tom/personal_site
1
0
Fork 0
mirror of https://github.com/TomHodson/tomhodson.github.com.git synced 2025-06-26 10:01:18 +02:00
Code Issues Packages Projects Releases Wiki Activity

Updates!

Browse Source
This commit is contained in:
Tom Hodson 2022-08-27 10:23:45 +02:00
parent 9000efddb9
commit 941b83e767
22 changed files with 972 additions and 1656 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

593
_thesis/1_Introduction/1_Intro.html
View File

@ -1,6 +1,6 @@
--- ---
title: 1_Intro title: Introduction
excerpt: excerpt: Why do we do Condensed Matter theory at all?
layout: none layout: none
image: image:
@ -11,7 +11,8 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>1_Intro</title> <meta name="description" content="Why do we do Condensed Matter theory at all?" />
<title>Introduction</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
@ -34,7 +35,6 @@ Body Systems</a></li>
Insulators</a></li> 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="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
@ -55,7 +55,6 @@ Body Systems</a></li>
Insulators</a></li> 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="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
@ -355,43 +354,61 @@ href="#ref-antipovCriticalExponentsStrongly2014"
role="doc-biblioref">38</a>–<a role="doc-biblioref">38</a>–<a
href="#ref-herrmannNonequilibriumDynamicalCluster2016" href="#ref-herrmannNonequilibriumDynamicalCluster2016"
role="doc-biblioref">41</a>]</span>.</p> role="doc-biblioref">41</a>]</span>.</p>
<p>In Chapter 3 I will introduce a generalized FK model in one <p>In Chapter 3 I will introduce a generalized Falikov-Kimball model in
dimension. With the addition of long-range interactions in the one dimension I call the Long-Range Falikov-Kimball model. With the
background field, the model shows a similarly rich phase diagram. I use addition of long-range interactions in the background field, the model
an exact Markov chain Monte Carlo method to map the phase diagram and shows a similarly rich phase diagram its higher dimensional cousins. I
compute the energy-resolved localization properties of the fermions. I use an exact Markov chain Monte Carlo method to map the phase diagram
then compare the behaviour of this transitionally invariant model to an and compute the energy-resolved localization properties of the fermions.
Anderson model of uncorrelated binary disorder about a background charge I then compare the behaviour of this transitionally invariant model to
density wave field which confirms that the fermionic sector only fully an Anderson model of uncorrelated binary disorder about a background
localizes for very large system sizes.</p> charge density wave field which confirms that the fermionic sector only
fully localizes for very large system sizes.</p>
</section> </section>
<section id="quantum-spin-liquids" class="level1"> <section id="quantum-spin-liquids" class="level1">
<h1>Quantum Spin Liquids</h1> <h1>Quantum Spin Liquids</h1>
<p>To turn to the other key topic of this thesis, we have discussed the <p>To turn to the other key topic of this thesis, we have already
question of the magnetic ordering of local moments in the Mott discussed the AFM ordering of local moments in the Mott insulating
insulating state. The local moments may form an AFM ground state. state. Landau-Ginzburg-Wilson theory characterises phases of matter as
Alternatively they may fail to order even at zero temperature <span
class="citation"
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">28</a>,<a
href="#ref-ribakGaplessExcitationsGround2017"
role="doc-biblioref">29</a>]</span>, giving rise to what is known as a
quantum spin liquid (QSL) state.</p>
<p>Landau-Ginzburg-Wilson theory characterises phases of matter as
inextricably linked to the emergence of long range order via a inextricably linked to the emergence of long range order via a
spontaneously broken symmetry. The fractional quantum Hall (FQH) state, spontaneously broken symmetry. So within this paradigm we would not
discovered in the 1980s is an explicit example of an electronic system expect any interesting phases of matter not associated with AFM or other
that falls outside of the Landau-Ginzburg-Wilson paradigm. FQH systems long-range order. However, Anderson first proposed in 1973 <span
exhibit fractionalised excitations linked to their ground state having class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a
long range entanglement and non-trivial topological properties <span
class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a
href="#ref-broholmQuantumSpinLiquids2020"
role="doc-biblioref">42</a>]</span>. Quantum spin liquids are the
analogous phase of matter for spin systems. Remarkably the existence of
QSLs was first suggested by Anderson in 1973 <span class="citation"
data-cites="andersonResonatingValenceBonds1973"> [<a
href="#ref-andersonResonatingValenceBonds1973" href="#ref-andersonResonatingValenceBonds1973"
role="doc-biblioref">43</a>]</span>.</p> role="doc-biblioref">42</a>]</span> that if long range order is
suppressed by some mechanism, it might lead to a liquid-like state even
at zero temperature, the Quantum Spin Liquid (QSL).</p>
<p>This QSL state would exist at zero or very low temperatures, so we
would expect quantum effects to be very strong, which will turn out to
have far reaching consequences. It was the discovery of a different
phase, however that really kickstarted interest in the topic. The
fractional quantum Hall (FQH) state, discovered in the 1980s is an
explicit example of an interacting electron system that falls outside of
the Landau-Ginzburg-Wilson paradigm. It shares many phenomenological
properties with the QSL state. They both exhibit fractionalised
excitations, braiding statistics and non-trivial topological
properties <span class="citation"
data-cites="broholmQuantumSpinLiquids2020"> [<a
href="#ref-broholmQuantumSpinLiquids2020"
role="doc-biblioref">43</a>]</span>. The many-body ground state of such
systems acts as a complex and highly entangled vacuum. This vacuum can
support quasiparticle excitations with properties unbound from that of
the Dirac fermions of the standard model.</p>
<p>How do we actually make a QSL? Frustration is one mechanism that we
can use to suppress magnetic order in spin models <span class="citation"
data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022"
role="doc-biblioref">44</a>]</span>. Frustration can be geometric,
triangular lattices for instance cannot support AFM order. It can also
come about as a result of spin-orbit coupling or other physics. There
are also other routes to QSLs besides frustrated spin systems that we
will not discuss here <span class="citation"
data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a
href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">45</a>–<a
href="#ref-linExactSymmetryWeaklyinteracting1998"
role="doc-biblioref">47</a>]</span>.</p>
<!-- Experimentally, Mott insulating systems without magnetic order have been proposed as QSL systems\ [@law1TTaS2QuantumSpin2017; @ribakGaplessExcitationsGround2017]. -->
<!-- Other exampels: Quantum spin liquids are the analogous phase of matter for spin systems. Spin ice support deconfined magnetic monopoles. -->
<div id="fig:correlation_spin_orbit_PT" class="fignos"> <div id="fig:correlation_spin_orbit_PT" class="fignos">
<figure> <figure>
<img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png" <img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png"
@ -403,97 +420,176 @@ href="#ref-TrebstPhysRep2022"
role="doc-biblioref">44</a>]</span>.</figcaption> role="doc-biblioref">44</a>]</span>.</figcaption>
</figure> </figure>
</div> </div>
<p>The main route to QSLs, though there are others <span <p>Spin-orbit coupling is a relativistic effect, that very roughly
class="citation" corresponds to the fact that in the frame of reference of a moving
data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a electron, the electric field of nearby nuclei look like magnetic fields
href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">45</a>–<a to which the electron spin couples. This effectively couples the spatial
href="#ref-linExactSymmetryWeaklyinteracting1998" and spin parts of the electron wavefunction, meaning that the lattice
role="doc-biblioref">47</a>]</span>, is via frustration of spin models structure can influence the form of the spin-spin interactions leading
that would otherwise order have AFM order. This frustration can come to spatial anisotropy. This anisotropy will be how we frustrate the Mott
geometrically, triangular lattices for instance cannot support AFM insulators <span class="citation"
order. It can also come about as a result of spin-orbit coupling.</p> data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a
<p>Electron spin naturally couples to magnetic fields. Spin-orbit href="#ref-jackeliMottInsulatorsStrong2009"
coupling is a relativistic effect, that very roughly corresponds to the role="doc-biblioref">48</a>,<a
fact that in the frame of reference of a moving electron, the electric href="#ref-khaliullinOrbitalOrderFluctuations2005"
field of nearby nuclei look like magnetic field to which the electron role="doc-biblioref">49</a>]</span>. As we saw with the Hubbard model,
spin couples. In certain transition metal based compounds, such as those interaction effects are only strong or weak in comparison to the
based on Iridium and Rutheniun, crystal field effects, strong spin-orbit bandwidth or hopping integral <span class="math inline">\(t\)</span> so
coupling and narrow bandwidths lead to effective spin-<span what we need to see strong frustration is a material with strong
spin-orbit coupling <span class="math inline">\(\lambda\)</span>
relative to its bandwidth <span class="math inline">\(t\)</span>.</p>
<p>In certain transition metal based compounds, such as those based on
Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling
and narrow bandwidths lead to effective spin-<span
class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with
strongly anisotropic spin-spin couplings known as Kitaev Materials <span strongly anisotropic spin-spin couplings. These transition metal
class="citation" compounds, known Kitaev Materials, draw their name from the celebrated
Kitaev Honeycomb Model which is expected to model their low temperature
behaviour <span class="citation"
data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a
href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>,<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>,<a
href="#ref-Jackeli2009" role="doc-biblioref">48</a>–<a href="#ref-Jackeli2009" role="doc-biblioref">50</a>–<a
href="#ref-Takagi2019" role="doc-biblioref">51</a>]</span>. Kitaev href="#ref-Takagi2019" role="doc-biblioref">53</a>]</span>.</p>
materials draw their name from the celebrated Kitaev Honeycomb Model as <p>At this point we can sketch out a phase diagram like that of fig. <a
it is believed they will realise the QSL state via the mechanisms of the href="#fig:correlation_spin_orbit_PT">3</a>. When both electron-electron
Kitaev Model.</p> interactions <span class="math inline">\(U\)</span> and spin-orbit
couplings <span class="math inline">\(\lambda\)</span> are small
relative to the bandwidth <span class="math inline">\(t\)</span> we
recover standard band theory of band insulators and metals. In the upper
left we have the simple Mott insulating state as described by the
Hubbard model. In the lower right, strong spin-orbit coupling gives rise
to Topological insulators (TIs) characterised by symmetry protected edge
modes and non-zero Chern number. Kitaev materials occur in the region
where strong electron-electron interaction and spin-orbit coupling
interact. See <span class="citation"
data-cites="witczak-krempaCorrelatedQuantumPhenomena2014"> [<a
href="#ref-witczak-krempaCorrelatedQuantumPhenomena2014"
role="doc-biblioref">54</a>]</span> for a much more expansive version of
this diagram.</p>
<p>The Kitaev Honeycomb model <span class="citation" <p>The Kitaev Honeycomb model <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006" href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">52</a>]</span> was the first concrete model with a role="doc-biblioref">55</a>]</span> was the first concrete spin model
QSL ground state. It is defined on the honeycomb lattice and provides an with a QSL ground state. It is defined on the two dimensional honeycomb
exactly solvable model whose ground state is a QSL characterized by a lattice and provides an exactly solvable model that can be reduced to a
static <span class="math inline">\(\mathbb Z_2\)</span> gauge field and free fermion problem via a mapping to Majorana fermions. This yields an
Majorana fermion excitations. It can be reduced to a free fermion extensive number of static <span class="math inline">\(\mathbb
problem via a mapping to Majorana fermions which yields an extensive Z_2\)</span> fluxes tied to an emergent gauge field. The model is
number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes remarkable not only for its QSL ground state but also for its
tied to an emergent gauge field. The model is remarkable not only for fractionalised excitations with non-trivial braiding statistics. It has
its QSL ground state, it supports a rich phase diagram hosting gapless, a rich phase diagram hosting gapless, Abelian and non-Abelian
Abelian and non-Abelian phases <span class="citation" phases <span class="citation"
data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a
href="#ref-knolleDynamicsFractionalizationQuantum2015" href="#ref-knolleDynamicsFractionalizationQuantum2015"
role="doc-biblioref">53</a>]</span> and a finite temperature phase role="doc-biblioref">56</a>]</span> and a finite temperature phase
transition to a thermal metal state <span class="citation" 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">54</a>]</span>. It been proposed that its role="doc-biblioref">57</a>]</span>. It been proposed that its
non-Abelian excitations could be used to support robust topological non-Abelian excitations could be used to support robust topological
quantum computing [<span class="citation" quantum computing <span class="citation"
data-cites="kitaev_fault-tolerant_2003"> [<a data-cites="kitaev_fault-tolerant_2003 freedmanTopologicalQuantumComputation2003 nayakNonAbelianAnyonsTopological2008"> [<a
href="#ref-kitaev_fault-tolerant_2003" href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">58</a>–<a
role="doc-biblioref">55</a>]</span>; <span class="citation" href="#ref-nayakNonAbelianAnyonsTopological2008"
data-cites="freedmanTopologicalQuantumComputation2003"> [<a role="doc-biblioref">60</a>]</span>.</p>
href="#ref-freedmanTopologicalQuantumComputation2003" <p>As Kitaev points out in his original paper, the model remains
role="doc-biblioref">56</a>]</span>; solvable on any tri-coordinated <span class="math inline">\(z=3\)</span>
nayakNonAbelianAnyonsTopological2008].</p> graph which can be 3-edge-coloured. Indeed many generalisations of the
<p>It is by now understood that the Kitaev model on any tri-coordinated model to  <span class="citation"
<span class="math inline">\(z=3\)</span> graph has conserved plaquette data-cites="Baskaran2007 Baskaran2008 Nussinov2009 OBrienPRB2016 hermanns2015weyl"> [<a
operators and local symmetries <span class="citation" href="#ref-Baskaran2007" role="doc-biblioref">61</a>–<a
data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007" href="#ref-hermanns2015weyl" role="doc-biblioref">65</a>]</span>.
role="doc-biblioref">57</a>,<a href="#ref-Baskaran2008" Notably, the Yao-Kivelson model <span class="citation"
role="doc-biblioref">58</a>]</span> which allow a mapping onto effective data-cites="yaoExactChiralSpin2007"> [<a
free Majorana fermion problems in a background of static <span href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">66</a>]</span>
class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation" introduces triangular plaquettes to the honeycomb lattice leading to
data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a spontaneous chiral symmetry breaking. These extensions all retain
href="#ref-Nussinov2009" role="doc-biblioref">59</a>–<a translation symmetry, likely because edge-colouring and finding the
href="#ref-hermanns2015weyl" role="doc-biblioref">62</a>]</span>. ground state become much harder without it. Finding the ground state
However, depending on lattice symmetries, finding the ground state flux 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">63</a>,<a href="#ref-eschmann2019thermodynamics" role="doc-biblioref">67</a>,<a
href="#ref-Peri2020" role="doc-biblioref">64</a>]</span>.</p> href="#ref-Peri2020" role="doc-biblioref">68</a>]</span>. Undeterred,
<p><strong>paragraph about amorphous lattices</strong></p> this gap lead us to wonder what might happen if we remove translation
<p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin symmetry from the Kitaev Model. This might would be a model of a
liquid by extending the Kitaev honeycomb model to random lattices with tri-coordinated, highly bond anisotropic but otherwise amorphous
fixed coordination number three. The model retains its exact solubility material.</p>
but the presence of plaquettes with an odd number of sides leads to a <p>Amorphous materials do no have long-range lattice regularities but
spontaneous breaking of time reversal symmetry. I unearth a rich phase covalent compounds can induce short-range regularities in the lattice
diagram displaying Abelian as well as a non-Abelian quantum spin liquid structure such as fixed coordination number <span
phases with a remarkably simple ground state flux pattern. Furthermore, class="math inline">\(z\)</span>. The best examples being amorphous
I show that the system undergoes a finite-temperature phase transition Silicon and Germanium with <span class="math inline">\(z=4\)</span>
to a conducting thermal metal state and discuss possible experimental which are used to make thin-film solar cells <span class="citation"
realisations.</p> data-cites="Weaire1971 betteridge1973possible"> [<a
</section> href="#ref-Weaire1971" role="doc-biblioref">69</a>,<a
<section id="outline" class="level1"> href="#ref-betteridge1973possible" role="doc-biblioref">70</a>]</span>.
<h1>Outline</h1> Recently is has been shown that topological insulating (TI) phases can
exist in amorphous systems. Amorphous TIs are characterized by similar
protected edge states to their translation invariant cousins and
generalised topological bulk invariants <span class="citation"
data-cites="mitchellAmorphousTopologicalInsulators2018 agarwala2019topological marsalTopologicalWeaireThorpeModels2020 costa2019toward agarwala2020higher spring2021amorphous corbae2019evidence"> [<a
href="#ref-mitchellAmorphousTopologicalInsulators2018"
role="doc-biblioref">71</a>–<a href="#ref-corbae2019evidence"
role="doc-biblioref">77</a>]</span>. However, research on amorphous
electronic systems has been mostly focused on non-interacting systems
with a few exceptions, for example, to account for the observation of
superconductivity <span class="citation"
data-cites="buckel1954einfluss mcmillan1981electron meisel1981eliashberg bergmann1976amorphous mannaNoncrystallineTopologicalSuperconductors2022"> [<a
href="#ref-buckel1954einfluss" role="doc-biblioref">78</a>–<a
href="#ref-mannaNoncrystallineTopologicalSuperconductors2022"
role="doc-biblioref">82</a>]</span> in amorphous materials or very
recently to understand the effect of strong electron repulsion in
TIs <span class="citation" data-cites="kim2022fractionalization"> [<a
href="#ref-kim2022fractionalization"
role="doc-biblioref">83</a>]</span>.</p>
<p>Amorphous <em>magnetic</em> systems has been investigated since the
1960s, mostly through the adaptation of theoretical tools developed for
disordered systems <span class="citation"
data-cites="aharony1975critical Petrakovski1981 kaneyoshi1992introduction Kaneyoshi2018"> [<a
href="#ref-aharony1975critical" role="doc-biblioref">84</a>–<a
href="#ref-Kaneyoshi2018" role="doc-biblioref">87</a>]</span> and with
numerical methods <span class="citation"
data-cites="fahnle1984monte plascak2000ising"> [<a
href="#ref-fahnle1984monte" role="doc-biblioref">88</a>,<a
href="#ref-plascak2000ising" role="doc-biblioref">89</a>]</span>.
Research on classical Heisenberg and Ising models has been shown to
account for observed behaviour of ferromagnetism, disordered
antiferromagnetism and widely observed spin glass behaviour <span
class="citation" data-cites="coey1978amorphous"> [<a
href="#ref-coey1978amorphous" role="doc-biblioref">90</a>]</span>.
However, the role of spin-anisotropic interactions and quantum effects
in amorphous magnets has not been addressed. It is an open question
whether frustrated magnetic interactions on amorphous lattices can give
rise genuine quantum phases, i.e. to long-range entangled quantum spin
liquids (QSL) <span class="citation"
data-cites="Anderson1973 Knolle2019 Savary2016 Lacroix2011"> [<a
href="#ref-Anderson1973" role="doc-biblioref">91</a>–<a
href="#ref-Lacroix2011" role="doc-biblioref">94</a>]</span>.</p>
<p>In Chapter 4 I will introduce the Amorphous Kitaev model, a
generalisation of the Kitaev honeycomb model to random lattices with
fixed coordination number three. We will show that this model is a
soluble chiral amorphous quantum spin liquid. The model retains its
exact solubility but, as with the Yao-Kivelson model <span
class="citation" data-cites="yaoExactChiralSpin2007"> [<a
href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">66</a>]</span>,
the presence of plaquettes with an odd number of sides leads to a
spontaneous breaking of time reversal symmetry. We will confirm prior
observations that the form of the ground state can be written in terms
of the number of sides of elementary plaquettes of the model <span
class="citation"
data-cites="OBrienPRB2016 eschmannThermodynamicClassificationThreedimensional2020"> [<a
href="#ref-OBrienPRB2016" role="doc-biblioref">64</a>,<a
href="#ref-eschmannThermodynamicClassificationThreedimensional2020"
role="doc-biblioref">95</a>]</span>. We unearth a rich phase diagram
displaying Abelian as well as a non-Abelian chiral spin liquid phases.
Furthermore, I show that the system undergoes a finite-temperature phase
transition to a conducting thermal metal state and discuss possible
experimental realisations.</p>
<p>The next chapter, Chapter 2, will introduce some necessary background <p>The next chapter, Chapter 2, will introduce some necessary background
to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and
localisation.</p> localisation. Then Chapter 3 introduces and studies the Long Range
<p>In Chapter 3 I introduce the Long Range Falikov-Kimball Model in Falikov-Kimball Model in one dimension while 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>
</section> </section>
<section id="bibliography" class="level1 unnumbered"> <section id="bibliography" class="level1 unnumbered">
@ -820,22 +916,22 @@ href="https://doi.org/10.1103/PhysRevB.94.245114">Nonequilibrium
Dynamical Cluster Approximation Study of the Falicov-Kimball Dynamical Cluster Approximation Study of the Falicov-Kimball
Model</a></em>, Phys. Rev. B <strong>94</strong>, 245114 (2016).</div> Model</a></em>, Phys. Rev. B <strong>94</strong>, 245114 (2016).</div>
</div> </div>
<div id="ref-broholmQuantumSpinLiquids2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[42] </div><div class="csl-right-inline">C.
Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T.
Senthil, <em><a href="https://doi.org/10.1126/science.aay0668">Quantum
Spin Liquids</a></em>, Science <strong>367</strong>, eaay0668
(2020).</div>
</div>
<div id="ref-andersonResonatingValenceBonds1973" class="csl-entry" <div id="ref-andersonResonatingValenceBonds1973" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[43] </div><div class="csl-right-inline">P. <div class="csl-left-margin">[42] </div><div class="csl-right-inline">P.
W. Anderson, <em><a W. Anderson, <em><a
href="https://doi.org/10.1016/0025-5408(73)90167-0">Resonating Valence href="https://doi.org/10.1016/0025-5408(73)90167-0">Resonating Valence
Bonds: A New Kind of Insulator?</a></em>, Materials Research Bulletin Bonds: A New Kind of Insulator?</a></em>, Materials Research Bulletin
<strong>8</strong>, 153 (1973).</div> <strong>8</strong>, 153 (1973).</div>
</div> </div>
<div id="ref-broholmQuantumSpinLiquids2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[43] </div><div class="csl-right-inline">C.
Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T.
Senthil, <em><a href="https://doi.org/10.1126/science.aay0668">Quantum
Spin Liquids</a></em>, Science <strong>367</strong>, eaay0668
(2020).</div>
</div>
<div id="ref-TrebstPhysRep2022" class="csl-entry" <div id="ref-TrebstPhysRep2022" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[44] </div><div class="csl-right-inline">S. <div class="csl-left-margin">[44] </div><div class="csl-right-inline">S.
@ -868,17 +964,33 @@ class="csl-right-inline">H.-H. Lin, L. Balents, and M. P. A. Fisher,
Symmetry in the Weakly-Interacting Two-Leg Ladder</a></em>, Phys. Rev. B Symmetry in the Weakly-Interacting Two-Leg Ladder</a></em>, Phys. Rev. B
<strong>58</strong>, 1794 (1998).</div> <strong>58</strong>, 1794 (1998).</div>
</div> </div>
<div id="ref-Jackeli2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-jackeliMottInsulatorsStrong2009" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[48] </div><div class="csl-right-inline">G. <div class="csl-left-margin">[48] </div><div class="csl-right-inline">G.
Jackeli and G. Khaliullin, <em><a Jackeli and G. Khaliullin, <em><a
href="https://doi.org/10.1103/PhysRevLett.102.017205">Mott Insulators in href="https://doi.org/10.1103/PhysRevLett.102.017205">Mott Insulators in
the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum
Compass and Kitaev Models</a></em>, Phys. Rev. Lett.
<strong>102</strong>, 017205 (2009).</div>
</div>
<div id="ref-khaliullinOrbitalOrderFluctuations2005" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[49] </div><div class="csl-right-inline">G.
Khaliullin, <em><a href="https://doi.org/10.1143/PTPS.160.155">Orbital
Order and Fluctuations in Mott Insulators</a></em>, Progress of
Theoretical Physics Supplement <strong>160</strong>, 155 (2005).</div>
</div>
<div id="ref-Jackeli2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[50] </div><div class="csl-right-inline">G.
Jackeli and G. Khaliullin, <em><a
href="https://doi.org/10.1103/PhysRevLett.102.017205">Mott Insulators in
the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum
Compass and Kitaev Models</a></em>, Physical Review Letters Compass and Kitaev Models</a></em>, Physical Review Letters
<strong>102</strong>, 017205 (2009).</div> <strong>102</strong>, 017205 (2009).</div>
</div> </div>
<div id="ref-HerrmannsAnRev2018" class="csl-entry" <div id="ref-HerrmannsAnRev2018" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[49] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[51] </div><div class="csl-right-inline">M.
Hermanns, I. Kimchi, and J. Knolle, <em><a Hermanns, I. Kimchi, and J. Knolle, <em><a
href="https://doi.org/10.1146/annurev-conmatphys-033117-053934">Physics href="https://doi.org/10.1146/annurev-conmatphys-033117-053934">Physics
of the Kitaev Model: Fractionalization, Dynamic Correlations, and of the Kitaev Model: Fractionalization, Dynamic Correlations, and
@ -886,28 +998,36 @@ Material Connections</a></em>, Annual Review of Condensed Matter Physics
<strong>9</strong>, 17 (2018).</div> <strong>9</strong>, 17 (2018).</div>
</div> </div>
<div id="ref-Winter2017" class="csl-entry" role="doc-biblioentry"> <div id="ref-Winter2017" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[50] </div><div class="csl-right-inline">S. <div class="csl-left-margin">[52] </div><div class="csl-right-inline">S.
M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P.
Gegenwart, and R. Valentí, <em>Models and Materials for Generalized Gegenwart, and R. Valentí, <em>Models and Materials for Generalized
Kitaev Magnetism</em>, Journal of Physics: Condensed Matter Kitaev Magnetism</em>, Journal of Physics: Condensed Matter
<strong>29</strong>, 493002 (2017).</div> <strong>29</strong>, 493002 (2017).</div>
</div> </div>
<div id="ref-Takagi2019" class="csl-entry" role="doc-biblioentry"> <div id="ref-Takagi2019" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[51] </div><div class="csl-right-inline">H. <div class="csl-left-margin">[53] </div><div class="csl-right-inline">H.
Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler,
<em>Concept and Realization of Kitaev Quantum Spin Liquids</em>, Nature <em>Concept and Realization of Kitaev Quantum Spin Liquids</em>, Nature
Reviews Physics <strong>1</strong>, 264 (2019).</div> Reviews Physics <strong>1</strong>, 264 (2019).</div>
</div> </div>
<div id="ref-witczak-krempaCorrelatedQuantumPhenomena2014"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[54] </div><div class="csl-right-inline">W.
Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, <em><a
href="https://doi.org/10.1146/annurev-conmatphys-020911-125138">Correlated
Quantum Phenomena in the Strong Spin-Orbit Regime</a></em>, Annual
Review of Condensed Matter Physics <strong>5</strong>, 57 (2014).</div>
</div>
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry" <div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[52] </div><div class="csl-right-inline">A. <div class="csl-left-margin">[55] </div><div class="csl-right-inline">A.
Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons 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" <div id="ref-knolleDynamicsFractionalizationQuantum2015"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[53] </div><div class="csl-right-inline">J. <div class="csl-left-margin">[56] </div><div class="csl-right-inline">J.
Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, <em><a Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, <em><a
href="https://doi.org/10.1103/PhysRevB.92.115127">Dynamics of href="https://doi.org/10.1103/PhysRevB.92.115127">Dynamics of
Fractionalization in Quantum Spin Liquids</a></em>, Phys. Rev. B Fractionalization in Quantum Spin Liquids</a></em>, Phys. Rev. B
@ -915,7 +1035,7 @@ Fractionalization in Quantum Spin Liquids</a></em>, Phys. Rev. B
</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">[54] </div><div class="csl-right-inline">C. <div class="csl-left-margin">[57] </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
@ -924,7 +1044,7 @@ the Kitaev Model with Parity Projection</a></em>, Phys. Rev. B
</div> </div>
<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">[55] </div><div class="csl-right-inline">A. <div class="csl-left-margin">[58] </div><div class="csl-right-inline">A.
Yu. 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
@ -932,21 +1052,29 @@ Quantum Computation by Anyons</a></em>, Annals of Physics
</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">[56] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[59] </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-nayakNonAbelianAnyonsTopological2008" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[60] </div><div class="csl-right-inline">C.
Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, <em><a
href="https://doi.org/10.1103/RevModPhys.80.1083">Non-Abelian Anyons and
Topological Quantum Computation</a></em>, Rev. Mod. Phys.
<strong>80</strong>, 1083 (2008).</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">[57] </div><div class="csl-right-inline">G. <div class="csl-left-margin">[61] </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">[58] </div><div class="csl-right-inline">G. <div class="csl-left-margin">[62] </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
@ -954,48 +1082,229 @@ 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">[59] </div><div class="csl-right-inline">Z. <div class="csl-left-margin">[63] </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">[60] </div><div class="csl-right-inline">K. <div class="csl-left-margin">[64] </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>,
Phys. Rev. B <strong>93</strong>, 085101 (2016).</div> Phys. Rev. B <strong>93</strong>, 085101 (2016).</div>
</div> </div>
<div id="ref-hermanns2015weyl" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[65] </div><div class="csl-right-inline">M.
Hermanns, K. O’Brien, and S. Trebst, <em>Weyl Spin Liquids</em>,
Physical Review Letters <strong>114</strong>, 157202 (2015).</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">[61] </div><div class="csl-right-inline">H. <div class="csl-left-margin">[66] </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 class="csl-left-margin">[62] </div><div class="csl-right-inline">M.
Hermanns, K. O’Brien, and S. Trebst, <em>Weyl Spin Liquids</em>,
Physical Review Letters <strong>114</strong>, 157202 (2015).</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">[63] </div><div class="csl-right-inline">T. <div class="csl-left-margin">[67] </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">[64] </div><div class="csl-right-inline">V. <div class="csl-left-margin">[68] </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
Spin Liquid on a Simple Non-Archimedean Lattice</a></em>, Phys. Rev. B Spin Liquid on a Simple Non-Archimedean Lattice</a></em>, Phys. Rev. B
<strong>101</strong>, 041114 (2020).</div> <strong>101</strong>, 041114 (2020).</div>
</div> </div>
<div id="ref-Weaire1971" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[69] </div><div class="csl-right-inline">D.
Weaire and M. F. Thorpe, <em><a
href="https://doi.org/10.1103/PhysRevB.4.2508">Electronic Properties of
an Amorphous Solid. I. A Simple Tight-Binding Theory</a></em>, Phys.
Rev. B <strong>4</strong>, 2508 (1971).</div>
</div>
<div id="ref-betteridge1973possible" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[70] </div><div class="csl-right-inline">G.
Betteridge, <em>A Possible Model of Amorphous Silicon and
Germanium</em>, Journal of Physics C: Solid State Physics
<strong>6</strong>, L427 (1973).</div>
</div>
<div id="ref-mitchellAmorphousTopologicalInsulators2018"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[71] </div><div class="csl-right-inline">N.
P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine,
<em><a href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous
topological insulators constructed from random point sets</a></em>,
Nature Phys <strong>14</strong>, 380 (2018).</div>
</div>
<div id="ref-agarwala2019topological" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[72] </div><div class="csl-right-inline">A.
Agarwala, <em>Topological Insulators in Amorphous Systems</em>, in
<em>Excursions in Ill-Condensed Quantum Matter</em> (Springer, 2019),
pp. 61–79.</div>
</div>
<div id="ref-marsalTopologicalWeaireThorpeModels2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[73] </div><div class="csl-right-inline">Q.
Marsal, D. Varjas, and A. G. Grushin, <em><a
href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire-Thorpe
Models of Amorphous Matter</a></em>, Proc. Natl. Acad. Sci. U.S.A.
<strong>117</strong>, 30260 (2020).</div>
</div>
<div id="ref-costa2019toward" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[74] </div><div class="csl-right-inline">M.
Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A.
Fazzio, <em>Toward Realistic Amorphous Topological Insulators</em>, Nano
Letters <strong>19</strong>, 8941 (2019).</div>
</div>
<div id="ref-agarwala2020higher" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[75] </div><div class="csl-right-inline">A.
Agarwala, V. Juričić, and B. Roy, <em>Higher-Order Topological
Insulators in Amorphous Solids</em>, Physical Review Research
<strong>2</strong>, 012067 (2020).</div>
</div>
<div id="ref-spring2021amorphous" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[76] </div><div class="csl-right-inline">H.
Spring, A. Akhmerov, and D. Varjas, <em>Amorphous Topological Phases
Protected by Continuous Rotation Symmetry</em>, SciPost Physics
<strong>11</strong>, 022 (2021).</div>
</div>
<div id="ref-corbae2019evidence" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[77] </div><div class="csl-right-inline">P.
Corbae et al., <em>Evidence for Topological Surface States in Amorphous
Bi _ {2} Se _ {3}</em>, arXiv Preprint arXiv:1910.13412 (2019).</div>
</div>
<div id="ref-buckel1954einfluss" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[78] </div><div class="csl-right-inline">W.
Buckel and R. Hilsch, <em>Einfluß Der Kondensation Bei Tiefen
Temperaturen Auf Den Elektrischen Widerstand Und Die Supraleitung Für
Verschiedene Metalle</em>, Zeitschrift Für Physik <strong>138</strong>,
109 (1954).</div>
</div>
<div id="ref-mcmillan1981electron" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[79] </div><div class="csl-right-inline">W.
McMillan and J. Mochel, <em>Electron Tunneling Experiments on Amorphous
Ge 1- x Au x</em>, Physical Review Letters <strong>46</strong>, 556
(1981).</div>
</div>
<div id="ref-meisel1981eliashberg" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[80] </div><div class="csl-right-inline">L.
V. Meisel and P. J. Cote, <em><a
href="https://doi.org/10.1103/PhysRevB.23.5834">Eliashberg Function in
Amorphous Metals</a></em>, Phys. Rev. B <strong>23</strong>, 5834
(1981).</div>
</div>
<div id="ref-bergmann1976amorphous" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[81] </div><div class="csl-right-inline">G.
Bergmann, <em>Amorphous Metals and Their Superconductivity</em>, Physics
Reports <strong>27</strong>, 159 (1976).</div>
</div>
<div id="ref-mannaNoncrystallineTopologicalSuperconductors2022"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[82] </div><div class="csl-right-inline">S.
Manna, S. K. Das, and B. Roy, <em><a
href="http://arxiv.org/abs/2207.02203">Noncrystalline Topological
Superconductors</a></em>, arXiv:2207.02203.</div>
</div>
<div id="ref-kim2022fractionalization" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[83] </div><div class="csl-right-inline">S.
Kim, A. Agarwala, and D. Chowdhury, <em>Fractionalization and Topology
in Amorphous Electronic Solids</em>, arXiv Preprint arXiv:2205.11523
(2022).</div>
</div>
<div id="ref-aharony1975critical" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[84] </div><div class="csl-right-inline">A.
Aharony, <em>Critical Behavior of Amorphous Magnets</em>, Physical
Review B <strong>12</strong>, 1038 (1975).</div>
</div>
<div id="ref-Petrakovski1981" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[85] </div><div class="csl-right-inline">G.
A. Petrakovskii, <em><a
href="https://doi.org/10.1070/pu1981v024n06abeh004850">Amorphous
Magnetic Materials</a></em>, Soviet Physics Uspekhi <strong>24</strong>,
511 (1981).</div>
</div>
<div id="ref-kaneyoshi1992introduction" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[86] </div><div class="csl-right-inline">T.
Kaneyoshi, <em>Introduction to Amorphous Magnets</em> (World Scientific
Publishing Company, 1992).</div>
</div>
<div id="ref-Kaneyoshi2018" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[87] </div><div class="csl-right-inline">T.
Kaneyoshi, editor, <em>Amorphous Magnetism</em> (CRC Press, Boca Raton,
2018).</div>
</div>
<div id="ref-fahnle1984monte" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[88] </div><div class="csl-right-inline">M.
Fähnle, <em>Monte Carlo Study of Phase Transitions in Bond-and
Site-Disordered Ising and Classical Heisenberg Ferromagnets</em>,
Journal of Magnetism and Magnetic Materials <strong>45</strong>, 279
(1984).</div>
</div>
<div id="ref-plascak2000ising" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[89] </div><div class="csl-right-inline">J.
Plascak, L. E. Zamora, and G. P. Alcazar, <em>Ising Model for Disordered
Ferromagnetic Fe- Al Alloys</em>, Physical Review B <strong>61</strong>,
3188 (2000).</div>
</div>
<div id="ref-coey1978amorphous" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[90] </div><div class="csl-right-inline">J.
Coey, <em>Amorphous Magnetic Order</em>, Journal of Applied Physics
<strong>49</strong>, 1646 (1978).</div>
</div>
<div id="ref-Anderson1973" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[91] </div><div class="csl-right-inline">P.
W. Anderson, <em>Resonating Valence Bonds: A New Kind of
Insulator?</em>, Mater. Res. Bull. <strong>8</strong>, 153 (1973).</div>
</div>
<div id="ref-Knolle2019" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[92] </div><div class="csl-right-inline">J.
Knolle and R. Moessner, <em><a
href="https://doi.org/10.1146/annurev-conmatphys-031218-013401">A Field
Guide to Spin Liquids</a></em>, Annual Review of Condensed Matter
Physics <strong>10</strong>, 451 (2019).</div>
</div>
<div id="ref-Savary2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[93] </div><div class="csl-right-inline">L.
Savary and L. Balents, <em>Quantum Spin Liquids: A Review</em>, Reports
on Progress in Physics <strong>80</strong>, 016502 (2017).</div>
</div>
<div id="ref-Lacroix2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[94] </div><div class="csl-right-inline">C.
Lacroix, P. Mendels, and F. Mila, editors, <em>Introduction to
Frustrated Magnetism</em>, Vol. 164 (Springer-Verlag, Berlin Heidelberg,
2011).</div>
</div>
<div id="ref-eschmannThermodynamicClassificationThreedimensional2020"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[95] </div><div class="csl-right-inline">T.
Eschmann, P. A. Mishchenko, K. O’Brien, T. A. Bojesen, Y. Kato, M.
Hermanns, Y. Motome, and S. Trebst, <em><a
href="https://doi.org/10.1103/PhysRevB.102.075125">Thermodynamic
Classification of Three-Dimensional Kitaev Spin Liquids</a></em>, Phys.
Rev. B <strong>102</strong>, 075125 (2020).</div>
</div>
</div> </div>
</section> </section>

593
_thesis/2_Background/2.1_FK_Model.html
View File

@ -1,5 +1,5 @@
--- ---
title: 2.1_FK_Model title: Background - The Falikov Kimball Model
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>2.1_FK_Model</title> <title>Background - The Falikov Kimball Model</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
@ -87,13 +87,15 @@ H_{\mathrm{FK}} = &amp; \;U \sum_{i} (d^\dagger_{i}d_{i} -
\tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle
i,j\rangle} c^\dagger_{i}c_{j}.\\ i,j\rangle} c^\dagger_{i}c_{j}.\\
\end{aligned}\]</span></p> \end{aligned}\]</span></p>
<p>The connection to the Hubbard model is that we have relabel the up <p>Here we will only discuss the hypercubic lattices, i.e the chain, the
and down spin electron states and removed the hopping term for one square lattice, the cubic lattice and so on. The connection to the
species, the equivalent of taking the limit of infinite mass ratio <span Hubbard model is that we have relabel the up and down spin electron
class="citation" data-cites="devriesSimplifiedHubbardModel1993"> [<a states and removed the hopping term for one species, the equivalent of
taking the limit of infinite mass ratio <span class="citation"
data-cites="devriesSimplifiedHubbardModel1993"> [<a
href="#ref-devriesSimplifiedHubbardModel1993" href="#ref-devriesSimplifiedHubbardModel1993"
role="doc-biblioref">5</a>]</span>.</p> role="doc-biblioref">5</a>]</span>.</p>
<p>Like other exactly solvable models <span class="citation" <p>Like other exactly solvable models <span class="citation"
data-cites="smithDisorderFreeLocalization2017"> [<a data-cites="smithDisorderFreeLocalization2017"> [<a
href="#ref-smithDisorderFreeLocalization2017" href="#ref-smithDisorderFreeLocalization2017"
role="doc-biblioref">6</a>]</span> and the Kitaev Model, the FK model role="doc-biblioref">6</a>]</span> and the Kitaev Model, the FK model
@ -108,15 +110,17 @@ model exactly solvable, in contrast to the Hubbard model.</p>
<p>Due to Pauli exclusion, maximum filling occurs when each lattice site <p>Due to Pauli exclusion, maximum filling occurs when each lattice site
is fully occupied, <span class="math inline">\(\langle n_c + n_d \rangle is fully occupied, <span class="math inline">\(\langle n_c + n_d \rangle
= 2\)</span>. Here we will focus on the half filled case <span = 2\)</span>. Here we will focus on the half filled case <span
class="math inline">\(\langle n_c + n_d \rangle = 1\)</span>. Doping the class="math inline">\(\langle n_c + n_d \rangle = 1\)</span>. The ground
model away from the half-filled point leads to rich physics including state phenomenology as the model is doped away from the half-filled
superconductivity <span class="citation" state can be rich <span class="citation"
data-cites="jedrzejewskiFalicovKimballModels2001"> [<a data-cites="jedrzejewskiFalicovKimballModels2001 gruberGroundStatesSpinless1990"> [<a
href="#ref-jedrzejewskiFalicovKimballModels2001" href="#ref-jedrzejewskiFalicovKimballModels2001"
role="doc-biblioref">7</a>]</span>.</p> role="doc-biblioref">7</a>,<a href="#ref-gruberGroundStatesSpinless1990"
role="doc-biblioref">8</a>]</span> but from this point we will only
consider the half-filled point.</p>
<p>At half-filling and on bipartite lattices, FK the model is <p>At half-filling and on bipartite lattices, FK the model is
particle-hole symmetric. That is, the Hamiltonian anticommutes with the particle-hole (PH) symmetric. That is, the Hamiltonian anticommutes with
particle hole operator <span the particle hole operator <span
class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a
consequence the energy spectrum is symmetric about <span consequence the energy spectrum is symmetric about <span
class="math inline">\(E = 0\)</span> and this is the Fermi energy. The class="math inline">\(E = 0\)</span> and this is the Fermi energy. The
@ -127,9 +131,11 @@ class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and
<span class="math inline">\(-1\)</span> for the even sublattice <span <span class="math inline">\(-1\)</span> for the even sublattice <span
class="citation" data-cites="gruberFalicovKimballModel2005"> [<a class="citation" data-cites="gruberFalicovKimballModel2005"> [<a
href="#ref-gruberFalicovKimballModel2005" href="#ref-gruberFalicovKimballModel2005"
role="doc-biblioref">8</a>]</span>. The absence of a hopping term for role="doc-biblioref">9</a>]</span>. The absence of a hopping term for
the heavy electrons means they do not need the factor of <span the heavy electrons means they do not need the factor of <span
class="math inline">\(\epsilon_i\)</span>.</p> class="math inline">\(\epsilon_i\)</span>. See appendix <a
href="../6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">A.1</a>
for a full derivation of the PH symmetry.</p>
<div id="fig:simple_DOS" class="fignos"> <div id="fig:simple_DOS" class="fignos">
<figure> <figure>
<img src="/assets/thesis/background_chapter/simple_DOS.svg" <img src="/assets/thesis/background_chapter/simple_DOS.svg"
@ -147,23 +153,21 @@ wave background with 2% binary disorder.</figcaption>
</figure> </figure>
</div> </div>
<p>We will later add a long range interaction between the localised <p>We will later add a long range interaction between the localised
electrons so we will replace the immobile fermions with a classical electrons at which point we will replace the immobile fermions with a
Ising field <span class="math inline">\(S_i = 1 - 2d^\dagger_id_i = classical Ising field <span class="math inline">\(S_i = 1 -
\pm\tfrac{1}{2}\)</span>.</p> 2d^\dagger_id_i = \pm\tfrac{1}{2}\)</span> which I will refer to as the
spins.</p>
<p><span class="math display">\[\begin{aligned} <p><span class="math display">\[\begin{aligned}
H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
\tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\ \tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
\end{aligned}\]</span></p> \end{aligned}\]</span></p>
<p>The FK model can be solved exaclty with dynamic mean field theory in <p>The FK model can be solved exactly with dynamic mean field theory in
the infinite dimensional <span class="citation" the infinite dimensional <span class="citation"
data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a
href="#ref-antipovCriticalExponentsStrongly2014" href="#ref-antipovCriticalExponentsStrongly2014"
role="doc-biblioref">9</a>–<a role="doc-biblioref">10</a>–<a
href="#ref-herrmannNonequilibriumDynamicalCluster2016" href="#ref-herrmannNonequilibriumDynamicalCluster2016"
role="doc-biblioref">12</a>]</span>.</p> role="doc-biblioref">13</a>]</span>.</p>
<ul>
<li>displays disorder free localisation</li>
</ul>
</section> </section>
<section id="phase-diagrams" class="level2"> <section id="phase-diagrams" class="level2">
<h2>Phase Diagrams</h2> <h2>Phase Diagrams</h2>
@ -172,161 +176,245 @@ role="doc-biblioref">12</a>]</span>.</p>
<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg" <img src="/assets/thesis/background_chapter/fk_phase_diagram.svg"
data-short-caption="Fermi-Hubbard and Falikov-Kimball Temperatue-Interaction Phase Diagrams" data-short-caption="Fermi-Hubbard and Falikov-Kimball Temperatue-Interaction Phase Diagrams"
style="width:100.0%" style="width:100.0%"
alt="Figure 2: Schematic Phase diagrams of the Fermi-Hubbard (left) and Falikov-Kimball model (right) showing temperature (T) and repulsive interaction strength (U). Hubbard model diagram adapted from  [13], Falikov-Kimball model from  [14,15]" /> alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from  [10,14]" />
<figcaption aria-hidden="true"><span>Figure 2:</span> Schematic Phase <figcaption aria-hidden="true"><span>Figure 2:</span> Schematic Phase
diagrams of the Fermi-Hubbard (left) and Falikov-Kimball model (right) diagram of the Falikov-Kimball model in dimensions greater than two. At
showing temperature (T) and repulsive interaction strength (U). Hubbard low temperature the classical fermions (spins) settle into an ordered
model diagram adapted from <span class="citation" charge density wave state (antiferromagnetic state). The schematic
data-cites="micnasSuperconductivityNarrowbandSystems1990"> [<a diagram for the Hubbard model is the same. Reproduced from <span
href="#ref-micnasSuperconductivityNarrowbandSystems1990"
role="doc-biblioref">13</a>]</span>, Falikov-Kimball model from <span
class="citation" class="citation"
data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014a"> [<a data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014"> [<a
href="#ref-antipovCriticalExponentsStrongly2014"
role="doc-biblioref">10</a>,<a
href="#ref-antipovInteractionTunedAndersonMott2016" href="#ref-antipovInteractionTunedAndersonMott2016"
role="doc-biblioref">14</a>,<a role="doc-biblioref">14</a>]</span></figcaption>
href="#ref-antipovCriticalExponentsStrongly2014a"
role="doc-biblioref">15</a>]</span></figcaption>
</figure> </figure>
</div> </div>
<ul> <p>In dimensions greater than one, the FK model exhibits a phase
<li>rich phase diagram in 2d Despite its simplicity, the FK model has a transition at some <span class="math inline">\(U\)</span> dependent
rich phase diagram in <span class="math inline">\(D \geq 2\)</span> critical temperature <span class="math inline">\(T_c(U)\)</span> to a
dimensions. For example, it shows an interaction-induced gap opening low temperature ordered phase <span class="citation"
even at high temperatures, similar to the corresponding Hubbard
Model <span class="citation"
data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
href="#ref-brandtThermodynamicsCorrelationFunctions1989"
role="doc-biblioref">16</a>]</span>.</li>
</ul>
<p>At half filling and in dimensions greater than one, the FK model
exhibits a phase transition at some <span
class="math inline">\(U\)</span> dependent critical temperature <span
class="math inline">\(T_c(U)\)</span> to a low temperature charge
density wave state in which the spins order antiferromagnetically. This
corresponds to the heavy electrons occupying one of the two sublattices
A and B <span class="citation"
data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
role="doc-biblioref">17</a>]</span>. In the disordered region above role="doc-biblioref">15</a>]</span>. In terms of the immobile electrons
<span class="math inline">\(T_c(U)\)</span> there is a transition this corresponds to them occupying only one of the two sublattices A and
between an Anderson insulator phase at weak interaction and a Mott B this is known as a charge density wave (CDW) phase. In terms of spins
insulator phase in the strongly interacting regime <span this is an AFM phase.</p>
class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a <p>In the disordered region above <span
class="math inline">\(T_c(U)\)</span> there are two insulating phases.
For weak interactions <span class="math inline">\(U &lt;&lt; t\)</span>,
thermal fluctuations in the spins act as an effective disorder potential
for the fermions, causing them to localise and giving rise to an
Anderson insulating state <span class="citation"
data-cites="andersonAbsenceDiffusionCertain1958"> [<a
href="#ref-andersonAbsenceDiffusionCertain1958" href="#ref-andersonAbsenceDiffusionCertain1958"
role="doc-biblioref">18</a>]</span>.</p> role="doc-biblioref">16</a>]</span> which we will discuss more in
<ul> section <a
<li>superconductivity when doped</li> href="../2_Background/2.3_Disorder.html#bg-disorder-and-localisation">2.3</a>.
</ul> For strong interactions <span class="math inline">\(U &gt;&gt;
<p>In 1D, the ground state phenomenology as the model is doped away from t\)</span>, the spins are not ordered but nevertheless their interaction
the half-filled state can be rich <span class="citation" with the electrons opens a gap, leading a Mott insulator analogous to
data-cites="gruberGroundStatesSpinless1990"> [<a that of the Hubbard model <span class="citation"
href="#ref-gruberGroundStatesSpinless1990" data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
role="doc-biblioref">19</a>]</span> but the system is disordered for all href="#ref-brandtThermodynamicsCorrelationFunctions1989"
<span class="math inline">\(T &gt; 0\)</span> <span class="citation" role="doc-biblioref">17</a>]</span>.</p>
data-cites="kennedyItinerantElectronModel1986"> [<a <p>By contrast, in the one dimensional FK model there is no
href="#ref-kennedyItinerantElectronModel1986" finite-temperature phase transition (FTPT) to an ordered CDW phase <span
role="doc-biblioref">20</a>]</span>.</p>
<p>In the one dimensional FK model there is no ordered CDW phase <span
class="citation" data-cites="liebAbsenceMottTransition1968"> [<a class="citation" data-cites="liebAbsenceMottTransition1968"> [<a
href="#ref-liebAbsenceMottTransition1968" href="#ref-liebAbsenceMottTransition1968"
role="doc-biblioref">21</a>]</span>. The supression of phase transition role="doc-biblioref">18</a>]</span>. Indeed dimensionality is crucial
is a common phenomena in one dimensional systems. It can be understood for the physics of both localisation and FTPTs. In one dimension,
via Peierls’ argument <span class="citation" disorder generally dominates: even the weakest disorder exponentially
data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a localises <em>all</em> single particle eigenstates. Only longer-range
href="#ref-kennedyItinerantElectronModel1986" correlations of the disorder potential can potentially induce
role="doc-biblioref">20</a>,<a localisation-delocalisation transitions in one dimension <span
class="citation"
data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a
href="#ref-aubryAnalyticityBreakingAnderson1980"
role="doc-biblioref">19</a>–<a
href="#ref-dunlapAbsenceLocalizationRandomdimer1990"
role="doc-biblioref">21</a>]</span>. Thermodynamically, short-range
interactions cannot overcome thermal defects in one dimension which
prevents ordered phases at non-zero temperature <span class="citation"
data-cites="goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a
href="#ref-goldshteinPurePointSpectrum1977"
role="doc-biblioref">22</a>–<a
href="#ref-kramerLocalizationTheoryExperiment1993"
role="doc-biblioref">24</a>]</span>.</p>
<p>However, the absence of an FTPT in the short ranged FK chain is far
from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY)
interaction mediated by the fermions <span class="citation"
data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a
href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">25</a>–<a
href="#ref-yosidaMagneticPropertiesCuMn1957"
role="doc-biblioref">28</a>]</span> decays as <span
class="math inline">\(r^{-1}\)</span> in one dimension <span
class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a
href="#ref-rusinCalculationRKKYRange2017"
role="doc-biblioref">29</a>]</span>. This could in principle induce the
necessary long-range interactions for the classical Ising background to
order at low temperatures <span class="citation"
data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a
href="#ref-thoulessLongRangeOrderOneDimensional1969"
role="doc-biblioref">30</a>,<a
href="#ref-peierlsIsingModelFerromagnetism1936" href="#ref-peierlsIsingModelFerromagnetism1936"
role="doc-biblioref">22</a>]</span> to be a consequence of the low role="doc-biblioref">31</a>]</span>. However, Kennedy and Lieb
established rigorously that at half-filling a CDW phase only exists at
<span class="math inline">\(T = 0\)</span> for the one dimensional FK
model <span class="citation"
data-cites="kennedyItinerantElectronModel1986"> [<a
href="#ref-kennedyItinerantElectronModel1986"
role="doc-biblioref">32</a>]</span>.</p>
<p>Based on this primacy of dimensionality, we will go digging into the
one dimensional case. In chapter <a
href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#fk-model">3</a>
we will construct a generalised one-dimensional FK model with long-range
interactions which induces the otherwise forbidden CDW phase at non-zero
temperature. To do this we will draw on theory of the Long Range Ising
Model which is the subject of the next section.</p>
</section>
<section id="long-ranged-ising-model" class="level2">
<h2>Long Ranged Ising model</h2>
<p>The suppression of phase transitions is a common phenomena in one
dimensional systems and the Ising model serves as a great illustration
of this. In terms of classical spins <span class="math inline">\(S_i =
\pm \frac{1}{2}\)</span> the standard Ising model reads</p>
<p><span class="math display">\[H_{\mathrm{I}} = \sum_{\langle ij
\rangle} S_i S_j\]</span></p>
<p>Like the FK model, the Ising model shows an FTPT to an ordered state
only in two dimensions and above. This can be understood via Peierls’
argument <span class="citation"
data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a
href="#ref-peierlsIsingModelFerromagnetism1936"
role="doc-biblioref">31</a>,<a
href="#ref-kennedyItinerantElectronModel1986"
role="doc-biblioref">32</a>]</span> to be a consequence of the low
energy penalty for domain walls in one dimensional systems.</p> energy penalty for domain walls in one dimensional systems.</p>
<p>Following Peierls’ argument, consider the difference in free energy <p>Following Peierls’ argument, consider the difference in free energy
<span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span> <span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span>
between an ordered state and a state with single domain wall in a between an ordered state and a state with single domain wall in a
discrete order parameter. Short range interactions produce a constant discrete order parameter. If this value is negative it implies that the
energy penalty for such a domain wall that does not scale with system ordered state is unstable with respect to domain wall defects, and they
size. In contrast, the number of such single domain wall states scales will thus proliferate, destroying the ordered phase. If we consider the
linearly so the entropy is <span class="math inline">\(\propto \ln scaling of the two terms with system size <span
L\)</span>. Thus the entropic contribution dominates (eventually) in the class="math inline">\(L\)</span> we see that short range interactions
thermodynamic limit and no finite temperature order is possible. In two produce a constant energy penalty <span class="math inline">\(\Delta
dimensions and above, the energy penalty of a domain wall scales like E\)</span> for a domain wall. In contrast, the number of such single
<span class="math inline">\(L^{d-1}\)</span> so they can support ordered domain wall states scales linearly with system size so the entropy is
phases.</p> <span class="math inline">\(\propto \ln L\)</span>. Thus the entropic
</section> contribution dominates (eventually) in the thermodynamic limit and no
<section id="long-ranged-ising-model" class="level2"> finite temperature order is possible. In two dimensions and above, the
<h2>Long Ranged Ising model</h2> energy penalty of a domain wall scales like <span
<p>Our extension to the FK model could now be though of as spinless class="math inline">\(L^{d-1}\)</span> which is why they can support
fermions coupled to a long range Ising (LRI) model. The LRI model has ordered phases. This argument does not quite apply to the FK model
been extensively studied and its behaviour may be bear relation to the because of the aforementioned RKKY interaction. Instead this argument
behaviour of our modified FK model.</p> will give us insight into how to recover an ordered phase in the one
dimensional FK model.</p>
<p>In contrast the long range Ising (LRI) model <span
class="math inline">\(H_{\mathrm{LRI}}\)</span> can have an FTPT in one
dimension.</p>
<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|) <p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|)
\tau_i \tau_j = J \sum_{i\neq j} |i - j|^{-\alpha} \tau_i S_i S_j = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\]</span></p>
\tau_j\]</span></p>
<p>Renormalisation group analyses show that the LRI model has an ordered <p>Renormalisation group analyses show that the LRI model has an ordered
phase in 1D for $1 &lt; &lt; 2 $ <span class="citation" phase in 1D for <span class="math inline">\(1 &lt; \alpha &lt;
2\)</span>  <span class="citation"
data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
href="#ref-dysonExistencePhasetransitionOnedimensional1969" href="#ref-dysonExistencePhasetransitionOnedimensional1969"
role="doc-biblioref">23</a>]</span>. Peierls’ argument can be role="doc-biblioref">33</a>]</span>. Peierls’ argument can be
extended <span class="citation" extended <span class="citation"
data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
href="#ref-thoulessLongRangeOrderOneDimensional1969" href="#ref-thoulessLongRangeOrderOneDimensional1969"
role="doc-biblioref">24</a>]</span> to long range interactions to role="doc-biblioref">30</a>]</span> to long range interactions to
provide intuition for why this is the case. Again considering the energy provide intuition for why this is the case. Again considering the energy
difference between the ordered state <span difference between the ordered state <span
class="math inline">\(\ket{\ldots\uparrow\uparrow\uparrow\uparrow\ldots}\)</span> class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span>
and a domain wall state <span and a domain wall state <span
class="math inline">\(\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}\)</span>. class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\)</span>.
In the case of the LRI model, careful counting shows that this energy In the case of the LRI model, careful counting shows that this energy
penalty is: <span class="math display">\[\Delta E \propto penalty is <span class="math display">\[\Delta E \propto
\sum_{n=1}^{\infty} n J(n)\]</span></p> \sum_{n=1}^{\infty} n J(n)\]</span></p>
<p>because each interaction between spins separated across the domain by <p>because each interaction between spins separated across the domain by
a bond length <span class="math inline">\(n\)</span> can be drawn a bond length <span class="math inline">\(n\)</span> can be drawn
between <span class="math inline">\(n\)</span> equivalent pairs of between <span class="math inline">\(n\)</span> equivalent pairs of
sites. Ruelle proved rigorously for a very general class of 1D systems, sites. The behaviour then depends crucially on the sum scales with
that if <span class="math inline">\(\Delta E\)</span> or its many-body system size. Ruelle proved rigorously for a very general class of 1D
generalisation converges in the thermodynamic limit then the free energy systems, that if <span class="math inline">\(\Delta E\)</span> or its
is analytic <span class="citation" many-body generalisation converges to a constant in the thermodynamic
limit then the free energy is analytic <span class="citation"
data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a
href="#ref-ruelleStatisticalMechanicsOnedimensional1968" href="#ref-ruelleStatisticalMechanicsOnedimensional1968"
role="doc-biblioref">25</a>]</span>. This rules out a finite order phase role="doc-biblioref">34</a>]</span>. This rules out a finite order phase
transition, though not one of the Kosterlitz-Thouless type. Dyson also transition, though not one of the Kosterlitz-Thouless type. Dyson also
proves this though with a slightly different condition on <span proves this though with a slightly different condition on <span
class="math inline">\(J(n)\)</span> <span class="citation" class="math inline">\(J(n)\)</span> <span class="citation"
data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
href="#ref-dysonExistencePhasetransitionOnedimensional1969" href="#ref-dysonExistencePhasetransitionOnedimensional1969"
role="doc-biblioref">23</a>]</span>.</p> role="doc-biblioref">33</a>]</span>.</p>
<p>With a power law form for <span class="math inline">\(J(n)\)</span>, <p>With a power law form for <span class="math inline">\(J(n)\)</span>,
there are three cases to consider:</p> there are a few cases to consider:</p>
<ol type="1"> <p>For <span class="math inline">\(\alpha = 0\)</span> i.e infinite
<li>$ = 0$ For infinite range interactions the Ising model is exactly range interactions, the Ising model is exactly solvable and mean field
solveable and mean field theory is exact <span class="citation" theory is exact <span class="citation"
data-cites="lipkinValidityManybodyApproximation1965"> [<a data-cites="lipkinValidityManybodyApproximation1965"> [<a
href="#ref-lipkinValidityManybodyApproximation1965" href="#ref-lipkinValidityManybodyApproximation1965"
role="doc-biblioref">26</a>]</span>.</li> role="doc-biblioref">35</a>]</span>. This limit is the same as the
<li>$ $ For slowly decaying interactions <span infinite dimensional limit.</p>
class="math inline">\(\sum_n J(n)\)</span> does not converge so the <p>For <span class="math inline">\(\alpha \leq 1\)</span> we have very
Hamiltonian is non-extensive, a case which won’t be further considered slowly decaying interactions. <span class="math inline">\(\Delta
here.</li> E\)</span> does not converge as a function of system size so the
<li>$ 1 &lt; &lt; 2 $ A phase transition to an ordered state at a finite Hamiltonian is non-extensive, a topic not without some considerable
temperature.</li> controversy <span class="citation"
<li>$ = 2 $ The energy of domain walls diverges logarithmically, and data-cites="grossNonextensiveHamiltonianSystems2002 lutskoQuestioningValidityNonextensive2011 wangCommentNonextensiveHamiltonian2003"> [<a
this turns out to be a Kostelitz-Thouless transition <span href="#ref-grossNonextensiveHamiltonianSystems2002"
class="citation" role="doc-biblioref">36</a>–<a
href="#ref-wangCommentNonextensiveHamiltonian2003"
role="doc-biblioref">38</a>]</span> that we will not consider further
here.</p>
<p>For <span class="math inline">\(1 &lt; \alpha &lt; 2\)</span>, we get
a phase transition to an ordered state at a finite temperature, this is
what we want!</p>
<p>For <span class="math inline">\(\alpha = 2\)</span>, the energy of
domain walls diverges logarithmically, and this turns out to be a
Kostelitz-Thouless transition <span class="citation"
data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
href="#ref-thoulessLongRangeOrderOneDimensional1969" href="#ref-thoulessLongRangeOrderOneDimensional1969"
role="doc-biblioref">24</a>]</span>.</li> role="doc-biblioref">30</a>]</span>.</p>
<li>$ 2 &lt; $ For quickly decaying interactions, domain walls have a <p>Finally, for <span class="math inline">\(2 &lt; \alpha\)</span> we
have very quickly decaying interactions and domain walls again have a
finite energy penalty, hence Peirels’ argument holds and there is no finite energy penalty, hence Peirels’ argument holds and there is no
phase transition.</li> phase transition.</p>
</ol> <p>One final complexity is that for <span
class="math inline">\(\tfrac{3}{2} &lt; \alpha &lt; 2\)</span>
renormalisation group methods show that the critical point has
non-universal critical exponents that depend on <span
class="math inline">\(\alpha\)</span>  <span class="citation"
data-cites="fisherCriticalExponentsLongRange1972"> [<a
href="#ref-fisherCriticalExponentsLongRange1972"
role="doc-biblioref">39</a>]</span>. To avoid this potential confounding
factors we will park ourselves at <span class="math inline">\(\alpha =
1.25\)</span> when we apply these ideas to the FK model.</p>
<p>Were we to extend this to arbitrary dimension <span
class="math inline">\(d\)</span> we would find that thermodynamics
properties generally both <span class="math inline">\(d\)</span> and
<span class="math inline">\(\alpha\)</span>, long range interactions can
modify the ‘effective dimension’ of thermodynamic systems <span
class="citation"
data-cites="angeliniRelationsShortrangeLongrange2014"> [<a
href="#ref-angeliniRelationsShortrangeLongrange2014"
role="doc-biblioref">40</a>]</span>.</p>
<div id="fig:alpha_diagram" class="fignos"> <div id="fig:alpha_diagram" class="fignos">
<figure> <figure>
<img src="/assets/thesis/background_chapter/alpha_diagram.svg" <img src="/assets/thesis/background_chapter/alpha_diagram.svg"
data-short-caption="Long Range Ising Model Behaviour" data-short-caption="Long Range Ising Model Behaviour"
style="width:100.0%" alt="Figure 3: " /> style="width:100.0%"
<figcaption aria-hidden="true"><span>Figure 3:</span> </figcaption> alt="Figure 3: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied." />
<figcaption aria-hidden="true"><span>Figure 3:</span> The thermodynamic
behaviour of the long range Ising model <span
class="math inline">\(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i -
j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span
class="math inline">\(\alpha\)</span> is varied.</figcaption>
</figure> </figure>
</div> </div>
<div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
</section> </section>
</section> </section>
<section id="bibliography" class="level1 unnumbered"> <section id="bibliography" class="level1 unnumbered">
@ -388,16 +476,24 @@ href="http://adsabs.harvard.edu/abs/2001AcPPB..32.3243J">Falicov--Kimball
Models of Collective Phenomena in Solids (A Concise Guide)</a></em>, Models of Collective Phenomena in Solids (A Concise Guide)</a></em>,
Acta Physica Polonica B <strong>32</strong>, 3243 (2001).</div> Acta Physica Polonica B <strong>32</strong>, 3243 (2001).</div>
</div> </div>
<div id="ref-gruberFalicovKimballModel2005" class="csl-entry" <div id="ref-gruberGroundStatesSpinless1990" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">C. <div class="csl-left-margin">[8] </div><div class="csl-right-inline">C.
Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a
href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the
Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B
<strong>41</strong>, 2198 (1990).</div>
</div>
<div id="ref-gruberFalicovKimballModel2005" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">C.
Gruber and D. Ueltschi, <em><a Gruber and D. Ueltschi, <em><a
href="http://arxiv.org/abs/math-ph/0502041">The Falicov-Kimball href="http://arxiv.org/abs/math-ph/0502041">The Falicov-Kimball
Model</a></em>, arXiv:math-Ph/0502041 (2005).</div> Model</a></em>, arXiv:math-Ph/0502041 (2005).</div>
</div> </div>
<div id="ref-antipovCriticalExponentsStrongly2014" class="csl-entry" <div id="ref-antipovCriticalExponentsStrongly2014" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">A. <div class="csl-left-margin">[10] </div><div class="csl-right-inline">A.
E. Antipov, E. Gull, and S. Kirchner, <em><a E. Antipov, E. Gull, and S. Kirchner, <em><a
href="https://doi.org/10.1103/PhysRevLett.112.226401">Critical Exponents href="https://doi.org/10.1103/PhysRevLett.112.226401">Critical Exponents
of Strongly Correlated Fermion Systems from Diagrammatic Multiscale of Strongly Correlated Fermion Systems from Diagrammatic Multiscale
@ -406,7 +502,7 @@ Methods</a></em>, Phys. Rev. Lett. <strong>112</strong>, 226401
</div> </div>
<div id="ref-ribicNonlocalCorrelationsSpectral2016" class="csl-entry" <div id="ref-ribicNonlocalCorrelationsSpectral2016" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">T. <div class="csl-left-margin">[11] </div><div class="csl-right-inline">T.
Ribic, G. Rohringer, and K. Held, <em><a Ribic, G. Rohringer, and K. Held, <em><a
href="https://doi.org/10.1103/PhysRevB.93.195105">Nonlocal Correlations href="https://doi.org/10.1103/PhysRevB.93.195105">Nonlocal Correlations
and Spectral Properties of the Falicov-Kimball Model</a></em>, Phys. and Spectral Properties of the Falicov-Kimball Model</a></em>, Phys.
@ -414,7 +510,7 @@ Rev. B <strong>93</strong>, 195105 (2016).</div>
</div> </div>
<div id="ref-freericksExactDynamicalMeanfield2003" class="csl-entry" <div id="ref-freericksExactDynamicalMeanfield2003" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">J. <div class="csl-left-margin">[12] </div><div class="csl-right-inline">J.
K. Freericks and V. Zlatić, <em><a K. Freericks and V. Zlatić, <em><a
href="https://doi.org/10.1103/RevModPhys.75.1333">Exact Dynamical href="https://doi.org/10.1103/RevModPhys.75.1333">Exact Dynamical
Mean-Field Theory of the Falicov-Kimball Model</a></em>, Rev. Mod. Phys. Mean-Field Theory of the Falicov-Kimball Model</a></em>, Rev. Mod. Phys.
@ -422,21 +518,12 @@ Mean-Field Theory of the Falicov-Kimball Model</a></em>, Rev. Mod. Phys.
</div> </div>
<div id="ref-herrmannNonequilibriumDynamicalCluster2016" <div id="ref-herrmannNonequilibriumDynamicalCluster2016"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">A. <div class="csl-left-margin">[13] </div><div class="csl-right-inline">A.
J. Herrmann, N. Tsuji, M. Eckstein, and P. Werner, <em><a J. Herrmann, N. Tsuji, M. Eckstein, and P. Werner, <em><a
href="https://doi.org/10.1103/PhysRevB.94.245114">Nonequilibrium href="https://doi.org/10.1103/PhysRevB.94.245114">Nonequilibrium
Dynamical Cluster Approximation Study of the Falicov-Kimball Dynamical Cluster Approximation Study of the Falicov-Kimball
Model</a></em>, Phys. Rev. B <strong>94</strong>, 245114 (2016).</div> Model</a></em>, Phys. Rev. B <strong>94</strong>, 245114 (2016).</div>
</div> </div>
<div id="ref-micnasSuperconductivityNarrowbandSystems1990"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">R.
Micnas, J. Ranninger, and S. Robaszkiewicz, <em><a
href="https://doi.org/10.1103/RevModPhys.62.113">Superconductivity in
Narrow-Band Systems with Local Nonretarded Attractive
Interactions</a></em>, Rev. Mod. Phys. <strong>62</strong>, 113
(1990).</div>
</div>
<div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry" <div id="ref-antipovInteractionTunedAndersonMott2016" 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.
@ -445,26 +532,9 @@ href="https://doi.org/10.1103/PhysRevLett.117.146601">Interaction-Tuned
Anderson Versus Mott Localization</a></em>, Phys. Rev. Lett. Anderson Versus Mott Localization</a></em>, Phys. Rev. Lett.
<strong>117</strong>, 146601 (2016).</div> <strong>117</strong>, 146601 (2016).</div>
</div> </div>
<div id="ref-antipovCriticalExponentsStrongly2014a" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">A.
E. Antipov, E. Gull, and S. Kirchner, <em><a
href="https://doi.org/10.1103/PhysRevLett.112.226401">Critical Exponents
of Strongly Correlated Fermion Systems from Diagrammatic Multiscale
Methods</a></em>, Phys. Rev. Lett. <strong>112</strong>, 226401
(2014).</div>
</div>
<div id="ref-brandtThermodynamicsCorrelationFunctions1989"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">U.
Brandt and C. Mielsch, <em><a
href="https://doi.org/10.1007/BF01321824">Thermodynamics and Correlation
Functions of the Falicov-Kimball Model in Large Dimensions</a></em>, Z.
Physik B - Condensed Matter <strong>75</strong>, 365 (1989).</div>
</div>
<div id="ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" <div id="ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[15] </div><div class="csl-right-inline">M.
M. Maśka and K. Czajka, <em><a M. Maśka and K. Czajka, <em><a
href="https://doi.org/10.1103/PhysRevB.74.035109">Thermodynamics of the href="https://doi.org/10.1103/PhysRevB.74.035109">Thermodynamics of the
Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo
@ -472,64 +542,152 @@ Study</a></em>, Phys. Rev. B <strong>74</strong>, 035109 (2006).</div>
</div> </div>
<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry" <div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">P. <div class="csl-left-margin">[16] </div><div class="csl-right-inline">P.
W. Anderson, <em><a W. Anderson, <em><a
href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in
Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492 Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492
(1958).</div> (1958).</div>
</div> </div>
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry" <div id="ref-brandtThermodynamicsCorrelationFunctions1989"
role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">C. <div class="csl-left-margin">[17] </div><div class="csl-right-inline">U.
Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a Brandt and C. Mielsch, <em><a
href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the href="https://doi.org/10.1007/BF01321824">Thermodynamics and Correlation
Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B Functions of the Falicov-Kimball Model in Large Dimensions</a></em>, Z.
<strong>41</strong>, 2198 (1990).</div> Physik B - Condensed Matter <strong>75</strong>, 365 (1989).</div>
</div>
<div id="ref-kennedyItinerantElectronModel1986" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[20] </div><div class="csl-right-inline">T.
Kennedy and E. H. Lieb, <em><a
href="https://doi.org/10.1016/0378-4371(86)90188-3">An Itinerant
Electron Model with Crystalline or Magnetic Long Range Order</a></em>,
Physica A: Statistical Mechanics and Its Applications
<strong>138</strong>, 320 (1986).</div>
</div> </div>
<div id="ref-liebAbsenceMottTransition1968" class="csl-entry" <div id="ref-liebAbsenceMottTransition1968" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[21] </div><div class="csl-right-inline">E. <div class="csl-left-margin">[18] </div><div class="csl-right-inline">E.
H. Lieb and F. Y. Wu, <em><a H. Lieb and F. Y. Wu, <em><a
href="https://doi.org/10.1103/PhysRevLett.20.1445">Absence of Mott href="https://doi.org/10.1103/PhysRevLett.20.1445">Absence of Mott
Transition in an Exact Solution of the Short-Range, One-Band Model in Transition in an Exact Solution of the Short-Range, One-Band Model in
One Dimension</a></em>, Phys. Rev. Lett. <strong>20</strong>, 1445 One Dimension</a></em>, Phys. Rev. Lett. <strong>20</strong>, 1445
(1968).</div> (1968).</div>
</div> </div>
<div id="ref-peierlsIsingModelFerromagnetism1936" class="csl-entry" <div id="ref-aubryAnalyticityBreakingAnderson1980" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[22] </div><div class="csl-right-inline">R. <div class="csl-left-margin">[19] </div><div class="csl-right-inline">S.
Peierls, <em><a href="https://doi.org/10.1017/S0305004100019174">On Aubry and G. André, <em>Analyticity Breaking and Anderson Localization
Ising’s Model of Ferromagnetism</a></em>, Mathematical Proceedings of in Incommensurate Lattices</em>, Proceedings, VIII International
the Cambridge Philosophical Society <strong>32</strong>, 477 Colloquium on Group-Theoretical Methods in Physics <strong>3</strong>,
(1936).</div> 18 (1980).</div>
</div> </div>
<div id="ref-dysonExistencePhasetransitionOnedimensional1969" <div id="ref-dassarmaLocalizationMobilityEdges1990" class="csl-entry"
class="csl-entry" role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[23] </div><div class="csl-right-inline">F. <div class="csl-left-margin">[20] </div><div class="csl-right-inline">S.
J. Dyson, <em><a href="https://doi.org/10.1007/BF01645907">Existence of Das Sarma, S. He, and X. C. Xie, <em><a
a Phase-Transition in a One-Dimensional Ising Ferromagnet</a></em>, href="https://doi.org/10.1103/PhysRevB.41.5544">Localization, Mobility
Commun.Math. Phys. <strong>12</strong>, 91 (1969).</div> Edges, and Metal-Insulator Transition in a Class of One-Dimensional
Slowly Varying Deterministic Potentials</a></em>, Phys. Rev. B
<strong>41</strong>, 5544 (1990).</div>
</div>
<div id="ref-dunlapAbsenceLocalizationRandomdimer1990" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[21] </div><div class="csl-right-inline">D.
H. Dunlap, H.-L. Wu, and P. W. Phillips, <em><a
href="https://doi.org/10.1103/PhysRevLett.65.88">Absence of Localization
in a Random-Dimer Model</a></em>, Phys. Rev. Lett. <strong>65</strong>,
88 (1990).</div>
</div>
<div id="ref-goldshteinPurePointSpectrum1977" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[22] </div><div class="csl-right-inline">I.
Ya. Gol’dshtein, S. A. Molchanov, and L. A. Pastur, <em><a
href="https://doi.org/10.1007/BF01135526">A Pure Point Spectrum of the
Stochastic One-Dimensional Schrödinger Operator</a></em>, Funct Anal Its
Appl <strong>11</strong>, 1 (1977).</div>
</div>
<div id="ref-abrahamsScalingTheoryLocalization1979" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[23] </div><div class="csl-right-inline">E.
Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan,
<em><a href="https://doi.org/10.1103/PhysRevLett.42.673">Scaling Theory
of Localization: Absence of Quantum Diffusion in Two
Dimensions</a></em>, Phys. Rev. Lett. <strong>42</strong>, 673
(1979).</div>
</div>
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[24] </div><div class="csl-right-inline">B.
Kramer and A. MacKinnon, <em><a
href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory
and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469
(1993).</div>
</div>
<div id="ref-kasuyaTheoryMetallicFerro1956" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[25] </div><div class="csl-right-inline">T.
Kasuya, <em><a href="https://doi.org/10.1143/PTP.16.45">A Theory of
Metallic Ferro- and Antiferromagnetism on Zener’s Model</a></em>, Prog
Theor Phys <strong>16</strong>, 45 (1956).</div>
</div>
<div id="ref-rudermanIndirectExchangeCoupling1954" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[26] </div><div class="csl-right-inline">M.
A. Ruderman and C. Kittel, <em><a
href="https://doi.org/10.1103/PhysRev.96.99">Indirect Exchange Coupling
of Nuclear Magnetic Moments by Conduction Electrons</a></em>, Phys. Rev.
<strong>96</strong>, 99 (1954).</div>
</div>
<div id="ref-vanvleckNoteInteractionsSpins1962" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[27] </div><div class="csl-right-inline">J.
H. Van Vleck, <em><a
href="https://doi.org/10.1103/RevModPhys.34.681">Note on the
Interactions Between the Spins of Magnetic Ions or Nuclei in
Metals</a></em>, Rev. Mod. Phys. <strong>34</strong>, 681 (1962).</div>
</div>
<div id="ref-yosidaMagneticPropertiesCuMn1957" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">K.
Yosida, <em><a href="https://doi.org/10.1103/PhysRev.106.893">Magnetic
Properties of Cu-Mn Alloys</a></em>, Phys. Rev. <strong>106</strong>,
893 (1957).</div>
</div>
<div id="ref-rusinCalculationRKKYRange2017" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">T.
M. Rusin and W. Zawadzki, <em><a
href="https://doi.org/10.1016/j.jmmm.2017.06.007">On Calculation of RKKY
Range Function in One Dimension</a></em>, Journal of Magnetism and
Magnetic Materials <strong>441</strong>, 387 (2017).</div>
</div> </div>
<div id="ref-thoulessLongRangeOrderOneDimensional1969" class="csl-entry" <div id="ref-thoulessLongRangeOrderOneDimensional1969" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[24] </div><div class="csl-right-inline">D. <div class="csl-left-margin">[30] </div><div class="csl-right-inline">D.
J. Thouless, <em><a J. Thouless, <em><a
href="https://doi.org/10.1103/PhysRev.187.732">Long-Range Order in href="https://doi.org/10.1103/PhysRev.187.732">Long-Range Order in
One-Dimensional Ising Systems</a></em>, Phys. Rev. <strong>187</strong>, One-Dimensional Ising Systems</a></em>, Phys. Rev. <strong>187</strong>,
732 (1969).</div> 732 (1969).</div>
</div> </div>
<div id="ref-peierlsIsingModelFerromagnetism1936" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">R.
Peierls, <em><a href="https://doi.org/10.1017/S0305004100019174">On
Ising’s Model of Ferromagnetism</a></em>, Mathematical Proceedings of
the Cambridge Philosophical Society <strong>32</strong>, 477
(1936).</div>
</div>
<div id="ref-kennedyItinerantElectronModel1986" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">T.
Kennedy and E. H. Lieb, <em><a
href="https://doi.org/10.1016/0378-4371(86)90188-3">An Itinerant
Electron Model with Crystalline or Magnetic Long Range Order</a></em>,
Physica A: Statistical Mechanics and Its Applications
<strong>138</strong>, 320 (1986).</div>
</div>
<div id="ref-dysonExistencePhasetransitionOnedimensional1969"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[33] </div><div class="csl-right-inline">F.
J. Dyson, <em><a href="https://doi.org/10.1007/BF01645907">Existence of
a Phase-Transition in a One-Dimensional Ising Ferromagnet</a></em>,
Commun.Math. Phys. <strong>12</strong>, 91 (1969).</div>
</div>
<div id="ref-ruelleStatisticalMechanicsOnedimensional1968" <div id="ref-ruelleStatisticalMechanicsOnedimensional1968"
class="csl-entry" role="doc-biblioentry"> class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[25] </div><div class="csl-right-inline">D. <div class="csl-left-margin">[34] </div><div class="csl-right-inline">D.
Ruelle, <em><a Ruelle, <em><a
href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical
Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys. Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys.
@ -537,13 +695,56 @@ Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys.
</div> </div>
<div id="ref-lipkinValidityManybodyApproximation1965" class="csl-entry" <div id="ref-lipkinValidityManybodyApproximation1965" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[26] </div><div class="csl-right-inline">H. <div class="csl-left-margin">[35] </div><div class="csl-right-inline">H.
J. Lipkin, N. Meshkov, and A. J. Glick, <em><a J. Lipkin, N. Meshkov, and A. J. Glick, <em><a
href="https://doi.org/10.1016/0029-5582(65)90862-X">Validity of href="https://doi.org/10.1016/0029-5582(65)90862-X">Validity of
Many-Body Approximation Methods for a Solvable Model. (I). Exact Many-Body Approximation Methods for a Solvable Model. (I). Exact
Solutions and Perturbation Theory</a></em>, Nuclear Physics Solutions and Perturbation Theory</a></em>, Nuclear Physics
<strong>62</strong>, 188 (1965).</div> <strong>62</strong>, 188 (1965).</div>
</div> </div>
<div id="ref-grossNonextensiveHamiltonianSystems2002" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[36] </div><div class="csl-right-inline">D.
H. E. Gross, <em><a
href="https://doi.org/10.1016/S0378-4371(01)00646-X">Non-Extensive
Hamiltonian Systems Follow Boltzmann’s Principle Not Tsallis Statistics.
-- Phase Transitions, Second Law of Thermodynamics</a></em>, Physica A:
Statistical Mechanics and Its Applications <strong>305</strong>, 99
(2002).</div>
</div>
<div id="ref-lutskoQuestioningValidityNonextensive2011"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[37] </div><div class="csl-right-inline">J.
F. Lutsko and J. P. Boon, <em><a
href="https://doi.org/10.1209/0295-5075/95/20006">Questioning the
Validity of Non-Extensive Thermodynamics for Classical Hamiltonian
Systems</a></em>, EPL <strong>95</strong>, 20006 (2011).</div>
</div>
<div id="ref-wangCommentNonextensiveHamiltonian2003" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[38] </div><div class="csl-right-inline">Q.
A. Wang, <em><a
href="https://hal.archives-ouvertes.fr/hal-00009462">Comment on
<span>“Nonextensive Hamiltonian Systems Follow Boltzmann’s Principle Not
Tsallis Statistics-Phase Transition, Second Law of
Thermodynamics”</span> by Gross</a></em>, (2003).</div>
</div>
<div id="ref-fisherCriticalExponentsLongRange1972" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[39] </div><div class="csl-right-inline">M.
E. Fisher, S. Ma, and B. G. Nickel, <em><a
href="https://doi.org/10.1103/PhysRevLett.29.917">Critical Exponents for
Long-Range Interactions</a></em>, Phys. Rev. Lett. <strong>29</strong>,
917 (1972).</div>
</div>
<div id="ref-angeliniRelationsShortrangeLongrange2014" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[40] </div><div class="csl-right-inline">M.
C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a
href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between
Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E
<strong>89</strong>, 062120 (2014).</div>
</div>
</div> </div>
</section> </section>

10
_thesis/2_Background/2.2_HKM_Model.html
View File

@ -1,5 +1,5 @@
--- ---
title: 2.2_HKM_Model title: Background - The Kitaev Honeycomb Model
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>2.2_HKM_Model</title> <title>Background - The Kitaev Honeycomb Model</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
@ -30,7 +30,7 @@ image:
<li><a href="#the-kitaev-honeycomb-model" <li><a href="#the-kitaev-honeycomb-model"
id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a> id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
<ul> <ul>
<li><a href="#the-model" id="toc-the-model">The Model</a></li> <li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Model</a></li>
<li><a href="#a-mapping-to-majorana-fermions" <li><a href="#a-mapping-to-majorana-fermions"
id="toc-a-mapping-to-majorana-fermions">A mapping to Majorana id="toc-a-mapping-to-majorana-fermions">A mapping to Majorana
Fermions</a></li> Fermions</a></li>
@ -57,7 +57,7 @@ Diagram</a></li>
<li><a href="#the-kitaev-honeycomb-model" <li><a href="#the-kitaev-honeycomb-model"
id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a> id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
<ul> <ul>
<li><a href="#the-model" id="toc-the-model">The Model</a></li> <li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Model</a></li>
<li><a href="#a-mapping-to-majorana-fermions" <li><a href="#a-mapping-to-majorana-fermions"
id="toc-a-mapping-to-majorana-fermions">A mapping to Majorana id="toc-a-mapping-to-majorana-fermions">A mapping to Majorana
Fermions</a></li> Fermions</a></li>
@ -88,7 +88,7 @@ with long range entanglement (not simple paramagnet)</p>
<li>experimental probes include inelastic neutron scattering, Raman <li>experimental probes include inelastic neutron scattering, Raman
scattering</li> scattering</li>
</ul> </ul>
<section id="the-model" class="level2"> <section id="bg-hkm-model" class="level2">
<h2>The Model</h2> <h2>The Model</h2>
<div id="fig:intro_figure_by_hand" class="fignos"> <div id="fig:intro_figure_by_hand" class="fignos">
<figure> <figure>

14
_thesis/2_Background/2.3_Disorder.html
View File

@ -1,5 +1,5 @@
--- ---
title: 2.3_Disorder title: Background - Disorder &amp; Localisation
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>2.3_Disorder</title> <title>Background - Disorder &amp; Localisation</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
@ -27,8 +27,8 @@ image:
<br> <br>
<nav aria-label="Table of Contents" class="page-table-of-contents"> <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#disorder-and-localisation" <li><a href="#bg-disorder-and-localisation"
id="toc-disorder-and-localisation">Disorder and Localisation</a> id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
<ul> <ul>
<li><a href="#localisation-anderson-many-body-and-disorder-free" <li><a href="#localisation-anderson-many-body-and-disorder-free"
id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: id="toc-localisation-anderson-many-body-and-disorder-free">Localisation:
@ -52,8 +52,8 @@ id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
<!-- Table of Contents --> <!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#disorder-and-localisation" <li><a href="#bg-disorder-and-localisation"
id="toc-disorder-and-localisation">Disorder and Localisation</a> id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
<ul> <ul>
<li><a href="#localisation-anderson-many-body-and-disorder-free" <li><a href="#localisation-anderson-many-body-and-disorder-free"
id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: id="toc-localisation-anderson-many-body-and-disorder-free">Localisation:
@ -70,7 +70,7 @@ id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
--> -->
<!-- Main Page Body --> <!-- Main Page Body -->
<section id="disorder-and-localisation" class="level1"> <section id="bg-disorder-and-localisation" class="level1">
<h1>Disorder and Localisation</h1> <h1>Disorder and Localisation</h1>
<section id="localisation-anderson-many-body-and-disorder-free" <section id="localisation-anderson-many-body-and-disorder-free"
class="level2"> class="level2">

8
_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html
View File

@ -27,7 +27,7 @@ image:
<br> <br>
<nav aria-label="Table of Contents" class="page-table-of-contents"> <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#sec:FK-Model" id="toc-sec:FK-Model">The Model</a></li> <li><a href="#fk-model" id="toc-fk-model">The Model</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
@ -41,7 +41,7 @@ image:
<!-- Table of Contents --> <!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#sec:FK-Model" id="toc-sec:FK-Model">The Model</a></li> <li><a href="#fk-model" id="toc-fk-model">The Model</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
@ -72,13 +72,13 @@ analysis presented here.</p>
present its phase diagram. Second, we present the methods used to solve present its phase diagram. Second, we present the methods used to solve
it numerically. Last, we investigate the model’s localisation properties it numerically. Last, we investigate the model’s localisation properties
and conclude.</p> and conclude.</p>
<section id="sec:FK-Model" class="level1"> <section id="fk-model" class="level1">
<h1>The Model</h1> <h1>The Model</h1>
<p>Dimensionality is crucial for the physics of both localisation and <p>Dimensionality is crucial for the physics of both localisation and
FTPTs. In 1D, disorder generally dominates, even the weakest disorder FTPTs. In 1D, disorder generally dominates, even the weakest disorder
exponentially localises <em>all</em> single particle eigenstates. Only exponentially localises <em>all</em> single particle eigenstates. Only
longer-range correlations of the disorder potential can potentially longer-range correlations of the disorder potential can potentially
induce delocalization <span class="citation" induce delocalisation <span class="citation"
data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a
href="#ref-aubryAnalyticityBreakingAnderson1980" href="#ref-aubryAnalyticityBreakingAnderson1980"
role="doc-biblioref">3</a>–<a role="doc-biblioref">3</a>–<a

10
_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html
View File

@ -1,5 +1,5 @@
--- ---
title: 3.2_LRFK_Methods title: The Long Range Falikov-Kimball Model - Methods
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>3.2_LRFK_Methods</title> <title>The Long Range Falikov-Kimball Model - Methods</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
@ -27,7 +27,7 @@ image:
<br> <br>
<nav aria-label="Table of Contents" class="page-table-of-contents"> <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#sec:FK-Methods" id="toc-sec:FK-Methods">Methods</a> <li><a href="#fk-methods" id="toc-fk-methods">Methods</a>
<ul> <ul>
<li><a href="#markov-chain-monte-carlo" <li><a href="#markov-chain-monte-carlo"
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li> id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
@ -116,7 +116,7 @@ Trick</a></li>
<!-- Table of Contents --> <!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#sec:FK-Methods" id="toc-sec:FK-Methods">Methods</a> <li><a href="#fk-methods" id="toc-fk-methods">Methods</a>
<ul> <ul>
<li><a href="#markov-chain-monte-carlo" <li><a href="#markov-chain-monte-carlo"
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li> id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
@ -198,7 +198,7 @@ Trick</a></li>
--> -->
<!-- Main Page Body --> <!-- Main Page Body -->
<section id="sec:FK-Methods" class="level1"> <section id="fk-methods" class="level1">
<h1>Methods</h1> <h1>Methods</h1>
<section id="markov-chain-monte-carlo" class="level2"> <section id="markov-chain-monte-carlo" class="level2">
<h2>Markov Chain Monte Carlo</h2> <h2>Markov Chain Monte Carlo</h2>

35
_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html
View File

@ -1,5 +1,5 @@
--- ---
title: 3.3_LRFK_Results title: The Long Range Falikov-Kimball Model - Results
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>3.3_LRFK_Results</title> <title>The Long Range Falikov-Kimball Model - Results</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
@ -27,16 +27,15 @@ image:
<br> <br>
<nav aria-label="Table of Contents" class="page-table-of-contents"> <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#sec:FK-results" id="toc-sec:FK-results">Results</a> <li><a href="#fk-results" id="toc-fk-results">Results</a>
<ul> <ul>
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase <li><a href="#lrfk-results-phase-diagram"
Diagram</a></li> id="toc-lrfk-results-phase-diagram">Phase Diagram</a></li>
<li><a href="#localisation-properties" <li><a href="#localisation-properties"
id="toc-localisation-properties">Localisation Properties</a></li> id="toc-localisation-properties">Localisation Properties</a></li>
</ul></li> </ul></li>
<li><a href="#discussion-and-conclusion-secamk-conclusion" <li><a href="#fk-conclusion" id="toc-fk-conclusion">Discussion and
id="toc-discussion-and-conclusion-secamk-conclusion">Discussion and Conclusion</a></li>
Conclusion {sec:AMK-Conclusion}</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
@ -50,23 +49,22 @@ Conclusion {sec:AMK-Conclusion}</a></li>
<!-- Table of Contents --> <!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#sec:FK-results" id="toc-sec:FK-results">Results</a> <li><a href="#fk-results" id="toc-fk-results">Results</a>
<ul> <ul>
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase <li><a href="#lrfk-results-phase-diagram"
Diagram</a></li> id="toc-lrfk-results-phase-diagram">Phase Diagram</a></li>
<li><a href="#localisation-properties" <li><a href="#localisation-properties"
id="toc-localisation-properties">Localisation Properties</a></li> id="toc-localisation-properties">Localisation Properties</a></li>
</ul></li> </ul></li>
<li><a href="#discussion-and-conclusion-secamk-conclusion" <li><a href="#fk-conclusion" id="toc-fk-conclusion">Discussion and
id="toc-discussion-and-conclusion-secamk-conclusion">Discussion and Conclusion</a></li>
Conclusion {sec:AMK-Conclusion}</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
</nav> </nav>
--> -->
<!-- Main Page Body --> <!-- Main Page Body -->
<section id="sec:FK-results" class="level1"> <section id="fk-results" class="level1">
<h1>Results</h1> <h1>Results</h1>
<div id="fig:phase_diagram" class="fignos"> <div id="fig:phase_diagram" class="fignos">
<figure> <figure>
@ -115,7 +113,7 @@ alt="Figure 2: Hello I am the figure caption!" />
figure caption!</figcaption> figure caption!</figcaption>
</figure> </figure>
</div> </div>
<section id="phase-diagram" class="level2"> <section id="lrfk-results-phase-diagram" class="level2">
<h2>Phase Diagram</h2> <h2>Phase Diagram</h2>
<p>Figs. [<a href="#fig:phase_diagram" data-reference-type="ref" <p>Figs. [<a href="#fig:phase_diagram" data-reference-type="ref"
data-reference="fig:phase_diagram">1</a>a] and [<a data-reference="fig:phase_diagram">1</a>a] and [<a
@ -402,9 +400,8 @@ model is that we can explore very large system sizes for a complete
understanding.</p> understanding.</p>
</section> </section>
</section> </section>
<section id="discussion-and-conclusion-secamk-conclusion" <section id="fk-conclusion" class="level1">
class="level1"> <h1>Discussion and Conclusion</h1>
<h1>Discussion and Conclusion {sec:AMK-Conclusion}</h1>
<p>The FK model is one of the simplest non-trivial models of interacting <p>The FK model is one of the simplest non-trivial models of interacting
fermions. We studied its thermodynamic and localisation properties fermions. We studied its thermodynamic and localisation properties
brought down in dimensionality to 1D by adding a novel long-ranged brought down in dimensionality to 1D by adding a novel long-ranged

29
_thesis/4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html
View File

@ -1048,21 +1048,21 @@ ground states in the non-Abelian phase <span class="math inline">\((+1,
data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"> [<a data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"> [<a
href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">2</a>,<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">2</a>,<a
href="#ref-chungTopologicalQuantumPhase2010" href="#ref-chungTopologicalQuantumPhase2010"
role="doc-biblioref">15</a>]</span>. Concretely, this is because the role="doc-biblioref"><strong>chungTopologicalQuantumPhase2010?</strong></a>]</span>.
projector enforces both flux and fermion parity. When we wind a vortex Concretely, this is because the projector enforces both flux and fermion
around both non-contractible loops of the torus, it flips the flux parity. When we wind a vortex around both non-contractible loops of the
parity. Therefore, we have to introduce a fermionic excitation to make torus, it flips the flux parity. Therefore, we have to introduce a
the state physical. Hence, the process does not give a fourth ground fermionic excitation to make the state physical. Hence, the process does
state.</p> not give a fourth ground state.</p>
<p>Recently, the topology has notably gained interest because of <p>Recently, the topology has notably gained interest because of
proposals to use this ground state degeneracy to implement both 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-poulinStabilizerFormalismOperator2005" href="#ref-poulinStabilizerFormalismOperator2005"
role="doc-biblioref">16</a>,<a role="doc-biblioref">15</a>,<a
href="#ref-hastingsDynamicallyGeneratedLogical2021" href="#ref-hastingsDynamicallyGeneratedLogical2021"
role="doc-biblioref">17</a>,<a role="doc-biblioref">16</a>,<a
href="#ref-kitaevFaulttolerantQuantumComputation2003" href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p> role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p>
</section> </section>
@ -1173,18 +1173,9 @@ 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"
role="doc-biblioentry">
<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
href="https://doi.org/10.1103/PhysRevB.81.060403">Topological Quantum
Phase Transition in an Exactly Solvable Model of a Chiral Spin Liquid at
Finite Temperature</a></em>, Phys. Rev. B <strong>81</strong>, 060403
(2010).</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">[16] </div><div class="csl-right-inline">D. <div class="csl-left-margin">[15] </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.
@ -1192,7 +1183,7 @@ 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">[17] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[16] </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>

72
_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html
View File

@ -28,7 +28,7 @@ image:
<br> <br>
<nav aria-label="Table of Contents" class="page-table-of-contents"> <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#sec:AMK-Model" id="toc-sec:AMK-Model">The Model</a> <li><a href="#amk-Model" id="toc-amk-Model">The Model</a>
<ul> <ul>
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous <li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
Systems</a></li> Systems</a></li>
@ -70,7 +70,7 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
<!-- Table of Contents --> <!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#sec:AMK-Model" id="toc-sec:AMK-Model">The Model</a> <li><a href="#amk-Model" id="toc-amk-Model">The Model</a>
<ul> <ul>
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous <li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
Systems</a></li> Systems</a></li>
@ -107,24 +107,32 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
<!-- Main Page Body --> <!-- Main Page Body -->
<p><strong>Contributions</strong></p> <p><strong>Contributions</strong></p>
<p>The material in this chapter expands on work presented in</p> <p>The material in this chapter expands on work presented in</p>
<p><strong>Insert citation of amorphous Kitaev paper here</strong></p> <p> <span class="citation"
<p>which was a joint project of the first three authors with advice and data-cites="cassellaExactChiralAmorphous2022"> [<a
href="#ref-cassellaExactChiralAmorphous2022"
role="doc-biblioref">1</a>]</span> Cassella, G., D’Ornellas, P., Hodson,
T., Natori, W. M., &amp; Knolle, J. (2022). An exact chiral amorphous
spin liquid. <em>arXiv preprint arXiv:2208.08246.</em></p>
<p>the code is available at <span class="citation"
data-cites="hodsonKoalaKitaevAmorphous2022"> [<a
href="#ref-hodsonKoalaKitaevAmorphous2022"
role="doc-biblioref">2</a>]</span>.</p>
<p>This was a joint project of Gino, Peru and myself with advice and
guidance from Willian and Johannes. The project grew out of an interest guidance from Willian and Johannes. The project grew out of an interest
Gino, Peru and I had in studying amorphous systems, coupled with the three of us had in studying amorphous systems, coupled with
Johannes’ expertise on the Kitaev model. The idea to use voronoi Johannes’ expertise on the Kitaev model. The idea to use voronoi
partitions came from <span class="citation" partitions came from <span class="citation"
data-cites="marsalTopologicalWeaireThorpe2020"> [<a data-cites="marsalTopologicalWeaireThorpe2020"> [<a
href="#ref-marsalTopologicalWeaireThorpe2020" href="#ref-marsalTopologicalWeaireThorpe2020"
role="doc-biblioref">1</a>]</span> and Gino did the implementation of role="doc-biblioref">3</a>]</span> and Gino did the implementation of
this. The idea and implementation of the edge colouring using SAT this. The idea and implementation of the edge colouring using SAT
solvers, the mapping from flux sector to bond sector using A* search solvers, the mapping from flux sector to bond sector using A* search
were both entirely my work. Peru came up with the ground state were both entirely my work. Peru found the ground state and implemented
conjecture and implemented the local markers. Gino and I did much of the the local markers. Gino and I did much of the rest of the programming
rest of the programming for Koala while pair programming and for Koala while pair programming and ’whiteboard’ing, this included the
’whiteboard’ing, this included the phase diagram, edge mode and finite phase diagram, edge mode and finite temperature analyses as well as the
temperature analyses as well as the derivation of the projector in the derivation of the projector in the amorphous case.</p>
amorphous case.</p> <section id="amk-Model" class="level1">
<section id="sec:AMK-Model" class="level1">
<h1>The Model</h1> <h1>The Model</h1>
<div id="fig:intro_figure_by_hand" class="fignos"> <div id="fig:intro_figure_by_hand" class="fignos">
<figure> <figure>
@ -160,7 +168,7 @@ structure to behave according to the Kitaev Honeycomb model with small
corrections <span class="citation" corrections <span class="citation"
data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"> [<a data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"> [<a
href="#ref-banerjeeProximateKitaevQuantum2016" href="#ref-banerjeeProximateKitaevQuantum2016"
role="doc-biblioref">2</a>,<a href="#ref-trebstKitaevMaterials2022" role="doc-biblioref">4</a>,<a href="#ref-trebstKitaevMaterials2022"
role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a>]</span>.</p> role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a>]</span>.</p>
<p><strong>expand later: Why do we need spin orbit coupling and what <p><strong>expand later: Why do we need spin orbit coupling and what
will the corrections be?</strong></p> will the corrections be?</strong></p>
@ -172,14 +180,14 @@ because, among other reasons, they can be braided through spacetime to
achieve noise tolerant quantum computations <span class="citation" achieve noise tolerant quantum computations <span class="citation"
data-cites="freedmanTopologicalQuantumComputation2003"> [<a data-cites="freedmanTopologicalQuantumComputation2003"> [<a
href="#ref-freedmanTopologicalQuantumComputation2003" href="#ref-freedmanTopologicalQuantumComputation2003"
role="doc-biblioref">3</a>]</span>.</p> role="doc-biblioref">5</a>]</span>.</p>
<p>Third, and perhaps most importantly, this model is a rare many body <p>Third, and perhaps most importantly, this model is a rare many body
interacting quantum system that can be treated analytically. It is interacting quantum system that can be treated analytically. It is
exactly solvable. We can explicitly write down its many body ground exactly solvable. We can explicitly write down its many body ground
states in terms of single particle states <span class="citation" states in terms of single particle states <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006" href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a>]</span>. The solubility of the Kitaev role="doc-biblioref">6</a>]</span>. The solubility of the Kitaev
Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes
about because the model has extensively many conserved degrees of about because the model has extensively many conserved degrees of
freedom. These conserved quantities can be factored out as classical freedom. These conserved quantities can be factored out as classical
@ -197,7 +205,7 @@ transformation to a Majorana hamiltonian. This discussion shows that,
for the the model to be solvable, it needs only be defined on a for the the model to be solvable, it needs only be defined on a
trivalent, tri-edge-colourable lattice <span class="citation" trivalent, tri-edge-colourable lattice <span class="citation"
data-cites="Nussinov2009"> [<a href="#ref-Nussinov2009" data-cites="Nussinov2009"> [<a href="#ref-Nussinov2009"
role="doc-biblioref">5</a>]</span>.</p> role="doc-biblioref">7</a>]</span>.</p>
<p>The methods section discusses how to generate such lattices and <p>The methods section discusses how to generate such lattices and
colour them. It also explain how to map back and forth between colour them. It also explain how to map back and forth between
configurations of the gauge field and configurations of the gauge configurations of the gauge field and configurations of the gauge
@ -395,7 +403,7 @@ class="math inline">\(\alpha\)</span>-bond with exchange coupling <span
class="math inline">\(J^\alpha\)</span> <span class="citation" class="math inline">\(J^\alpha\)</span> <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006" href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a>]</span>. For notational brevity, it is useful role="doc-biblioref">6</a>]</span>. For notational brevity, it is useful
to introduce the bond operators <span class="math inline">\(K_{ij} = to introduce the bond operators <span class="math inline">\(K_{ij} =
\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> where <span \sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> where <span
class="math inline">\(\alpha\)</span> is a function of <span class="math inline">\(\alpha\)</span> is a function of <span
@ -635,7 +643,7 @@ have paired along bonds to become a classical gauge field <span
class="math inline">\(u_{ij}\)</span>. What follows is relatively class="math inline">\(u_{ij}\)</span>. What follows is relatively
standard theory for quadratic Majorana Hamiltonians <span standard theory for quadratic Majorana Hamiltonians <span
class="citation" data-cites="BlaizotRipka1986"> [<a class="citation" data-cites="BlaizotRipka1986"> [<a
href="#ref-BlaizotRipka1986" role="doc-biblioref">6</a>]</span>.</p> href="#ref-BlaizotRipka1986" role="doc-biblioref">8</a>]</span>.</p>
<p>Because of the antisymmetry of the matrix with entries <span <p>Because of the antisymmetry of the matrix with entries <span
class="math inline">\(J^{\alpha} u_{ij}\)</span>, the eigenvalues of the class="math inline">\(J^{\alpha} u_{ij}\)</span>, the eigenvalues of the
Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> come in Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> come in
@ -762,9 +770,23 @@ class="math inline">\(b^\alpha\)</span> operators could be performed.
<section id="bibliography" class="level1 unnumbered"> <section id="bibliography" class="level1 unnumbered">
<h1 class="unnumbered">Bibliography</h1> <h1 class="unnumbered">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-cassellaExactChiralAmorphous2022" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">G.
Cassella, P. D’Ornellas, T. Hodson, W. M. H. Natori, and J. Knolle,
<em><a href="https://doi.org/10.48550/arXiv.2208.08246">An Exact Chiral
Amorphous Spin Liquid</a></em>, arXiv:2208.08246.</div>
</div>
<div id="ref-hodsonKoalaKitaevAmorphous2022" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">T.
Hodson, P. D’Ornellas, and G. Cassella, <em><a
href="https://doi.org/10.5281/zenodo.6303275">Koala: Kitaev on Amorphous
Lattices</a></em>, (2022).</div>
</div>
<div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry" <div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">Q. <div class="csl-left-margin">[3] </div><div class="csl-right-inline">Q.
Marsal, D. Varjas, and A. G. Grushin, <em><a Marsal, D. Varjas, and A. G. Grushin, <em><a
href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire–Thorpe href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire–Thorpe
Models of Amorphous Matter</a></em>, Proceedings of the National Academy Models of Amorphous Matter</a></em>, Proceedings of the National Academy
@ -772,7 +794,7 @@ of Sciences <strong>117</strong>, 30260 (2020).</div>
</div> </div>
<div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry" <div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">A. <div class="csl-left-margin">[4] </div><div class="csl-right-inline">A.
Banerjee et al., <em><a Banerjee et al., <em><a
href="https://doi.org/10.1038/nmat4604">Proximate Kitaev Quantum Spin href="https://doi.org/10.1038/nmat4604">Proximate Kitaev Quantum Spin
Liquid Behaviour in {\Alpha}-RuCl$_3$</a></em>, Nature Mater Liquid Behaviour in {\Alpha}-RuCl$_3$</a></em>, Nature Mater
@ -780,7 +802,7 @@ Liquid Behaviour in {\Alpha}-RuCl$_3$</a></em>, Nature Mater
</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">[3] </div><div class="csl-right-inline">M. <div class="csl-left-margin">[5] </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
@ -788,20 +810,20 @@ Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31
</div> </div>
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry" <div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry"> role="doc-biblioentry">
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">A. <div class="csl-left-margin">[6] </div><div class="csl-right-inline">A.
Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons 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-Nussinov2009" class="csl-entry" role="doc-biblioentry"> <div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">Z. <div class="csl-left-margin">[7] </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-BlaizotRipka1986" class="csl-entry" role="doc-biblioentry"> <div id="ref-BlaizotRipka1986" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[6] </div><div <div class="csl-left-margin">[8] </div><div
class="csl-right-inline">J.-P. Blaizot and G. Ripka, <em>Quantum Theory class="csl-right-inline">J.-P. Blaizot and G. Ripka, <em>Quantum Theory
of Finite Systems</em> (The MIT Press, 1986).</div> of Finite Systems</em> (The MIT Press, 1986).</div>
</div> </div>

6
_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html
View File

@ -28,7 +28,7 @@ image:
<br> <br>
<nav aria-label="Table of Contents" class="page-table-of-contents"> <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#sec:AMK-Methods" id="toc-sec:AMK-Methods">Methods</a> <li><a href="#amk-methods" id="toc-amk-methods">Methods</a>
<ul> <ul>
<li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li> <li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li>
<li><a href="#graph-representation" id="toc-graph-representation">Graph <li><a href="#graph-representation" id="toc-graph-representation">Graph
@ -65,7 +65,7 @@ Markers</a></li>
<!-- Table of Contents --> <!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#sec:AMK-Methods" id="toc-sec:AMK-Methods">Methods</a> <li><a href="#amk-methods" id="toc-amk-methods">Methods</a>
<ul> <ul>
<li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li> <li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li>
<li><a href="#graph-representation" id="toc-graph-representation">Graph <li><a href="#graph-representation" id="toc-graph-representation">Graph
@ -97,7 +97,7 @@ Markers</a></li>
<!-- Main Page Body --> <!-- Main Page Body -->
<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>
<section id="sec:AMK-Methods" class="level1"> <section id="amk-methods" class="level1">
<h1>Methods</h1> <h1>Methods</h1>
<p>The practical implementation of what is described in this section is <p>The practical implementation of what is described in this section is
available as a Python package called Koala (Kitaev On Amorphous available as a Python package called Koala (Kitaev On Amorphous

20
_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html
View File

@ -28,7 +28,7 @@ image:
<br> <br>
<nav aria-label="Table of Contents" class="page-table-of-contents"> <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#sec:AMK-Results" id="toc-sec:AMK-Results">Results</a> <li><a href="#amk-results" id="toc-amk-results">Results</a>
<ul> <ul>
<li><a href="#the-ground-state-flux-sector" <li><a href="#the-ground-state-flux-sector"
id="toc-the-ground-state-flux-sector">The Ground State Flux id="toc-the-ground-state-flux-sector">The Ground State Flux
@ -52,8 +52,9 @@ Thermal Metal</a></li>
and Conclusion</a> and Conclusion</a>
<ul> <ul>
<li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li> <li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li>
<li><a href="#discussion" id="toc-discussion">Discussion</a></li> <li><a href="#amk-discussion"
<li><a href="#outlook" id="toc-outlook">Outlook</a></li> id="toc-amk-discussion">Discussion</a></li>
<li><a href="#amk-outlook" id="toc-amk-outlook">Outlook</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
@ -68,7 +69,7 @@ and Conclusion</a>
<!-- Table of Contents --> <!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#sec:AMK-Results" id="toc-sec:AMK-Results">Results</a> <li><a href="#amk-results" id="toc-amk-results">Results</a>
<ul> <ul>
<li><a href="#the-ground-state-flux-sector" <li><a href="#the-ground-state-flux-sector"
id="toc-the-ground-state-flux-sector">The Ground State Flux id="toc-the-ground-state-flux-sector">The Ground State Flux
@ -92,8 +93,9 @@ Thermal Metal</a></li>
and Conclusion</a> and Conclusion</a>
<ul> <ul>
<li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li> <li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li>
<li><a href="#discussion" id="toc-discussion">Discussion</a></li> <li><a href="#amk-discussion"
<li><a href="#outlook" id="toc-outlook">Outlook</a></li> id="toc-amk-discussion">Discussion</a></li>
<li><a href="#amk-outlook" id="toc-amk-outlook">Outlook</a></li>
</ul></li> </ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li> <li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul> </ul>
@ -101,7 +103,7 @@ and Conclusion</a>
--> -->
<!-- Main Page Body --> <!-- Main Page Body -->
<section id="sec:AMK-Results" class="level1"> <section id="amk-results" class="level1">
<h1>Results</h1> <h1>Results</h1>
<section id="the-ground-state-flux-sector" class="level2"> <section id="the-ground-state-flux-sector" class="level2">
<h2>The Ground State Flux Sector</h2> <h2>The Ground State Flux Sector</h2>
@ -562,7 +564,7 @@ spin liquid phase.</p>
Anderson transition to a thermal metal phase, driven by the Anderson transition to a thermal metal phase, driven by the
proliferation of vortices with increasing temperature.</p> proliferation of vortices with increasing temperature.</p>
</section> </section>
<section id="discussion" class="level2"> <section id="amk-discussion" class="level2">
<h2>Discussion</h2> <h2>Discussion</h2>
<p><strong>Limits of the ground state conjecture</strong></p> <p><strong>Limits of the ground state conjecture</strong></p>
<p>We found a small number of lattices for which the ground state <p>We found a small number of lattices for which the ground state
@ -581,7 +583,7 @@ code A phase. It is possible that some property of the particular
colouring chosen is what leads to failure of the ground state conjecture colouring chosen is what leads to failure of the ground state conjecture
here.</p> here.</p>
</section> </section>
<section id="outlook" class="level2"> <section id="amk-outlook" class="level2">
<h2>Outlook</h2> <h2>Outlook</h2>
<p>This exactly solvable chiral QSL provides a first example of a <p>This exactly solvable chiral QSL provides a first example of a
topological quantum many-body phase in amorphous magnets, which raises a topological quantum many-body phase in amorphous magnets, which raises a

205
_thesis/6_Appendices/A.1_Markov_Chain_Monte_Carlo.html
View File

@ -1,205 +0,0 @@
---
title: A.1_Markov_Chain_Monte_Carlo
excerpt:
layout: none
image:
---
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>A.1_Markov_Chain_Monte_Carlo</title>
<!-- <style>
html {
line-height: 1.5;
font-family: Georgia, serif;
font-size: 20px;
color: #1a1a1a;
background-color: #fdfdfd;
}
body {
margin: 0 auto;
max-width: 36em;
padding-left: 50px;
padding-right: 50px;
padding-top: 50px;
padding-bottom: 50px;
hyphens: auto;
overflow-wrap: break-word;
text-rendering: optimizeLegibility;
font-kerning: normal;
}
@media (max-width: 600px) {
body {
font-size: 0.9em;
padding: 1em;
}
h1 {
font-size: 1.8em;
}
}
@media print {
body {
background-color: transparent;
color: black;
font-size: 12pt;
}
p, h2, h3 {
orphans: 3;
widows: 3;
}
h2, h3, h4 {
page-break-after: avoid;
}
}
p {
margin: 1em 0;
}
a {
color: #1a1a1a;
}
a:visited {
color: #1a1a1a;
}
img {
max-width: 100%;
}
h1, h2, h3, h4, h5, h6 {
margin-top: 1.4em;
}
h5, h6 {
font-size: 1em;
font-style: italic;
}
h6 {
font-weight: normal;
}
ol, ul {
padding-left: 1.7em;
margin-top: 1em;
}
li > ol, li > ul {
margin-top: 0;
}
blockquote {
margin: 1em 0 1em 1.7em;
padding-left: 1em;
border-left: 2px solid #e6e6e6;
color: #606060;
}
code {
font-family: Menlo, Monaco, 'Lucida Console', Consolas, monospace;
font-size: 85%;
margin: 0;
}
pre {
margin: 1em 0;
overflow: auto;
}
pre code {
padding: 0;
overflow: visible;
overflow-wrap: normal;
}
.sourceCode {
background-color: transparent;
overflow: visible;
}
hr {
background-color: #1a1a1a;
border: none;
height: 1px;
margin: 1em 0;
}
table {
margin: 1em 0;
border-collapse: collapse;
width: 100%;
overflow-x: auto;
display: block;
font-variant-numeric: lining-nums tabular-nums;
}
table caption {
margin-bottom: 0.75em;
}
tbody {
margin-top: 0.5em;
border-top: 1px solid #1a1a1a;
border-bottom: 1px solid #1a1a1a;
}
th {
border-top: 1px solid #1a1a1a;
padding: 0.25em 0.5em 0.25em 0.5em;
}
td {
padding: 0.125em 0.5em 0.25em 0.5em;
}
header {
margin-bottom: 4em;
text-align: center;
}
#TOC li {
list-style: none;
}
#TOC ul {
padding-left: 1.3em;
}
#TOC > ul {
padding-left: 0;
}
#TOC a:not(:hover) {
text-decoration: none;
}
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
span.underline{text-decoration: underline;}
div.column{display: inline-block; vertical-align: top; width: 50%;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
</style> -->
<!-- -->
<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
-->
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<!--[if lt IE 9]>
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
<![endif]-->
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#markov-chain-monte-carlo"
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#markov-chain-monte-carlo"
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
</ul>
</nav>
-->
<h1 id="markov-chain-monte-carlo">Markov Chain Monte Carlo</h1>
</main>
</body>
</html>

4
_thesis/6_Appendices/A.1_Particle_Hole_Symmetry.html
View File

@ -1,5 +1,5 @@
--- ---
title: A.1_Particle_Hole_Symmetry title: Particle-Hole Symmetry
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>A.1_Particle_Hole_Symmetry</title> <title>Particle-Hole Symmetry</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>

270
_thesis/6_Appendices/A.2_Lattice_Generation.html
View File

@ -1,270 +0,0 @@
---
title: A.2_Lattice_Generation
excerpt:
layout: none
image:
---
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>A.2_Lattice_Generation</title>
<!-- <style>
html {
line-height: 1.5;
font-family: Georgia, serif;
font-size: 20px;
color: #1a1a1a;
background-color: #fdfdfd;
}
body {
margin: 0 auto;
max-width: 36em;
padding-left: 50px;
padding-right: 50px;
padding-top: 50px;
padding-bottom: 50px;
hyphens: auto;
overflow-wrap: break-word;
text-rendering: optimizeLegibility;
font-kerning: normal;
}
@media (max-width: 600px) {
body {
font-size: 0.9em;
padding: 1em;
}
h1 {
font-size: 1.8em;
}
}
@media print {
body {
background-color: transparent;
color: black;
font-size: 12pt;
}
p, h2, h3 {
orphans: 3;
widows: 3;
}
h2, h3, h4 {
page-break-after: avoid;
}
}
p {
margin: 1em 0;
}
a {
color: #1a1a1a;
}
a:visited {
color: #1a1a1a;
}
img {
max-width: 100%;
}
h1, h2, h3, h4, h5, h6 {
margin-top: 1.4em;
}
h5, h6 {
font-size: 1em;
font-style: italic;
}
h6 {
font-weight: normal;
}
ol, ul {
padding-left: 1.7em;
margin-top: 1em;
}
li > ol, li > ul {
margin-top: 0;
}
blockquote {
margin: 1em 0 1em 1.7em;
padding-left: 1em;
border-left: 2px solid #e6e6e6;
color: #606060;
}
code {
font-family: Menlo, Monaco, 'Lucida Console', Consolas, monospace;
font-size: 85%;
margin: 0;
}
pre {
margin: 1em 0;
overflow: auto;
}
pre code {
padding: 0;
overflow: visible;
overflow-wrap: normal;
}
.sourceCode {
background-color: transparent;
overflow: visible;
}
hr {
background-color: #1a1a1a;
border: none;
height: 1px;
margin: 1em 0;
}
table {
margin: 1em 0;
border-collapse: collapse;
width: 100%;
overflow-x: auto;
display: block;
font-variant-numeric: lining-nums tabular-nums;
}
table caption {
margin-bottom: 0.75em;
}
tbody {
margin-top: 0.5em;
border-top: 1px solid #1a1a1a;
border-bottom: 1px solid #1a1a1a;
}
th {
border-top: 1px solid #1a1a1a;
padding: 0.25em 0.5em 0.25em 0.5em;
}
td {
padding: 0.125em 0.5em 0.25em 0.5em;
}
header {
margin-bottom: 4em;
text-align: center;
}
#TOC li {
list-style: none;
}
#TOC ul {
padding-left: 1.3em;
}
#TOC > ul {
padding-left: 0;
}
#TOC a:not(:hover) {
text-decoration: none;
}
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
span.underline{text-decoration: underline;}
div.column{display: inline-block; vertical-align: top; width: 50%;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ff0000; font-weight: bold; } /* Alert */
code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { color: #7d9029; } /* Attribute */
code span.bn { color: #40a070; } /* BaseN */
code span.bu { } /* BuiltIn */
code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
code span.ch { color: #4070a0; } /* Char */
code span.cn { color: #880000; } /* Constant */
code span.co { color: #60a0b0; font-style: italic; } /* Comment */
code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */
code span.do { color: #ba2121; font-style: italic; } /* Documentation */
code span.dt { color: #902000; } /* DataType */
code span.dv { color: #40a070; } /* DecVal */
code span.er { color: #ff0000; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #40a070; } /* Float */
code span.fu { color: #06287e; } /* Function */
code span.im { } /* Import */
code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #007020; font-weight: bold; } /* Keyword */
code span.op { color: #666666; } /* Operator */
code span.ot { color: #007020; } /* Other */
code span.pp { color: #bc7a00; } /* Preprocessor */
code span.sc { color: #4070a0; } /* SpecialChar */
code span.ss { color: #bb6688; } /* SpecialString */
code span.st { color: #4070a0; } /* String */
code span.va { color: #19177c; } /* Variable */
code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
</style> -->
<!-- -->
<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
-->
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<!--[if lt IE 9]>
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
<![endif]-->
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#lattice-generation" id="toc-lattice-generation">Lattice
Generation</a></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#lattice-generation" id="toc-lattice-generation">Lattice
Generation</a></li>
</ul>
</nav>
-->
<h1 id="lattice-generation">Lattice Generation</h1>
<div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
</main>
</body>
</html>

4
_thesis/6_Appendices/A.2_Markov_Chain_Monte_Carlo.html
View File

@ -1,5 +1,5 @@
--- ---
title: A.2_Markov_Chain_Monte_Carlo title: Markov Chain Monte Carlo
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>A.2_Markov_Chain_Monte_Carlo</title> <title>Markov Chain Monte Carlo</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>

270
_thesis/6_Appendices/A.3_Lattice_Colouring.html
View File

@ -1,270 +0,0 @@
---
title: A.3_Lattice_Colouring
excerpt:
layout: none
image:
---
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>A.3_Lattice_Colouring</title>
<!-- <style>
html {
line-height: 1.5;
font-family: Georgia, serif;
font-size: 20px;
color: #1a1a1a;
background-color: #fdfdfd;
}
body {
margin: 0 auto;
max-width: 36em;
padding-left: 50px;
padding-right: 50px;
padding-top: 50px;
padding-bottom: 50px;
hyphens: auto;
overflow-wrap: break-word;
text-rendering: optimizeLegibility;
font-kerning: normal;
}
@media (max-width: 600px) {
body {
font-size: 0.9em;
padding: 1em;
}
h1 {
font-size: 1.8em;
}
}
@media print {
body {
background-color: transparent;
color: black;
font-size: 12pt;
}
p, h2, h3 {
orphans: 3;
widows: 3;
}
h2, h3, h4 {
page-break-after: avoid;
}
}
p {
margin: 1em 0;
}
a {
color: #1a1a1a;
}
a:visited {
color: #1a1a1a;
}
img {
max-width: 100%;
}
h1, h2, h3, h4, h5, h6 {
margin-top: 1.4em;
}
h5, h6 {
font-size: 1em;
font-style: italic;
}
h6 {
font-weight: normal;
}
ol, ul {
padding-left: 1.7em;
margin-top: 1em;
}
li > ol, li > ul {
margin-top: 0;
}
blockquote {
margin: 1em 0 1em 1.7em;
padding-left: 1em;
border-left: 2px solid #e6e6e6;
color: #606060;
}
code {
font-family: Menlo, Monaco, 'Lucida Console', Consolas, monospace;
font-size: 85%;
margin: 0;
}
pre {
margin: 1em 0;
overflow: auto;
}
pre code {
padding: 0;
overflow: visible;
overflow-wrap: normal;
}
.sourceCode {
background-color: transparent;
overflow: visible;
}
hr {
background-color: #1a1a1a;
border: none;
height: 1px;
margin: 1em 0;
}
table {
margin: 1em 0;
border-collapse: collapse;
width: 100%;
overflow-x: auto;
display: block;
font-variant-numeric: lining-nums tabular-nums;
}
table caption {
margin-bottom: 0.75em;
}
tbody {
margin-top: 0.5em;
border-top: 1px solid #1a1a1a;
border-bottom: 1px solid #1a1a1a;
}
th {
border-top: 1px solid #1a1a1a;
padding: 0.25em 0.5em 0.25em 0.5em;
}
td {
padding: 0.125em 0.5em 0.25em 0.5em;
}
header {
margin-bottom: 4em;
text-align: center;
}
#TOC li {
list-style: none;
}
#TOC ul {
padding-left: 1.3em;
}
#TOC > ul {
padding-left: 0;
}
#TOC a:not(:hover) {
text-decoration: none;
}
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
span.underline{text-decoration: underline;}
div.column{display: inline-block; vertical-align: top; width: 50%;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ff0000; font-weight: bold; } /* Alert */
code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { color: #7d9029; } /* Attribute */
code span.bn { color: #40a070; } /* BaseN */
code span.bu { } /* BuiltIn */
code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
code span.ch { color: #4070a0; } /* Char */
code span.cn { color: #880000; } /* Constant */
code span.co { color: #60a0b0; font-style: italic; } /* Comment */
code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */
code span.do { color: #ba2121; font-style: italic; } /* Documentation */
code span.dt { color: #902000; } /* DataType */
code span.dv { color: #40a070; } /* DecVal */
code span.er { color: #ff0000; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #40a070; } /* Float */
code span.fu { color: #06287e; } /* Function */
code span.im { } /* Import */
code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #007020; font-weight: bold; } /* Keyword */
code span.op { color: #666666; } /* Operator */
code span.ot { color: #007020; } /* Other */
code span.pp { color: #bc7a00; } /* Preprocessor */
code span.sc { color: #4070a0; } /* SpecialChar */
code span.ss { color: #bb6688; } /* SpecialString */
code span.st { color: #4070a0; } /* String */
code span.va { color: #19177c; } /* Variable */
code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
</style> -->
<!-- -->
<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
-->
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<!--[if lt IE 9]>
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
<![endif]-->
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#lattice-colouring" id="toc-lattice-colouring">Lattice
Colouring</a></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#lattice-colouring" id="toc-lattice-colouring">Lattice
Colouring</a></li>
</ul>
</nav>
-->
<h1 id="lattice-colouring">Lattice Colouring</h1>
<div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
</main>
</body>
</html>

4
_thesis/6_Appendices/A.3_Lattice_Generation.html
View File

@ -1,5 +1,5 @@
--- ---
title: A.3_Lattice_Generation title: Lattice Generation
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>A.3_Lattice_Generation</title> <title>Lattice Generation</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>

4
_thesis/6_Appendices/A.4_Lattice_Colouring.html
View File

@ -1,5 +1,5 @@
--- ---
title: A.4_Lattice_Colouring title: Lattice Colouring
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>A.4_Lattice_Colouring</title> <title>Lattice Colouring</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>

205
_thesis/6_Appendices/A.4_The_Projector.html
View File

@ -1,205 +0,0 @@
---
title: A.4_The_Projector
excerpt:
layout: none
image:
---
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>A.4_The_Projector</title>
<!-- <style>
html {
line-height: 1.5;
font-family: Georgia, serif;
font-size: 20px;
color: #1a1a1a;
background-color: #fdfdfd;
}
body {
margin: 0 auto;
max-width: 36em;
padding-left: 50px;
padding-right: 50px;
padding-top: 50px;
padding-bottom: 50px;
hyphens: auto;
overflow-wrap: break-word;
text-rendering: optimizeLegibility;
font-kerning: normal;
}
@media (max-width: 600px) {
body {
font-size: 0.9em;
padding: 1em;
}
h1 {
font-size: 1.8em;
}
}
@media print {
body {
background-color: transparent;
color: black;
font-size: 12pt;
}
p, h2, h3 {
orphans: 3;
widows: 3;
}
h2, h3, h4 {
page-break-after: avoid;
}
}
p {
margin: 1em 0;
}
a {
color: #1a1a1a;
}
a:visited {
color: #1a1a1a;
}
img {
max-width: 100%;
}
h1, h2, h3, h4, h5, h6 {
margin-top: 1.4em;
}
h5, h6 {
font-size: 1em;
font-style: italic;
}
h6 {
font-weight: normal;
}
ol, ul {
padding-left: 1.7em;
margin-top: 1em;
}
li > ol, li > ul {
margin-top: 0;
}
blockquote {
margin: 1em 0 1em 1.7em;
padding-left: 1em;
border-left: 2px solid #e6e6e6;
color: #606060;
}
code {
font-family: Menlo, Monaco, 'Lucida Console', Consolas, monospace;
font-size: 85%;
margin: 0;
}
pre {
margin: 1em 0;
overflow: auto;
}
pre code {
padding: 0;
overflow: visible;
overflow-wrap: normal;
}
.sourceCode {
background-color: transparent;
overflow: visible;
}
hr {
background-color: #1a1a1a;
border: none;
height: 1px;
margin: 1em 0;
}
table {
margin: 1em 0;
border-collapse: collapse;
width: 100%;
overflow-x: auto;
display: block;
font-variant-numeric: lining-nums tabular-nums;
}
table caption {
margin-bottom: 0.75em;
}
tbody {
margin-top: 0.5em;
border-top: 1px solid #1a1a1a;
border-bottom: 1px solid #1a1a1a;
}
th {
border-top: 1px solid #1a1a1a;
padding: 0.25em 0.5em 0.25em 0.5em;
}
td {
padding: 0.125em 0.5em 0.25em 0.5em;
}
header {
margin-bottom: 4em;
text-align: center;
}
#TOC li {
list-style: none;
}
#TOC ul {
padding-left: 1.3em;
}
#TOC > ul {
padding-left: 0;
}
#TOC a:not(:hover) {
text-decoration: none;
}
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
span.underline{text-decoration: underline;}
div.column{display: inline-block; vertical-align: top; width: 50%;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
</style> -->
<!-- -->
<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
-->
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<!--[if lt IE 9]>
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
<![endif]-->
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
Contents:
<ul>
<li><a href="#the-projector" id="toc-the-projector">The
Projector</a></li>
</ul>
{% endcapture %}
<!-- Give the table of contents to header as a variable -->
{% include header.html extra=tableOfContents %}
<main>
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#the-projector" id="toc-the-projector">The
Projector</a></li>
</ul>
</nav>
-->
<h1 id="the-projector">The Projector</h1>
</main>
</body>
</html>

10
_thesis/6_Appendices/A.5_The_Projector.html
View File

@ -1,5 +1,5 @@
--- ---
title: A.5_The_Projector title: The Projector
excerpt: excerpt:
layout: none layout: none
image: image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" /> <meta charset="utf-8" />
<meta name="generator" content="pandoc" /> <meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>A.5_The_Projector</title> <title>The Projector</title>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script> <script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
@ -27,7 +27,7 @@ image:
<br> <br>
<nav aria-label="Table of Contents" class="page-table-of-contents"> <nav aria-label="Table of Contents" class="page-table-of-contents">
<ul> <ul>
<li><a href="#the-projector" id="toc-the-projector">The <li><a href="#app-the-projector" id="toc-app-the-projector">The
Projector</a></li> Projector</a></li>
</ul> </ul>
</nav> </nav>
@ -41,14 +41,14 @@ Projector</a></li>
<!-- Table of Contents --> <!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc"> <!-- <nav id="TOC" role="doc-toc">
<ul> <ul>
<li><a href="#the-projector" id="toc-the-projector">The <li><a href="#app-the-projector" id="toc-app-the-projector">The
Projector</a></li> Projector</a></li>
</ul> </ul>
</nav> </nav>
--> -->
<!-- Main Page Body --> <!-- Main Page Body -->
<section id="the-projector" class="level1"> <section id="app-the-projector" class="level1">
<h1>The Projector</h1> <h1>The Projector</h1>
</section> </section>

251
_thesis/6_Appendices/Untitled.html
View File

@ -1,251 +0,0 @@
---
title: Introduction
excerpt: Why do we do Condensed Matter theory at all?
layout: none
image:
---
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<meta name="description" content="Why do we do Condensed Matter theory at all?" />
<title>Introduction</title>
<!-- <style>
html {
line-height: 1.5;
font-family: Georgia, serif;
font-size: 20px;
color: #1a1a1a;
background-color: #fdfdfd;
}
body {
margin: 0 auto;
max-width: 36em;
padding-left: 50px;
padding-right: 50px;
padding-top: 50px;
padding-bottom: 50px;
hyphens: auto;
overflow-wrap: break-word;
text-rendering: optimizeLegibility;
font-kerning: normal;
}
@media (max-width: 600px) {
body {
font-size: 0.9em;
padding: 1em;
}
h1 {
font-size: 1.8em;
}
}
@media print {
body {
background-color: transparent;
color: black;
font-size: 12pt;
}
p, h2, h3 {
orphans: 3;
widows: 3;
}
h2, h3, h4 {
page-break-after: avoid;
}
}
p {
margin: 1em 0;
}
a {
color: #1a1a1a;
}
a:visited {
color: #1a1a1a;
}
img {
max-width: 100%;
}
h1, h2, h3, h4, h5, h6 {
margin-top: 1.4em;
}
h5, h6 {
font-size: 1em;
font-style: italic;
}
h6 {
font-weight: normal;
}
ol, ul {
padding-left: 1.7em;
margin-top: 1em;
}
li > ol, li > ul {
margin-top: 0;
}
blockquote {
margin: 1em 0 1em 1.7em;
padding-left: 1em;
border-left: 2px solid #e6e6e6;
color: #606060;
}
code {
font-family: Menlo, Monaco, 'Lucida Console', Consolas, monospace;
font-size: 85%;
margin: 0;
}
pre {
margin: 1em 0;
overflow: auto;
}
pre code {
padding: 0;
overflow: visible;
overflow-wrap: normal;
}
.sourceCode {
background-color: transparent;
overflow: visible;
}
hr {
background-color: #1a1a1a;
border: none;
height: 1px;
margin: 1em 0;
}
table {
margin: 1em 0;
border-collapse: collapse;
width: 100%;
overflow-x: auto;
display: block;
font-variant-numeric: lining-nums tabular-nums;
}
table caption {
margin-bottom: 0.75em;
}
tbody {
margin-top: 0.5em;
border-top: 1px solid #1a1a1a;
border-bottom: 1px solid #1a1a1a;
}
th {
border-top: 1px solid #1a1a1a;
padding: 0.25em 0.5em 0.25em 0.5em;
}
td {
padding: 0.125em 0.5em 0.25em 0.5em;
}
header {
margin-bottom: 4em;
text-align: center;
}
#TOC li {
list-style: none;
}
#TOC ul {
padding-left: 1.3em;
}
#TOC > ul {
padding-left: 0;
}
#TOC a:not(:hover) {
text-decoration: none;
}
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
span.underline{text-decoration: underline;}
div.column{display: inline-block; vertical-align: top; width: 50%;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ff0000; font-weight: bold; } /* Alert */
code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { color: #7d9029; } /* Attribute */
code span.bn { color: #40a070; } /* BaseN */
code span.bu { } /* BuiltIn */
code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
code span.ch { color: #4070a0; } /* Char */
code span.cn { color: #880000; } /* Constant */
code span.co { color: #60a0b0; font-style: italic; } /* Comment */
code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */
code span.do { color: #ba2121; font-style: italic; } /* Documentation */
code span.dt { color: #902000; } /* DataType */
code span.dv { color: #40a070; } /* DecVal */
code span.er { color: #ff0000; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #40a070; } /* Float */
code span.fu { color: #06287e; } /* Function */
code span.im { } /* Import */
code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #007020; font-weight: bold; } /* Keyword */
code span.op { color: #666666; } /* Operator */
code span.ot { color: #007020; } /* Other */
code span.pp { color: #bc7a00; } /* Preprocessor */
code span.sc { color: #4070a0; } /* SpecialChar */
code span.ss { color: #bb6688; } /* SpecialString */
code span.st { color: #4070a0; } /* String */
code span.va { color: #19177c; } /* Variable */
code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
</style> -->
<!-- -->
<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
-->
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<!--[if lt IE 9]>
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
<![endif]-->
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
{% include header.html %}
<main>
<div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="op">%%</span>html</span></code></pre></div>
</main>
</body>
</html>

7
_thesis/toc.html
View File

@ -4,7 +4,6 @@
<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">Mott Insulators</a></li> <li><a href="./1_Introduction/1_Intro.html#mott-insulators">Mott Insulators</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#quantum-spin-liquids">Quantum Spin Liquids</a></li>
<li><a href="./1_Introduction/1_Intro.html#outline">Outline</a></li>
</ul> </ul>
<li><a href="./2_Background/2.1_FK_Model.html#the-falikov-kimball-model">Background</a></li> <li><a href="./2_Background/2.1_FK_Model.html#the-falikov-kimball-model">Background</a></li>
<ul> <ul>
@ -27,14 +26,10 @@
<li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#discussion-and-conclusion">Discussion and Conclusion</a></li> <li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#discussion-and-conclusion">Discussion and Conclusion</a></li>
</ul> </ul>
<li><a href="./5_Conclusion/5_Conclusion.html#discussion">Conclusion</a></li> <li><a href="./5_Conclusion/5_Conclusion.html#discussion">Conclusion</a></li>
<li><a href="./6_Appendices/A.1_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Appendices</a></li> <li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">Appendices</a></li>
<ul> <ul>
<li><a href="./6_Appendices/A.1_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
<li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">Particle-Hole Symmetry</a></li> <li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">Particle-Hole Symmetry</a></li>
<li><a href="./6_Appendices/A.2_Lattice_Generation.html#lattice-generation">Lattice Generation</a></li>
<li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li> <li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
<li><a href="./6_Appendices/A.3_Lattice_Colouring.html#lattice-colouring">Lattice Colouring</a></li>
<li><a href="./6_Appendices/A.3_Lattice_Generation.html#lattice-generation">Lattice Generation</a></li> <li><a href="./6_Appendices/A.3_Lattice_Generation.html#lattice-generation">Lattice Generation</a></li>
<li><a href="./6_Appendices/A.4_Lattice_Colouring.html#lattice-colouring">Lattice Colouring</a></li> <li><a href="./6_Appendices/A.4_Lattice_Colouring.html#lattice-colouring">Lattice Colouring</a></li>
<li><a href="./6_Appendices/A.4_The_Projector.html#the-projector">The Projector</a></li>
<li><a href="./6_Appendices/A.5_The_Projector.html#the-projector">The Projector</a></li> <li><a href="./6_Appendices/A.5_The_Projector.html#the-projector">The Projector</a></li>