond</a></em>, Annals of Physics
<strong>321</strong>, 2 (2006).</div> <strong>321</strong>, 2 (2006).</div>
</div> </div>
<div id="ref-knolleDynamicsFractionalizationQuantum2015"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[51] </div><div class="csl-right-inline">J.
Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, <em><a
href="https://doi.org/10.1103/PhysRevB.92.115127">Dynamics of
Fractionalization in Quantum Spin Liquids</a></em>, Phys. Rev. B
<strong>92</strong>, 115127 (2015).</div>
</div>
<div id="ref-selfThermallyInducedMetallic2019" class="csl-entry" <div id="ref-selfThermallyInducedMetallic2019" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[51] </div><div class="csl-right-inline">C. <div class="csl-left-margin">[52] </div><div class="csl-right-inline">C.
N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, <em><a N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, <em><a
href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally Induced href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally Induced
Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of
the Kitaev Model with Parity Projection</a></em>, Phys. Rev. B the Kitaev Model with Parity Projection</a></em>, Phys. Rev. B
<strong>99</strong>, 045142 (2019).</div> <strong>99</strong>, 045142 (2019).</div>
</div> </div>
<div id="ref-kitaev_fault-tolerant_2003" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[53] </div><div class="csl-right-inline">A.
Yu. Kitaev, <em><a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
Quantum Computation by Anyons</a></em>, Annals of Physics
<strong>303</strong>, 2 (2003).</div>
</div>
<div id="ref-freedmanTopologicalQuantumComputation2003" <div id="ref-freedmanTopologicalQuantumComputation2003"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[52] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[54] </div><div class="csl-right-inline">M.
Freedman, A. Kitaev, M. Larsen, and Z. Wang, <em><a Freedman, A. Kitaev, M. Larsen, and Z. Wang, <em><a
href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological Quantum href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological Quantum
Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31 Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31
(2003).</div> (2003).</div>
</div> </div>
<div id="ref-Baskaran2007" class="csl-entry" role="doc-biblioentry"> <div id="ref-Baskaran2007" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[53] </div><div class="csl-right-inline">G. <div class="csl-left-margin">[55] </div><div class="csl-right-inline">G.
Baskaran, S. Mandal, and R. Shankar, <em><a Baskaran, S. Mandal, and R. Shankar, <em><a
href="https://doi.org/10.1103/PhysRevLett.98.247201">Exact Results for href="https://doi.org/10.1103/PhysRevLett.98.247201">Exact Results for
Spin Dynamics and Fractionalization in the Kitaev Model</a></em>, Phys. Spin Dynamics and Fractionalization in the Kitaev Model</a></em>, Phys.
Rev. Lett. <strong>98</strong>, 247201 (2007).</div> Rev. Lett. <strong>98</strong>, 247201 (2007).</div>
</div> </div>
<div id="ref-Baskaran2008" class="csl-entry" role="doc-biblioentry"> <div id="ref-Baskaran2008" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[54] </div><div class="csl-right-inline">G. <div class="csl-left-margin">[56] </div><div class="csl-right-inline">G.
Baskaran, D. Sen, and R. Shankar, <em><a Baskaran, D. Sen, and R. Shankar, <em><a
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev Model: href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev Model:
Classical Ground States, Order from Disorder, and Exact Correlation Classical Ground States, Order from Disorder, and Exact Correlation
@ -1065,14 +1092,14 @@ Functions</a></em>, Phys. Rev. B <strong>78</strong>, 115116
(2008).</div> (2008).</div>
</div> </div>
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[55] </div><div class="csl-right-inline">Z. <div class="csl-left-margin">[57] </div><div class="csl-right-inline">Z.
Nussinov and G. Ortiz, <em><a Nussinov and G. Ortiz, <em><a
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and
Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>, Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>,
Physical Review B <strong>79</strong>, 214440 (2009).</div> Physical Review B <strong>79</strong>, 214440 (2009).</div>
</div> </div>
<div id="ref-OBrienPRB2016" class="csl-entry" role="doc-biblioentry"> <div id="ref-OBrienPRB2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[56] </div><div class="csl-right-inline">K. <div class="csl-left-margin">[58] </div><div class="csl-right-inline">K.
O’Brien, M. Hermanns, and S. Trebst, <em><a O’Brien, M. Hermanns, and S. Trebst, <em><a
href="https://doi.org/10.1103/PhysRevB.93.085101">Classification of href="https://doi.org/10.1103/PhysRevB.93.085101">Classification of
Gapless Z₂ Spin Liquids in Three-Dimensional Kitaev Models</a></em>, Gapless Z₂ Spin Liquids in Three-Dimensional Kitaev Models</a></em>,
@ -1080,27 +1107,27 @@ Phys. Rev. B <strong>93</strong>, 085101 (2016).</div>
</div> </div>
<div id="ref-yaoExactChiralSpin2007" class="csl-entry" <div id="ref-yaoExactChiralSpin2007" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[57] </div><div class="csl-right-inline">H. <div class="csl-left-margin">[59] </div><div class="csl-right-inline">H.
Yao and S. A. Kivelson, <em><a Yao and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral
Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett. Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett.
<strong>99</strong>, 247203 (2007).</div> <strong>99</strong>, 247203 (2007).</div>
</div> </div>
<div id="ref-hermanns2015weyl" class="csl-entry" role="doc-biblioentry"> <div id="ref-hermanns2015weyl" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[58] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[60] </div><div class="csl-right-inline">M.
Hermanns, K. O’Brien, and S. Trebst, <em>Weyl Spin Liquids</em>, Hermanns, K. O’Brien, and S. Trebst, <em>Weyl Spin Liquids</em>,
Physical Review Letters <strong>114</strong>, 157202 (2015).</div> Physical Review Letters <strong>114</strong>, 157202 (2015).</div>
</div> </div>
<div id="ref-eschmann2019thermodynamics" class="csl-entry" <div id="ref-eschmann2019thermodynamics" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[59] </div><div class="csl-right-inline">T. <div class="csl-left-margin">[61] </div><div class="csl-right-inline">T.
Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y. Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y.
Motome, and S. Trebst, <em>Thermodynamics of a Gauge-Frustrated Kitaev Motome, and S. Trebst, <em>Thermodynamics of a Gauge-Frustrated Kitaev
Spin Liquid</em>, Physical Review Research <strong>1</strong>, 032011(R) Spin Liquid</em>, Physical Review Research <strong>1</strong>, 032011(R)
(2019).</div> (2019).</div>
</div> </div>
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry"> <div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[60] </div><div class="csl-right-inline">V. <div class="csl-left-margin">[62] </div><div class="csl-right-inline">V.
Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R. Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R.
Moessner, and R. Thomale, <em><a Moessner, and R. Thomale, <em><a
href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian Chiral href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian Chiral

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_thesis/2_Background/2.2_HKM_Model.html
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@ -282,6 +282,7 @@ Chern number</a></li>
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase <li><a href="#phase-diagram" id="toc-phase-diagram">Phase
Diagram</a></li> Diagram</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
{% endcapture %} {% endcapture %}
@ -305,10 +306,13 @@ Chern number</a></li>
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase <li><a href="#phase-diagram" id="toc-phase-diagram">Phase
Diagram</a></li> Diagram</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
<h1 id="the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</h1> <h1 id="the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</h1>
<p><strong>papers</strong> Jos on dynamics
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.115127</p>
<p><strong>intro</strong> - strong spin orbit coupling leads to <p><strong>intro</strong> - strong spin orbit coupling leads to
anisotropic spin exchange (as opposed to isotropic exchange like the anisotropic spin exchange (as opposed to isotropic exchange like the
Heisenberg model) - geometrical frustration leads to QSL ground state Heisenberg model) - geometrical frustration leads to QSL ground state
@ -360,6 +364,7 @@ Chern number</h2>
<h2 id="phase-diagram">Phase Diagram</h2> <h2 id="phase-diagram">Phase Diagram</h2>
<div class="sourceCode" id="cb1"><pre <div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div> class="sourceCode python"><code class="sourceCode python"></code></pre></div>
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry" <div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">

3
_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html
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@ -318,6 +318,7 @@ Review</a></li>
<li><a href="#the-falikov-kimball-model-1" <li><a href="#the-falikov-kimball-model-1"
id="toc-the-falikov-kimball-model-1">The Falikov-Kimball model</a></li> id="toc-the-falikov-kimball-model-1">The Falikov-Kimball model</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
{% endcapture %} {% endcapture %}
@ -377,6 +378,7 @@ Review</a></li>
<li><a href="#the-falikov-kimball-model-1" <li><a href="#the-falikov-kimball-model-1"
id="toc-the-falikov-kimball-model-1">The Falikov-Kimball model</a></li> id="toc-the-falikov-kimball-model-1">The Falikov-Kimball model</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
@ -1419,6 +1421,7 @@ electons that can move and those that can’t.</p>
<div class="sourceCode" id="cb2"><pre <div class="sourceCode" id="cb2"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div> class="sourceCode python"><code class="sourceCode python"></code></pre></div>
<p></ij></ij></ij></ij></ij></ij></p> <p></ij></ij></ij></ij></ij></ij></p>
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-hodsonOnedimensionalLongrangeFalikovKimball2021" <div id="ref-hodsonOnedimensionalLongrangeFalikovKimball2021"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">

3
_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html
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@ -346,6 +346,7 @@ distribution</a></li>
<li><a href="#two-step-trick-2" id="toc-two-step-trick-2">Two Step <li><a href="#two-step-trick-2" id="toc-two-step-trick-2">Two Step
Trick</a></li> Trick</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
{% endcapture %} {% endcapture %}
@ -433,6 +434,7 @@ distribution</a></li>
<li><a href="#two-step-trick-2" id="toc-two-step-trick-2">Two Step <li><a href="#two-step-trick-2" id="toc-two-step-trick-2">Two Step
Trick</a></li> Trick</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
@ -1605,6 +1607,7 @@ class="sourceCode python"><code class="sourceCode python"><span id="cb6-1"><a hr
<div class="sourceCode" id="cb7"><pre <div class="sourceCode" id="cb7"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div> class="sourceCode python"><code class="sourceCode python"></code></pre></div>
<p></ij></ij></p> <p></ij></ij></p>
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-devroyeRandomSampling1986" class="csl-entry" <div id="ref-devroyeRandomSampling1986" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">

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_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html
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@ -279,6 +279,7 @@ id="toc-acknowledgments">Acknowledgments</a></li>
<li><a href="#uncorrelated-disorder-model" <li><a href="#uncorrelated-disorder-model"
id="toc-uncorrelated-disorder-model"><span id="app:disorder_model" id="toc-uncorrelated-disorder-model"><span id="app:disorder_model"
label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li> label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
{% endcapture %} {% endcapture %}
@ -299,6 +300,7 @@ id="toc-acknowledgments">Acknowledgments</a></li>
<li><a href="#uncorrelated-disorder-model" <li><a href="#uncorrelated-disorder-model"
id="toc-uncorrelated-disorder-model"><span id="app:disorder_model" id="toc-uncorrelated-disorder-model"><span id="app:disorder_model"
label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li> label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
@ -696,6 +698,7 @@ H_{\mathrm{DM}} = &amp; \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} -
\nonumber\end{aligned}\]</span></p> \nonumber\end{aligned}\]</span></p>
<div class="sourceCode" id="cb1"><pre <div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div> class="sourceCode python"><code class="sourceCode python"></code></pre></div>
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" <div id="ref-binderFiniteSizeScaling1981" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">

87
_thesis/4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html
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@ -251,6 +251,7 @@ modes</a></li>
<li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic <li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic
Statistics</a></li> Statistics</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
{% endcapture %} {% endcapture %}
@ -305,6 +306,7 @@ modes</a></li>
<li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic <li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic
Statistics</a></li> Statistics</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
@ -598,7 +600,7 @@ topologically protected qubits since the four sectors can only be mixed
by a highly non-local perturbations <span class="citation" by a highly non-local perturbations <span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003"> [<a data-cites="kitaevFaulttolerantQuantumComputation2003"> [<a
href="#ref-kitaevFaulttolerantQuantumComputation2003" href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref">1</a>]</span>.</p> role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p>
<p>Takeaway: The Extended Hilbert Space decomposes into a direct product <p>Takeaway: The Extended Hilbert Space decomposes into a direct product
of Flux Sectors, four Topological Sectors and a set of gauge of Flux Sectors, four Topological Sectors and a set of gauge
symmetries.</p> symmetries.</p>
@ -735,7 +737,7 @@ reduces to a determinant of the Q matrix and the fermion parity,
see <span class="citation" see <span class="citation"
data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011" href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
role="doc-biblioref">2</a>]</span>. The only difference from the role="doc-biblioref">1</a>]</span>. The only difference from the
honeycomb case is that we cannot explicitly compute the factors <span honeycomb case is that we cannot explicitly compute the factors <span
class="math inline">\(p_x,p_y,p_z = \pm\;1\)</span> that arise from class="math inline">\(p_x,p_y,p_z = \pm\;1\)</span> that arise from
reordering the b operators such that pairs of vertices linked by the reordering the b operators such that pairs of vertices linked by the
@ -761,7 +763,7 @@ determined by fermionic occupation numbers <span
class="math inline">\(n_i\)</span>. As discussed in <span class="math inline">\(n_i\)</span>. As discussed in <span
class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011" href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
role="doc-biblioref">2</a>]</span>, <span role="doc-biblioref">1</a>]</span>, <span
class="math inline">\(\hat{\pi}\)</span> is gauge invariant in the sense class="math inline">\(\hat{\pi}\)</span> is gauge invariant in the sense
that <span class="math inline">\([\hat{\pi}, D_i] = 0\)</span>.</p> that <span class="math inline">\([\hat{\pi}, D_i] = 0\)</span>.</p>
<p>This implies that <span class="math inline">\(det(Q^u) \prod -i <p>This implies that <span class="math inline">\(det(Q^u) \prod -i
@ -770,7 +772,7 @@ invariant models this quantity which can be related to the parity of the
number of vortex pairs in the system <span class="citation" number of vortex pairs in the system <span class="citation"
data-cites="yaoAlgebraicSpinLiquid2009"> [<a data-cites="yaoAlgebraicSpinLiquid2009"> [<a
href="#ref-yaoAlgebraicSpinLiquid2009" href="#ref-yaoAlgebraicSpinLiquid2009"
role="doc-biblioref">3</a>]</span>.</p> role="doc-biblioref">2</a>]</span>.</p>
<p>All these factors take values <span class="math inline">\(\pm <p>All these factors take values <span class="math inline">\(\pm
1\)</span> so <span class="math inline">\(\mathcal{P}_0\)</span> is 0 or 1\)</span> so <span class="math inline">\(\mathcal{P}_0\)</span> is 0 or
1 for a particular state. Since <span 1 for a particular state. Since <span
@ -802,9 +804,9 @@ diameters of the torus and, then, annihilating them again.</figcaption>
<p>More general arguments <span class="citation" <p>More general arguments <span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
href="#ref-chungExplicitMonodromyMoore2007" href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">4</a>,<a role="doc-biblioref">3</a>,<a
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007" href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
role="doc-biblioref">5</a>]</span> imply that <span role="doc-biblioref">4</a>]</span> imply that <span
class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> has an class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> has an
interesting relationship to the topological fluxes. In the non-Abelian interesting relationship to the topological fluxes. In the non-Abelian
phase, we expect that it will change sign in exactly one of the four phase, we expect that it will change sign in exactly one of the four
@ -894,7 +896,7 @@ definition, the vortex free sector.</p>
corresponds to the state where all <span class="math inline">\(u_{jk} = corresponds to the state where all <span class="math inline">\(u_{jk} =
1\)</span>. This implies that the flux free sector is the ground state 1\)</span>. This implies that the flux free sector is the ground state
sector <span class="citation" data-cites="lieb_flux_1994"> [<a sector <span class="citation" data-cites="lieb_flux_1994"> [<a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span>.</p> href="#ref-lieb_flux_1994" role="doc-biblioref">5</a>]</span>.</p>
<p>Lieb’s theorem does not generalise easily to the amorphous case. <p>Lieb’s theorem does not generalise easily to the amorphous case.
However, we can get some intuition by examining the problem that will However, we can get some intuition by examining the problem that will
lead to a guess for the ground state. We will then provide numerical lead to a guess for the ground state. We will then provide numerical
@ -976,7 +978,7 @@ that form each plaquette and the choice of sign gives a twofold chiral
ground state degeneracy.</p> ground state degeneracy.</p>
<p>This conjecture is consistent with Lieb’s theorem on regular <p>This conjecture is consistent with Lieb’s theorem on regular
lattices <span class="citation" data-cites="lieb_flux_1994"> [<a lattices <span class="citation" data-cites="lieb_flux_1994"> [<a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span> and is href="#ref-lieb_flux_1994" role="doc-biblioref">5</a>]</span> and is
supported by numerical evidence. As noted before, any flux that differs supported by numerical evidence. As noted before, any flux that differs
from the ground state is an excitation which we call a vortex.</p> from the ground state is an excitation which we call a vortex.</p>
<h3 id="finite-size-effects">Finite size effects</h3> <h3 id="finite-size-effects">Finite size effects</h3>
@ -1031,8 +1033,8 @@ of the odd plaquettes does not matter.</p>
<p>This happens because we have broken the time reversal symmetry of the <p>This happens because we have broken the time reversal symmetry of the
original model by adding odd plaquettes <span class="citation" original model by adding odd plaquettes <span class="citation"
data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"> [<a data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"> [<a
href="#ref-Chua2011" role="doc-biblioref">7</a>–<a href="#ref-Chua2011" role="doc-biblioref">6</a>–<a
href="#ref-WangHaoranPRB2021" role="doc-biblioref">14</a>]</span>.</p> href="#ref-WangHaoranPRB2021" role="doc-biblioref">13</a>]</span>.</p>
<p>Similarly to the behaviour of the original Kitaev model in response <p>Similarly to the behaviour of the original Kitaev model in response
to a magnetic field, we get two degenerate ground states of different to a magnetic field, we get two degenerate ground states of different
handedness. Practically speaking, one ground state is related to the handedness. Practically speaking, one ground state is related to the
@ -1040,7 +1042,7 @@ other by inverting the imaginary <span
class="math inline">\(\phi\)</span> fluxes <span class="citation" class="math inline">\(\phi\)</span> fluxes <span class="citation"
data-cites="yaoExactChiralSpin2007"> [<a data-cites="yaoExactChiralSpin2007"> [<a
href="#ref-yaoExactChiralSpin2007" href="#ref-yaoExactChiralSpin2007"
role="doc-biblioref">8</a>]</span>.</p> role="doc-biblioref">7</a>]</span>.</p>
<h2 id="phases-of-the-kitaev-model">Phases of the Kitaev Model</h2> <h2 id="phases-of-the-kitaev-model">Phases of the Kitaev Model</h2>
<p>discuss the Abelian A phase / toric code phase / anisotropic <p>discuss the Abelian A phase / toric code phase / anisotropic
phase</p> phase</p>
@ -1167,23 +1169,23 @@ and construct the set <span class="math inline">\((+1, +1), (+1, -1),
<figure> <figure>
<img src="/assets/thesis/amk_chapter/topological_fluxes.png" <img src="/assets/thesis/amk_chapter/topological_fluxes.png"
data-short-caption="Topological Fluxes" style="width:57.0%" data-short-caption="Topological Fluxes" style="width:57.0%"
alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make  [15]." /> alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make  [14]." />
<figcaption aria-hidden="true"><span>Figure 14:</span> Wilson loops that <figcaption aria-hidden="true"><span>Figure 14:</span> Wilson loops that
wind the major or minor diameters of the torus measure flux winding wind the major or minor diameters of the torus measure flux winding
through the hole of the doughnut/torus or through the filling. If they through the hole of the doughnut/torus or through the filling. If they
made doughnuts that both had a jam filling and a hole, this analogy made doughnuts that both had a jam filling and a hole, this analogy
would be a lot easier to make <span class="citation" would be a lot easier to make <span class="citation"
data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis" data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis"
role="doc-biblioref">15</a>]</span>.</figcaption> role="doc-biblioref">14</a>]</span>.</figcaption>
</figure> </figure>
</div> </div>
<p>However, in the non-Abelian phase we have to wrangle with <p>However, in the non-Abelian phase we have to wrangle with
monodromy <span class="citation" monodromy <span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
href="#ref-chungExplicitMonodromyMoore2007" href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">4</a>,<a role="doc-biblioref">3</a>,<a
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007" href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
role="doc-biblioref">5</a>]</span>. Monodromy is the behaviour of role="doc-biblioref">4</a>]</span>. Monodromy is the behaviour of
objects as they move around a singularity. This manifests here in that objects as they move around a singularity. This manifests here in that
the identity of a vortex and cloud of Majoranas can change as we wind the identity of a vortex and cloud of Majoranas can change as we wind
them around the torus in such a way that, rather than annihilating to them around the torus in such a way that, rather than annihilating to
@ -1192,9 +1194,9 @@ ground state. This means that we end up with only three degenerate
ground states in the non-Abelian phase <span class="math inline">\((+1, ground states in the non-Abelian phase <span class="math inline">\((+1,
+1), (+1, -1), (-1, +1)\)</span> <span class="citation" +1), (+1, -1), (-1, +1)\)</span> <span class="citation"
data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"> [<a data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"> [<a
href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">3</a>,<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">2</a>,<a
href="#ref-chungTopologicalQuantumPhase2010" href="#ref-chungTopologicalQuantumPhase2010"
role="doc-biblioref">16</a>]</span>. Concretely, this is because the role="doc-biblioref">15</a>]</span>. Concretely, this is because the
projector enforces both flux and fermion parity. When we wind a vortex projector enforces both flux and fermion parity. When we wind a vortex
around both non-contractible loops of the torus, it flips the flux around both non-contractible loops of the torus, it flips the flux
parity. Therefore, we have to introduce a fermionic excitation to make parity. Therefore, we have to introduce a fermionic excitation to make
@ -1205,24 +1207,17 @@ proposals to use this ground state degeneracy to implement both
passively fault tolerant and actively stabilised quantum passively fault tolerant and actively stabilised quantum
computations <span class="citation" computations <span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a data-cites="kitaevFaulttolerantQuantumComputation2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a
href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref">1</a>,<a
href="#ref-poulinStabilizerFormalismOperator2005" href="#ref-poulinStabilizerFormalismOperator2005"
role="doc-biblioref">17</a>,<a role="doc-biblioref">16</a>,<a
href="#ref-hastingsDynamicallyGeneratedLogical2021" href="#ref-hastingsDynamicallyGeneratedLogical2021"
role="doc-biblioref">18</a>]</span>.</p> role="doc-biblioref">17</a>,<a
href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p>
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-kitaevFaulttolerantQuantumComputation2003"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">A.
Yu. Kitaev, <em><a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
Quantum Computation by Anyons</a></em>, Annals of Physics
<strong>303</strong>, 2 (2003).</div>
</div>
<div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry" <div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">F. <div class="csl-left-margin">[1] </div><div class="csl-right-inline">F.
L. Pedrocchi, S. Chesi, and D. Loss, <em><a L. Pedrocchi, S. Chesi, and D. Loss, <em><a
href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of
the Kitaev honeycomb model</a></em>, Phys. Rev. B <strong>84</strong>, the Kitaev honeycomb model</a></em>, Phys. Rev. B <strong>84</strong>,
@ -1230,7 +1225,7 @@ the Kitaev honeycomb model</a></em>, Phys. Rev. B <strong>84</strong>,
</div> </div>
<div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry" <div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">H. <div class="csl-left-margin">[2] </div><div class="csl-right-inline">H.
Yao, S.-C. Zhang, and S. A. Kivelson, <em><a Yao, S.-C. Zhang, and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin
Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett. Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett.
@ -1238,7 +1233,7 @@ Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett.
</div> </div>
<div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry" <div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">S. <div class="csl-left-margin">[3] </div><div class="csl-right-inline">S.
B. Chung and M. Stone, <em><a B. Chung and M. Stone, <em><a
href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit Monodromy of href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit Monodromy of
Moore–Read Wavefunctions on a Torus</a></em>, J. Phys. A: Math. Theor. Moore–Read Wavefunctions on a Torus</a></em>, J. Phys. A: Math. Theor.
@ -1246,20 +1241,20 @@ Moore–Read Wavefunctions on a Torus</a></em>, J. Phys. A: Math. Theor.
</div> </div>
<div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007" <div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007"
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<div class="csl-left-margin">[5] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[4] </div><div class="csl-right-inline">M.
Oshikawa, Y. B. Kim, K. Shtengel, C. Nayak, and S. Tewari, <em><a Oshikawa, Y. B. Kim, K. Shtengel, C. Nayak, and S. Tewari, <em><a
href="https://doi.org/10.1016/j.aop.2006.08.001">Topological Degeneracy href="https://doi.org/10.1016/j.aop.2006.08.001">Topological Degeneracy
of Non-Abelian States for Dummies</a></em>, Annals of Physics of Non-Abelian States for Dummies</a></em>, Annals of Physics
<strong>322</strong>, 1477 (2007).</div> <strong>322</strong>, 1477 (2007).</div>
</div> </div>
<div id="ref-lieb_flux_1994" class="csl-entry" role="doc-biblioentry"> <div id="ref-lieb_flux_1994" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">E. <div class="csl-left-margin">[5] </div><div class="csl-right-inline">E.
H. Lieb, <em><a href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux H. Lieb, <em><a href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux
Phase of the Half-Filled Band</a></em>, Physical Review Letters Phase of the Half-Filled Band</a></em>, Physical Review Letters
<strong>73</strong>, 2158 (1994).</div> <strong>73</strong>, 2158 (1994).</div>
</div> </div>
<div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry"> <div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">V. <div class="csl-left-margin">[6] </div><div class="csl-right-inline">V.
Chua, H. Yao, and G. A. Fiete, <em><a Chua, H. Yao, and G. A. Fiete, <em><a
href="https://doi.org/10.1103/PhysRevB.83.180412">Exact Chiral Spin href="https://doi.org/10.1103/PhysRevB.83.180412">Exact Chiral Spin
Liquid with Stable Spin Fermi Surface on the Kagome Lattice</a></em>, Liquid with Stable Spin Fermi Surface on the Kagome Lattice</a></em>,
@ -1267,21 +1262,21 @@ Phys. Rev. B <strong>83</strong>, 180412 (2011).</div>
</div> </div>
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Yao and S. A. Kivelson, <em><a Yao and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral
Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett. Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett.
<strong>99</strong>, 247203 (2007).</div> <strong>99</strong>, 247203 (2007).</div>
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Chua and G. A. Fiete, <em><a Chua and G. A. Fiete, <em><a
href="https://doi.org/10.1103/PhysRevB.84.195129">Exactly Solvable href="https://doi.org/10.1103/PhysRevB.84.195129">Exactly Solvable
Topological Chiral Spin Liquid with Random Exchange</a></em>, Phys. Rev. Topological Chiral Spin Liquid with Random Exchange</a></em>, Phys. Rev.
B <strong>84</strong>, 195129 (2011).</div> B <strong>84</strong>, 195129 (2011).</div>
</div> </div>
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<div class="csl-left-margin">[10] </div><div class="csl-right-inline">G. <div class="csl-left-margin">[9] </div><div class="csl-right-inline">G.
A. Fiete, V. Chua, M. Kargarian, R. Lundgren, A. Rüegg, J. Wen, and V. A. Fiete, V. Chua, M. Kargarian, R. Lundgren, A. Rüegg, J. Wen, and V.
Zyuzin, <em><a Zyuzin, <em><a
href="https://doi.org/10.1016/j.physe.2011.11.011">Topological href="https://doi.org/10.1016/j.physe.2011.11.011">Topological
@ -1289,19 +1284,19 @@ Insulators and Quantum Spin Liquids</a></em>, Physica E: Low-Dimensional
Systems and Nanostructures <strong>44</strong>, 845 (2012).</div> Systems and Nanostructures <strong>44</strong>, 845 (2012).</div>
</div> </div>
<div id="ref-Natori2016" class="csl-entry" role="doc-biblioentry"> <div id="ref-Natori2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">W. <div class="csl-left-margin">[10] </div><div class="csl-right-inline">W.
M. H. Natori, E. C. Andrade, E. Miranda, and R. G. Pereira, <em><a M. H. Natori, E. C. Andrade, E. Miranda, and R. G. Pereira, <em><a
href="https://link.aps.org/doi/10.1103/PhysRevLett.117.017204">Chiral href="https://link.aps.org/doi/10.1103/PhysRevLett.117.017204">Chiral
Spin-Orbital Liquids with Nodal Lines</a></em>, Phys. Rev. Lett. Spin-Orbital Liquids with Nodal Lines</a></em>, Phys. Rev. Lett.
<strong>117</strong>, 017204 (2016).</div> <strong>117</strong>, 017204 (2016).</div>
</div> </div>
<div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">C. <div class="csl-left-margin">[11] </div><div class="csl-right-inline">C.
Wu, D. Arovas, and H.-H. Hung, <em>Γ-Matrix Generalization of the Kitaev Wu, D. Arovas, and H.-H. Hung, <em>Γ-Matrix Generalization of the Kitaev
Model</em>, Physical Review B <strong>79</strong>, 134427 (2009).</div> Model</em>, Physical Review B <strong>79</strong>, 134427 (2009).</div>
</div> </div>
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry"> <div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">V. <div class="csl-left-margin">[12] </div><div class="csl-right-inline">V.
Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R. Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R.
Moessner, and R. Thomale, <em><a Moessner, and R. Thomale, <em><a
href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian Chiral href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian Chiral
@ -1310,7 +1305,7 @@ Spin Liquid on a Simple Non-Archimedean Lattice</a></em>, Phys. Rev. B
</div> </div>
<div id="ref-WangHaoranPRB2021" class="csl-entry" <div id="ref-WangHaoranPRB2021" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">H. <div class="csl-left-margin">[13] </div><div class="csl-right-inline">H.
Wang and A. Principi, <em><a Wang and A. Principi, <em><a
href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana Edge and href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana Edge and
Corner States in Square and Kagome Quantum Spin-3/2 Liquids</a></em>, Corner States in Square and Kagome Quantum Spin-3/2 Liquids</a></em>,
@ -1318,14 +1313,14 @@ Phys. Rev. B <strong>104</strong>, 214422 (2021).</div>
</div> </div>
<div id="ref-parkerWhyDoesThis" class="csl-entry" <div id="ref-parkerWhyDoesThis" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[15] </div><div <div class="csl-left-margin">[14] </div><div
class="csl-right-inline"><em><a class="csl-right-inline"><em><a
href="https://www.youtube.com/watch?v=ymF1bp-qrjU">Why Does This Balloon href="https://www.youtube.com/watch?v=ymF1bp-qrjU">Why Does This Balloon
Have -1 Holes?</a></em> (n.d.).</div> Have -1 Holes?</a></em> (n.d.).</div>
</div> </div>
<div id="ref-chungTopologicalQuantumPhase2010" class="csl-entry" <div id="ref-chungTopologicalQuantumPhase2010" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">S. <div class="csl-left-margin">[15] </div><div class="csl-right-inline">S.
B. Chung, H. Yao, T. L. Hughes, and E.-A. Kim, <em><a B. Chung, H. Yao, T. L. Hughes, and E.-A. Kim, <em><a
href="https://doi.org/10.1103/PhysRevB.81.060403">Topological Quantum href="https://doi.org/10.1103/PhysRevB.81.060403">Topological Quantum
Phase Transition in an Exactly Solvable Model of a Chiral Spin Liquid at Phase Transition in an Exactly Solvable Model of a Chiral Spin Liquid at
@ -1334,7 +1329,7 @@ Finite Temperature</a></em>, Phys. Rev. B <strong>81</strong>, 060403
</div> </div>
<div id="ref-poulinStabilizerFormalismOperator2005" class="csl-entry" <div id="ref-poulinStabilizerFormalismOperator2005" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">D. <div class="csl-left-margin">[16] </div><div class="csl-right-inline">D.
Poulin, <em><a Poulin, <em><a
href="https://doi.org/10.1103/PhysRevLett.95.230504">Stabilizer href="https://doi.org/10.1103/PhysRevLett.95.230504">Stabilizer
Formalism for Operator Quantum Error Correction</a></em>, Phys. Rev. Formalism for Operator Quantum Error Correction</a></em>, Phys. Rev.
@ -1342,7 +1337,7 @@ Lett. <strong>95</strong>, 230504 (2005).</div>
</div> </div>
<div id="ref-hastingsDynamicallyGeneratedLogical2021" class="csl-entry" <div id="ref-hastingsDynamicallyGeneratedLogical2021" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[17] </div><div class="csl-right-inline">M.
B. Hastings and J. Haah, <em><a B. Hastings and J. Haah, <em><a
href="https://doi.org/10.22331/q-2021-10-19-564">Dynamically Generated href="https://doi.org/10.22331/q-2021-10-19-564">Dynamically Generated
Logical Qubits</a></em>, Quantum <strong>5</strong>, 564 (2021).</div> Logical Qubits</a></em>, Quantum <strong>5</strong>, 564 (2021).</div>

3
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@ -237,6 +237,7 @@ back from Bond Sectors to the Physical Subspace</a></li>
id="toc-open-boundary-conditions">Open boundary conditions</a></li> id="toc-open-boundary-conditions">Open boundary conditions</a></li>
</ul></li> </ul></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
{% endcapture %} {% endcapture %}
@ -277,6 +278,7 @@ back from Bond Sectors to the Physical Subspace</a></li>
id="toc-open-boundary-conditions">Open boundary conditions</a></li> id="toc-open-boundary-conditions">Open boundary conditions</a></li>
</ul></li> </ul></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
@ -905,6 +907,7 @@ which we set to 1 when calculating the projector.</p>
anyway, an arbitrary pairing of the unpaired <span anyway, an arbitrary pairing of the unpaired <span
class="math inline">\(b^\alpha\)</span> operators could be performed. class="math inline">\(b^\alpha\)</span> operators could be performed.
&lt;/i,j&gt;&lt;/i,j&gt;</p> &lt;/i,j&gt;&lt;/i,j&gt;</p>
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry" <div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">

3
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@ -293,6 +293,7 @@ flux sectors and bond sectors</a></li>
<li><a href="#chern-markers" id="toc-chern-markers">Chern <li><a href="#chern-markers" id="toc-chern-markers">Chern
Markers</a></li> Markers</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
{% endcapture %} {% endcapture %}
@ -326,6 +327,7 @@ flux sectors and bond sectors</a></li>
<li><a href="#chern-markers" id="toc-chern-markers">Chern <li><a href="#chern-markers" id="toc-chern-markers">Chern
Markers</a></li> Markers</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
@ -757,6 +759,7 @@ system.</p>
<p><strong>Expand on definition here</strong></p> <p><strong>Expand on definition here</strong></p>
<p><strong>Discuss link between Chern number and Anyonic <p><strong>Discuss link between Chern number and Anyonic
Statistics</strong></p> Statistics</strong></p>
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-mitchellAmorphousTopologicalInsulators2018" <div id="ref-mitchellAmorphousTopologicalInsulators2018"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">

5
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@ -241,6 +241,7 @@ Realisations and Signatures</a></li>
<li><a href="#generalisations" <li><a href="#generalisations"
id="toc-generalisations">Generalisations</a></li> id="toc-generalisations">Generalisations</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
{% endcapture %} {% endcapture %}
@ -285,6 +286,7 @@ Realisations and Signatures</a></li>
<li><a href="#generalisations" <li><a href="#generalisations"
id="toc-generalisations">Generalisations</a></li> id="toc-generalisations">Generalisations</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
@ -829,6 +831,7 @@ href="#ref-Wu2009" role="doc-biblioref">47</a>]</span></p>
quantum many body phases albeit material candidates aplenty. We expect quantum many body phases albeit material candidates aplenty. We expect
our exact chiral amorphous spin liquid to find many generalisation to our exact chiral amorphous spin liquid to find many generalisation to
realistic amorphous quantum magnets and beyond.</p> realistic amorphous quantum magnets and beyond.</p>
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography"> <div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry" <div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
@ -935,7 +938,7 @@ Conductivity as a Local Chern Marker</a></em>, arXiv Preprint
<div id="ref-kitaev_fault-tolerant_2003" class="csl-entry" <div id="ref-kitaev_fault-tolerant_2003" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">A. <div class="csl-left-margin">[14] </div><div class="csl-right-inline">A.
Y. Kitaev, <em><a Yu. Kitaev, <em><a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
Quantum Computation by Anyons</a></em>, Annals of Physics Quantum Computation by Anyons</a></em>, Annals of Physics
<strong>303</strong>, 2 (2003).</div> <strong>303</strong>, 2 (2003).</div>

6
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@ -2,9 +2,8 @@
<li>Introduction</li> <li>Introduction</li>
<ul> <ul>
<li><a href="./1_Introduction/1_Intro.html#interacting-quantum-many-body-systems">Interacting Quantum Many Body Systems</a></li> <li><a href="./1_Introduction/1_Intro.html#interacting-quantum-many-body-systems">Interacting Quantum Many Body Systems</a></li>
<li><a href="./1_Introduction/1_Intro.html#mott-insulators-and-the-hubbard-model">Mott Insulators and The Hubbard Model</a></li> <li><a href="./1_Introduction/1_Intro.html#mott-insulators">Mott Insulators</a></li>
<li><a href="./1_Introduction/1_Intro.html#spin-liquids">Spin Liquids</a></li> <li><a href="./1_Introduction/1_Intro.html#quantum-spin-liquids">Quantum Spin Liquids</a></li>
<li><a href="./1_Introduction/1_Intro.html#exactly-solvable-models">Exactly Solvable Models</a></li>
<li><a href="./1_Introduction/1_Intro.html#outline">Outline</a></li> <li><a href="./1_Introduction/1_Intro.html#outline">Outline</a></li>
</ul> </ul>
<li>Background</li> <li>Background</li>
@ -31,7 +30,6 @@
</ul></ul> </ul></ul>
<li>Chapter 3: The Long Range Falikov-Kimball Model</li> <li>Chapter 3: The Long Range Falikov-Kimball Model</li>
<ul><ul> <ul><ul>
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#contributions">Contributions</a></li>
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#chapter-summary">Chapter Summary</a></li> <li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#chapter-summary">Chapter Summary</a></li>
</ul> </ul>
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#the-model">The Model</a></li> <li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#the-model">The Model</a></li>

106
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