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

updateo

Browse Source
This commit is contained in:
Tom Hodson 2022-08-24 17:30:38 +02:00
parent 1eea7745d4
commit 37d9ce1699
10 changed files with 1388 additions and 1319 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

18
_sass/thesis.scss
View File

@ -35,13 +35,25 @@ main > ul > ul > li {
margin-top: 0.5em;
}
// Mess with the formatting of the citations
// Pull the citations a little closer in to the previous word
span.citation {
margin-left: -1em;
a {
text-decoration: none;
color: darkblue;
}
}
// Mess with the formatting of the bibliography
div.csl-entry {
margin-bottom: 0.5em;
}
// div.csl-entry a {
div.csl-entry a {
// text-decoration: none;
// }
text-decoration: none;
color: darkblue;
}
div.csl-entry div {
display: inline;

771
_thesis/1_Introduction/1_Intro.html
View File

@ -204,43 +204,35 @@ image:
<main>
<nav id="TOC" role="doc-toc">
<ul>
<li><a href="#interacting-quantum-many-body-systems"
id="toc-interacting-quantum-many-body-systems">Interacting Quantum Many
Body Systems</a></li>
<li><a href="#mott-insulators-and-the-hubbard-model"
id="toc-mott-insulators-and-the-hubbard-model">Mott Insulators and The
Hubbard Model</a></li>
<li><a href="#outline" id="toc-outline">Outline</a></li>
</ul>
</nav>
<h1 id="interacting-quantum-many-body-systems">Interacting Quantum Many
Body Systems</h1>
<p><strong>Interacting Quantum Many Body Systems</strong></p>
<p>When you take many objects and let them interact together, it is
often simpler to describe the behaviour of the group differently than
one would describe the individual objects. Consider a flock (technically
called a <em>murmuration</em>) of starlings like fig. <a
href="#fig:Studland_Starlings">1</a>. Watching the flock you’ll see that
it has a distinct outline, that waves of density will sometimes
propagate through the closely packed birds and that the flock seems to
respond to predators as a distinct object. The natural description of
this phenomena is couched in terms of the flock rather than the
individual birds.</p>
<p>The behaviours of the flock are an emergent phenomena. The starlings
are only interacting with their immediate six or seven neighbours<span
class="citation"
data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"><sup><a
often simpler to describe the behaviour of the group differently from
the way one would describe the individual objects. Consider a flock of
starlings like that of fig. <a href="#fig:Studland_Starlings">1</a>.
Watching the flock you’ll see that it has a distinct outline, that waves
of density will sometimes propagate through the closely packed birds and
that the flock seems to respond to predators as a distinct object. The
natural description of this phenomena is couched in terms of the flock
rather than of the individual birds.</p>
<p>The behaviours of the flock are an <em>emergent phenomena</em>. The
starlings are only interacting with their immediate six or seven
neighbours <span class="citation"
data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"> [<a
href="#ref-king2012murmurations" role="doc-biblioref">1</a>,<a
href="#ref-balleriniInteractionRulingAnimal2008"
role="doc-biblioref">2</a></sup></span>. This is what a physicist would
call a <em>local interaction</em>. There is much philosophical debate
about how exactly to define emergence<span class="citation"
data-cites="andersonMoreDifferent1972 kivelsonDefiningEmergencePhysics2016"><sup><a
role="doc-biblioref">2</a>]</span>, what a physicist would call a
<em>local interaction</em>. There is much philosophical debate about how
exactly to define emergence <span class="citation"
data-cites="andersonMoreDifferent1972 kivelsonDefiningEmergencePhysics2016"> [<a
href="#ref-andersonMoreDifferent1972" role="doc-biblioref">3</a>,<a
href="#ref-kivelsonDefiningEmergencePhysics2016"
role="doc-biblioref">4</a></sup></span> but for our purposes it enough
to say that emergence is the fact that the aggregate behaviour of many
interacting objects may be very different from the individual behaviour
of those objects.</p>
role="doc-biblioref">4</a>]</span> but for our purposes it enough to say
that emergence is the fact that the aggregate behaviour of many
interacting objects may necessitate a description very different from
that of the individual objects.</p>
<div id="fig:Studland_Starlings" class="fignos">
<figure>
<img src="/assets/thesis/intro_chapter/Studland_Starlings.jpeg"
@ -253,17 +245,17 @@ href="creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA
3.0</a></figcaption>
</figure>
</div>
<p>To give another example, our understanding of thermodynamics began
with bulk properties like heat, energy, pressure and temperature<span
class="citation"
data-cites="saslowHistoryThermodynamicsMissing2020"><sup><a
<p>To give an example closer to the topic at hand, our understanding of
thermodynamics began with bulk properties like heat, energy, pressure
and temperature <span class="citation"
data-cites="saslowHistoryThermodynamicsMissing2020"> [<a
href="#ref-saslowHistoryThermodynamicsMissing2020"
role="doc-biblioref">5</a></sup></span>. It was only later that we
gained an understanding of how these properties emerge from microscopic
interactions between very large numbers of particles<span
class="citation" data-cites="flammHistoryOutlookStatistical1998"><sup><a
role="doc-biblioref">5</a>]</span>. It was only later that we gained an
understanding of how these properties emerge from microscopic
interactions between very large numbers of particles <span
class="citation" data-cites="flammHistoryOutlookStatistical1998"> [<a
href="#ref-flammHistoryOutlookStatistical1998"
role="doc-biblioref">6</a></sup></span>.</p>
role="doc-biblioref">6</a>]</span>.</p>
<p>Condensed Matter is, at its heart, the study of what behaviours
emerge from large numbers of interacting quantum objects at low energy.
When these three properties are present together: a large number of
@ -273,60 +265,65 @@ these three ingredients nature builds all manner of weird and wonderful
materials.</p>
<p>Historically, we made initial headway in the study of many-body
systems, ignoring interactions and quantum properties. The ideal gas law
and the Drude classical electron gas<span class="citation"
data-cites="ashcroftSolidStatePhysics1976"><sup><a
and the Drude classical electron gas <span class="citation"
data-cites="ashcroftSolidStatePhysics1976"> [<a
href="#ref-ashcroftSolidStatePhysics1976"
role="doc-biblioref">7</a></sup></span> are good examples. Including
interactions into many-body physics leads to the Ising model<span
class="citation" data-cites="isingBeitragZurTheorie1925"><sup><a
role="doc-biblioref">7</a>]</span> are good examples. Including
interactions into many-body physics leads to the Ising model <span
class="citation" data-cites="isingBeitragZurTheorie1925"> [<a
href="#ref-isingBeitragZurTheorie1925"
role="doc-biblioref">8</a></sup></span>, Landau theory<span
class="citation" data-cites="landau2013fluid"><sup><a
href="#ref-landau2013fluid" role="doc-biblioref">9</a></sup></span> and
the classical theory of phase transitions<span class="citation"
data-cites="jaegerEhrenfestClassificationPhase1998"><sup><a
role="doc-biblioref">8</a>]</span>, Landau theory <span class="citation"
data-cites="landau2013fluid"> [<a href="#ref-landau2013fluid"
role="doc-biblioref">9</a>]</span> and the classical theory of phase
transitions <span class="citation"
data-cites="jaegerEhrenfestClassificationPhase1998"> [<a
href="#ref-jaegerEhrenfestClassificationPhase1998"
role="doc-biblioref">10</a></sup></span>. In contrast, condensed matter
theory got it state in quantum many-body theory. Bloch’s theorem<span
role="doc-biblioref">10</a>]</span>. In contrast, condensed matter
theory got it state in quantum many-body theory. Bloch’s theorem <span
class="citation"
data-cites="blochÜberQuantenmechanikElektronen1929"><sup><a
data-cites="blochÜberQuantenmechanikElektronen1929"> [<a
href="#ref-blochÜberQuantenmechanikElektronen1929"
role="doc-biblioref">11</a></sup></span> predicted the properties of
role="doc-biblioref">11</a>]</span> predicted the properties of
non-interacting electrons in crystal lattices, leading to band theory.
In the same vein, advances were made in understanding the quantum
origins of magnetism, including ferromagnetism and
antiferromagnetism<span class="citation"
data-cites="MagnetismCondensedMatter"><sup><a
origins of magnetism, including ferromagnetism and antiferromagnetism
<span class="citation" data-cites="MagnetismCondensedMatter"> [<a
href="#ref-MagnetismCondensedMatter"
role="doc-biblioref">12</a></sup></span>.</p>
role="doc-biblioref">12</a>]</span>.</p>
<p>However, at some point we had to start on the interacting quantum
many body systems. Some phenomena cannot be understood without a taking
into account all three effects. The canonical examples are
superconductivity<span class="citation"
data-cites="MicroscopicTheorySuperconductivity"><sup><a
many body systems. The properties of some materials cannot be understood
without a taking into account all three effects and these are
collectively called strongly correlated materials. The canonical
examples are superconductivity <span class="citation"
data-cites="MicroscopicTheorySuperconductivity"> [<a
href="#ref-MicroscopicTheorySuperconductivity"
role="doc-biblioref">13</a></sup></span>, the fractional quantum hall
effect<span class="citation"
data-cites="feldmanFractionalChargeFractional2021"><sup><a
role="doc-biblioref">13</a>]</span>, the fractional quantum hall effect
<span class="citation"
data-cites="feldmanFractionalChargeFractional2021"> [<a
href="#ref-feldmanFractionalChargeFractional2021"
role="doc-biblioref">14</a></sup></span> and the Mott insulators<span
role="doc-biblioref">14</a>]</span> and the Mott insulators <span
class="citation"
data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"><sup><a
data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a
href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">15</a>,<a
href="#ref-fisherMottInsulatorsSpin1999"
role="doc-biblioref">16</a></sup></span>. We will discuss the latter in
more detail.</p>
<p>Electrical conductivity, the bulk movement of electrons, requires
both that there are electronic states very close in energy to the ground
state and that those states are delocalised so that they can contribute
to macroscopic transport. Band insulators are systems whose Fermi level
falls within a gap in the density of states and thus fail the first
criteria. Anderson Insulators have only localised electronic states near
the fermi level and therefore fail the second criteria. We will discuss
Anderson insulators and disorder in a later section.</p>
role="doc-biblioref">16</a>]</span>. We’ll start by looking at the
latter but shall see that there are many links between three topics.</p>
<p><strong>Mott Insulators</strong></p>
<p>Mott Insulators are remarkable because their electrical insulator
properties come from electron-electron interactions. Electrical
conductivity, the bulk movement of electrons, requires both that there
are electronic states very close in energy to the ground state and that
those states are delocalised so that they can contribute to macroscopic
transport. Band insulators are systems whose Fermi level falls within a
gap in the density of states and thus fail the first criteria. Band
insulators derive their character from the characteristics of the
underlying lattice. Anderson Insulators have only localised electronic
states near the fermi level and therefore fail the second criteria. We
will discuss Anderson insulators and disorder in a later section.</p>
<p>Both band and Anderson insulators occur without electron-electron
interactions. Mott insulators, by contrast, are by these interactions
and hence elude band theory and single-particle methods.</p>
interactions. Mott insulators, by contrast, require a many body picture
to understand and thus elude band theory and single-particle
methods.</p>
<div id="fig:venn_diagram" class="fignos">
<figure>
<img src="/assets/thesis/intro_chapter/venn_diagram.svg"
@ -342,146 +339,202 @@ or indirectly. When taken together, these three properties can give rise
to what are called strongly correlated materials.</figcaption>
</figure>
</div>
<h1 id="mott-insulators-and-the-hubbard-model">Mott Insulators and The
Hubbard Model</h1>
<p>The theory of Mott insulators developed out of the observation that
many transition metal oxides are erroneously predicted by band theory to
be conductive<span class="citation"
data-cites="boerSemiconductorsPartiallyCompletely1937"><sup><a
be conductive <span class="citation"
data-cites="boerSemiconductorsPartiallyCompletely1937"> [<a
href="#ref-boerSemiconductorsPartiallyCompletely1937"
role="doc-biblioref">17</a></sup></span> leading to the suggestion that
electron-electron interactions were the cause of this effect<span
class="citation" data-cites="mottDiscussionPaperBoer1937"><sup><a
role="doc-biblioref">17</a>]</span> leading to the suggestion that
electron-electron interactions were the cause of this effect <span
class="citation" data-cites="mottDiscussionPaperBoer1937"> [<a
href="#ref-mottDiscussionPaperBoer1937"
role="doc-biblioref">18</a></sup></span>. Interest grew with the
discovery of high temperature superconductivity in the cuprates in
1986<span class="citation"
data-cites="bednorzPossibleHighTcSuperconductivity1986"><sup><a
role="doc-biblioref">18</a>]</span>. Interest grew with the discovery of
high temperature superconductivity in the cuprates in 1986 <span
class="citation"
data-cites="bednorzPossibleHighTcSuperconductivity1986"> [<a
href="#ref-bednorzPossibleHighTcSuperconductivity1986"
role="doc-biblioref">19</a></sup></span> which is believed to arise as
the result of doping a Mott insulator state<span class="citation"
data-cites="leeDopingMottInsulator2006"><sup><a
role="doc-biblioref">19</a>]</span> which is believed to arise as the
result of a doped Mott insulator state <span class="citation"
data-cites="leeDopingMottInsulator2006"> [<a
href="#ref-leeDopingMottInsulator2006"
role="doc-biblioref">20</a></sup></span>.</p>
<p>The canonical toy model of the Mott insulator is the Hubbard
model<span class="citation"
data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"><sup><a
role="doc-biblioref">20</a>]</span>.</p>
<p>The canonical toy model of the Mott insulator is the Hubbard model
<span class="citation"
data-cites="gutzwillerEffectCorrelationFerromagnetism1963 kanamoriElectronCorrelationFerromagnetism1963 hubbardj.ElectronCorrelationsNarrow1963"> [<a
href="#ref-gutzwillerEffectCorrelationFerromagnetism1963"
role="doc-biblioref">21</a>–<a
href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
role="doc-biblioref">23</a></sup></span> of <span
role="doc-biblioref">23</a>]</span> of <span
class="math inline">\(1/2\)</span> fermions hopping on the lattice with
hopping parameter <span class="math inline">\(t\)</span> and
electron-electron repulsion <span class="math inline">\(U\)</span></p>
<p><span class="math display">\[ H = -t \sum_{\langle i,j \rangle
\alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow}
<p><span class="math display">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j
\rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow}
n_{i\downarrow} - \mu \sum_{i,\alpha} n_{i\alpha}\]</span></p>
<p>where <span class="math inline">\(c^\dagger_{i\alpha}\)</span>
creates a spin <span class="math inline">\(\alpha\)</span> electron at
site <span class="math inline">\(i\)</span> and the number operator
<span class="math inline">\(n_{i\alpha}\)</span> measures the number of
electrons with spin <span class="math inline">\(\alpha\)</span> at site
<span class="math inline">\(i\)</span>. In the non-interacting limit
<span class="math inline">\(U &lt;&lt; t\)</span>, the model reduces to
free fermions and the many-body ground state is a separable product of
Bloch waves filled up to the Fermi level. In the interacting limit <span
class="math inline">\(U &gt;&gt; t\)</span> on the other hand, the
system breaks up into a product of local moments, each in one the four
states <span class="math inline">\(|0\rangle, |\uparrow\rangle,
|\downarrow\rangle, |\uparrow\downarrow\rangle\)</span> depending on the
filing.</p>
<span class="math inline">\(i\)</span>. The sum runs over lattice
neighbours <span class="math inline">\(\langle i,j \rangle\)</span>
including both <span class="math inline">\(\langle i,j \rangle\)</span>
and <span class="math inline">\(\langle j,i \rangle\)</span> so that the
model is Hermition.</p>
<p>In the non-interacting limit <span class="math inline">\(U &lt;&lt;
t\)</span>, the model reduces to free fermions and the many-body ground
state is a separable product of Bloch waves filled up to the Fermi
level. In the interacting limit <span class="math inline">\(U &gt;&gt;
t\)</span> on the other hand, the system breaks up into a product of
local moments, each in one the four states <span
class="math inline">\(|0\rangle, |\uparrow\rangle, |\downarrow\rangle,
|\uparrow\downarrow\rangle\)</span> depending on the filing.</p>
<p>The Mott insulating phase occurs at half filling <span
class="math inline">\(\mu = \tfrac{U}{2}\)</span> where there is one
electron per lattice site<span class="citation"
data-cites="hubbardElectronCorrelationsNarrow1964"><sup><a
electron per lattice site <span class="citation"
data-cites="hubbardElectronCorrelationsNarrow1964"> [<a
href="#ref-hubbardElectronCorrelationsNarrow1964"
role="doc-biblioref">24</a></sup></span>. Here the model can be
rewritten in a symmetric form <span class="math display">\[ H = -t
role="doc-biblioref">24</a>]</span>. Here the model can be rewritten in
a symmetric form <span class="math display">\[ H_{\mathrm{H}} = -t
\sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U
\sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} -
\tfrac{1}{2})\]</span></p>
<p>The basic reason that the half filled state is insulating seems is
trivial. Any excitation must include states of double occupancy that
cost energy <span class="math inline">\(U\)</span>, hence the system has
a finite bandgap and is an interaction driven Mott insulator. Originally
it was proposed that antiferromagnetic order was a necessary condition
for the Mott insulator transition<span class="citation"
data-cites="mottMetalInsulatorTransitions1990"><sup><a
a finite bandgap and is an interaction driven Mott insulator. Depending
on the lattice, the local moments may then order antiferromagnetically.
Originally it was proposed that this antiferromagnetic order was the
cause of the gap opening <span class="citation"
data-cites="mottMetalInsulatorTransitions1990"> [<a
href="#ref-mottMetalInsulatorTransitions1990"
role="doc-biblioref">25</a></sup></span> but later examples were found
without magnetic order <strong>cite</strong>.</p>
role="doc-biblioref">25</a>]</span>. However, Mott insulators have been
found <span class="citation"
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">26</a>,<a
href="#ref-ribakGaplessExcitationsGround2017"
role="doc-biblioref">27</a>]</span> without magnetic order. Instead the
local moments may form a highly entangled state known as a quantum spin
liquid, which will be discussed shortly.</p>
<p>Various theoretical treatments of the Hubbard model have been made,
including those based on Fermi liquid theory, mean field treatments, the
local density approximation (LDA)<span class="citation"
data-cites="slaterMagneticEffectsHartreeFock1951"><sup><a
local density approximation (LDA) <span class="citation"
data-cites="slaterMagneticEffectsHartreeFock1951"> [<a
href="#ref-slaterMagneticEffectsHartreeFock1951"
role="doc-biblioref">26</a></sup></span> and dynamical mean-field
theory<span class="citation"
data-cites="greinerQuantumPhaseTransition2002"><sup><a
role="doc-biblioref">28</a>]</span> and dynamical mean-field theory
<span class="citation"
data-cites="greinerQuantumPhaseTransition2002"> [<a
href="#ref-greinerQuantumPhaseTransition2002"
role="doc-biblioref">27</a></sup></span>. None of these approaches is
role="doc-biblioref">29</a>]</span>. None of these approaches are
perfect. Strong correlations are poorly described by the Fermi liquid
theory and the LDA approaches while mean field approximations do poorly
in low dimensional systems. This theoretical difficulty has made the
Hubbard model a target for cold atom simulations<span class="citation"
data-cites="mazurenkoColdatomFermiHubbard2017"><sup><a
Hubbard model a target for cold atom simulations <span class="citation"
data-cites="mazurenkoColdatomFermiHubbard2017"> [<a
href="#ref-mazurenkoColdatomFermiHubbard2017"
role="doc-biblioref">28</a></sup></span>.</p>
role="doc-biblioref">30</a>]</span>.</p>
<p>From here the discussion will branch two directions. First, we will
discuss a limit of the Hubbard model called the Falikov Kimball Model.
Second, we will go down the rabbit hole of strongly correlated systems
without magnetic order. This will lead us to Quantum spin liquids and
the Kitaev honeycomb model.</p>
<p><strong>An exactly solvable model of the Mott Insulator</strong> -
demonstrate mott insulator in hubbard model, briefly tease the falikov
kimball model in order to lay the ground work to talk about the falikov
kimball model later</p>
<ul>
<li>FK model has extensively many conserved charges which makes it
tractable</li>
<li>Disorder free localisation</li>
</ul>
<p><strong>An exactly solvable Quantum Spin Liquid</strong> -
relationship between mott insulators and spin liquids: the electrons in
a mott insulator form local moments that normally form an AFM ground
state but if they don’t, due to frustration or other reason, the local
moments may form a QSL at T=0 instead.<span class="citation"
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"><sup><a
href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">29</a>,<a
discuss a limit of the Hubbard model called the Falikov-Kimball Model.
Second, we will look at quantum spin liquids and the Kitaev honeycomb
model.</p>
<p><strong>The Falikov-Kimball Model</strong></p>
<p>Though not the original reason for its introduction, the
Falikov-Kimball (FK) model is the limit of the Hubbard model as the mass
ratio of the spin up and spin down electron is taken to infinity. This
gives a model with two fermion species, one itinerant and one entirely
immobile. The number operators for the immobile fermions are therefore
conserved quantities and can be be treated like classical degrees of
freedom. For our purposes it will be useful to replace the immobile
fermions with a classical Ising background field <span
class="math inline">\(S_i = \pm1\)</span>.</p>
<p><span class="math display">\[\begin{aligned}
H_{\mathrm{FK}} = &amp; -\;t \sum_{\langle i,j \rangle}
c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
\tfrac{1}{2}). \\
\end{aligned}\]</span></p>
<p>Given that the physics of states near the metal-insulator (MI)
transition is still poorly understood <span class="citation"
data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a
href="#ref-belitzAndersonMottTransition1994"
role="doc-biblioref">31</a>,<a
href="#ref-baskoMetalInsulatorTransition2006"
role="doc-biblioref">32</a>]</span> the FK model provides a rich test
bed to explore interaction driven MI transition physics. Despite its
simplicity, the model has a rich phase diagram in <span
class="math inline">\(D \geq 2\)</span> dimensions. It shows an Mott
insulator transition even at high temperature, similar to the
corresponding Hubbard Model <span class="citation"
data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
href="#ref-brandtThermodynamicsCorrelationFunctions1989"
role="doc-biblioref">33</a>]</span>. In 1D, the ground state
phenomenology as a function of filling can be rich <span
class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a
href="#ref-gruberGroundStatesSpinless1990"
role="doc-biblioref">34</a>]</span> but the system is disordered for all
<span class="math inline">\(T &gt; 0\)</span> <span class="citation"
data-cites="kennedyItinerantElectronModel1986"> [<a
href="#ref-kennedyItinerantElectronModel1986"
role="doc-biblioref">35</a>]</span>. The model has also been a test-bed
for many-body methods, interest took off when an exact dynamical
mean-field theory solution in the infinite dimensional case was
found <span class="citation"
data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a
href="#ref-antipovCriticalExponentsStrongly2014"
role="doc-biblioref">36</a>–<a
href="#ref-herrmannNonequilibriumDynamicalCluster2016"
role="doc-biblioref">39</a>]</span>.</p>
<p>In Chapter 3 I will introduce a generalized FK model in one
dimension. With the addition of long-range interactions in the
background field, the model shows a similarly rich phase diagram. I use
an exact Markov chain Monte Carlo method to map the phase diagram and
compute the energy-resolved localization properties of the fermions. I
then compare the behaviour of this transitionally invariant model to an
Anderson model of uncorrelated binary disorder about a background charge
density wave field which confirms that the fermionic sector only fully
localizes for very large system sizes.</p>
<p><strong>An exactly solvable Quantum Spin Liquid</strong></p>
<p>To turn to the other key topic of this thesis, we have discussed the
question of the magnetic ordering of local moments in the Mott
insulating state. The local moments may form an AFM ground state.
Alternatively they may fail to order even at zero temperature <span
class="citation"
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">26</a>,<a
href="#ref-ribakGaplessExcitationsGround2017"
role="doc-biblioref">30</a></sup></span></p>
role="doc-biblioref">27</a>]</span>, giving rise to what is known as a
quantum spin liquid (QSL) state.</p>
<p>QSLs are a long range entangled ground state of a highly
frustated</p>
<ul>
<li><p>QSLs introduced by anderson 1973<span class="citation"
data-cites="andersonResonatingValenceBonds1973"><sup><a
<li><p>QSLs introduced by anderson 1973 <span class="citation"
data-cites="andersonResonatingValenceBonds1973"> [<a
href="#ref-andersonResonatingValenceBonds1973"
role="doc-biblioref">31</a></sup></span></p></li>
role="doc-biblioref">40</a>]</span></p></li>
<li><p>Geometric frustration that prevents magnetic ordering is an
important part of getting a QSL, suggests exploring the lattice and
avenue of interest.</p></li>
<li><p>Spin orbit effect is a relativistic effect that couples electron
spin to orbital angular moment. Very roughly, an electron sees the
electric field of the nucleus as a magnetic field due to its movement
and the electron spin couples to this.</p></li>
<li><p>can be string in heavy elements</p></li>
<li><p>The Kitaev Model</p></li>
and the electron spin couples to this. Can be strong in heavy
elements</p></li>
<li><p>The Kitaev Model as a canonical QSL</p></li>
<li><p>Kitaev model has extensively many conserved charges too</p></li>
<li><p>Frustration</p></li>
<li><p>anyons</p></li>
<li><p>fractionalisation</p></li>
<li><p>Topology -&gt; GS degeneracy depends on the genus of the
surface</p></li>
<li><p>the chern number</p></li>
<li><p>quasiparticles</p></li>
<li><p>topological order</p></li>
<li><p>protected edge states</p></li>
<li><p>Abelian and non-Abelian anyons</p></li>
</ul>
<div id="fig:correlation_spin_orbit_PT" class="fignos">
<figure>
<img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png"
data-short-caption="Phase Diagram" style="width:100.0%"
alt="Figure 3: From32." />
<figcaption aria-hidden="true"><span>Figure 3:</span> From<span
class="citation" data-cites="TrebstPhysRep2022"><sup><a
alt="Figure 3: From  [41]." />
<figcaption aria-hidden="true"><span>Figure 3:</span> From <span
class="citation" data-cites="TrebstPhysRep2022"> [<a
href="#ref-TrebstPhysRep2022"
role="doc-biblioref">32</a></sup></span>.</figcaption>
role="doc-biblioref">41</a>]</span>.</figcaption>
</figure>
</div>
<p>kinds of mott insulators: Mott-Heisenberg (AFM order below Néel
@ -507,263 +560,327 @@ designed to fill this gap and present the results.</p>
<p>Finally in chapter 4 I will summarise the results and discuss what
implications they have for our understanding interacting many-body
quantum systems.</p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-king2012murmurations" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div
class="csl-right-inline">King, A. J. &amp; Sumpter, D. J. Murmurations.
<em>Current Biology</em> <strong>22</strong>, R112–R114 (2012).</div>
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">A.
J. King and D. J. Sumpter, <em>Murmurations</em>, Current Biology
<strong>22</strong>, R112 (2012).</div>
</div>
<div id="ref-balleriniInteractionRulingAnimal2008" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div
class="csl-right-inline">Ballerini, M. <em>et al.</em> <a
href="https://doi.org/10.1073/pnas.0711437105">Interaction ruling animal
collective behavior depends on topological rather than metric distance:
Evidence from a field study</a>. <em>Proceedings of the National Academy
of Sciences</em> <strong>105</strong>, 1232–1237 (2008).</div>
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">M.
Ballerini et al., <em><a
href="https://doi.org/10.1073/pnas.0711437105">Interaction Ruling Animal
Collective Behavior Depends on Topological Rather Than Metric Distance:
Evidence from a Field Study</a></em>, Proceedings of the National
Academy of Sciences <strong>105</strong>, 1232 (2008).</div>
</div>
<div id="ref-andersonMoreDifferent1972" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div
class="csl-right-inline">Anderson, P. W. <a
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">P.
W. Anderson, <em><a
href="https://doi.org/10.1126/science.177.4047.393">More Is
Different</a>. <em>Science</em> <strong>177</strong>, 393–396
(1972).</div>
Different</a></em>, Science <strong>177</strong>, 393 (1972).</div>
</div>
<div id="ref-kivelsonDefiningEmergencePhysics2016" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div
class="csl-right-inline">Kivelson, S. &amp; Kivelson, S. A. <a
href="https://doi.org/10.1038/npjquantmats.2016.24">Defining emergence
in physics</a>. <em>npj Quant Mater</em> <strong>1</strong>, 1–2
(2016).</div>
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">S.
Kivelson and S. A. Kivelson, <em><a
href="https://doi.org/10.1038/npjquantmats.2016.24">Defining Emergence
in Physics</a></em>, Npj Quant Mater <strong>1</strong>, 1 (2016).</div>
</div>
<div id="ref-saslowHistoryThermodynamicsMissing2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div
class="csl-right-inline">Saslow, W. M. <a
href="https://doi.org/10.3390/e22010077">A History of Thermodynamics:
The Missing Manual</a>. <em>Entropy (Basel)</em> <strong>22</strong>, 77
(2020).</div>
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">W.
M. Saslow, <em><a href="https://doi.org/10.3390/e22010077">A History of
Thermodynamics: The Missing Manual</a></em>, Entropy (Basel)
<strong>22</strong>, 77 (2020).</div>
</div>
<div id="ref-flammHistoryOutlookStatistical1998" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div
class="csl-right-inline">Flamm, D. History and outlook of statistical
physics. Preprint at <a
href="https://doi.org/10.48550/arXiv.physics/9803005">https://doi.org/10.48550/arXiv.physics/9803005</a>
(1998).</div>
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">D.
Flamm, <em><a
href="https://doi.org/10.48550/arXiv.physics/9803005">History and
Outlook of Statistical Physics</a></em>, arXiv:physics/9803005.</div>
</div>
<div id="ref-ashcroftSolidStatePhysics1976" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div
class="csl-right-inline">Ashcroft, N. W. &amp; Mermin, N. D. <em>Solid
State Physics</em>. (Holt, Rinehart and Winston, 1976).</div>
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">N.
W. Ashcroft and N. D. Mermin, <em>Solid State Physics</em> (Holt,
Rinehart and Winston, 1976).</div>
</div>
<div id="ref-isingBeitragZurTheorie1925" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div
class="csl-right-inline">Ising, E. <a
href="https://doi.org/10.1007/BF02980577">Beitrag zur Theorie des
Ferromagnetismus</a>. <em>Z. Physik</em> <strong>31</strong>, 253–258
(1925).</div>
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">E.
Ising, <em><a href="https://doi.org/10.1007/BF02980577">Beitrag zur
Theorie des Ferromagnetismus</a></em>, Z. Physik <strong>31</strong>,
253 (1925).</div>
</div>
<div id="ref-landau2013fluid" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div
class="csl-right-inline">Landau, L. D. &amp; Lifshitz, E. M. <em>Fluid
mechanics: Landau and lifshitz: Course of theoretical physics, volume
6</em>. vol. 6 (Elsevier, 2013).</div>
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">L.
D. Landau and E. M. Lifshitz, <em>Fluid Mechanics: Landau and Lifshitz:
Course of Theoretical Physics, Volume 6</em>, Vol. 6 (Elsevier,
2013).</div>
</div>
<div id="ref-jaegerEhrenfestClassificationPhase1998" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div
class="csl-right-inline">Jaeger, G. <a
href="https://doi.org/10.1007/s004070050021">The Ehrenfest
Classification of Phase Transitions: Introduction and Evolution</a>.
<em>Arch Hist Exact Sc.</em> <strong>53</strong>, 51–81 (1998).</div>
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">G.
Jaeger, <em><a href="https://doi.org/10.1007/s004070050021">The
Ehrenfest Classification of Phase Transitions: Introduction and
Evolution</a></em>, Arch Hist Exact Sc. <strong>53</strong>, 51
(1998).</div>
</div>
<div id="ref-blochÜberQuantenmechanikElektronen1929" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div
class="csl-right-inline">Bloch, F. <a
href="https://doi.org/10.1007/BF01339455">Über die Quantenmechanik der
Elektronen in Kristallgittern</a>. <em>Z. Physik</em>
<strong>52</strong>, 555–600 (1929).</div>
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">F.
Bloch, <em><a href="https://doi.org/10.1007/BF01339455">Über die
Quantenmechanik der Elektronen in Kristallgittern</a></em>, Z. Physik
<strong>52</strong>, 555 (1929).</div>
</div>
<div id="ref-MagnetismCondensedMatter" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div
class="csl-right-inline">Blundell, S. <em>Magnetism in Condensed
Matter</em>. (OUP Oxford, 2001).</div>
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">S.
Blundell, <em>Magnetism in Condensed Matter</em> (OUP Oxford,
2001).</div>
</div>
<div id="ref-MicroscopicTheorySuperconductivity" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div
class="csl-right-inline">Bardeen, J., Cooper, L. N. &amp; Schrieffer, J.
R. <a href="https://doi.org/10.1103/PhysRev.106.162">Microscopic Theory
of Superconductivity</a>. <em>Phys. Rev.</em> <strong>106</strong>,
162–164 (1957).</div>
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">J.
Bardeen, L. N. Cooper, and J. R. Schrieffer, <em><a
href="https://doi.org/10.1103/PhysRev.106.162">Microscopic Theory of
Superconductivity</a></em>, Phys. Rev. <strong>106</strong>, 162
(1957).</div>
</div>
<div id="ref-feldmanFractionalChargeFractional2021" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div
class="csl-right-inline">Feldman, D. E. &amp; Halperin, B. I. <a
href="https://doi.org/10.1088/1361-6633/ac03aa">Fractional charge and
fractional statistics in the quantum Hall effects</a>. <em>Rep. Prog.
Phys.</em> <strong>84</strong>, 076501 (2021).</div>
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">D.
E. Feldman and B. I. Halperin, <em><a
href="https://doi.org/10.1088/1361-6633/ac03aa">Fractional Charge and
Fractional Statistics in the Quantum Hall Effects</a></em>, Rep. Prog.
Phys. <strong>84</strong>, 076501 (2021).</div>
</div>
<div id="ref-mottBasisElectronTheory1949" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div
class="csl-right-inline">Mott, N. F. <a
href="https://doi.org/10.1088/0370-1298/62/7/303">The Basis of the
Electron Theory of Metals, with Special Reference to the Transition
Metals</a>. <em>Proc. Phys. Soc. A</em> <strong>62</strong>, 416–422
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">N.
F. Mott, <em><a href="https://doi.org/10.1088/0370-1298/62/7/303">The
Basis of the Electron Theory of Metals, with Special Reference to the
Transition Metals</a></em>, Proc. Phys. Soc. A <strong>62</strong>, 416
(1949).</div>
</div>
<div id="ref-fisherMottInsulatorsSpin1999" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div
class="csl-right-inline">Fisher, M. P. A. <a
href="https://doi.org/10.1007/3-540-46637-1_8">Mott insulators, Spin
liquids and Quantum Disordered Superconductivity</a>. in <em>Aspects
topologiques de la physique en basse dimension. Topological aspects of
low dimensional systems</em> (eds. Comtet, A., Jolicœur, T., Ouvry, S.
&amp; David, F.) vol. 69 575–641 (Springer Berlin Heidelberg,
1999).</div>
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">M.
P. A. Fisher, <em><a href="https://doi.org/10.1007/3-540-46637-1_8">Mott
Insulators, Spin Liquids and Quantum Disordered
Superconductivity</a></em>, in <em>Aspects Topologiques de La Physique
En Basse Dimension. Topological Aspects of Low Dimensional Systems</em>,
edited by A. Comtet, T. Jolicœur, S. Ouvry, and F. David, Vol. 69
(Springer Berlin Heidelberg, Berlin, Heidelberg, 1999), pp.
575–641.</div>
</div>
<div id="ref-boerSemiconductorsPartiallyCompletely1937"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div
class="csl-right-inline">Boer, J. H. de &amp; Verwey, E. J. W. <a
href="https://doi.org/10.1088/0959-5309/49/4S/307">Semi-conductors with
partially and with completely filled &lt;script&gt;3d-lattice
bands&lt;/script&gt;</a>. <em>Proc. Phys. Soc.</em> <strong>49</strong>,
59–71 (1937).</div>
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">J.
H. de Boer and E. J. W. Verwey, <em><a
href="https://doi.org/10.1088/0959-5309/49/4S/307">Semi-Conductors with
Partially and with Completely Filled &lt;Script&gt;3d-Lattice
Bands&lt;/Script&gt;</a></em>, Proc. Phys. Soc. <strong>49</strong>, 59
(1937).</div>
</div>
<div id="ref-mottDiscussionPaperBoer1937" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div
class="csl-right-inline">Mott, N. F. &amp; Peierls, R. <a
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">N.
F. Mott and R. Peierls, <em><a
href="https://doi.org/10.1088/0959-5309/49/4S/308">Discussion of the
paper by de Boer and Verwey</a>. <em>Proc. Phys. Soc.</em>
<strong>49</strong>, 72–73 (1937).</div>
Paper by de Boer and Verwey</a></em>, Proc. Phys. Soc.
<strong>49</strong>, 72 (1937).</div>
</div>
<div id="ref-bednorzPossibleHighTcSuperconductivity1986"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">19. </div><div
class="csl-right-inline">Bednorz, J. G. &amp; Müller, K. A. <a
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">J.
G. Bednorz and K. A. Müller, <em><a
href="https://doi.org/10.1007/BF01303701">Possible highTc
superconductivity in the Ba−La−Cu−O system</a>. <em>Z. Physik B -
Condensed Matter</em> <strong>64</strong>, 189–193 (1986).</div>
Superconductivity in the Ba−La−Cu−O System</a></em>, Z. Physik B -
Condensed Matter <strong>64</strong>, 189 (1986).</div>
</div>
<div id="ref-leeDopingMottInsulator2006" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">20. </div><div
class="csl-right-inline">Lee, P. A., Nagaosa, N. &amp; Wen, X.-G. <a
href="https://doi.org/10.1103/RevModPhys.78.17">Doping a Mott insulator:
Physics of high-temperature superconductivity</a>. <em>Rev. Mod.
Phys.</em> <strong>78</strong>, 17–85 (2006).</div>
<div class="csl-left-margin">[20] </div><div class="csl-right-inline">P.
A. Lee, N. Nagaosa, and X.-G. Wen, <em><a
href="https://doi.org/10.1103/RevModPhys.78.17">Doping a Mott Insulator:
Physics of High-Temperature Superconductivity</a></em>, Rev. Mod. Phys.
<strong>78</strong>, 17 (2006).</div>
</div>
<div id="ref-gutzwillerEffectCorrelationFerromagnetism1963"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">21. </div><div
class="csl-right-inline">Gutzwiller, M. C. <a
<div class="csl-left-margin">[21] </div><div class="csl-right-inline">M.
C. Gutzwiller, <em><a
href="https://doi.org/10.1103/PhysRevLett.10.159">Effect of Correlation
on the Ferromagnetism of Transition Metals</a>. <em>Phys. Rev.
Lett.</em> <strong>10</strong>, 159–162 (1963).</div>
on the Ferromagnetism of Transition Metals</a></em>, Phys. Rev. Lett.
<strong>10</strong>, 159 (1963).</div>
</div>
<div id="ref-kanamoriElectronCorrelationFerromagnetism1963"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">22. </div><div
class="csl-right-inline">Kanamori, J. <a
href="https://doi.org/10.1143/PTP.30.275">Electron Correlation and
Ferromagnetism of Transition Metals</a>. <em>Progress of Theoretical
Physics</em> <strong>30</strong>, 275–289 (1963).</div>
<div class="csl-left-margin">[22] </div><div class="csl-right-inline">J.
Kanamori, <em><a href="https://doi.org/10.1143/PTP.30.275">Electron
Correlation and Ferromagnetism of Transition Metals</a></em>, Progress
of Theoretical Physics <strong>30</strong>, 275 (1963).</div>
</div>
<div id="ref-hubbardj.ElectronCorrelationsNarrow1963" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">23. </div><div
class="csl-right-inline">Hubbard, J. <a
href="https://doi.org/10.1098/rspa.1963.0204">Electron correlations in
narrow energy bands</a>. <em>Proceedings of the Royal Society of London.
Series A. Mathematical and Physical Sciences</em> <strong>276</strong>,
238–257 (1963).</div>
<div class="csl-left-margin">[23] </div><div
class="csl-right-inline">Hubbard, J., <em><a
href="https://doi.org/10.1098/rspa.1963.0204">Electron Correlations in
Narrow Energy Bands</a></em>, Proceedings of the Royal Society of
London. Series A. Mathematical and Physical Sciences
<strong>276</strong>, 238 (1963).</div>
</div>
<div id="ref-hubbardElectronCorrelationsNarrow1964" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">24. </div><div
class="csl-right-inline">Hubbard, J. &amp; Flowers, B. H. <a
href="https://doi.org/10.1098/rspa.1964.0190">Electron correlations in
narrow energy bands III. An improved solution</a>. <em>Proceedings of
<div class="csl-left-margin">[24] </div><div class="csl-right-inline">J.
Hubbard and B. H. Flowers, <em><a
href="https://doi.org/10.1098/rspa.1964.0190">Electron Correlations in
Narrow Energy Bands III. An Improved Solution</a></em>, Proceedings of
the Royal Society of London. Series A. Mathematical and Physical
Sciences</em> <strong>281</strong>, 401–419 (1964).</div>
Sciences <strong>281</strong>, 401 (1964).</div>
</div>
<div id="ref-mottMetalInsulatorTransitions1990" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">25. </div><div
class="csl-right-inline">Mott, N. <em>Metal-Insulator Transitions</em>.
(CRC Press, 1990). doi:<a
href="https://doi.org/10.1201/b12795">10.1201/b12795</a>.</div>
</div>
<div id="ref-slaterMagneticEffectsHartreeFock1951" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">26. </div><div
class="csl-right-inline">Slater, J. C. <a
href="https://doi.org/10.1103/PhysRev.82.538">Magnetic Effects and the
Hartree-Fock Equation</a>. <em>Phys. Rev.</em> <strong>82</strong>,
538–541 (1951).</div>
</div>
<div id="ref-greinerQuantumPhaseTransition2002" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">27. </div><div
class="csl-right-inline">Greiner, M., Mandel, O., Esslinger, T., Hänsch,
T. W. &amp; Bloch, I. <a href="https://doi.org/10.1038/415039a">Quantum
phase transition from a superfluid to a Mott insulator in a gas of
ultracold atoms</a>. <em>Nature</em> <strong>415</strong>, 39
(2002).</div>
</div>
<div id="ref-mazurenkoColdatomFermiHubbard2017" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">28. </div><div
class="csl-right-inline">Mazurenko, A. <em>et al.</em> <a
href="https://doi.org/10.1038/nature22362">A cold-atom Fermi–Hubbard
antiferromagnet</a>. <em>Nature</em> <strong>545</strong>, 462–466
(2017).</div>
<div class="csl-left-margin">[25] </div><div class="csl-right-inline">N.
Mott, <em><a href="https://doi.org/10.1201/b12795">Metal-Insulator
Transitions</a></em> (CRC Press, London, 1990).</div>
</div>
<div id="ref-law1TTaS2QuantumSpin2017" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">29. </div><div
class="csl-right-inline">Law, K. T. &amp; Lee, P. A. <a
href="https://doi.org/10.1073/pnas.1706769114">1T-TaS2 as a quantum spin
liquid</a>. <em>Proceedings of the National Academy of Sciences</em>
<strong>114</strong>, 6996–7000 (2017).</div>
<div class="csl-left-margin">[26] </div><div class="csl-right-inline">K.
T. Law and P. A. Lee, <em><a
href="https://doi.org/10.1073/pnas.1706769114">1t-TaS2 as a Quantum Spin
Liquid</a></em>, Proceedings of the National Academy of Sciences
<strong>114</strong>, 6996 (2017).</div>
</div>
<div id="ref-ribakGaplessExcitationsGround2017" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">30. </div><div
class="csl-right-inline">Ribak, A. <em>et al.</em> <a
href="https://doi.org/10.1103/PhysRevB.96.195131">Gapless excitations in
the ground state of
&lt;script&gt;$1T\text{\ensuremath{-}}{\mathrm{TaS}}_{2}$&lt;/script&gt;</a>.
<em>Phys. Rev. B</em> <strong>96</strong>, 195131 (2017).</div>
<div class="csl-left-margin">[27] </div><div class="csl-right-inline">A.
Ribak, I. Silber, C. Baines, K. Chashka, Z. Salman, Y. Dagan, and A.
Kanigel, <em><a
href="https://doi.org/10.1103/PhysRevB.96.195131">Gapless Excitations in
the Ground State of
&lt;Script&gt;$1T\text{\ensuremath{-}}{\Mathrm{TaS}}_{2}$&lt;/Script&gt;</a></em>,
Phys. Rev. B <strong>96</strong>, 195131 (2017).</div>
</div>
<div id="ref-slaterMagneticEffectsHartreeFock1951" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">J.
C. Slater, <em><a href="https://doi.org/10.1103/PhysRev.82.538">Magnetic
Effects and the Hartree-Fock Equation</a></em>, Phys. Rev.
<strong>82</strong>, 538 (1951).</div>
</div>
<div id="ref-greinerQuantumPhaseTransition2002" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">M.
Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, <em><a
href="https://doi.org/10.1038/415039a">Quantum Phase Transition from a
Superfluid to a Mott Insulator in a Gas of Ultracold Atoms</a></em>,
Nature <strong>415</strong>, 39 (2002).</div>
</div>
<div id="ref-mazurenkoColdatomFermiHubbard2017" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[30] </div><div class="csl-right-inline">A.
Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt,
F. Grusdt, E. Demler, D. Greif, and M. Greiner, <em><a
href="https://doi.org/10.1038/nature22362">A Cold-Atom Fermi–Hubbard
Antiferromagnet</a></em>, Nature <strong>545</strong>, 462 (2017).</div>
</div>
<div id="ref-belitzAndersonMottTransition1994" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">D.
Belitz and T. R. Kirkpatrick, <em><a
href="https://doi.org/10.1103/RevModPhys.66.261">The Anderson-Mott
Transition</a></em>, Rev. Mod. Phys. <strong>66</strong>, 261
(1994).</div>
</div>
<div id="ref-baskoMetalInsulatorTransition2006" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">D.
M. Basko, I. L. Aleiner, and B. L. Altshuler, <em><a
href="https://doi.org/10.1016/j.aop.2005.11.014">Metal–Insulator
Transition in a Weakly Interacting Many-Electron System with Localized
Single-Particle States</a></em>, Annals of Physics <strong>321</strong>,
1126 (2006).</div>
</div>
<div id="ref-brandtThermodynamicsCorrelationFunctions1989"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[33] </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-gruberGroundStatesSpinless1990" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[34] </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-kennedyItinerantElectronModel1986" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[35] </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-antipovCriticalExponentsStrongly2014" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[36] </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-ribicNonlocalCorrelationsSpectral2016" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[37] </div><div class="csl-right-inline">T.
Ribic, G. Rohringer, and K. Held, <em><a
href="https://doi.org/10.1103/PhysRevB.93.195105">Nonlocal Correlations
and Spectral Properties of the Falicov-Kimball Model</a></em>, Phys.
Rev. B <strong>93</strong>, 195105 (2016).</div>
</div>
<div id="ref-freericksExactDynamicalMeanfield2003" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[38] </div><div class="csl-right-inline">J.
K. Freericks and V. Zlatić, <em><a
href="https://doi.org/10.1103/RevModPhys.75.1333">Exact Dynamical
Mean-Field Theory of the Falicov-Kimball Model</a></em>, Rev. Mod. Phys.
<strong>75</strong>, 1333 (2003).</div>
</div>
<div id="ref-herrmannNonequilibriumDynamicalCluster2016"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[39] </div><div class="csl-right-inline">A.
J. Herrmann, N. Tsuji, M. Eckstein, and P. Werner, <em><a
href="https://doi.org/10.1103/PhysRevB.94.245114">Nonequilibrium
Dynamical Cluster Approximation Study of the Falicov-Kimball
Model</a></em>, Phys. Rev. B <strong>94</strong>, 245114 (2016).</div>
</div>
<div id="ref-andersonResonatingValenceBonds1973" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">31. </div><div
class="csl-right-inline">Anderson, P. W. <a
href="https://doi.org/10.1016/0025-5408(73)90167-0">Resonating valence
bonds: A new kind of insulator?</a> <em>Materials Research Bulletin</em>
<strong>8</strong>, 153–160 (1973).</div>
<div class="csl-left-margin">[40] </div><div class="csl-right-inline">P.
W. Anderson, <em><a
href="https://doi.org/10.1016/0025-5408(73)90167-0">Resonating Valence
Bonds: A New Kind of Insulator?</a></em>, Materials Research Bulletin
<strong>8</strong>, 153 (1973).</div>
</div>
<div id="ref-TrebstPhysRep2022" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">32. </div><div
class="csl-right-inline">Trebst, S. &amp; Hickey, C. <a
<div class="csl-left-margin">[41] </div><div class="csl-right-inline">S.
Trebst and C. Hickey, <em><a
href="https://doi.org/10.1016/j.physrep.2021.11.003">Kitaev
materials</a>. <em>Physics Reports</em> <strong>950</strong>, 1–37
Materials</a></em>, Physics Reports <strong>950</strong>, 1
(2022).</div>
</div>
</div>

17
_thesis/2_Background/2.2_HKM_Model.html
View File

@ -319,10 +319,10 @@ Majorana <span class="math inline">\(c_i\)</span> per site.</figcaption>
</div>
<ul>
<li>strong spin orbit coupling yields spatial anisotropic spin exchange
leading to compass models<span class="citation"
data-cites="kugelJahnTellerEffectMagnetism1982"><sup><a
leading to compass models <span class="citation"
data-cites="kugelJahnTellerEffectMagnetism1982"> [<a
href="#ref-kugelJahnTellerEffectMagnetism1982"
role="doc-biblioref">1</a></sup></span></li>
role="doc-biblioref">1</a>]</span></li>
<li>spin model of the Kitaev model is one</li>
<li>has extensively many conserved fluxes</li>
<li></li>
@ -335,15 +335,14 @@ Chern number</h2>
<h2 id="phase-diagram">Phase Diagram</h2>
<div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div
class="csl-right-inline">Kugel’, K. I. &amp; Khomskiĭ, D. I. <a
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">K.
I. Kugel’ and D. I. Khomskiĭ, <em><a
href="https://doi.org/10.1070/PU1982v025n04ABEH004537">The Jahn-Teller
effect and magnetism: transition metal compounds</a>. <em>Sov. Phys.
Usp.</em> <strong>25</strong>, 231 (1982).</div>
Effect and Magnetism: Transition Metal Compounds</a></em>, Sov. Phys.
Usp. <strong>25</strong>, 231 (1982).</div>
</div>
</div>
</main>

319
_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html
View File

@ -360,10 +360,10 @@ The fact they’re uncorrelated is key as we’ll see later. Examples of
direct sampling methods range from the trivial: take n random bits to
generate integers uniformly between 0 and <span
class="math inline">\(2^n\)</span> to more complex methods such as
inverse transform sampling and rejection sampling<span class="citation"
data-cites="devroyeRandomSampling1986"><sup><a
inverse transform sampling and rejection sampling <span class="citation"
data-cites="devroyeRandomSampling1986"> [<a
href="#ref-devroyeRandomSampling1986"
role="doc-biblioref">1</a></sup></span>.</p>
role="doc-biblioref">1</a>]</span>.</p>
<p>In physics the distribution we usually want to sample from is the
Boltzmann probability over states of the system <span
class="math inline">\(S\)</span>: <span class="math display">\[
@ -383,9 +383,9 @@ with system size. Even if we could calculate <span
class="math inline">\(\mathcal{Z}\)</span>, sampling from an
exponentially large number of options quickly become tricky. This kind
of problem happens in many other disciplines too, particularly when
fitting statistical models using Bayesian inference<span
class="citation" data-cites="BMCP2021"><sup><a href="#ref-BMCP2021"
role="doc-biblioref">2</a></sup></span>.</p>
fitting statistical models using Bayesian inference <span
class="citation" data-cites="BMCP2021"> [<a href="#ref-BMCP2021"
role="doc-biblioref">2</a>]</span>.</p>
<h2 id="markov-chains">Markov Chains</h2>
<p>So what can we do? Well it turns out that if we’re willing to give up
in the requirement that the samples be uncorrelated then we can use MCMC
@ -393,11 +393,11 @@ instead.</p>
<p>MCMC defines a weighted random walk over the states <span
class="math inline">\((S_0, S_1, S_2, ...)\)</span>, such that in the
long time limit, states are visited according to their probability <span
class="math inline">\(p(S)\)</span>.<span class="citation"
data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"><sup><a
class="math inline">\(p(S)\)</span>. <span class="citation"
data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"> [<a
href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">3</a>–<a
href="#ref-wolffMonteCarloErrors2004"
role="doc-biblioref">5</a></sup></span>.</p>
role="doc-biblioref">5</a>]</span>.</p>
<p>In a physics context this lets us evaluate any observable with a mean
over the states visited by the walk. <span
class="math display">\[\begin{aligned}
@ -407,9 +407,9 @@ class="math display">\[\begin{aligned}
<p>The choice of the transition function for MCMC is under-determined as
one only needs to satisfy a set of balance conditions for which there
are many solutions <span class="citation"
data-cites="kellyReversibilityStochasticNetworks1981"><sup><a
data-cites="kellyReversibilityStochasticNetworks1981"> [<a
href="#ref-kellyReversibilityStochasticNetworks1981"
role="doc-biblioref">6</a></sup></span>.</p>
role="doc-biblioref">6</a>]</span>.</p>
<h2 id="application-to-the-fk-model">Application to the FK Model</h2>
<p>We will work in the grand canonical ensemble of fixed temperature,
chemical potential and volume.</p>
@ -447,11 +447,11 @@ F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}
expectation values <span class="math inline">\(\expval{O}\)</span> with
respect to some physical system defined by a set of states <span
class="math inline">\(\{x: x \in S\}\)</span> and a free energy <span
class="math inline">\(F(x)\)</span><span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"><sup><a
class="math inline">\(F(x)\)</span> <span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a></sup></span>. The thermal expectation value
is defined via a Boltzmann weighted sum over the entire states: <span
role="doc-biblioref">7</a>]</span>. The thermal expectation value is
defined via a Boltzmann weighted sum over the entire states: <span
class="math display">\[
\begin{aligned}
\expval{O} &amp;= \frac{1}{\mathcal{Z}} \sum_{x \in S} O(x) P(x) \\
@ -526,10 +526,10 @@ P(x) \mathcal{T}(x \rightarrow x&#39;) = P(x&#39;) \mathcal{T}(x&#39;
\rightarrow x)
\]</span> % In practice most algorithms are constructed to satisfy
detailed balance though there are arguments that relaxing the condition
can lead to faster algorithms<span class="citation"
data-cites="kapferSamplingPolytopeHarddisk2013"><sup><a
can lead to faster algorithms <span class="citation"
data-cites="kapferSamplingPolytopeHarddisk2013"> [<a
href="#ref-kapferSamplingPolytopeHarddisk2013"
role="doc-biblioref">8</a></sup></span>.</p>
role="doc-biblioref">8</a>]</span>.</p>
<p>The goal of MCMC is then to choose <span
class="math inline">\(\mathcal{T}\)</span> so that it has the desired
thermal distribution <span class="math inline">\(P(x)\)</span> as its
@ -558,10 +558,10 @@ x_{i}\)</span>. Now <span class="math inline">\(\mathcal{T}(x\to x&#39;)
<p>The Metropolis-Hasting algorithm is a slight extension of the
original Metropolis algorithm that allows for non-symmetric proposal
distributions $q(xx’) q(x’x) $. It can be derived starting from detailed
balance<span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"><sup><a
balance <span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a></sup></span>: <span
role="doc-biblioref">7</a>]</span>: <span
class="math display">\[\begin{aligned}
P(x)\mathcal{T}(x \to x&#39;) &amp;= P(x&#39;)\mathcal{T}(x&#39; \to x)
\\
@ -671,11 +671,11 @@ problematic because it means very few new samples will be generated. If
it is too high it implies the steps are too small, a problem because
then the walk will take longer to explore the state space and the
samples will be highly correlated. Ideal values for the acceptance rate
can be calculated under certain assumptions<span class="citation"
data-cites="robertsWeakConvergenceOptimal1997"><sup><a
can be calculated under certain assumptions <span class="citation"
data-cites="robertsWeakConvergenceOptimal1997"> [<a
href="#ref-robertsWeakConvergenceOptimal1997"
role="doc-biblioref">9</a></sup></span>. Here we monitor the acceptance
rate and if it is too high we re-run the MCMC with a modified proposal
role="doc-biblioref">9</a>]</span>. Here we monitor the acceptance rate
and if it is too high we re-run the MCMC with a modified proposal
distribution that has a chance to propose moves that flip multiple sites
at a time.</p>
<p>In addition we exploit the particle-hole symmetry of the problem by
@ -686,10 +686,10 @@ produce a state at or near the energy of the current one.</p>
<p>The matrix diagonalisation is the most computationally expensive step
of the process, a speed up can be obtained by modifying the proposal
distribution to depend on the classical part of the energy, a trick
gleaned from Ref.<span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"><sup><a
gleaned from Ref. <span class="citation"
data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a></sup></span>: <span class="math display">\[
role="doc-biblioref">7</a>]</span>: <span class="math display">\[
\begin{aligned}
q(k \to k&#39;) &amp;= \min\left(1, e^{\beta (H^{k&#39;} - H^k)}\right)
\\
@ -700,12 +700,11 @@ without performing the diagonalisation at no cost to the accuracy of the
MCMC method.</p>
<p>An extension of this idea is to try to define a classical model with
a similar free energy dependence on the classical state as the full
quantum, Ref.<span class="citation"
data-cites="huangAcceleratedMonteCarlo2017"><sup><a
quantum, Ref. <span class="citation"
data-cites="huangAcceleratedMonteCarlo2017"> [<a
href="#ref-huangAcceleratedMonteCarlo2017"
role="doc-biblioref">10</a></sup></span> does this with restricted
Boltzmann machines whose form is very similar to a classical spin
model.</p>
role="doc-biblioref">10</a>]</span> does this with restricted Boltzmann
machines whose form is very similar to a classical spin model.</p>
<h2 id="scaling">Scaling</h2>
<p>In order to reduce the effects of the boundary conditions and the
finite size of the system we redefine and normalise the coupling matrix
@ -726,12 +725,12 @@ central moments of the order parameter m: <span class="math display">\[m
= \sum_i (-1)^i (2n_i - 1) / N\]</span> % The Binder cumulant evaluated
against temperature can be used as a diagnostic for the existence of a
phase transition. If multiple such curves are plotted for different
system sizes, a crossing indicates the location of a critical point<span
class="citation"
data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"><sup><a
system sizes, a crossing indicates the location of a critical point
<span class="citation"
data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a
href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">11</a>,<a
href="#ref-musialMonteCarloSimulations2002"
role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a></sup></span>.</p>
role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a>]</span>.</p>
<h2 id="markov-chain-monte-carlo-in-practice">Markov Chain Monte-Carlo
in Practice</h2>
<h3 id="quick-intro-to-mcmc">Quick Intro to MCMC</h3>
@ -758,13 +757,13 @@ very expensive operation!~\footnote{The effort involved in exact
diagonalisation scales like <span class="math inline">\(N^2\)</span> for
systems with a tri-diagonal matrix representation (open boundary
conditions and nearest neighbour hopping) and like <span
class="math inline">\(N^3\)</span> for a generic matrix<span
class="math inline">\(N^3\)</span> for a generic matrix <span
class="citation"
data-cites="bolchQueueingNetworksMarkov2006 usmaniInversionTridiagonalJacobi1994"><sup><a
data-cites="bolchQueueingNetworksMarkov2006 usmaniInversionTridiagonalJacobi1994"> [<a
href="#ref-bolchQueueingNetworksMarkov2006"
role="doc-biblioref">12</a>,<a
href="#ref-usmaniInversionTridiagonalJacobi1994"
role="doc-biblioref">13</a></sup></span>.</p>
role="doc-biblioref">13</a>]</span>.</p>
<p>c</p>
<p>MCMC sidesteps these issues by defining a random walk that focuses on
the states with the greatest Boltzmann weight. At low temperatures this
@ -878,10 +877,10 @@ auto-correlation time <span class="math inline">\(\tau(O)\)</span>
informally as the number of MCMC samples of some observable O that are
statistically equal to one independent sample or equivalently as the
number of MCMC steps after which the samples are correlated below some
cutoff, see<span class="citation"
data-cites="krauthIntroductionMonteCarlo1996"><sup><a
cutoff, see <span class="citation"
data-cites="krauthIntroductionMonteCarlo1996"> [<a
href="#ref-krauthIntroductionMonteCarlo1996"
role="doc-biblioref">14</a></sup></span> for a more rigorous definition
role="doc-biblioref">14</a>]</span> for a more rigorous definition
involving a sum over the auto-correlation function. The auto-correlation
time is generally shorter than the convergence time so it therefore
makes sense from an efficiency standpoint to run a single walker for
@ -960,28 +959,28 @@ than the current state.</p>
<h2 id="two-step-trick">Two Step Trick</h2>
<p>Here, we incorporate a modification to the standard
Metropolis-Hastings algorithm <span class="citation"
data-cites="hastingsMonteCarloSampling1970"><sup><a
data-cites="hastingsMonteCarloSampling1970"> [<a
href="#ref-hastingsMonteCarloSampling1970"
role="doc-biblioref">15</a></sup></span> gleaned from Krauth <span
class="citation" data-cites="krauthIntroductionMonteCarlo1998"><sup><a
role="doc-biblioref">15</a>]</span> gleaned from Krauth <span
class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a
href="#ref-krauthIntroductionMonteCarlo1998"
role="doc-biblioref">7</a></sup></span>.</p>
role="doc-biblioref">7</a>]</span>.</p>
<p>In our computations <span class="citation"
data-cites="hodsonMCMCFKModel2021"><sup><a
href="#ref-hodsonMCMCFKModel2021"
role="doc-biblioref">16</a></sup></span> we employ a modification of the
algorithm which is based on the observation that the free energy of the
FK system is composed of a classical part which is much quicker to
compute than the quantum part. Hence, we can obtain a computational
speedup by first considering the value of the classical energy
difference <span class="math inline">\(\Delta H_s\)</span> and rejecting
the transition if the former is too high. We only compute the quantum
energy difference <span class="math inline">\(\Delta F_c\)</span> if the
transition is accepted. We then perform a second rejection sampling step
based upon it. This corresponds to two nested comparisons with the
majority of the work only occurring if the first test passes and has the
acceptance function <span class="math display">\[\mathcal{A}(a \to b) =
\min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta
data-cites="hodsonMCMCFKModel2021"> [<a
href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">16</a>]</span> we
employ a modification of the algorithm which is based on the observation
that the free energy of the FK system is composed of a classical part
which is much quicker to compute than the quantum part. Hence, we can
obtain a computational speedup by first considering the value of the
classical energy difference <span class="math inline">\(\Delta
H_s\)</span> and rejecting the transition if the former is too high. We
only compute the quantum energy difference <span
class="math inline">\(\Delta F_c\)</span> if the transition is accepted.
We then perform a second rejection sampling step based upon it. This
corresponds to two nested comparisons with the majority of the work only
occurring if the first test passes and has the acceptance function <span
class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta
\Delta H_s}\right)\min\left(1, e^{-\beta \Delta
F_c}\right)\;.\]</span></p>
<p>For the model parameters used in Fig. <a href="#fig:indiv_IPR"
data-reference-type="ref" data-reference="fig:indiv_IPR">2</a>, we find
@ -1008,9 +1007,9 @@ distribution, a problem which MCMC was employed to solve in the first
place. For example, recent work trains restricted Boltzmann machines
(RBMs) to generate samples for the proposal distribution of the FK
model <span class="citation"
data-cites="huangAcceleratedMonteCarlo2017"><sup><a
data-cites="huangAcceleratedMonteCarlo2017"> [<a
href="#ref-huangAcceleratedMonteCarlo2017"
role="doc-biblioref">10</a></sup></span>. The RBMs are chosen as a
role="doc-biblioref">10</a>]</span>. The RBMs are chosen as a
parametrisation of the proposal distribution that can be efficiently
sampled from while offering sufficient flexibility that they can be
adjusted to match the target distribution. Our proposed method is
@ -1021,11 +1020,11 @@ the two step method</h2>
<p>Given a MCMC algorithm with target distribution <span
class="math inline">\(\pi(a)\)</span> and transition function <span
class="math inline">\(\mathcal{T}\)</span> the detailed balance
condition is sufficient (along with some technical constraints<span
class="citation" data-cites="wolffMonteCarloErrors2004"><sup><a
condition is sufficient (along with some technical constraints <span
class="citation" data-cites="wolffMonteCarloErrors2004"> [<a
href="#ref-wolffMonteCarloErrors2004"
role="doc-biblioref">5</a></sup></span>) to guarantee that in the long
time limit the algorithm produces samples from <span
role="doc-biblioref">5</a>]</span>) to guarantee that in the long time
limit the algorithm produces samples from <span
class="math inline">\(\pi\)</span>. <span
class="math display">\[\pi(a)\mathcal{T}(a \to b) = \pi(b)\mathcal{T}(b
\to a)\]</span></p>
@ -1141,10 +1140,10 @@ for the additional complexity it would require.</p>
<h3 id="inverse-participation-ratio">Inverse Participation Ratio</h3>
<p>The inverse participation ratio is defined for a normalised wave
function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i
\abs{\psi_i}^2 = 1\)</span> as its fourth moment<span class="citation"
data-cites="kramerLocalizationTheoryExperiment1993"><sup><a
\abs{\psi_i}^2 = 1\)</span> as its fourth moment <span class="citation"
data-cites="kramerLocalizationTheoryExperiment1993"> [<a
href="#ref-kramerLocalizationTheoryExperiment1993"
role="doc-biblioref">17</a></sup></span>: <span class="math display">\[
role="doc-biblioref">17</a>]</span>: <span class="math display">\[
P^{-1} = \sum_i \abs{\psi_i}^4
\]</span> % It acts as a measure of the portion of space occupied by the
wave function. For localised states it will be independent of system
@ -1155,11 +1154,10 @@ fractal dimensionality <span class="math inline">\(d &gt; d* &gt;
P(L) \goeslike L^{d*}
\]</span> % For extended states <span class="math inline">\(d* =
0\)</span> while for localised ones <span class="math inline">\(d* =
0\)</span>. In this work we take use an energy resolved IPR<span
class="citation"
data-cites="andersonAbsenceDiffusionCertain1958"><sup><a
0\)</span>. In this work we take use an energy resolved IPR <span
class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a
href="#ref-andersonAbsenceDiffusionCertain1958"
role="doc-biblioref">18</a></sup></span>: <span class="math display">\[
role="doc-biblioref">18</a>]</span>: <span class="math display">\[
DOS(\omega) = \sum_n \delta(\omega - \epsilon_n)
IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n)
\abs{\psi_{n,i}}^4
@ -1518,148 +1516,141 @@ class="sourceCode python"><code class="sourceCode python"><span id="cb6-1"><a hr
<div class="sourceCode" id="cb7"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
<p></ij></ij></p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-devroyeRandomSampling1986" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div
class="csl-right-inline">Devroye, L. Random Sampling. in <em>Non-Uniform
Random Variate Generation</em> (ed. Devroye, L.) 611–641 (Springer,
1986). doi:<a
href="https://doi.org/10.1007/978-1-4613-8643-8_12">10.1007/978-1-4613-8643-8_12</a>.</div>
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">L.
Devroye, <em><a
href="https://doi.org/10.1007/978-1-4613-8643-8_12">Random
Sampling</a></em>, in <em>Non-Uniform Random Variate Generation</em>,
edited by L. Devroye (Springer, New York, NY, 1986), pp. 611–641.</div>
</div>
<div id="ref-BMCP2021" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div
class="csl-right-inline">Martin, O. A., Kumar, R. &amp; Lao, J.
<em>Bayesian modeling and computation in python</em>. (2021).</div>
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">O.
A. Martin, R. Kumar, and J. Lao, <em>Bayesian Modeling and Computation
in Python</em> (Boca Raton, 2021).</div>
</div>
<div id="ref-binderGuidePracticalWork1988" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div
class="csl-right-inline">Binder, K. &amp; Heermann, D. W. Guide to
Practical Work with the Monte Carlo Method. in <em>Monte Carlo
Simulation in Statistical Physics: An Introduction</em> (eds. Binder, K.
&amp; Heermann, D. W.) 68–112 (Springer Berlin Heidelberg, 1988). doi:<a
href="https://doi.org/10.1007/978-3-662-08854-8_3">10.1007/978-3-662-08854-8_3</a>.</div>
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">K.
Binder and D. W. Heermann, <em><a
href="https://doi.org/10.1007/978-3-662-08854-8_3">Guide to Practical
Work with the Monte Carlo Method</a></em>, in <em>Monte Carlo Simulation
in Statistical Physics: An Introduction</em>, edited by K. Binder and D.
W. Heermann (Springer Berlin Heidelberg, Berlin, Heidelberg, 1988), pp.
68–112.</div>
</div>
<div id="ref-kerteszAdvancesComputerSimulation1998" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div
class="csl-right-inline"><em>Advances in Computer Simulation: Lectures
Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July
1996</em>. (Springer-Verlag, 1998). doi:<a
href="https://doi.org/10.1007/BFb0105456">10.1007/BFb0105456</a>.</div>
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">J.
Kertesz and I. Kondor, editors, <em><a
href="https://doi.org/10.1007/BFb0105456">Advances in Computer
Simulation: Lectures Held at the Eötvös Summer School in Budapest,
Hungary, 16–20 July 1996</a></em> (Springer-Verlag, Berlin Heidelberg,
1998).</div>
</div>
<div id="ref-wolffMonteCarloErrors2004" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div
class="csl-right-inline">Wolff, U. <a
href="https://doi.org/10.1016/S0010-4655(03)00467-3">Monte Carlo errors
with less errors</a>. <em>Computer Physics Communications</em>
<strong>156</strong>, 143–153 (2004).</div>
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">U.
Wolff, <em><a href="https://doi.org/10.1016/S0010-4655(03)00467-3">Monte
Carlo Errors with Less Errors</a></em>, Computer Physics Communications
<strong>156</strong>, 143 (2004).</div>
</div>
<div id="ref-kellyReversibilityStochasticNetworks1981" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div
class="csl-right-inline">Kelly, F. P. <a
href="https://doi.org/10.2307/2287860">Reversibility and stochastic
networks / F.P. Kelly</a>. <em>SERBIULA (sistema Librum 2.0)</em>
<strong>76</strong>, (1981).</div>
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">F.
P. Kelly, <em><a href="https://doi.org/10.2307/2287860">Reversibility
and Stochastic Networks / F.P. Kelly</a></em>, SERBIULA (Sistema Librum
2.0) <strong>76</strong>, (1981).</div>
</div>
<div id="ref-krauthIntroductionMonteCarlo1998" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div
class="csl-right-inline">Krauth, W. Introduction To Monte Carlo
Algorithms. in <em>Advances in Computer Simulation: Lectures Held at the
Eötvös Summer School in Budapest, Hungary, 16–20 July 1996</em>
(Springer-Verlag, 1998). doi:<a
href="https://doi.org/10.1007/BFb0105456">10.1007/BFb0105456</a>.</div>
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">W.
Krauth, <em><a href="https://doi.org/10.1007/BFb0105456">Introduction To
Monte Carlo Algorithms</a></em>, in <em>Advances in Computer Simulation:
Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20
July 1996</em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
</div>
<div id="ref-kapferSamplingPolytopeHarddisk2013" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div
class="csl-right-inline">Kapfer, S. C. &amp; Krauth, W. <a
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">S.
C. Kapfer and W. Krauth, <em><a
href="https://doi.org/10.1088/1742-6596/454/1/012031">Sampling from a
polytope and hard-disk Monte Carlo</a>. <em>J. Phys.: Conf. Ser.</em>
Polytope and Hard-Disk Monte Carlo</a></em>, J. Phys.: Conf. Ser.
<strong>454</strong>, 012031 (2013).</div>
</div>
<div id="ref-robertsWeakConvergenceOptimal1997" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div
class="csl-right-inline">Roberts, G. O., Gelman, A. &amp; Gilks, W. R.
<a href="https://doi.org/10.1214/aoap/1034625254">Weak convergence and
optimal scaling of random walk Metropolis algorithms</a>. <em>Ann. Appl.
Probab.</em> <strong>7</strong>, 110–120 (1997).</div>
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">G.
O. Roberts, A. Gelman, and W. R. Gilks, <em><a
href="https://doi.org/10.1214/aoap/1034625254">Weak Convergence and
Optimal Scaling of Random Walk Metropolis Algorithms</a></em>, Ann.
Appl. Probab. <strong>7</strong>, 110 (1997).</div>
</div>
<div id="ref-huangAcceleratedMonteCarlo2017" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div
class="csl-right-inline">Huang, L. &amp; Wang, L. <a
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">L.
Huang and L. Wang, <em><a
href="https://doi.org/10.1103/PhysRevB.95.035105">Accelerated Monte
Carlo simulations with restricted Boltzmann machines</a>. <em>Phys. Rev.
B</em> <strong>95</strong>, 035105 (2017).</div>
Carlo Simulations with Restricted Boltzmann Machines</a></em>, Phys.
Rev. B <strong>95</strong>, 035105 (2017).</div>
</div>
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div
class="csl-right-inline">Binder, K. <a
href="https://doi.org/10.1007/BF01293604">Finite size scaling analysis
of ising model block distribution functions</a>. <em>Z. Physik B -
Condensed Matter</em> <strong>43</strong>, 119–140 (1981).</div>
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">K.
Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size
Scaling Analysis of Ising Model Block Distribution Functions</a></em>,
Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
</div>
<div id="ref-bolchQueueingNetworksMarkov2006" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div
class="csl-right-inline">Bolch, G., Greiner, S., Meer, H. de &amp;
Trivedi, K. S. <em>Queueing Networks and Markov Chains: Modeling and
Performance Evaluation with Computer Science Applications</em>. (John
Wiley &amp; Sons, 2006).</div>
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">G.
Bolch, S. Greiner, H. de Meer, and K. S. Trivedi, <em>Queueing Networks
and Markov Chains: Modeling and Performance Evaluation with Computer
Science Applications</em> (John Wiley &amp; Sons, 2006).</div>
</div>
<div id="ref-usmaniInversionTridiagonalJacobi1994" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div
class="csl-right-inline">Usmani, R. A. <a
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">R.
A. Usmani, <em><a
href="https://doi.org/10.1016/0024-3795(94)90414-6">Inversion of a
tridiagonal jacobi matrix</a>. <em>Linear Algebra and its
Applications</em> <strong>212-213</strong>, 413–414 (1994).</div>
Tridiagonal Jacobi Matrix</a></em>, Linear Algebra and Its Applications
<strong>212-213</strong>, 413 (1994).</div>
</div>
<div id="ref-krauthIntroductionMonteCarlo1996" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div
class="csl-right-inline">Krauth, W. <a
href="http://arxiv.org/abs/cond-mat/9612186">Introduction To Monte Carlo
Algorithms</a>. <em>arXiv:cond-mat/9612186</em> (1996).</div>
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">W.
Krauth, <em><a href="http://arxiv.org/abs/cond-mat/9612186">Introduction
To Monte Carlo Algorithms</a></em>, arXiv:cond-Mat/9612186 (1996).</div>
</div>
<div id="ref-hastingsMonteCarloSampling1970" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div
class="csl-right-inline">Hastings, W. K. <a
href="https://doi.org/10.1093/biomet/57.1.97">Monte Carlo sampling
methods using Markov chains and their applications</a>.
<em>Biometrika</em> <strong>57</strong>, 97–109 (1970).</div>
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">W.
K. Hastings, <em><a href="https://doi.org/10.1093/biomet/57.1.97">Monte
Carlo Sampling Methods Using Markov Chains and Their
Applications</a></em>, Biometrika <strong>57</strong>, 97 (1970).</div>
</div>
<div id="ref-hodsonMCMCFKModel2021" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div
class="csl-right-inline">Hodson, T. Markov Chain Monte Carlo for the
Kitaev Model. (2021) doi:<a
href="https://doi.org/10.5281/zenodo.4593904">10.5281/zenodo.4593904</a>.</div>
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">T.
Hodson, <em><a href="https://doi.org/10.5281/zenodo.4593904">Markov
Chain Monte Carlo for the Kitaev Model</a></em>, (2021).</div>
</div>
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div
class="csl-right-inline">Kramer, B. &amp; MacKinnon, A. <a
href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: theory
and experiment</a>. <em>Rep. Prog. Phys.</em> <strong>56</strong>,
1469–1564 (1993).</div>
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">B.
Kramer and A. MacKinnon, <em><a
href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory
and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469
(1993).</div>
</div>
<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div
class="csl-right-inline">Anderson, P. W. <a
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">P.
W. Anderson, <em><a
href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in
Certain Random Lattices</a>. <em>Phys. Rev.</em> <strong>109</strong>,
1492–1505 (1958).</div>
Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492
(1958).</div>
</div>
</div>
<section class="footnotes footnotes-end-of-document"
@ -1675,11 +1666,11 @@ systems with a tri-diagonal matrix representation (open boundary
conditions and nearest neighbour hopping) and like <span
class="math inline">\(N^3\)</span> for a generic matrix <span
class="citation"
data-cites="bolchQueueingNetworksMarkov2006 usmaniInversionTridiagonalJacobi1994"><sup><a
data-cites="bolchQueueingNetworksMarkov2006 usmaniInversionTridiagonalJacobi1994"> [<a
href="#ref-bolchQueueingNetworksMarkov2006"
role="doc-biblioref">12</a>,<a
href="#ref-usmaniInversionTridiagonalJacobi1994"
role="doc-biblioref">13</a></sup></span>.<a href="#fnref2"
role="doc-biblioref">13</a>]</span>.<a href="#fnref2"
class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn3" role="doc-endnote"><p>or, in the general case, any desired
distribution. MCMC has found a lot of use in sampling from the
@ -1689,9 +1680,9 @@ role="doc-backlink">↩︎</a></p></li>
<li id="fn4" role="doc-endnote"><p>or equivalently as the number of MCMC
steps after which the samples are correlated below some cutoff,
see <span class="citation"
data-cites="krauthIntroductionMonteCarlo1996"><sup><a
data-cites="krauthIntroductionMonteCarlo1996"> [<a
href="#ref-krauthIntroductionMonteCarlo1996"
role="doc-biblioref">14</a></sup></span> for a more rigorous definition
role="doc-biblioref">14</a>]</span> for a more rigorous definition
involving a sum over the auto-correlation function.<a href="#fnref4"
class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>

198
_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html
View File

@ -327,9 +327,9 @@ constant <span class="math inline">\(U=5\)</span> and constant <span
class="math inline">\(J=5\)</span>, respectively. We determined the
transition temperatures from the crossings of the Binder cumulants <span
class="math inline">\(B_4 = \tex{m^4}/\tex{m^2}^2\)</span> <span
class="citation" data-cites="binderFiniteSizeScaling1981"><sup><a
class="citation" data-cites="binderFiniteSizeScaling1981"> [<a
href="#ref-binderFiniteSizeScaling1981"
role="doc-biblioref">1</a></sup></span>. For a representative set of
role="doc-biblioref">1</a>]</span>. For a representative set of
parameters, Fig. [<a href="#fig:phase_diagram" data-reference-type="ref"
data-reference="fig:phase_diagram">1</a>c] shows the order parameter
<span class="math inline">\(\tex{m}^2\)</span>. Fig. [<a
@ -350,12 +350,12 @@ fermion mediated RKKY interaction between the Ising spins is absent.</p>
<p>Our main interest concerns the additional structure of the fermionic
sector in the high temperature phase. Following Ref. <span
class="citation"
data-cites="antipovInteractionTunedAndersonMott2016"><sup><a
data-cites="antipovInteractionTunedAndersonMott2016"> [<a
href="#ref-antipovInteractionTunedAndersonMott2016"
role="doc-biblioref">2</a></sup></span>, we can distinguish between the
Mott and Anderson insulating phases. The former is characterised by a
gapped DOS in the absence of a CDW. Thus, the opening of a gap for large
<span class="math inline">\(U\)</span> is distinct from the gap-opening
role="doc-biblioref">2</a>]</span>, we can distinguish between the Mott
and Anderson insulating phases. The former is characterised by a gapped
DOS in the absence of a CDW. Thus, the opening of a gap for large <span
class="math inline">\(U\)</span> is distinct from the gap-opening
induced by the translational symmetry breaking in the CDW state below
<span class="math inline">\(T_c\)</span>, see also Fig. [<a
href="#fig:band_opening" data-reference-type="ref"
@ -381,12 +381,11 @@ ka)^2}\;.\]</span></p>
<p>At infinite temperature, all the spin configurations become equally
likely and the fermionic model reduces to one of binary uncorrelated
disorder in which all eigenstates are Anderson localised <span
class="citation"
data-cites="abrahamsScalingTheoryLocalization1979"><sup><a
class="citation" data-cites="abrahamsScalingTheoryLocalization1979"> [<a
href="#ref-abrahamsScalingTheoryLocalization1979"
role="doc-biblioref">3</a></sup></span>. An Anderson localised state
centered around <span class="math inline">\(r_0\)</span> has magnitude
that drops exponentially over some localisation length <span
role="doc-biblioref">3</a>]</span>. An Anderson localised state centered
around <span class="math inline">\(r_0\)</span> has magnitude that drops
exponentially over some localisation length <span
class="math inline">\(\xi\)</span> i.e <span
class="math inline">\(|\psi(r)|^2 \sim \exp{-\abs{r -
r_0}/\xi}\)</span>. Calculating <span class="math inline">\(\xi\)</span>
@ -417,12 +416,12 @@ additional complication arises from the fact that the scaling exponent
may display intermediate behaviours for correlated disorder and in the
vicinity of a localisation-delocalisation transition <span
class="citation"
data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"><sup><a
data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"> [<a
href="#ref-kramerLocalizationTheoryExperiment1993"
role="doc-biblioref">4</a>,<a href="#ref-eversAndersonTransitions2008"
role="doc-biblioref">5</a></sup></span>. The thermal defects of the CDW
phase lead to a binary disorder potential with a finite correlation
length, which in principle could result in delocalized eigenstates.</p>
role="doc-biblioref">5</a>]</span>. The thermal defects of the CDW phase
lead to a binary disorder potential with a finite correlation length,
which in principle could result in delocalized eigenstates.</p>
<p>The key question for our system is then: How is the <span
class="math inline">\(T=0\)</span> CDW phase with fully delocalized
fermionic states connected to the fully localized phase at high
@ -488,7 +487,7 @@ alt="The DOS (a) and scaling exponent \tau (b) as a function of energy for the C
<div id="fig:indiv_IPR_disorder" class="fignos">
<figure>
<img src="pdf_figs/indiv_IPR_disorder.svg"
alt="Figure 4: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 &lt; \rho &lt; 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation 2, hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N &gt; 400" />
alt="Figure 4: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 &lt; \rho &lt; 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation  [2], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N &gt; 400" />
<figcaption aria-hidden="true"><span>Figure 4:</span> A comparison of
the full FK model to a simple binary disorder model (DM) with a CDW wave
background perturbed by uncorrelated defects at density <span
@ -500,9 +499,9 @@ Energy resolved DOS(<span class="math inline">\(\omega\)</span>) and
and this data makes clear that the apparent scaling of IPR with system
size is a finite size effect due to weak localisation <span
class="citation"
data-cites="antipovInteractionTunedAndersonMott2016"><sup><a
data-cites="antipovInteractionTunedAndersonMott2016"> [<a
href="#ref-antipovInteractionTunedAndersonMott2016"
role="doc-biblioref">2</a></sup></span>, hence all the states are indeed
role="doc-biblioref">2</a>]</span>, hence all the states are indeed
localised as one would expect in 1D. The disorder model <span
class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a)
<span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span
@ -534,22 +533,21 @@ class="math inline">\(\tau = 0.30\pm0.03\)</span> and <span
class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This
surprising finding suggests that the CDW bands are partially delocalised
with multi-fractal behaviour of the wavefunctions <span class="citation"
data-cites="eversAndersonTransitions2008"><sup><a
data-cites="eversAndersonTransitions2008"> [<a
href="#ref-eversAndersonTransitions2008"
role="doc-biblioref">5</a></sup></span>. This phenomenon would be
unexpected in a 1D model as they generally do not support delocalisation
in the presence of disorder except as the result of correlations in the
role="doc-biblioref">5</a>]</span>. This phenomenon would be unexpected
in a 1D model as they generally do not support delocalisation in the
presence of disorder except as the result of correlations in the
emergent disorder potential <span class="citation"
data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"><sup><a
data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"> [<a
href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">6</a>,<a
href="#ref-goldshteinPurePointSpectrum1977"
role="doc-biblioref">7</a></sup></span>. However, we later show by
comparison to an uncorrelated Anderson model that these nonzero
exponents are a finite size effect and the states are localised with a
finite <span class="math inline">\(\xi\)</span> similar to the system
size. As a result, the IPR does not scale correctly until the system
size has grown much larger than <span
class="math inline">\(\xi\)</span>. Fig. [<a
role="doc-biblioref">7</a>]</span>. However, we later show by comparison
to an uncorrelated Anderson model that these nonzero exponents are a
finite size effect and the states are localised with a finite <span
class="math inline">\(\xi\)</span> similar to the system size. As a
result, the IPR does not scale correctly until the system size has grown
much larger than <span class="math inline">\(\xi\)</span>. Fig. [<a
href="#fig:indiv_IPR_disorder" data-reference-type="ref"
data-reference="fig:indiv_IPR_disorder">4</a>] shows that the scaling of
the IPR in the CDW phase does flatten out eventually.</p>
@ -562,21 +560,21 @@ white, which highlights the distinction between the gapped Mott phase
and the ungapped Anderson phase. In-gap states appear just below the
critical point, smoothly filling the bandgap in the Anderson phase and
forming islands in the Mott phase. As in the finite <span
class="citation" data-cites="zondaGaplessRegimeCharge2019"><sup><a
class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a
href="#ref-zondaGaplessRegimeCharge2019"
role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a></sup></span>
role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a>]</span>
and infinite dimensional <span class="citation"
data-cites="hassanSpectralPropertiesChargedensitywave2007"><sup><a
data-cites="hassanSpectralPropertiesChargedensitywave2007"> [<a
href="#ref-hassanSpectralPropertiesChargedensitywave2007"
role="doc-biblioref">8</a></sup></span> cases, the in-gap states merge
and are pushed to lower energy for decreasing U as the <span
role="doc-biblioref">8</a>]</span> cases, the in-gap states merge and
are pushed to lower energy for decreasing U as the <span
class="math inline">\(T=0\)</span> CDW gap closes. Intuitively, the
presence of in-gap states can be understood as a result of domain wall
fluctuations away from the AFM ordered background. These domain walls
act as local potentials for impurity-like bound states <span
class="citation" data-cites="zondaGaplessRegimeCharge2019"><sup><a
class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a
href="#ref-zondaGaplessRegimeCharge2019"
role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a></sup></span>.</p>
role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a>]</span>.</p>
<p>In order to understand the localization properties we can compare the
behaviour of our model with that of a simpler Anderson disorder model
(DM) in which the spins are replaced by a CDW background with
@ -623,18 +621,18 @@ modify the localisation behaviour? Similar to other soluble models of
disorder-free localisation, we expect intriguing out-of equilibrium
physics, for example slow entanglement dynamics akin to more generic
interacting systems <span class="citation"
data-cites="hartLogarithmicEntanglementGrowth2020"><sup><a
data-cites="hartLogarithmicEntanglementGrowth2020"> [<a
href="#ref-hartLogarithmicEntanglementGrowth2020"
role="doc-biblioref">9</a></sup></span>. One could also investigate
whether the rich ground state phenomenology of the FK model as a
function of filling <span class="citation"
data-cites="gruberGroundStatesSpinless1990"><sup><a
role="doc-biblioref">9</a>]</span>. One could also investigate whether
the rich ground state phenomenology of the FK model as a function of
filling <span class="citation"
data-cites="gruberGroundStatesSpinless1990"> [<a
href="#ref-gruberGroundStatesSpinless1990"
role="doc-biblioref">10</a></sup></span> such as the devil’s
staircase <span class="citation"
data-cites="michelettiCompleteDevilTextquotesingles1997"><sup><a
role="doc-biblioref">10</a>]</span> such as the devil’s staircase <span
class="citation"
data-cites="michelettiCompleteDevilTextquotesingles1997"> [<a
href="#ref-michelettiCompleteDevilTextquotesingles1997"
role="doc-biblioref">11</a></sup></span> could be stabilised at finite
role="doc-biblioref">11</a>]</span> could be stabilised at finite
temperature. In a broader context, we envisage that long-range
interactions can also be used to gain a deeper understanding of the
temperature evolution of topological phases. One example would be a
@ -676,98 +674,94 @@ H_{\mathrm{DM}} = &amp; \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} -
\nonumber\end{aligned}\]</span></p>
<div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div
class="csl-right-inline">Binder, K. <a
href="https://doi.org/10.1007/BF01293604">Finite size scaling analysis
of ising model block distribution functions</a>. <em>Z. Physik B -
Condensed Matter</em> <strong>43</strong>, 119–140 (1981).</div>
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">K.
Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size
Scaling Analysis of Ising Model Block Distribution Functions</a></em>,
Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
</div>
<div id="ref-antipovInteractionTunedAndersonMott2016" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div
class="csl-right-inline">Antipov, A. E., Javanmard, Y., Ribeiro, P.
&amp; Kirchner, S. <a
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">A.
E. Antipov, Y. Javanmard, P. Ribeiro, and S. Kirchner, <em><a
href="https://doi.org/10.1103/PhysRevLett.117.146601">Interaction-Tuned
Anderson versus Mott Localization</a>. <em>Phys. Rev. Lett.</em>
Anderson Versus Mott Localization</a></em>, Phys. Rev. Lett.
<strong>117</strong>, 146601 (2016).</div>
</div>
<div id="ref-abrahamsScalingTheoryLocalization1979" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div
class="csl-right-inline">Abrahams, E., Anderson, P. W., Licciardello, D.
C. &amp; Ramakrishnan, T. V. <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.</em> <strong>42</strong>, 673–676 (1979).</div>
<div class="csl-left-margin">[3] </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">4. </div><div
class="csl-right-inline">Kramer, B. &amp; MacKinnon, A. <a
href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: theory
and experiment</a>. <em>Rep. Prog. Phys.</em> <strong>56</strong>,
1469–1564 (1993).</div>
<div class="csl-left-margin">[4] </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-eversAndersonTransitions2008" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div
class="csl-right-inline">Evers, F. &amp; Mirlin, A. D. <a
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">F.
Evers and A. D. Mirlin, <em><a
href="https://doi.org/10.1103/RevModPhys.80.1355">Anderson
Transitions</a>. <em>Rev. Mod. Phys.</em> <strong>80</strong>, 1355–1417
Transitions</a></em>, Rev. Mod. Phys. <strong>80</strong>, 1355
(2008).</div>
</div>
<div id="ref-croyAndersonLocalization1D2011" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div
class="csl-right-inline">Croy, A., Cain, P. &amp; Schreiber, M. <a
href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson localization
in 1D systems with correlated disorder</a>. <em>Eur. Phys. J. B</em>
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">A.
Croy, P. Cain, and M. Schreiber, <em><a
href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization
in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B
<strong>82</strong>, 107 (2011).</div>
</div>
<div id="ref-goldshteinPurePointSpectrum1977" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div
class="csl-right-inline">Gol’dshtein, I. Ya., Molchanov, S. A. &amp;
Pastur, L. A. <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</em> <strong>11</strong>, 1–8 (1977).</div>
<div class="csl-left-margin">[7] </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-hassanSpectralPropertiesChargedensitywave2007"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div
class="csl-right-inline">Hassan, S. R. &amp; Krishnamurthy, H. R. <a
href="https://doi.org/10.1103/PhysRevB.76.205109">Spectral properties in
the charge-density-wave phase of the half-filled Falicov-Kimball
model</a>. <em>Phys. Rev. B</em> <strong>76</strong>, 205109
(2007).</div>
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">S.
R. Hassan and H. R. Krishnamurthy, <em><a
href="https://doi.org/10.1103/PhysRevB.76.205109">Spectral Properties in
the Charge-Density-Wave Phase of the Half-Filled Falicov-Kimball
Model</a></em>, Phys. Rev. B <strong>76</strong>, 205109 (2007).</div>
</div>
<div id="ref-hartLogarithmicEntanglementGrowth2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div
class="csl-right-inline">Hart, O., Gopalakrishnan, S. &amp; Castelnovo,
C. <a href="http://arxiv.org/abs/2009.00618">Logarithmic entanglement
growth from disorder-free localisation in the two-leg compass
ladder</a>. <em>arXiv:2009.00618 [cond-mat]</em> (2020).</div>
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">O.
Hart, S. Gopalakrishnan, and C. Castelnovo, <em><a
href="http://arxiv.org/abs/2009.00618">Logarithmic Entanglement Growth
from Disorder-Free Localisation in the Two-Leg Compass Ladder</a></em>,
arXiv:2009.00618 [Cond-Mat] (2020).</div>
</div>
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div
class="csl-right-inline">Gruber, C., Iwanski, J., Jedrzejewski, J. &amp;
Lemberger, P. <a href="https://doi.org/10.1103/PhysRevB.41.2198">Ground
states of the spinless Falicov-Kimball model</a>. <em>Phys. Rev. B</em>
<strong>41</strong>, 2198–2209 (1990).</div>
<div class="csl-left-margin">[10] </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-michelettiCompleteDevilTextquotesingles1997"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div
class="csl-right-inline">Micheletti, C., Harris, A. B. &amp; Yeomans, J.
M. <a href="https://doi.org/10.1088/0305-4470/30/21/002">A complete
devil\textquotesingles staircase in the Falicov - Kimball model</a>.
<em>J. Phys. A: Math. Gen.</em> <strong>30</strong>, L711–L717
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">C.
Micheletti, A. B. Harris, and J. M. Yeomans, <em><a
href="https://doi.org/10.1088/0305-4470/30/21/002">A Complete
Devil\textquotesingles Staircase in the Falicov - Kimball
Model</a></em>, J. Phys. A: Math. Gen. <strong>30</strong>, L711
(1997).</div>
</div>
</div>

264
_thesis/4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html
View File

@ -539,10 +539,10 @@ symmetries</strong> and <strong><span class="math inline">\(2^2 =
4\)</span> topological sectors</strong>.</p>
<p>The topological sector forms the basis of proposals to construct
topologically protected qubits since the four sectors can only be mixed
by a highly non-local perturbations<span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003"><sup><a
by a highly non-local perturbations <span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003"> [<a
href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref">1</a></sup></span>.</p>
role="doc-biblioref">1</a>]</span>.</p>
<p>Takeaway: The Extended Hilbert Space decomposes into a direct product
of Flux Sectors, four Topological Sectors and a set of gauge
symmetries.</p>
@ -675,11 +675,11 @@ any information about the underlying lattice.</p>
<p><span class="math display">\[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i
\prod_i^{2N} b^y_i \prod_i^{2N} b^z_i \prod_i^{2N} c_i\]</span></p>
<p>The product over <span class="math inline">\(c_i\)</span> operators
reduces to a determinant of the Q matrix and the fermion parity,
see<span class="citation"
data-cites="pedrocchiPhysicalSolutionsKitaev2011"><sup><a
reduces to a determinant of the Q matrix and the fermion parity, see
<span class="citation"
data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
role="doc-biblioref">2</a></sup></span>. The only difference from the
role="doc-biblioref">2</a>]</span>. The only difference from the
honeycomb case is that we cannot explicitly compute the factors <span
class="math inline">\(p_x,p_y,p_z = \pm\;1\)</span> that arise from
reordering the b operators such that pairs of vertices linked by the
@ -702,20 +702,19 @@ depend only on the lattice structure.</p>
<p><span class="math inline">\(\hat{\pi} = \prod{i}^{N} (1 -
2\hat{n}_i)\)</span> is the parity of the particular many body state
determined by fermionic occupation numbers <span
class="math inline">\(n_i\)</span>. As discussed in<span
class="citation"
data-cites="pedrocchiPhysicalSolutionsKitaev2011"><sup><a
class="math inline">\(n_i\)</span>. As discussed in <span
class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
role="doc-biblioref">2</a></sup></span>, <span
role="doc-biblioref">2</a>]</span>, <span
class="math inline">\(\hat{\pi}\)</span> is gauge invariant in the sense
that <span class="math inline">\([\hat{\pi}, D_i] = 0\)</span>.</p>
<p>This implies that <span class="math inline">\(det(Q^u) \prod -i
u_{ij}\)</span> is also a gauge invariant quantity. In translation
invariant models this quantity which can be related to the parity of the
number of vortex pairs in the system<span class="citation"
data-cites="yaoAlgebraicSpinLiquid2009"><sup><a
number of vortex pairs in the system <span class="citation"
data-cites="yaoAlgebraicSpinLiquid2009"> [<a
href="#ref-yaoAlgebraicSpinLiquid2009"
role="doc-biblioref">3</a></sup></span>.</p>
role="doc-biblioref">3</a>]</span>.</p>
<p>All these factors take values <span class="math inline">\(\pm
1\)</span> so <span class="math inline">\(\mathcal{P}_0\)</span> is 0 or
1 for a particular state. Since <span
@ -744,12 +743,12 @@ vortex pair, transporting one of them around the major or minor
diameters of the torus and, then, annihilating them again.</figcaption>
</figure>
</div>
<p>More general arguments<span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"><sup><a
<p>More general arguments <span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">4</a>,<a
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
role="doc-biblioref">5</a></sup></span> imply that <span
role="doc-biblioref">5</a>]</span> imply that <span
class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> has an
interesting relationship to the topological fluxes. In the non-Abelian
phase, we expect that it will change sign in exactly one of the four
@ -838,8 +837,8 @@ definition, the vortex free sector.</p>
<p>On the Honeycomb, Lieb’s theorem implies that the ground state
corresponds to the state where all <span class="math inline">\(u_{jk} =
1\)</span>. This implies that the flux free sector is the ground state
sector<span class="citation" data-cites="lieb_flux_1994"><sup><a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a></sup></span>.</p>
sector <span class="citation" data-cites="lieb_flux_1994"> [<a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span>.</p>
<p>Lieb’s theorem does not generalise easily to the amorphous case.
However, we can get some intuition by examining the problem that will
lead to a guess for the ground state. We will then provide numerical
@ -919,12 +918,11 @@ i)^{n_{\mathrm{sides}}},
class="math inline">\(n_{\mathrm{sides}}\)</span> is the number of edges
that form each plaquette and the choice of sign gives a twofold chiral
ground state degeneracy.</p>
<p>This conjecture is consistent with Lieb’s theorem on regular
lattices<span class="citation" data-cites="lieb_flux_1994"><sup><a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a></sup></span> and
is supported by numerical evidence. As noted before, any flux that
differs from the ground state is an excitation which we call a
vortex.</p>
<p>This conjecture is consistent with Lieb’s theorem on regular lattices
<span class="citation" data-cites="lieb_flux_1994"> [<a
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span> and is
supported by numerical evidence. As noted before, any flux that differs
from the ground state is an excitation which we call a vortex.</p>
<h3 id="finite-size-effects">Finite size effects</h3>
<p>This guess only works for larger lattices. To rigorously test it, we
would want to directly enumerate the <span
@ -975,19 +973,18 @@ around the predicted ground state never yield a lower energy state.</p>
<strong>chiral</strong> degeneracy which arises because the global sign
of the odd plaquettes does not matter.</p>
<p>This happens because we have broken the time reversal symmetry of the
original model by adding odd plaquettes<span class="citation"
data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"><sup><a
original model by adding odd plaquettes <span class="citation"
data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"> [<a
href="#ref-Chua2011" role="doc-biblioref">7</a>–<a
href="#ref-WangHaoranPRB2021"
role="doc-biblioref">14</a></sup></span>.</p>
href="#ref-WangHaoranPRB2021" role="doc-biblioref">14</a>]</span>.</p>
<p>Similarly to the behaviour of the original Kitaev model in response
to a magnetic field, we get two degenerate ground states of different
handedness. Practically speaking, one ground state is related to the
other by inverting the imaginary <span
class="math inline">\(\phi\)</span> fluxes<span class="citation"
data-cites="yaoExactChiralSpin2007"><sup><a
class="math inline">\(\phi\)</span> fluxes <span class="citation"
data-cites="yaoExactChiralSpin2007"> [<a
href="#ref-yaoExactChiralSpin2007"
role="doc-biblioref">8</a></sup></span>.</p>
role="doc-biblioref">8</a>]</span>.</p>
<h2 id="phases-of-the-kitaev-model">Phases of the Kitaev Model</h2>
<p>discuss the Abelian A phase / toric code phase / anisotropic
phase</p>
@ -1114,190 +1111,185 @@ and construct the set <span class="math inline">\((+1, +1), (+1, -1),
<figure>
<img src="/assets/thesis/amk_chapter/topological_fluxes.png"
data-short-caption="Topological Fluxes" style="width:57.0%"
alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make15." />
alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make  [15]." />
<figcaption aria-hidden="true"><span>Figure 14:</span> Wilson loops that
wind the major or minor diameters of the torus measure flux winding
through the hole of the doughnut/torus or through the filling. If they
made doughnuts that both had a jam filling and a hole, this analogy
would be a lot easier to make<span class="citation"
data-cites="parkerWhyDoesThis"><sup><a href="#ref-parkerWhyDoesThis"
role="doc-biblioref">15</a></sup></span>.</figcaption>
would be a lot easier to make <span class="citation"
data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis"
role="doc-biblioref">15</a>]</span>.</figcaption>
</figure>
</div>
<p>However, in the non-Abelian phase we have to wrangle with
monodromy<span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"><sup><a
<p>However, in the non-Abelian phase we have to wrangle with monodromy
<span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">4</a>,<a
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
role="doc-biblioref">5</a></sup></span>. Monodromy is the behaviour of
role="doc-biblioref">5</a>]</span>. Monodromy is the behaviour of
objects as they move around a singularity. This manifests here in that
the identity of a vortex and cloud of Majoranas can change as we wind
them around the torus in such a way that, rather than annihilating to
the vacuum, we annihilate them to create an excited state instead of a
ground state. This means that we end up with only three degenerate
ground states in the non-Abelian phase <span class="math inline">\((+1,
+1), (+1, -1), (-1, +1)\)</span><span class="citation"
data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"><sup><a
+1), (+1, -1), (-1, +1)\)</span> <span class="citation"
data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"> [<a
href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">3</a>,<a
href="#ref-chungTopologicalQuantumPhase2010"
role="doc-biblioref">16</a></sup></span>. Concretely, this is because
the projector enforces both flux and fermion parity. When we wind a
vortex around both non-contractible loops of the torus, it flips the
flux parity. Therefore, we have to introduce a fermionic excitation to
make the state physical. Hence, the process does not give a fourth
ground state.</p>
role="doc-biblioref">16</a>]</span>. Concretely, this is because the
projector enforces both flux and fermion parity. When we wind a vortex
around both non-contractible loops of the torus, it flips the flux
parity. Therefore, we have to introduce a fermionic excitation to make
the state physical. Hence, the process does not give a fourth ground
state.</p>
<p>Recently, the topology has notably gained interest because of
proposals to use this ground state degeneracy to implement both
passively fault tolerant and actively stabilised quantum
computations<span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"><sup><a
passively fault tolerant and actively stabilised quantum computations
<span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a
href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref">1</a>,<a
href="#ref-poulinStabilizerFormalismOperator2005"
role="doc-biblioref">17</a>,<a
href="#ref-hastingsDynamicallyGeneratedLogical2021"
role="doc-biblioref">18</a></sup></span>.</p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
role="doc-biblioref">18</a>]</span>.</p>
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-kitaevFaulttolerantQuantumComputation2003"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div
class="csl-right-inline">Kitaev, A. Yu. <a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-tolerant
quantum computation by anyons</a>. <em>Annals of Physics</em>
<strong>303</strong>, 2–30 (2003).</div>
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">A.
Yu. Kitaev, <em><a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
Quantum Computation by Anyons</a></em>, Annals of Physics
<strong>303</strong>, 2 (2003).</div>
</div>
<div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div
class="csl-right-inline">Pedrocchi, F. L., Chesi, S. &amp; Loss, D. <a
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">F.
L. Pedrocchi, S. Chesi, and D. Loss, <em><a
href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of
the Kitaev honeycomb model</a>. <em>Phys. Rev. B</em>
<strong>84</strong>, 165414 (2011).</div>
the Kitaev honeycomb model</a></em>, Phys. Rev. B <strong>84</strong>,
165414 (2011).</div>
</div>
<div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div class="csl-right-inline">Yao,
H., Zhang, S.-C. &amp; Kivelson, S. A. <a
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">H.
Yao, S.-C. Zhang, and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin
Liquid in an Exactly Solvable Spin Model</a>. <em>Phys. Rev. Lett.</em>
Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett.
<strong>102</strong>, 217202 (2009).</div>
</div>
<div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div
class="csl-right-inline">Chung, S. B. &amp; Stone, M. <a
href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit monodromy of
Moore–Read wavefunctions on a torus</a>. <em>J. Phys. A: Math.
Theor.</em> <strong>40</strong>, 4923–4947 (2007).</div>
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">S.
B. Chung and M. Stone, <em><a
href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit Monodromy of
Moore–Read Wavefunctions on a Torus</a></em>, J. Phys. A: Math. Theor.
<strong>40</strong>, 4923 (2007).</div>
</div>
<div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div
class="csl-right-inline">Oshikawa, M., Kim, Y. B., Shtengel, K., Nayak,
C. &amp; Tewari, S. <a
href="https://doi.org/10.1016/j.aop.2006.08.001">Topological degeneracy
of non-Abelian states for dummies</a>. <em>Annals of Physics</em>
<strong>322</strong>, 1477–1498 (2007).</div>
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">M.
Oshikawa, Y. B. Kim, K. Shtengel, C. Nayak, and S. Tewari, <em><a
href="https://doi.org/10.1016/j.aop.2006.08.001">Topological Degeneracy
of Non-Abelian States for Dummies</a></em>, Annals of Physics
<strong>322</strong>, 1477 (2007).</div>
</div>
<div id="ref-lieb_flux_1994" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div
class="csl-right-inline">Lieb, E. H. <a
href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux Phase of the
Half-Filled Band</a>. <em>Physical Review Letters</em>
<strong>73</strong>, 2158–2161 (1994).</div>
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">E.
H. Lieb, <em><a href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux
Phase of the Half-Filled Band</a></em>, Physical Review Letters
<strong>73</strong>, 2158 (1994).</div>
</div>
<div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div
class="csl-right-inline">Chua, V., Yao, H. &amp; Fiete, G. A. <a
href="https://doi.org/10.1103/PhysRevB.83.180412">Exact chiral spin
liquid with stable spin Fermi surface on the kagome lattice</a>.
<em>Phys. Rev. B</em> <strong>83</strong>, 180412 (2011).</div>
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">V.
Chua, H. Yao, and G. A. Fiete, <em><a
href="https://doi.org/10.1103/PhysRevB.83.180412">Exact Chiral Spin
Liquid with Stable Spin Fermi Surface on the Kagome Lattice</a></em>,
Phys. Rev. B <strong>83</strong>, 180412 (2011).</div>
</div>
<div id="ref-yaoExactChiralSpin2007" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div class="csl-right-inline">Yao,
H. &amp; Kivelson, S. A. <a
href="https://doi.org/10.1103/PhysRevLett.99.247203">An exact chiral
spin liquid with non-Abelian anyons</a>. <em>Phys. Rev. Lett.</em>
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">H.
Yao and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral
Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett.
<strong>99</strong>, 247203 (2007).</div>
</div>
<div id="ref-ChuaPRB2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div
class="csl-right-inline">Chua, V. &amp; Fiete, G. A. <a
href="https://doi.org/10.1103/PhysRevB.84.195129">Exactly solvable
topological chiral spin liquid with random exchange</a>. <em>Phys. Rev.
B</em> <strong>84</strong>, 195129 (2011).</div>
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">V.
Chua and G. A. Fiete, <em><a
href="https://doi.org/10.1103/PhysRevB.84.195129">Exactly Solvable
Topological Chiral Spin Liquid with Random Exchange</a></em>, Phys. Rev.
B <strong>84</strong>, 195129 (2011).</div>
</div>
<div id="ref-Fiete2012" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div
class="csl-right-inline">Fiete, G. A. <em>et al.</em> <a
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">G.
A. Fiete, V. Chua, M. Kargarian, R. Lundgren, A. Rüegg, J. Wen, and V.
Zyuzin, <em><a
href="https://doi.org/10.1016/j.physe.2011.11.011">Topological
insulators and quantum spin liquids</a>. <em>Physica E: Low-dimensional
Systems and Nanostructures</em> <strong>44</strong>, 845–859
(2012).</div>
Insulators and Quantum Spin Liquids</a></em>, Physica E: Low-Dimensional
Systems and Nanostructures <strong>44</strong>, 845 (2012).</div>
</div>
<div id="ref-Natori2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div
class="csl-right-inline">Natori, W. M. H., Andrade, E. C., Miranda, E.
&amp; Pereira, R. G. <a
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">W.
M. H. Natori, E. C. Andrade, E. Miranda, and R. G. Pereira, <em><a
href="https://link.aps.org/doi/10.1103/PhysRevLett.117.017204">Chiral
spin-orbital liquids with nodal lines</a>. <em>Phys. Rev. Lett.</em>
Spin-Orbital Liquids with Nodal Lines</a></em>, Phys. Rev. Lett.
<strong>117</strong>, 017204 (2016).</div>
</div>
<div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div class="csl-right-inline">Wu,
C., Arovas, D. &amp; Hung, H.-H. Γ-matrix generalization of the Kitaev
model. <em>Physical Review B</em> <strong>79</strong>, 134427
(2009).</div>
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">C.
Wu, D. Arovas, and H.-H. Hung, <em>Γ-Matrix Generalization of the Kitaev
Model</em>, Physical Review B <strong>79</strong>, 134427 (2009).</div>
</div>
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div
class="csl-right-inline">Peri, V. <em>et al.</em> <a
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</em> <strong>101</strong>, 041114 (2020).</div>
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">V.
Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R.
Moessner, and R. Thomale, <em><a
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
<strong>101</strong>, 041114 (2020).</div>
</div>
<div id="ref-WangHaoranPRB2021" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div
class="csl-right-inline">Wang, H. &amp; Principi, A. <a
href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana edge and
corner states in square and kagome quantum spin-3/2 liquids</a>.
<em>Phys. Rev. B</em> <strong>104</strong>, 214422 (2021).</div>
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">H.
Wang and A. Principi, <em><a
href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana Edge and
Corner States in Square and Kagome Quantum Spin-3/2 Liquids</a></em>,
Phys. Rev. B <strong>104</strong>, 214422 (2021).</div>
</div>
<div id="ref-parkerWhyDoesThis" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div
<div class="csl-left-margin">[15] </div><div
class="csl-right-inline"><em><a
href="https://www.youtube.com/watch?v=ymF1bp-qrjU">Why does this balloon
have -1 holes?</a></em></div>
href="https://www.youtube.com/watch?v=ymF1bp-qrjU">Why Does This Balloon
Have -1 Holes?</a></em> (n.d.).</div>
</div>
<div id="ref-chungTopologicalQuantumPhase2010" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div
class="csl-right-inline">Chung, S. B., Yao, H., Hughes, T. L. &amp; Kim,
E.-A. <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</em>
<strong>81</strong>, 060403 (2010).</div>
<div class="csl-left-margin">[16] </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"
role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div
class="csl-right-inline">Poulin, D. <a
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">D.
Poulin, <em><a
href="https://doi.org/10.1103/PhysRevLett.95.230504">Stabilizer
Formalism for Operator Quantum Error Correction</a>. <em>Phys. Rev.
Lett.</em> <strong>95</strong>, 230504 (2005).</div>
Formalism for Operator Quantum Error Correction</a></em>, Phys. Rev.
Lett. <strong>95</strong>, 230504 (2005).</div>
</div>
<div id="ref-hastingsDynamicallyGeneratedLogical2021" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div
class="csl-right-inline">Hastings, M. B. &amp; Haah, J. <a
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">M.
B. Hastings and J. Haah, <em><a
href="https://doi.org/10.22331/q-2021-10-19-564">Dynamically Generated
Logical Qubits</a>. <em>Quantum</em> <strong>5</strong>, 564
(2021).</div>
Logical Qubits</a></em>, Quantum <strong>5</strong>, 564 (2021).</div>
</div>
</div>
</main>

103
_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html
View File

@ -245,11 +245,11 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
guidance from Willian and Johannes. The project grew out of an interest
Gino, Peru and I had in studying amorphous systems, coupled with
Johannes’ expertise on the Kitaev model. The idea to use voronoi
partitions came from<span class="citation"
data-cites="marsalTopologicalWeaireThorpe2020"><sup><a
partitions came from <span class="citation"
data-cites="marsalTopologicalWeaireThorpe2020"> [<a
href="#ref-marsalTopologicalWeaireThorpe2020"
role="doc-biblioref">1</a></sup></span> and Gino did the implementation
of this. The idea and implementation of the edge colouring using SAT
role="doc-biblioref">1</a>]</span> and Gino did the implementation of
this. The idea and implementation of the edge colouring using SAT
solvers, the mapping from flux sector to bond sector using A* search
were both entirely my work. Peru came up with the ground state
conjecture and implemented the local markers. Gino and I did much of the
@ -289,11 +289,11 @@ material. Candidate materials, such as <span
class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have
sufficiently strong spin-orbit coupling and the correct lattice
structure to behave according to the Kitaev Honeycomb model with small
corrections<span class="citation"
data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"><sup><a
corrections <span class="citation"
data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"> [<a
href="#ref-banerjeeProximateKitaevQuantum2016"
role="doc-biblioref">2</a>,<a href="#ref-trebstKitaevMaterials2022"
role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a></sup></span>.</p>
role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a>]</span>.</p>
<p><strong>expand later: Why do we need spin orbit coupling and what
will the corrections be?</strong></p>
<p>Second, its ground state is the canonical example of the long sought
@ -301,17 +301,17 @@ after quantum spin liquid state. Its excitations are anyons, particles
that can only exist in two dimensions that break the normal
fermion/boson dichotomy. Anyons have been the subject of much attention
because, among other reasons, they can be braided through spacetime to
achieve noise tolerant quantum computations<span class="citation"
data-cites="freedmanTopologicalQuantumComputation2003"><sup><a
achieve noise tolerant quantum computations <span class="citation"
data-cites="freedmanTopologicalQuantumComputation2003"> [<a
href="#ref-freedmanTopologicalQuantumComputation2003"
role="doc-biblioref">3</a></sup></span>.</p>
role="doc-biblioref">3</a>]</span>.</p>
<p>Third, and perhaps most importantly, this model is a rare many body
interacting quantum system that can be treated analytically. It is
exactly solvable. We can explicitly write down its many body ground
states in terms of single particle states<span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"><sup><a
states in terms of single particle states <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a></sup></span>. The solubility of the Kitaev
role="doc-biblioref">4</a>]</span>. The solubility of the Kitaev
Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes
about because the model has extensively many conserved degrees of
freedom. These conserved quantities can be factored out as classical
@ -326,9 +326,9 @@ lattices.</p>
look at the gauge symmetries of the model as well as its solution via a
transformation to a Majorana hamiltonian. This discussion shows that,
for the the model to be solvable, it needs only be defined on a
trivalent, tri-edge-colourable lattice<span class="citation"
data-cites="Nussinov2009"><sup><a href="#ref-Nussinov2009"
role="doc-biblioref">5</a></sup></span>.</p>
trivalent, tri-edge-colourable lattice <span class="citation"
data-cites="Nussinov2009"> [<a href="#ref-Nussinov2009"
role="doc-biblioref">5</a>]</span>.</p>
<p>The methods section discusses how to generate such lattices and
colour them. It also explain how to map back and forth between
configurations of the gauge field and configurations of the gauge
@ -512,12 +512,11 @@ on site <span class="math inline">\(j\)</span> and <span
class="math inline">\(\langle j,k\rangle_\alpha\)</span> is a pair of
nearest-neighbour indices connected by an <span
class="math inline">\(\alpha\)</span>-bond with exchange coupling <span
class="math inline">\(J^\alpha\)</span><span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"><sup><a
class="math inline">\(J^\alpha\)</span> <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a></sup></span>. For notational brevity, it is
useful to introduce the bond operators <span
class="math inline">\(K_{ij} =
role="doc-biblioref">4</a>]</span>. For notational brevity, it is useful
to introduce the bond operators <span class="math inline">\(K_{ij} =
\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> where <span
class="math inline">\(\alpha\)</span> is a function of <span
class="math inline">\(i,j\)</span> that picks the correct bond type.</p>
@ -744,10 +743,9 @@ theory of the Majorana Hamiltonian further.</p>
u_{ij} c_i c_j\]</span> in which most of the Majorana degrees of freedom
have paired along bonds to become a classical gauge field <span
class="math inline">\(u_{ij}\)</span>. What follows is relatively
standard theory for quadratic Majorana Hamiltonians<span
class="citation" data-cites="BlaizotRipka1986"><sup><a
href="#ref-BlaizotRipka1986"
role="doc-biblioref">6</a></sup></span>.</p>
standard theory for quadratic Majorana Hamiltonians <span
class="citation" data-cites="BlaizotRipka1986"> [<a
href="#ref-BlaizotRipka1986" role="doc-biblioref">6</a>]</span>.</p>
<p>Because of the antisymmetry of the matrix with entries <span
class="math inline">\(J^{\alpha} u_{ij}\)</span>, the eigenvalues of the
Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> come in
@ -865,52 +863,49 @@ which we set to 1 when calculating the projector.</p>
anyway, an arbitrary pairing of the unpaired <span
class="math inline">\(b^\alpha\)</span> operators could be performed.
&lt;/i,j&gt;&lt;/i,j&gt;</p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div
class="csl-right-inline">Marsal, Q., Varjas, D. &amp; Grushin, A. G. <a
<div class="csl-left-margin">[1] </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>Proceedings of the National Academy
of Sciences</em> <strong>117</strong>, 30260–30265 (2020).</div>
Models of Amorphous Matter</a></em>, Proceedings of the National Academy
of Sciences <strong>117</strong>, 30260 (2020).</div>
</div>
<div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div
class="csl-right-inline">Banerjee, A. <em>et al.</em> <a
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">A.
Banerjee et al., <em><a
href="https://doi.org/10.1038/nmat4604">Proximate Kitaev Quantum Spin
Liquid Behaviour in {\alpha}-RuCl$_3$</a>. <em>Nature Mater</em>
<strong>15</strong>, 733–740 (2016).</div>
Liquid Behaviour in {\Alpha}-RuCl$_3$</a></em>, Nature Mater
<strong>15</strong>, 733 (2016).</div>
</div>
<div id="ref-freedmanTopologicalQuantumComputation2003"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div
class="csl-right-inline">Freedman, M., Kitaev, A., Larsen, M. &amp;
Wang, Z. <a
href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological quantum
computation</a>. <em>Bull. Amer. Math. Soc.</em> <strong>40</strong>,
31–38 (2003).</div>
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">M.
Freedman, A. Kitaev, M. Larsen, and Z. Wang, <em><a
href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological Quantum
Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31
(2003).</div>
</div>
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div
class="csl-right-inline">Kitaev, A. <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</em>
<strong>321</strong>, 2–111 (2006).</div>
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">A.
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
<strong>321</strong>, 2 (2006).</div>
</div>
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div
class="csl-right-inline">Nussinov, Z. &amp; Ortiz, G. <a
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond algebras and
exact solvability of Hamiltonians: spin S=½ multilayer systems</a>.
<em>Physical Review B</em> <strong>79</strong>, 214440 (2009).</div>
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">Z.
Nussinov and G. Ortiz, <em><a
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and
Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>,
Physical Review B <strong>79</strong>, 214440 (2009).</div>
</div>
<div id="ref-BlaizotRipka1986" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div
class="csl-right-inline">Blaizot, J.-P. &amp; Ripka, G. <em>Quantum
theory of finite systems</em>. (The MIT Press, 1986).</div>
<div class="csl-left-margin">[6] </div><div
class="csl-right-inline">J.-P. Blaizot and G. Ripka, <em>Quantum Theory
of Finite Systems</em> (The MIT Press, 1986).</div>
</div>
</div>
</main>

302
_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html
View File

@ -234,20 +234,20 @@ Markers</a></li>
<h1 id="methods">Methods</h1>
<p>The practical implementation of what is described in this section is
available as a Python package called Koala (Kitaev On Amorphous
LAttices)<span class="citation"
data-cites="tomImperialCMTHKoalaFirst2022"><sup><a
LAttices) <span class="citation"
data-cites="tomImperialCMTHKoalaFirst2022"> [<a
href="#ref-tomImperialCMTHKoalaFirst2022"
role="doc-biblioref"><strong>tomImperialCMTHKoalaFirst2022?</strong></a></sup></span>.
role="doc-biblioref"><strong>tomImperialCMTHKoalaFirst2022?</strong></a>]</span>.
All results and figures were generated with Koala.</p>
<h2 id="voronisation">Voronisation</h2>
<p>To study the properties of the amorphous Kitaev model, we need to
sample from the space of possible trivalent graphs.</p>
<p>A simple method is to use a Voronoi partition of the torus<span
<p>A simple method is to use a Voronoi partition of the torus <span
class="citation"
data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020 florescu_designer_2009"><sup><a
data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020 florescu_designer_2009"> [<a
href="#ref-mitchellAmorphousTopologicalInsulators2018"
role="doc-biblioref">1</a>–<a href="#ref-florescu_designer_2009"
role="doc-biblioref">3</a></sup></span>. We start by sampling <em>seed
role="doc-biblioref">3</a>]</span>. We start by sampling <em>seed
points</em> uniformly (or otherwise) on the torus. Then, we compute the
partition of the torus into regions closest (with a Euclidean metric) to
each seed point. The straight lines (if the torus is flattened out) at
@ -259,23 +259,23 @@ the graph is embedded into the plane. It is also trivalent in that every
vertex is connected to exactly three edges <strong>cite</strong>.</p>
<p>Ideally, we would sample uniformly from the space of possible
trivalent graphs. Indeed, there has been some work on how to do this
using a Markov Chain Monte Carlo approach<span class="citation"
data-cites="alyamiUniformSamplingDirected2016"><sup><a
using a Markov Chain Monte Carlo approach <span class="citation"
data-cites="alyamiUniformSamplingDirected2016"> [<a
href="#ref-alyamiUniformSamplingDirected2016"
role="doc-biblioref">4</a></sup></span>. However, it does not guarantee
that the resulting graph is planar, which we must ensure so that the
edges can be 3-coloured.</p>
<p>In practice, we use a standard algorithm<span class="citation"
data-cites="barberQuickhullAlgorithmConvex1996"><sup><a
role="doc-biblioref">4</a>]</span>. However, it does not guarantee that
the resulting graph is planar, which we must ensure so that the edges
can be 3-coloured.</p>
<p>In practice, we use a standard algorithm <span class="citation"
data-cites="barberQuickhullAlgorithmConvex1996"> [<a
href="#ref-barberQuickhullAlgorithmConvex1996"
role="doc-biblioref">5</a></sup></span> from Scipy<span class="citation"
data-cites="virtanenSciPyFundamentalAlgorithms2020"><sup><a
role="doc-biblioref">5</a>]</span> from Scipy <span class="citation"
data-cites="virtanenSciPyFundamentalAlgorithms2020"> [<a
href="#ref-virtanenSciPyFundamentalAlgorithms2020"
role="doc-biblioref">6</a></sup></span> which computes the Voronoi
partition of the plane. To compute the Voronoi partition of the torus,
we take the seed points and replicate them into a repeating grid. This
will be either 3x3 or, for very small numbers of seed points, 5x5. Then,
we identify edges in the output to construct a lattice on the torus.</p>
role="doc-biblioref">6</a>]</span> which computes the Voronoi partition
of the plane. To compute the Voronoi partition of the torus, we take the
seed points and replicate them into a repeating grid. This will be
either 3x3 or, for very small numbers of seed points, 5x5. Then, we
identify edges in the output to construct a lattice on the torus.</p>
<div id="fig:lattice_construction_animated" class="fignos">
<figure>
<img
@ -368,47 +368,46 @@ onto the plane without any edges crossing. Bridgeless graphs do not
contain any edges that, when removed, would partition the graph into
disconnected components.</p>
<p>This problem must be distinguished from that considered by the famous
four-colour theorem<span class="citation"
data-cites="appelEveryPlanarMap1989"><sup><a
href="#ref-appelEveryPlanarMap1989"
role="doc-biblioref">7</a></sup></span>. The 4-colour theorem is
concerned with assigning colours to the <strong>vertices</strong> of a
graph, such that no vertices that share an edge have the same colour.
Here we are concerned with an edge colouring.</p>
four-colour theorem <span class="citation"
data-cites="appelEveryPlanarMap1989"> [<a
href="#ref-appelEveryPlanarMap1989" role="doc-biblioref">7</a>]</span>.
The 4-colour theorem is concerned with assigning colours to the
<strong>vertices</strong> of a graph, such that no vertices that share
an edge have the same colour. Here we are concerned with an edge
colouring.</p>
<p>The four-colour theorem applies to planar graphs, those that can be
embedded onto the plane without any edges crossing. Here we are
concerned with Toroidal graphs, which can be embedded onto the torus
without any edges crossing. In fact, toroidal graphs require up to seven
colours<span class="citation"
data-cites="heawoodMapColouringTheorems"><sup><a
colours <span class="citation"
data-cites="heawoodMapColouringTheorems"> [<a
href="#ref-heawoodMapColouringTheorems"
role="doc-biblioref">8</a></sup></span>. The complete graph <span
role="doc-biblioref">8</a>]</span>. The complete graph <span
class="math inline">\(K_7\)</span> is a good example of a toroidal graph
that requires seven colours.</p>
<p><span class="math inline">\(\Delta + 1\)</span> colours are enough to
edge-colour any graph. An <span
class="math inline">\(\mathcal{O}(mn)\)</span> algorithm exists to do it
for a graph with <span class="math inline">\(m\)</span> edges and <span
class="math inline">\(n\)</span> vertices<span class="citation"
data-cites="gEstimateChromaticClass1964"><sup><a
class="math inline">\(n\)</span> vertices <span class="citation"
data-cites="gEstimateChromaticClass1964"> [<a
href="#ref-gEstimateChromaticClass1964"
role="doc-biblioref">9</a></sup></span>. Restricting ourselves to graphs
with <span class="math inline">\(\Delta = 3\)</span> like ours, those
can be four-edge-coloured in linear time<span class="citation"
data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"><sup><a
role="doc-biblioref">9</a>]</span>. Restricting ourselves to graphs with
<span class="math inline">\(\Delta = 3\)</span> like ours, those can be
four-edge-coloured in linear time <span class="citation"
data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"> [<a
href="#ref-skulrattanakulchai4edgecoloringGraphsMaximum2002"
role="doc-biblioref">10</a></sup></span>.</p>
role="doc-biblioref">10</a>]</span>.</p>
<p>However, three-edge-colouring them is more difficult. Cubic, planar,
bridgeless graphs can be three-edge-coloured if and only if they can be
four-face-coloured<span class="citation"
data-cites="tait1880remarks"><sup><a href="#ref-tait1880remarks"
role="doc-biblioref">11</a></sup></span>. An <span
class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists
here<span class="citation" data-cites="robertson1996efficiently"><sup><a
four-face-coloured <span class="citation"
data-cites="tait1880remarks"> [<a href="#ref-tait1880remarks"
role="doc-biblioref">11</a>]</span>. An <span
class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists here
<span class="citation" data-cites="robertson1996efficiently"> [<a
href="#ref-robertson1996efficiently"
role="doc-biblioref">12</a></sup></span>. However, it is not clear
whether this extends to cubic, <strong>toroidal</strong> bridgeless
graphs.</p>
role="doc-biblioref">12</a>]</span>. However, it is not clear whether
this extends to cubic, <strong>toroidal</strong> bridgeless graphs.</p>
<div id="fig:multiple_colourings" class="fignos">
<figure>
<img
@ -467,22 +466,22 @@ solver. A SAT problem is a set of statements about some number of
boolean variables , such as “<span class="math inline">\(x_1\)</span> or
not <span class="math inline">\(x_3\)</span> is true”, and looks for an
assignment <span class="math inline">\(x_i \in {0,1}\)</span> that
satisfies all the statements<span class="citation"
data-cites="Karp1972"><sup><a href="#ref-Karp1972"
role="doc-biblioref">13</a></sup></span>.</p>
satisfies all the statements <span class="citation"
data-cites="Karp1972"> [<a href="#ref-Karp1972"
role="doc-biblioref">13</a>]</span>.</p>
<p>General purpose, high performance programs for solving SAT problems
have been an area of active research for decades<span class="citation"
data-cites="alounehComprehensiveStudyAnalysis2019"><sup><a
have been an area of active research for decades <span class="citation"
data-cites="alounehComprehensiveStudyAnalysis2019"> [<a
href="#ref-alounehComprehensiveStudyAnalysis2019"
role="doc-biblioref">14</a></sup></span>. Such programs are useful
because, by the Cook-Levin theorem, any NP problem can be encoded in
polynomial time as an instance of a SAT problem . This property is what
makes SAT one of the subset of NP problems called NP-Complete<span
role="doc-biblioref">14</a>]</span>. Such programs are useful because,
by the Cook-Levin theorem, any NP problem can be encoded in polynomial
time as an instance of a SAT problem . This property is what makes SAT
one of the subset of NP problems called NP-Complete <span
class="citation"
data-cites="cookComplexityTheoremprovingProcedures1971 levin1973universal"><sup><a
data-cites="cookComplexityTheoremprovingProcedures1971 levin1973universal"> [<a
href="#ref-cookComplexityTheoremprovingProcedures1971"
role="doc-biblioref">15</a>,<a href="#ref-levin1973universal"
role="doc-biblioref">16</a></sup></span>.</p>
role="doc-biblioref">16</a>]</span>.</p>
<p>Thus, it is a relatively standard technique in the computer science
community to solve NP problems by first transforming them to SAT
instances and then using an off the shelf SAT solver. The output of this
@ -495,9 +494,9 @@ could be used to speed up its solution, using a SAT solver appears to be
a reasonable first method to try. As will be discussed later, this
turned out to work well enough and looking for a better solution was not
necessary.</p>
<p>We use a solver called <code>MiniSAT</code><span class="citation"
data-cites="imms-sat18"><sup><a href="#ref-imms-sat18"
role="doc-biblioref">17</a></sup></span>. Like most modern SAT solvers,
<p>We use a solver called <code>MiniSAT</code> <span class="citation"
data-cites="imms-sat18"> [<a href="#ref-imms-sat18"
role="doc-biblioref">17</a>]</span>. Like most modern SAT solvers,
<code>MiniSAT</code> requires the input problem to be specified in
Conjunctive Normal Form (CNF). CNF requires that the constraints be
encoded as a set of <em>clauses</em> of the form <span
@ -555,11 +554,11 @@ a graph and assigns them a colour that is not already disallowed. This
does not work for our purposes because it is not designed to look for a
particular n-colouring. However, it does include the option of using a
heuristic function that determine the order in which vertices will be
coloured<span class="citation"
data-cites="kosowski2004classical matulaSmallestlastOrderingClustering1983"><sup><a
coloured <span class="citation"
data-cites="kosowski2004classical matulaSmallestlastOrderingClustering1983"> [<a
href="#ref-kosowski2004classical" role="doc-biblioref">18</a>,<a
href="#ref-matulaSmallestlastOrderingClustering1983"
role="doc-biblioref">19</a></sup></span>. Perhaps</p>
role="doc-biblioref">19</a>]</span>. Perhaps</p>
<div id="fig:times" class="fignos">
<figure>
<img src="/assets/thesis/amk_chapter/methods/times/times.svg"
@ -658,154 +657,147 @@ system.</p>
<p><strong>Expand on definition here</strong></p>
<p><strong>Discuss link between Chern number and Anyonic
Statistics</strong></p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-mitchellAmorphousTopologicalInsulators2018"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div
class="csl-right-inline">Mitchell, N. P., Nash, L. M., Hexner, D.,
Turner, A. M. &amp; Irvine, W. T. M. <a
href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous topological
insulators constructed from random point sets</a>. <em>Nature Phys</em>
<strong>14</strong>, 380–385 (2018).</div>
<div class="csl-left-margin">[1] </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-marsalTopologicalWeaireThorpeModels2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div
class="csl-right-inline">Marsal, Q., Varjas, D. &amp; Grushin, A. G. <a
<div class="csl-left-margin">[2] </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.</em>
<strong>117</strong>, 30260–30265 (2020).</div>
Models of Amorphous Matter</a></em>, Proc. Natl. Acad. Sci. U.S.A.
<strong>117</strong>, 30260 (2020).</div>
</div>
<div id="ref-florescu_designer_2009" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div
class="csl-right-inline">Florescu, M., Torquato, S. &amp; Steinhardt, P.
J. <a href="https://doi.org/10.1073/pnas.0907744106">Designer disordered
materials with large, complete photonic band gaps</a>. <em>Proceedings
of the National Academy of Sciences</em> <strong>106</strong>,
20658–20663 (2009).</div>
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">M.
Florescu, S. Torquato, and P. J. Steinhardt, <em><a
href="https://doi.org/10.1073/pnas.0907744106">Designer Disordered
Materials with Large, Complete Photonic Band Gaps</a></em>, Proceedings
of the National Academy of Sciences <strong>106</strong>, 20658
(2009).</div>
</div>
<div id="ref-alyamiUniformSamplingDirected2016" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div
class="csl-right-inline">Alyami, S. A., Azad, A. K. M. &amp; Keith, J.
M. <a href="https://doi.org/10.1016/j.endm.2016.05.005">Uniform Sampling
of Directed and Undirected Graphs Conditional on Vertex
Connectivity</a>. <em>Electronic Notes in Discrete Mathematics</em>
<strong>53</strong>, 43–55 (2016).</div>
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">S.
A. Alyami, A. K. M. Azad, and J. M. Keith, <em><a
href="https://doi.org/10.1016/j.endm.2016.05.005">Uniform Sampling of
Directed and Undirected Graphs Conditional on Vertex
Connectivity</a></em>, Electronic Notes in Discrete Mathematics
<strong>53</strong>, 43 (2016).</div>
</div>
<div id="ref-barberQuickhullAlgorithmConvex1996" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div
class="csl-right-inline">Barber, C. B., Dobkin, D. P. &amp; Huhdanpaa,
H. <a href="https://doi.org/10.1145/235815.235821">The quickhull
algorithm for convex hulls</a>. <em>ACM Trans. Math. Softw.</em>
<strong>22</strong>, 469–483 (1996).</div>
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">C.
B. Barber, D. P. Dobkin, and H. Huhdanpaa, <em><a
href="https://doi.org/10.1145/235815.235821">The Quickhull Algorithm for
Convex Hulls</a></em>, ACM Trans. Math. Softw. <strong>22</strong>, 469
(1996).</div>
</div>
<div id="ref-virtanenSciPyFundamentalAlgorithms2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div
class="csl-right-inline">Virtanen, P. <em>et al.</em> <a
href="https://doi.org/10.1038/s41592-019-0686-2">SciPy 1.0: fundamental
algorithms for scientific computing in Python</a>. <em>Nature
Methods</em> <strong>17</strong>, 261–272 (2020).</div>
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">P.
Virtanen et al., <em><a
href="https://doi.org/10.1038/s41592-019-0686-2">SciPy 1.0: Fundamental
Algorithms for Scientific Computing in Python</a></em>, Nature Methods
<strong>17</strong>, 3 (2020).</div>
</div>
<div id="ref-appelEveryPlanarMap1989" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div
class="csl-right-inline">Appel, K. &amp; Haken, W. Every Planar Map Is
Four Colorable. in (1989). doi:<a
href="https://doi.org/10.1090/conm/098">10.1090/conm/098</a>.</div>
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">K.
Appel and W. Haken, <em><a href="https://doi.org/10.1090/conm/098">Every
Planar Map Is Four Colorable</a></em>, in (1989).</div>
</div>
<div id="ref-heawoodMapColouringTheorems" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div
class="csl-right-inline">Heawood, P. J. Map colouring theorems.
<em>Quarterly Journal of Mathematics</em> 322–339.</div>
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">P.
J. Heawood, <em>Map Colouring Theorems</em>, Quarterly Journal of
Mathematics 322 (n.d.).</div>
</div>
<div id="ref-gEstimateChromaticClass1964" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div class="csl-right-inline">G,
V. V. <a href="https://cir.nii.ac.jp/crid/1571980075458819456">On an
estimate of the chromatic class of a p-graph</a>. <em>Discret
Analiz</em> <strong>3</strong>, 25–30 (1964).</div>
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">V.
V. G, <em><a href="https://cir.nii.ac.jp/crid/1571980075458819456">On an
Estimate of the Chromatic Class of a p-Graph</a></em>, Discret Analiz
<strong>3</strong>, 25 (1964).</div>
</div>
<div id="ref-skulrattanakulchai4edgecoloringGraphsMaximum2002"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div
class="csl-right-inline">Skulrattanakulchai, S. <a
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">S.
Skulrattanakulchai, <em><a
href="https://doi.org/10.1016/S0020-0190(01)00221-6">4-edge-coloring
graphs of maximum degree 3 in linear time</a>. <em>Inf. Process.
Lett.</em> <strong>81</strong>, 191–195 (2002).</div>
graphs of maximum degree 3 in linear time</a></em>, Inf. Process. Lett.
<strong>81</strong>, 191 (2002).</div>
</div>
<div id="ref-tait1880remarks" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div
class="csl-right-inline">Tait, P. G. Remarks on the colouring of maps.
in <em>Proc. Roy. Soc. Edinburgh</em> vol. 10 501–503 (1880).</div>
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">P.
G. Tait, <em>Remarks on the Colouring of Maps</em>, in <em>Proc. Roy.
Soc. Edinburgh</em>, Vol. 10 (1880), pp. 501–503.</div>
</div>
<div id="ref-robertson1996efficiently" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div
class="csl-right-inline">Robertson, N., Sanders, D. P., Seymour, P.
&amp; Thomas, R. Efficiently four-coloring planar graphs. in
<em>Proceedings of the twenty-eighth annual ACM symposium on Theory of
computing</em> 571–575 (1996).</div>
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">N.
Robertson, D. P. Sanders, P. Seymour, and R. Thomas, <em>Efficiently
Four-Coloring Planar Graphs</em>, in <em>Proceedings of the
Twenty-Eighth Annual ACM Symposium on Theory of Computing</em> (1996),
pp. 571–575.</div>
</div>
<div id="ref-Karp1972" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div
class="csl-right-inline">Karp, R. M. Reducibility among combinatorial
problems. in <em>Complexity of computer computations</em> (eds. Miller,
R. E., Thatcher, J. W. &amp; Bohlinger, J. D.) 85–103 (Springer US,
1972). doi:<a
href="https://doi.org/10.1007/978-1-4684-2001-2_9">10.1007/978-1-4684-2001-2_9</a>.</div>
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">R.
M. Karp, <em><a
href="https://doi.org/10.1007/978-1-4684-2001-2_9">Reducibility Among
Combinatorial Problems</a></em>, in <em>Complexity of Computer
Computations</em>, edited by R. E. Miller, J. W. Thatcher, and J. D.
Bohlinger (Springer US, Boston, MA, 1972), pp. 85–103.</div>
</div>
<div id="ref-alounehComprehensiveStudyAnalysis2019" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div
class="csl-right-inline">Alouneh, S., Abed, S., Al Shayeji, M. H. &amp;
Mesleh, R. <a href="https://doi.org/10.1007/s10462-018-9628-0">A
comprehensive study and analysis on SAT-solvers: advances, usages and
achievements</a>. <em>Artif Intell Rev</em> <strong>52</strong>,
2575–2601 (2019).</div>
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">S.
Alouneh, S. Abed, M. H. Al Shayeji, and R. Mesleh, <em><a
href="https://doi.org/10.1007/s10462-018-9628-0">A Comprehensive Study
and Analysis on SAT-Solvers: Advances, Usages and Achievements</a></em>,
Artif Intell Rev <strong>52</strong>, 2575 (2019).</div>
</div>
<div id="ref-cookComplexityTheoremprovingProcedures1971"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div
class="csl-right-inline">Cook, S. A. The complexity of theorem-proving
procedures. in <em>Proceedings of the third annual ACM symposium on
Theory of computing</em> 151–158 (Association for Computing Machinery,
1971). doi:<a
href="https://doi.org/10.1145/800157.805047">10.1145/800157.805047</a>.</div>
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">S.
A. Cook, <em><a href="https://doi.org/10.1145/800157.805047">The
Complexity of Theorem-Proving Procedures</a></em>, in <em>Proceedings of
the Third Annual ACM Symposium on Theory of Computing</em> (Association
for Computing Machinery, New York, NY, USA, 1971), pp. 151–158.</div>
</div>
<div id="ref-levin1973universal" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div
class="csl-right-inline">Levin, L. A. Universal sequential search
problems. <em>Problemy peredachi informatsii</em> <strong>9</strong>,
115–116 (1973).</div>
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">L.
A. Levin, <em>Universal Sequential Search Problems</em>, Problemy
Peredachi Informatsii <strong>9</strong>, 115 (1973).</div>
</div>
<div id="ref-imms-sat18" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div
class="csl-right-inline">Ignatiev, A., Morgado, A. &amp; Marques-Silva,
J. PySAT: A Python toolkit for prototyping with SAT oracles. in
<em>SAT</em> 428–437 (2018). doi:<a
href="https://doi.org/10.1007/978-3-319-94144-8_26">10.1007/978-3-319-94144-8_26</a>.</div>
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">A.
Ignatiev, A. Morgado, and J. Marques-Silva, <em><a
href="https://doi.org/10.1007/978-3-319-94144-8_26">PySAT: A Python
Toolkit for Prototyping with SAT Oracles</a></em>, in <em>SAT</em>
(2018), pp. 428–437.</div>
</div>
<div id="ref-kosowski2004classical" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div
class="csl-right-inline">Kosowski, A. &amp; Manuszewski, K. Classical
coloring of graphs. <em>Contemporary Mathematics</em>
<strong>352</strong>, 1–20 (2004).</div>
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">A.
Kosowski and K. Manuszewski, <em>Classical Coloring of Graphs</em>,
Contemporary Mathematics <strong>352</strong>, 1 (2004).</div>
</div>
<div id="ref-matulaSmallestlastOrderingClustering1983" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">19. </div><div
class="csl-right-inline">Matula, D. W. &amp; Beck, L. L. <a
href="https://doi.org/10.1145/2402.322385">Smallest-last ordering and
clustering and graph coloring algorithms</a>. <em>J. ACM</em>
<strong>30</strong>, 417–427 (1983).</div>
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">D.
W. Matula and L. L. Beck, <em><a
href="https://doi.org/10.1145/2402.322385">Smallest-Last Ordering and
Clustering and Graph Coloring Algorithms</a></em>, J. ACM
<strong>30</strong>, 417 (1983).</div>
</div>
</div>
</main>

715
_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html
View File

@ -249,10 +249,10 @@ ground state flux sector is correct. We will do this by enumerating all
the flux sectors of many separate system realisations. However there are
some issues we will need to address to make this argument work.</p>
<p>We have two seemingly irreconcilable problems. Finite size effects
have a large energetic contribution for small systems<span
class="citation" data-cites="kitaevAnyonsExactlySolved2006"><sup><a
have a large energetic contribution for small systems <span
class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">1</a></sup></span> so we would like to perform our
role="doc-biblioref">1</a>]</span> so we would like to perform our
analysis for very large lattices. However for an amorphous system with
<span class="math inline">\(N\)</span> plaquettes, <span
class="math inline">\(2N\)</span> edges and <span
@ -308,15 +308,15 @@ relatively regular pattern for the imaginary fluxes with only a global
two-fold chiral degeneracy.</p>
<p>Thus, states with a fixed flux sector spontaneously break time
reversal symmetry. This was first described by Yao and Kivelson for a
translation invariant Kitaev model with odd sided plaquettes<span
class="citation" data-cites="Yao2011"><sup><a href="#ref-Yao2011"
role="doc-biblioref">2</a></sup></span>.</p>
translation invariant Kitaev model with odd sided plaquettes <span
class="citation" data-cites="Yao2011"> [<a href="#ref-Yao2011"
role="doc-biblioref">2</a>]</span>.</p>
<p>So we have flux sectors that come in degenerate pairs, where time
reversal is equivalent to inverting the flux through every odd
plaquette, a general feature for lattices with odd plaquettes <span
class="citation" data-cites="yaoExactChiralSpin2007 Peri2020"><sup><a
class="citation" data-cites="yaoExactChiralSpin2007 Peri2020"> [<a
href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>,<a
href="#ref-Peri2020" role="doc-biblioref">4</a></sup></span>. This
href="#ref-Peri2020" role="doc-biblioref">4</a>]</span>. This
spontaneously broken symmetry avoids the need to explicitly break TRS
with a magnetic field term as is done in the original honeycomb
model.</p>
@ -348,12 +348,11 @@ straight lines <span class="math inline">\(|J^x| = |J^y| +
class="math inline">\(x,y,z\)</span>, shown as dotted line on ~<a
href="#fig:phase_diagram">1</a> (Right). We find that on the amorphous
lattice these boundaries exhibit an inward curvature, similar to
honeycomb Kitaev models with flux<span class="citation"
data-cites="Nasu_Thermal_2015"><sup><a href="#ref-Nasu_Thermal_2015"
role="doc-biblioref">5</a></sup></span> or bond<span class="citation"
data-cites="knolle_dynamics_2016"><sup><a
href="#ref-knolle_dynamics_2016" role="doc-biblioref">6</a></sup></span>
disorder.</p>
honeycomb Kitaev models with flux <span class="citation"
data-cites="Nasu_Thermal_2015"> [<a href="#ref-Nasu_Thermal_2015"
role="doc-biblioref">5</a>]</span> or bond <span class="citation"
data-cites="knolle_dynamics_2016"> [<a href="#ref-knolle_dynamics_2016"
role="doc-biblioref">6</a>]</span> disorder.</p>
<div id="fig:phase_diagram" class="fignos">
<figure>
<img
@ -388,11 +387,11 @@ class="math inline">\(0\)</span> to <span class="math inline">\(\pm
later I’ll double check this with finite size scaling.</p>
<p>The next question is: do these phases support excitations with
Abelian or non-Abelian statistics? To answer that we turn to Chern
numbers<span class="citation"
data-cites="berryQuantalPhaseFactors1984 simonHolonomyQuantumAdiabatic1983 thoulessQuantizedHallConductance1982"><sup><a
numbers <span class="citation"
data-cites="berryQuantalPhaseFactors1984 simonHolonomyQuantumAdiabatic1983 thoulessQuantizedHallConductance1982"> [<a
href="#ref-berryQuantalPhaseFactors1984" role="doc-biblioref">7</a>–<a
href="#ref-thoulessQuantizedHallConductance1982"
role="doc-biblioref">9</a></sup></span>. As discussed earlier the Chern
role="doc-biblioref">9</a>]</span>. As discussed earlier the Chern
number is a quantity intimately linked to both the topological
properties and the anyonic statistics of a model. Here we will make use
of the fact that the Abelian/non-Abelian character of a model is linked
@ -400,28 +399,27 @@ to its Chern number <strong>[citation]</strong>. However the Chern
number is only defined for the translation invariant case because it
relies on integrals defined in k-space.</p>
<p>A family of real space generalisations of the Chern number that work
for amorphous systems exist called local topological markers<span
for amorphous systems exist called local topological markers <span
class="citation"
data-cites="bianco_mapping_2011 Hastings_Almost_2010 mitchellAmorphousTopologicalInsulators2018"><sup><a
data-cites="bianco_mapping_2011 Hastings_Almost_2010 mitchellAmorphousTopologicalInsulators2018"> [<a
href="#ref-bianco_mapping_2011" role="doc-biblioref">10</a>–<a
href="#ref-mitchellAmorphousTopologicalInsulators2018"
role="doc-biblioref">12</a></sup></span> and indeed Kitaev defines one
in his original paper on the model<span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"><sup><a
role="doc-biblioref">12</a>]</span> and indeed Kitaev defines one in his
original paper on the model <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">1</a></sup></span>.</p>
<p>Here we use the crosshair marker of<span class="citation"
data-cites="peru_preprint"><sup><a href="#ref-peru_preprint"
role="doc-biblioref">13</a></sup></span> because it works well on
smaller systems. We calculate the projector <span
class="math inline">\(P = \sum_i |\psi_i\rangle \langle \psi_i|\)</span>
onto the occupied fermion eigenstates of the system in open boundary
conditions. The projector encodes local information about the occupied
eigenstates of the system and is typically exponentially localised
<strong>[cite]</strong>. The name <em>crosshair</em> comes from the fact
that the marker is defined with respect to a particular point <span
class="math inline">\((x_0, y_0)\)</span> by step functions in x and
y</p>
role="doc-biblioref">1</a>]</span>.</p>
<p>Here we use the crosshair marker of <span class="citation"
data-cites="peru_preprint"> [<a href="#ref-peru_preprint"
role="doc-biblioref">13</a>]</span> because it works well on smaller
systems. We calculate the projector <span class="math inline">\(P =
\sum_i |\psi_i\rangle \langle \psi_i|\)</span> onto the occupied fermion
eigenstates of the system in open boundary conditions. The projector
encodes local information about the occupied eigenstates of the system
and is typically exponentially localised <strong>[cite]</strong>. The
name <em>crosshair</em> comes from the fact that the marker is defined
with respect to a particular point <span class="math inline">\((x_0,
y_0)\)</span> by step functions in x and y</p>
<p><span class="math display">\[\begin{aligned}
\nu (x, y) = 4\pi \; \Im\; \mathrm{Tr}_{\mathrm{B}}
\left (
@ -440,54 +438,51 @@ character of the phases.</p>
<p>In the A phase of the amorphous model we find that <span
class="math inline">\(\nu=0\)</span> and hence the excitations have
Abelian character, similar to the honeycomb model. This phase is thus
the amorphous analogue of the Abelian toric-code quantum spin
liquid<span class="citation"
data-cites="kitaev_fault-tolerant_2003"><sup><a
the amorphous analogue of the Abelian toric-code quantum spin liquid
<span class="citation" data-cites="kitaev_fault-tolerant_2003"> [<a
href="#ref-kitaev_fault-tolerant_2003"
role="doc-biblioref">14</a></sup></span>.</p>
role="doc-biblioref">14</a>]</span>.</p>
<p>The B phase has <span class="math inline">\(\nu=\pm1\)</span> so is a
non-Abelian <em>chiral spin liquid</em> (CSL) similar to that of the
Yao-Kivelson model<span class="citation"
data-cites="yaoExactChiralSpin2007"><sup><a
href="#ref-yaoExactChiralSpin2007"
role="doc-biblioref">3</a></sup></span>. The CSL state is the the
magnetic analogue of the fractional quantum Hall state
<strong>[cite]</strong>. Hereafter we focus our attention on this
phase.</p>
Yao-Kivelson model <span class="citation"
data-cites="yaoExactChiralSpin2007"> [<a
href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>]</span>.
The CSL state is the the magnetic analogue of the fractional quantum
Hall state <strong>[cite]</strong>. Hereafter we focus our attention on
this phase.</p>
<div id="fig:phase_diagram_chern" class="fignos">
<figure>
<img
src="/assets/thesis/amk_chapter/results/phase_diagram_chern/phase_diagram_chern.svg"
data-short-caption="Local Chern Markers" style="width:100.0%"
alt="Figure 2: (Center) The crosshair marker13, a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = \pm 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian." />
alt="Figure 2: (Center) The crosshair marker  [13], a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = \pm 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian." />
<figcaption aria-hidden="true"><span>Figure 2:</span> (Center) The
crosshair marker<span class="citation"
data-cites="peru_preprint"><sup><a href="#ref-peru_preprint"
role="doc-biblioref">13</a></sup></span>, a local topological marker,
evaluated on the Amorphous Kitaev Model. The marker is defined around a
point, denoted by the dotted crosshair. Information about the local
topological properties of the system are encoded within a region around
that point. (Left) Summing these contributions up to some finite radius
(dotted line here, dotted circle in the centre) gives a generalised
version of the Chern number for the system which becomes quantised in
the thermodynamic limit. The radius must be chosen large enough to
capture information about the local properties of the lattice while not
so large as to include contributions from the edge states. The isotropic
regime <span class="math inline">\(J_\alpha = 1\)</span> in red has
<span class="math inline">\(\nu = \pm 1\)</span> implying it supports
excitations with non-Abelian statistics, while the anisotropic regime in
orange has <span class="math inline">\(\nu = \pm 0\)</span> implying it
has Abelian statistics. (Right) Extending this analysis to the whole
<span class="math inline">\(J_\alpha\)</span> phase diagram with fixed
<span class="math inline">\(r = 0.3\)</span> nicely confirms that the
isotropic phase is non-Abelian.</figcaption>
crosshair marker <span class="citation" data-cites="peru_preprint"> [<a
href="#ref-peru_preprint" role="doc-biblioref">13</a>]</span>, a local
topological marker, evaluated on the Amorphous Kitaev Model. The marker
is defined around a point, denoted by the dotted crosshair. Information
about the local topological properties of the system are encoded within
a region around that point. (Left) Summing these contributions up to
some finite radius (dotted line here, dotted circle in the centre) gives
a generalised version of the Chern number for the system which becomes
quantised in the thermodynamic limit. The radius must be chosen large
enough to capture information about the local properties of the lattice
while not so large as to include contributions from the edge states. The
isotropic regime <span class="math inline">\(J_\alpha = 1\)</span> in
red has <span class="math inline">\(\nu = \pm 1\)</span> implying it
supports excitations with non-Abelian statistics, while the anisotropic
regime in orange has <span class="math inline">\(\nu = \pm 0\)</span>
implying it has Abelian statistics. (Right) Extending this analysis to
the whole <span class="math inline">\(J_\alpha\)</span> phase diagram
with fixed <span class="math inline">\(r = 0.3\)</span> nicely confirms
that the isotropic phase is non-Abelian.</figcaption>
</figure>
</div>
<h3 id="edge-modes">Edge Modes</h3>
<p>Chiral Spin Liquids support topological protected edge modes on open
boundary conditions<span class="citation"
data-cites="qi_general_2006"><sup><a href="#ref-qi_general_2006"
role="doc-biblioref">15</a></sup></span>. fig. <a
boundary conditions <span class="citation"
data-cites="qi_general_2006"> [<a href="#ref-qi_general_2006"
role="doc-biblioref">15</a>]</span>. fig. <a
href="#fig:edge_modes">3</a> shows the probability density of one such
edge mode. It is near zero energy and exponentially localised to the
boundary of the system. While the model is gapped in periodic boundary
@ -522,35 +517,34 @@ states.</figcaption>
Thermal Metal</h2>
<p>Previous work on the honeycomb model at finite temperature has shown
that the B phase undergoes a thermal transition from a quantum spin
liquid phase a to a <strong>thermal metal</strong> phase<span
class="citation" data-cites="selfThermallyInducedMetallic2019"><sup><a
liquid phase a to a <strong>thermal metal</strong> phase <span
class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a></sup></span>.</p>
role="doc-biblioref">16</a>]</span>.</p>
<p>This happens because at finite temperature, thermal fluctuations lead
to spontaneous vortex-pair formation. As discussed previously these
fluxes are dressed by Majorana bounds states and the composite object is
an Ising-type non-Abelian anyon<span class="citation"
data-cites="Beenakker2013"><sup><a href="#ref-Beenakker2013"
role="doc-biblioref">17</a></sup></span>. The interactions between these
an Ising-type non-Abelian anyon <span class="citation"
data-cites="Beenakker2013"> [<a href="#ref-Beenakker2013"
role="doc-biblioref">17</a>]</span>. The interactions between these
anyons are oscillatory similar to the RKKY exchange and decay
exponentially with separation<span class="citation"
data-cites="Laumann2012 Lahtinen_2011 lahtinenTopologicalLiquidNucleation2012"><sup><a
exponentially with separation <span class="citation"
data-cites="Laumann2012 Lahtinen_2011 lahtinenTopologicalLiquidNucleation2012"> [<a
href="#ref-Laumann2012" role="doc-biblioref">18</a>–<a
href="#ref-lahtinenTopologicalLiquidNucleation2012"
role="doc-biblioref">20</a></sup></span>. At sufficient density, the
anyons hybridise to a macroscopically degenerate state known as
<em>thermal metal</em><span class="citation"
data-cites="Laumann2012"><sup><a href="#ref-Laumann2012"
role="doc-biblioref">18</a></sup></span>. At close range the oscillatory
behaviour of the interactions can be modelled by a random sign which
forms the basis for a random matrix theory description of the thermal
metal state.</p>
role="doc-biblioref">20</a>]</span>. At sufficient density, the anyons
hybridise to a macroscopically degenerate state known as <em>thermal
metal</em> <span class="citation" data-cites="Laumann2012"> [<a
href="#ref-Laumann2012" role="doc-biblioref">18</a>]</span>. At close
range the oscillatory behaviour of the interactions can be modelled by a
random sign which forms the basis for a random matrix theory description
of the thermal metal state.</p>
<p>The amorphous chiral spin liquid undergoes the same form of Anderson
transition to a thermal metal state. Markov Chain Monte Carlo would be
necessary to simulate this in full detail<span class="citation"
data-cites="selfThermallyInducedMetallic2019"><sup><a
necessary to simulate this in full detail <span class="citation"
data-cites="selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a></sup></span> but in order to avoid that
role="doc-biblioref">16</a>]</span> but in order to avoid that
complexity in the current work we instead opted to use vortex density
<span class="math inline">\(\rho\)</span> as a proxy for
temperature.</p>
@ -641,11 +635,11 @@ model onto a Majorana model with interactions that take random signs
which can itself be mapped onto a coarser lattice with lower energy
excitations and so on. This can be repeating indefinitely, showing the
model must have excitations at arbitrarily low energies in the
thermodynamic limit<span class="citation"
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"><sup><a
thermodynamic limit <span class="citation"
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000"
role="doc-biblioref">21</a></sup></span>.</p>
role="doc-biblioref">21</a>]</span>.</p>
<p>These signatures for our model and for the honeycomb model are shown
in fig. <a href="#fig:DOS_oscillations">6</a>. They do not occur in the
honeycomb model unless the chiral symmetry is broken by a magnetic
@ -656,21 +650,21 @@ field.</p>
src="/assets/thesis/amk_chapter/results/DOS_oscillations/DOS_oscillations.svg"
data-short-caption="Distinctive Oscillations in the Density of States"
style="width:100.0%"
alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state16,21. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is \rho = 0.5 corresponding to the T = \infty limit." />
alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state  [16,21]. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is \rho = 0.5 corresponding to the T = \infty limit." />
<figcaption aria-hidden="true"><span>Figure 6:</span> Density of states
at high temperature showing the logarithmic divergence at zero energy
and oscillations characteristic of the thermal metal state<span
and oscillations characteristic of the thermal metal state <span
class="citation"
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"><sup><a
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000"
role="doc-biblioref">21</a></sup></span>. (a) shows the honeycomb
lattice model in the B phase with magnetic field, while (b) shows that
our model transitions to a thermal metal phase without an external
magnetic field but rather due to the spontaneous chiral symmetry
breaking. In both plots the density of vortices is <span
class="math inline">\(\rho = 0.5\)</span> corresponding to the <span
class="math inline">\(T = \infty\)</span> limit.</figcaption>
role="doc-biblioref">21</a>]</span>. (a) shows the honeycomb lattice
model in the B phase with magnetic field, while (b) shows that our model
transitions to a thermal metal phase without an external magnetic field
but rather due to the spontaneous chiral symmetry breaking. In both
plots the density of vortices is <span class="math inline">\(\rho =
0.5\)</span> corresponding to the <span class="math inline">\(T =
\infty\)</span> limit.</figcaption>
</figure>
</div>
<h1 id="conclusion">Conclusion</h1>
@ -719,46 +713,45 @@ Realisations and Signatures</h2>
<p>The obvious question is whether amorphous Kitaev materials could be
physically realised.</p>
<p>Most crystals can as exists in a metastable amorphous state if they
are cooled rapidly, freezing them into a disordered configuration<span
are cooled rapidly, freezing them into a disordered configuration <span
class="citation"
data-cites="Weaire1976 Petrakovski1981 Kaneyoshi2018"><sup><a
data-cites="Weaire1976 Petrakovski1981 Kaneyoshi2018"> [<a
href="#ref-Weaire1976" role="doc-biblioref">22</a>–<a
href="#ref-Kaneyoshi2018" role="doc-biblioref">24</a></sup></span>.
Indeed quenching has been used by humans to control the hardness of
steel or iron for thousands of years. It would therefore be interesting
to study amorphous version of candidate Kitaev materials<span
class="citation" data-cites="trebstKitaevMaterials2022"><sup><a
href="#ref-Kaneyoshi2018" role="doc-biblioref">24</a>]</span>. Indeed
quenching has been used by humans to control the hardness of steel or
iron for thousands of years. It would therefore be interesting to study
amorphous version of candidate Kitaev materials <span class="citation"
data-cites="trebstKitaevMaterials2022"> [<a
href="#ref-trebstKitaevMaterials2022"
role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a></sup></span>
role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a>]</span>
such as <span class="math inline">\(\alpha-\textrm{RuCl}_3\)</span> to
see whether they maintain even approximate fixed coordination number
locally as is the case with amorphous Silicon and Germanium<span
class="citation" data-cites="Weaire1971 betteridge1973possible"><sup><a
locally as is the case with amorphous Silicon and Germanium <span
class="citation" data-cites="Weaire1971 betteridge1973possible"> [<a
href="#ref-Weaire1971" role="doc-biblioref">25</a>,<a
href="#ref-betteridge1973possible"
role="doc-biblioref">26</a></sup></span>.</p>
role="doc-biblioref">26</a>]</span>.</p>
<p>Looking instead at more engineered realisation, metal organic
frameworks have been shown to be capable of forming amorphous
lattices <span class="citation"
data-cites="bennett2014amorphous"><sup><a
href="#ref-bennett2014amorphous"
role="doc-biblioref">27</a></sup></span> and there are recent proposals
for realizing strong Kitaev interactions <span class="citation"
data-cites="yamadaDesigningKitaevSpin2017"><sup><a
lattices <span class="citation" data-cites="bennett2014amorphous"> [<a
href="#ref-bennett2014amorphous" role="doc-biblioref">27</a>]</span> and
there are recent proposals for realizing strong Kitaev
interactions <span class="citation"
data-cites="yamadaDesigningKitaevSpin2017"> [<a
href="#ref-yamadaDesigningKitaevSpin2017"
role="doc-biblioref">28</a></sup></span> as well as reports of QSL
role="doc-biblioref">28</a>]</span> as well as reports of QSL
behavior <span class="citation"
data-cites="misumiQuantumSpinLiquid2020"><sup><a
data-cites="misumiQuantumSpinLiquid2020"> [<a
href="#ref-misumiQuantumSpinLiquid2020"
role="doc-biblioref">29</a></sup></span>.</p>
role="doc-biblioref">29</a>]</span>.</p>
<h2 id="generalisations">Generalisations</h2>
<p>The model presented here could be generalized in several ways.</p>
<p>First, it would be interesting to study the stability of the chiral
amorphous Kitaev QSL with respect to perturbations <span
class="citation"
data-cites="Rau2014 Chaloupka2010 Chaloupka2013 Chaloupka2015 Winter2016"><sup><a
data-cites="Rau2014 Chaloupka2010 Chaloupka2013 Chaloupka2015 Winter2016"> [<a
href="#ref-Rau2014" role="doc-biblioref">30</a>–<a
href="#ref-Winter2016" role="doc-biblioref">34</a></sup></span>.</p>
href="#ref-Winter2016" role="doc-biblioref">34</a>]</span>.</p>
<p>Second, one could investigate whether a QSL phase may exist for for
other models defined on amorphous lattices. For example, in real
materials, there will generally be an additional small Heisenberg term
@ -767,398 +760,382 @@ j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} +
\sigma_j\sigma_k\]</span> With a view to more realistic prospects of
observation, it would be interesting to see if the properties of the
Kitaev-Heisenberg model generalise from the honeycomb to the amorphous
case[<span class="citation" data-cites="Chaloupka2010"><sup><a
href="#ref-Chaloupka2010" role="doc-biblioref">31</a></sup></span>;<span
class="citation" data-cites="Chaloupka2015"><sup><a
href="#ref-Chaloupka2015" role="doc-biblioref">33</a></sup></span>;<span
class="citation" data-cites="Jackeli2009"><sup><a
href="#ref-Jackeli2009" role="doc-biblioref">35</a></sup></span>;<span
class="citation" data-cites="Kalmeyer1989"><sup><a
href="#ref-Kalmeyer1989" role="doc-biblioref">36</a></sup></span>;<span
class="citation"
data-cites="manousakisSpinTextonehalfHeisenberg1991"><sup><a
case[<span class="citation" data-cites="Chaloupka2010"> [<a
href="#ref-Chaloupka2010" role="doc-biblioref">31</a>]</span>; <span
class="citation" data-cites="Chaloupka2015"> [<a
href="#ref-Chaloupka2015" role="doc-biblioref">33</a>]</span>; <span
class="citation" data-cites="Jackeli2009"> [<a href="#ref-Jackeli2009"
role="doc-biblioref">35</a>]</span>; <span class="citation"
data-cites="Kalmeyer1989"> [<a href="#ref-Kalmeyer1989"
role="doc-biblioref">36</a>]</span>; <span class="citation"
data-cites="manousakisSpinTextonehalfHeisenberg1991"> [<a
href="#ref-manousakisSpinTextonehalfHeisenberg1991"
role="doc-biblioref">37</a></sup></span>;].</p>
role="doc-biblioref">37</a>]</span>;].</p>
<p>Finally it might be possible to look at generalizations to
higher-spin models or those on random networks with different
coordination numbers<span class="citation"
data-cites="Baskaran2008 Yao2009 Nussinov2009 Yao2011 Chua2011 Natori2020 Chulliparambil2020 Chulliparambil2021 Seifert2020 WangHaoranPRB2021 Wu2009"><sup><a
coordination numbers <span class="citation"
data-cites="Baskaran2008 Yao2009 Nussinov2009 Yao2011 Chua2011 Natori2020 Chulliparambil2020 Chulliparambil2021 Seifert2020 WangHaoranPRB2021 Wu2009"> [<a
href="#ref-Yao2011" role="doc-biblioref">2</a>,<a
href="#ref-Baskaran2008" role="doc-biblioref">38</a>–<a
href="#ref-Wu2009" role="doc-biblioref">47</a></sup></span></p>
href="#ref-Wu2009" role="doc-biblioref">47</a>]</span></p>
<p>Overall, there has been surprisingly little research on amorphous
quantum many body phases albeit material candidates aplenty. We expect
our exact chiral amorphous spin liquid to find many generalisation to
realistic amorphous quantum magnets and beyond.</p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">1. </div><div
class="csl-right-inline">Kitaev, A. <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</em>
<strong>321</strong>, 2–111 (2006).</div>
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">A.
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
<strong>321</strong>, 2 (2006).</div>
</div>
<div id="ref-Yao2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div class="csl-right-inline">Yao,
H. &amp; Lee, D.-H. <a
href="https://doi.org/10.1103/PhysRevLett.107.087205">Fermionic magnons,
non-abelian spinons, and the spin quantum hall effect from an exactly
solvable spin-1/2 kitaev model with SU(2) symmetry</a>. <em>Phys. Rev.
Lett.</em> <strong>107</strong>, 087205 (2011).</div>
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">H.
Yao and D.-H. Lee, <em><a
href="https://doi.org/10.1103/PhysRevLett.107.087205">Fermionic Magnons,
Non-Abelian Spinons, and the Spin Quantum Hall Effect from an Exactly
Solvable Spin-1/2 Kitaev Model with SU(2) Symmetry</a></em>, Phys. Rev.
Lett. <strong>107</strong>, 087205 (2011).</div>
</div>
<div id="ref-yaoExactChiralSpin2007" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div class="csl-right-inline">Yao,
H. &amp; Kivelson, S. A. <a
href="https://doi.org/10.1103/PhysRevLett.99.247203">An exact chiral
spin liquid with non-Abelian anyons</a>. <em>Phys. Rev. Lett.</em>
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">H.
Yao and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral
Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett.
<strong>99</strong>, 247203 (2007).</div>
</div>
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div
class="csl-right-inline">Peri, V. <em>et al.</em> <a
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</em> <strong>101</strong>, 041114 (2020).</div>
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">V.
Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R.
Moessner, and R. Thomale, <em><a
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
<strong>101</strong>, 041114 (2020).</div>
</div>
<div id="ref-Nasu_Thermal_2015" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div
class="csl-right-inline">Nasu, J., Udagawa, M. &amp; Motome, Y. <a
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">J.
Nasu, M. Udagawa, and Y. Motome, <em><a
href="https://doi.org/10.1103/PhysRevB.92.115122">Thermal
fractionalization of quantum spins in a Kitaev model: Temperature-linear
specific heat and coherent transport of Majorana fermions</a>. <em>Phys.
Rev. B</em> <strong>92</strong>, 115122 (2015).</div>
Fractionalization of Quantum Spins in a Kitaev Model: Temperature-Linear
Specific Heat and Coherent Transport of Majorana Fermions</a></em>,
Phys. Rev. B <strong>92</strong>, 115122 (2015).</div>
</div>
<div id="ref-knolle_dynamics_2016" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div
class="csl-right-inline">Knolle, J. Dynamics of a quantum spin liquid.
(Max Planck Institute for the Physics of Complex Systems, Dresden,
2016).</div>
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">J.
Knolle, Dynamics of a Quantum Spin Liquid, Max Planck Institute for the
Physics of Complex Systems, Dresden, 2016.</div>
</div>
<div id="ref-berryQuantalPhaseFactors1984" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div
class="csl-right-inline">Berry, M. V. <a
href="https://doi.org/10.1098/rspa.1984.0023">Quantal phase factors
accompanying adiabatic changes</a>. <em>Proceedings of the Royal Society
of London. A. Mathematical and Physical Sciences</em>
<strong>392</strong>, 45–57 (1984).</div>
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">M.
V. Berry, <em><a href="https://doi.org/10.1098/rspa.1984.0023">Quantal
Phase Factors Accompanying Adiabatic Changes</a></em>, Proceedings of
the Royal Society of London. A. Mathematical and Physical Sciences
<strong>392</strong>, 45 (1984).</div>
</div>
<div id="ref-simonHolonomyQuantumAdiabatic1983" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div
class="csl-right-inline">Simon, B. <a
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">B.
Simon, <em><a
href="https://doi.org/10.1103/PhysRevLett.51.2167">Holonomy, the Quantum
Adiabatic Theorem, and Berry’s Phase</a>. <em>Phys. Rev. Lett.</em>
<strong>51</strong>, 2167–2170 (1983).</div>
Adiabatic Theorem, and Berry’s Phase</a></em>, Phys. Rev. Lett.
<strong>51</strong>, 2167 (1983).</div>
</div>
<div id="ref-thoulessQuantizedHallConductance1982" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div
class="csl-right-inline">Thouless, D. J., Kohmoto, M., Nightingale, M.
P. &amp; den Nijs, M. <a
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">D.
J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, <em><a
href="https://doi.org/10.1103/PhysRevLett.49.405">Quantized Hall
Conductance in a Two-Dimensional Periodic Potential</a>. <em>Phys. Rev.
Lett.</em> <strong>49</strong>, 405–408 (1982).</div>
Conductance in a Two-Dimensional Periodic Potential</a></em>, Phys. Rev.
Lett. <strong>49</strong>, 405 (1982).</div>
</div>
<div id="ref-bianco_mapping_2011" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div
class="csl-right-inline">Bianco, R. &amp; Resta, R. <a
href="https://doi.org/10.1103/PhysRevB.84.241106">Mapping topological
order in coordinate space</a>. <em>Physical Review B</em>
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">R.
Bianco and R. Resta, <em><a
href="https://doi.org/10.1103/PhysRevB.84.241106">Mapping Topological
Order in Coordinate Space</a></em>, Physical Review B
<strong>84</strong>, 241106 (2011).</div>
</div>
<div id="ref-Hastings_Almost_2010" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div
class="csl-right-inline">Hastings, M. B. &amp; Loring, T. A. <a
href="https://doi.org/10.1063/1.3274817">Almost commuting matrices,
localized Wannier functions, and the quantum Hall effect</a>.
<em>Journal of Mathematical Physics</em> <strong>51</strong>, 015214
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">M.
B. Hastings and T. A. Loring, <em><a
href="https://doi.org/10.1063/1.3274817">Almost Commuting Matrices,
Localized Wannier Functions, and the Quantum Hall Effect</a></em>,
Journal of Mathematical Physics <strong>51</strong>, 015214
(2010).</div>
</div>
<div id="ref-mitchellAmorphousTopologicalInsulators2018"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div
class="csl-right-inline">Mitchell, N. P., Nash, L. M., Hexner, D.,
Turner, A. M. &amp; Irvine, W. T. M. <a
href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous topological
insulators constructed from random point sets</a>. <em>Nature Phys</em>
<strong>14</strong>, 380–385 (2018).</div>
<div class="csl-left-margin">[12] </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-peru_preprint" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div
class="csl-right-inline">d’Ornellas, P., Barnett, R. &amp; Lee, D. K. K.
Quantised bulk conductivity as a local chern marker. <em>arXiv
preprint</em> (2022) doi:<a
href="https://doi.org/10.48550/ARXIV.2207.01389">10.48550/ARXIV.2207.01389</a>.</div>
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">P.
d’Ornellas, R. Barnett, and D. K. K. Lee, <em><a
href="https://doi.org/10.48550/ARXIV.2207.01389">Quantised Bulk
Conductivity as a Local Chern Marker</a></em>, arXiv Preprint
(2022).</div>
</div>
<div id="ref-kitaev_fault-tolerant_2003" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div
class="csl-right-inline">Kitaev, A. Y. <a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-tolerant
quantum computation by anyons</a>. <em>Annals of Physics</em>
<strong>303</strong>, 2–30 (2003).</div>
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">A.
Y. Kitaev, <em><a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
Quantum Computation by Anyons</a></em>, Annals of Physics
<strong>303</strong>, 2 (2003).</div>
</div>
<div id="ref-qi_general_2006" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div class="csl-right-inline">Qi,
X.-L., Wu, Y.-S. &amp; Zhang, S.-C. <a
href="https://doi.org/10.1103/PhysRevB.74.045125">General theorem
relating the bulk topological number to edge states in two-dimensional
insulators</a>. <em>Physical Review B</em> <strong>74</strong>, 045125
<div class="csl-left-margin">[15] </div><div
class="csl-right-inline">X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, <em><a
href="https://doi.org/10.1103/PhysRevB.74.045125">General Theorem
Relating the Bulk Topological Number to Edge States in Two-Dimensional
Insulators</a></em>, Physical Review B <strong>74</strong>, 045125
(2006).</div>
</div>
<div id="ref-selfThermallyInducedMetallic2019" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div
class="csl-right-inline">Self, C. N., Knolle, J., Iblisdir, S. &amp;
Pachos, J. K. <a
href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally induced
metallic phase in a gapped quantum spin liquid - a Monte Carlo study of
the Kitaev model with parity projection</a>. <em>Phys. Rev. B</em>
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">C.
N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, <em><a
href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally Induced
Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of
the Kitaev Model with Parity Projection</a></em>, Phys. Rev. B
<strong>99</strong>, 045142 (2019).</div>
</div>
<div id="ref-Beenakker2013" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">17. </div><div
class="csl-right-inline">Beenakker, C. W. J. <a
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">C.
W. J. Beenakker, <em><a
href="https://doi.org/10.1146/annurev-conmatphys-030212-184337">Search
for majorana fermions in superconductors</a>. <em>Annual Review of
Condensed Matter Physics</em> <strong>4</strong>, 113–136 (2013).</div>
for Majorana Fermions in Superconductors</a></em>, Annual Review of
Condensed Matter Physics <strong>4</strong>, 113 (2013).</div>
</div>
<div id="ref-Laumann2012" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">18. </div><div
class="csl-right-inline">Laumann, C. R., Ludwig, A. W. W., Huse, D. A.
&amp; Trebst, S. <a
href="https://doi.org/10.1103/PhysRevB.85.161301">Disorder-induced
Majorana metal in interacting non-Abelian anyon systems</a>. <em>Phys.
Rev. B</em> <strong>85</strong>, 161301 (2012).</div>
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">C.
R. Laumann, A. W. W. Ludwig, D. A. Huse, and S. Trebst, <em><a
href="https://doi.org/10.1103/PhysRevB.85.161301">Disorder-Induced
Majorana Metal in Interacting Non-Abelian Anyon Systems</a></em>, Phys.
Rev. B <strong>85</strong>, 161301 (2012).</div>
</div>
<div id="ref-Lahtinen_2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">19. </div><div
class="csl-right-inline">Lahtinen, V. <a
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">V.
Lahtinen, <em><a
href="https://doi.org/10.1088/1367-2630/13/7/075009">Interacting
non-Abelian anyons as Majorana fermions in the honeycomb lattice
model</a>. <em>New Journal of Physics</em> <strong>13</strong>, 075009
Non-Abelian Anyons as Majorana Fermions in the Honeycomb Lattice
Model</a></em>, New Journal of Physics <strong>13</strong>, 075009
(2011).</div>
</div>
<div id="ref-lahtinenTopologicalLiquidNucleation2012" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">20. </div><div
class="csl-right-inline">Lahtinen, V., Ludwig, A. W. W., Pachos, J. K.
&amp; Trebst, S. <a
href="https://doi.org/10.1103/PhysRevB.86.075115">Topological liquid
nucleation induced by vortex-vortex interactions in Kitaev’s honeycomb
model</a>. <em>Phys. Rev. B</em> <strong>86</strong>, 075115
(2012).</div>
<div class="csl-left-margin">[20] </div><div class="csl-right-inline">V.
Lahtinen, A. W. W. Ludwig, J. K. Pachos, and S. Trebst, <em><a
href="https://doi.org/10.1103/PhysRevB.86.075115">Topological Liquid
Nucleation Induced by Vortex-Vortex Interactions in Kitaev’s Honeycomb
Model</a></em>, Phys. Rev. B <strong>86</strong>, 075115 (2012).</div>
</div>
<div id="ref-bocquet_disordered_2000" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">21. </div><div
class="csl-right-inline">Bocquet, M., Serban, D. &amp; Zirnbauer, M. R.
<a href="https://doi.org/10.1016/S0550-3213(00)00208-X">Disordered 2d
quasiparticles in class D: Dirac fermions with random mass, and dirty
superconductors</a>. <em>Nuclear Physics B</em> <strong>578</strong>,
628–680 (2000).</div>
<div class="csl-left-margin">[21] </div><div class="csl-right-inline">M.
Bocquet, D. Serban, and M. R. Zirnbauer, <em><a
href="https://doi.org/10.1016/S0550-3213(00)00208-X">Disordered 2d
Quasiparticles in Class D: Dirac Fermions with Random Mass, and Dirty
Superconductors</a></em>, Nuclear Physics B <strong>578</strong>, 628
(2000).</div>
</div>
<div id="ref-Weaire1976" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">22. </div><div
class="csl-right-inline">Weaire, D. &amp; Thorpe, M. F. <a
href="https://doi.org/10.1080/00107517608210851">The structure of
amorphous solids</a>. <em>Contemporary Physics</em> <strong>17</strong>,
173–191 (1976).</div>
<div class="csl-left-margin">[22] </div><div class="csl-right-inline">D.
Weaire and M. F. Thorpe, <em><a
href="https://doi.org/10.1080/00107517608210851">The Structure of
Amorphous Solids</a></em>, Contemporary Physics <strong>17</strong>, 173
(1976).</div>
</div>
<div id="ref-Petrakovski1981" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">23. </div><div
class="csl-right-inline">Petrakovskii, G. A. <a
<div class="csl-left-margin">[23] </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</em>
<strong>24</strong>, 511–525 (1981).</div>
Magnetic Materials</a></em>, Soviet Physics Uspekhi <strong>24</strong>,
511 (1981).</div>
</div>
<div id="ref-Kaneyoshi2018" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">24. </div><div
class="csl-right-inline"><em>Amorphous magnetism</em>. (CRC Press,
<div class="csl-left-margin">[24] </div><div class="csl-right-inline">T.
Kaneyoshi, editor, <em>Amorphous Magnetism</em> (CRC Press, Boca Raton,
2018).</div>
</div>
<div id="ref-Weaire1971" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">25. </div><div
class="csl-right-inline">Weaire, D. &amp; Thorpe, M. F. <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</em> <strong>4</strong>, 2508–2520 (1971).</div>
<div class="csl-left-margin">[25] </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">26. </div><div
class="csl-right-inline">Betteridge, G. A possible model of amorphous
silicon and germanium. <em>Journal of Physics C: Solid State
Physics</em> <strong>6</strong>, L427 (1973).</div>
<div class="csl-left-margin">[26] </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-bennett2014amorphous" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">27. </div><div
class="csl-right-inline">Bennett, T. D. &amp; Cheetham, A. K. Amorphous
metal–organic frameworks. <em>Accounts of chemical research</em>
<strong>47</strong>, 1555–1562 (2014).</div>
<div class="csl-left-margin">[27] </div><div class="csl-right-inline">T.
D. Bennett and A. K. Cheetham, <em>Amorphous Metal–Organic
Frameworks</em>, Accounts of Chemical Research <strong>47</strong>, 1555
(2014).</div>
</div>
<div id="ref-yamadaDesigningKitaevSpin2017" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">28. </div><div
class="csl-right-inline">Yamada, M. G., Fujita, H. &amp; Oshikawa, M. <a
<div class="csl-left-margin">[28] </div><div class="csl-right-inline">M.
G. Yamada, H. Fujita, and M. Oshikawa, <em><a
href="https://doi.org/10.1103/PhysRevLett.119.057202">Designing Kitaev
Spin Liquids in Metal-Organic Frameworks</a>. <em>Phys. Rev. Lett.</em>
Spin Liquids in Metal-Organic Frameworks</a></em>, Phys. Rev. Lett.
<strong>119</strong>, 057202 (2017).</div>
</div>
<div id="ref-misumiQuantumSpinLiquid2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">29. </div><div
class="csl-right-inline">Misumi, Y. <em>et al.</em> <a
href="https://doi.org/10.1021/jacs.0c05472">Quantum Spin Liquid State in
a Two-Dimensional Semiconductive Metal–Organic Framework</a>. <em>J. Am.
Chem. Soc.</em> <strong>142</strong>, 16513–16517 (2020).</div>
<div class="csl-left-margin">[29] </div><div class="csl-right-inline">Y.
Misumi, A. Yamaguchi, Z. Zhang, T. Matsushita, N. Wada, M. Tsuchiizu,
and K. Awaga, <em><a href="https://doi.org/10.1021/jacs.0c05472">Quantum
Spin Liquid State in a Two-Dimensional Semiconductive Metal–Organic
Framework</a></em>, J. Am. Chem. Soc. <strong>142</strong>, 16513
(2020).</div>
</div>
<div id="ref-Rau2014" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">30. </div><div
class="csl-right-inline">Rau, J. G., Lee, E. K.-H. &amp; Kee, H.-Y. <a
href="https://doi.org/10.1103/PhysRevLett.112.077204">Generic spin model
for the honeycomb iridates beyond the kitaev limit</a>. <em>Phys. Rev.
Lett.</em> <strong>112</strong>, 077204 (2014).</div>
<div class="csl-left-margin">[30] </div><div class="csl-right-inline">J.
G. Rau, E. K.-H. Lee, and H.-Y. Kee, <em><a
href="https://doi.org/10.1103/PhysRevLett.112.077204">Generic Spin Model
for the Honeycomb Iridates Beyond the Kitaev Limit</a></em>, Phys. Rev.
Lett. <strong>112</strong>, 077204 (2014).</div>
</div>
<div id="ref-Chaloupka2010" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">31. </div><div
class="csl-right-inline">Chaloupka, J., Jackeli, G. &amp; Khaliullin, G.
Kitaev-Heisenberg model on a honeycomb lattice: possible exotic phases
in iridium oxides A₂IrO₃. <em>Phys. Rev. Lett.</em>
<strong>105</strong>, 027204 (2010).</div>
<div class="csl-left-margin">[31] </div><div class="csl-right-inline">J.
Chaloupka, G. Jackeli, and G. Khaliullin, <em>Kitaev-Heisenberg Model on
a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides
A₂IrO₃</em>, Phys. Rev. Lett. <strong>105</strong>, 027204 (2010).</div>
</div>
<div id="ref-Chaloupka2013" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">32. </div><div
class="csl-right-inline">Chaloupka, J., Jackeli, G. &amp; Khaliullin, G.
<a href="https://doi.org/10.1103/PhysRevLett.110.097204">Zigzag magnetic
order in the iridium oxide Na₂IrO₃</a>. <em>Phys. Rev. Lett.</em>
<div class="csl-left-margin">[32] </div><div class="csl-right-inline">J.
Chaloupka, G. Jackeli, and G. Khaliullin, <em><a
href="https://doi.org/10.1103/PhysRevLett.110.097204">Zigzag Magnetic
Order in the Iridium Oxide Na₂IrO₃</a></em>, Phys. Rev. Lett.
<strong>110</strong>, 097204 (2013).</div>
</div>
<div id="ref-Chaloupka2015" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">33. </div><div
class="csl-right-inline">Chaloupka, J. &amp; Khaliullin, G. Hidden
symmetries of the extended Kitaev-Heisenberg model: Implications for
honeycomb lattice iridates A₂IrO₃. <em>Phys. Rev. B</em>
<strong>92</strong>, 024413 (2015).</div>
<div class="csl-left-margin">[33] </div><div class="csl-right-inline">J.
Chaloupka and G. Khaliullin, <em>Hidden Symmetries of the Extended
Kitaev-Heisenberg Model: Implications for Honeycomb Lattice Iridates
A₂IrO₃</em>, Phys. Rev. B <strong>92</strong>, 024413 (2015).</div>
</div>
<div id="ref-Winter2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">34. </div><div
class="csl-right-inline">Winter, S. M., Li, Y., Jeschke, H. O. &amp;
Valentí, R. <a
href="https://doi.org/10.1103/PhysRevB.93.214431">Challenges in design
of Kitaev materials: Magnetic interactions from competing energy
scales</a>. <em>Phys. Rev. B</em> <strong>93</strong>, 214431
(2016).</div>
<div class="csl-left-margin">[34] </div><div class="csl-right-inline">S.
M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, <em><a
href="https://doi.org/10.1103/PhysRevB.93.214431">Challenges in Design
of Kitaev Materials: Magnetic Interactions from Competing Energy
Scales</a></em>, Phys. Rev. B <strong>93</strong>, 214431 (2016).</div>
</div>
<div id="ref-Jackeli2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">35. </div><div
class="csl-right-inline">Jackeli, G. &amp; Khaliullin, G. <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</em>
<div class="csl-left-margin">[35] </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
<strong>102</strong>, 017205 (2009).</div>
</div>
<div id="ref-Kalmeyer1989" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">36. </div><div
class="csl-right-inline">Kalmeyer, V. &amp; Laughlin, R. B. <a
href="https://doi.org/10.1103/PhysRevB.39.11879">Theory of the spin
liquid state of the Heisenberg antiferromagnet</a>. <em>Phys. Rev.
B</em> <strong>39</strong>, 11879–11899 (1989).</div>
<div class="csl-left-margin">[36] </div><div class="csl-right-inline">V.
Kalmeyer and R. B. Laughlin, <em><a
href="https://doi.org/10.1103/PhysRevB.39.11879">Theory of the Spin
Liquid State of the Heisenberg Antiferromagnet</a></em>, Phys. Rev. B
<strong>39</strong>, 11879 (1989).</div>
</div>
<div id="ref-manousakisSpinTextonehalfHeisenberg1991" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">37. </div><div
class="csl-right-inline">Manousakis, E. <a
href="https://doi.org/10.1103/RevModPhys.63.1">The spin-\textonehalf{}
Heisenberg antiferromagnet on a square lattice and its application to
the cuprous oxides</a>. <em>Rev. Mod. Phys.</em> <strong>63</strong>,
1–62 (1991).</div>
<div class="csl-left-margin">[37] </div><div class="csl-right-inline">E.
Manousakis, <em><a href="https://doi.org/10.1103/RevModPhys.63.1">The
Spin-\Textonehalf{} Heisenberg Antiferromagnet on a Square Lattice and
Its Application to the Cuprous Oxides</a></em>, Rev. Mod. Phys.
<strong>63</strong>, 1 (1991).</div>
</div>
<div id="ref-Baskaran2008" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">38. </div><div
class="csl-right-inline">Baskaran, G., Sen, D. &amp; Shankar, R. <a
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev model:
Classical ground states, order from disorder, and exact correlation
functions</a>. <em>Phys. Rev. B</em> <strong>78</strong>, 115116
<div class="csl-left-margin">[38] </div><div class="csl-right-inline">G.
Baskaran, D. Sen, and R. Shankar, <em><a
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev Model:
Classical Ground States, Order from Disorder, and Exact Correlation
Functions</a></em>, Phys. Rev. B <strong>78</strong>, 115116
(2008).</div>
</div>
<div id="ref-Yao2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">39. </div><div
class="csl-right-inline">Yao, H., Zhang, S.-C. &amp; Kivelson, S. A. <a
href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic spin
liquid in an exactly solvable spin model</a>. <em>Phys. Rev. Lett.</em>
<div class="csl-left-margin">[39] </div><div class="csl-right-inline">H.
Yao, S.-C. Zhang, and S. A. Kivelson, <em><a
href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin
Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett.
<strong>102</strong>, 217202 (2009).</div>
</div>
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">40. </div><div
class="csl-right-inline">Nussinov, Z. &amp; Ortiz, G. <a
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond algebras and
exact solvability of Hamiltonians: spin S=½ multilayer systems</a>.
<em>Physical Review B</em> <strong>79</strong>, 214440 (2009).</div>
<div class="csl-left-margin">[40] </div><div class="csl-right-inline">Z.
Nussinov and G. Ortiz, <em><a
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and
Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>,
Physical Review B <strong>79</strong>, 214440 (2009).</div>
</div>
<div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">41. </div><div
class="csl-right-inline">Chua, V., Yao, H. &amp; Fiete, G. A. <a
href="https://doi.org/10.1103/PhysRevB.83.180412">Exact chiral spin
liquid with stable spin Fermi surface on the kagome lattice</a>.
<em>Phys. Rev. B</em> <strong>83</strong>, 180412 (2011).</div>
<div class="csl-left-margin">[41] </div><div class="csl-right-inline">V.
Chua, H. Yao, and G. A. Fiete, <em><a
href="https://doi.org/10.1103/PhysRevB.83.180412">Exact Chiral Spin
Liquid with Stable Spin Fermi Surface on the Kagome Lattice</a></em>,
Phys. Rev. B <strong>83</strong>, 180412 (2011).</div>
</div>
<div id="ref-Natori2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">42. </div><div
class="csl-right-inline">Natori, W. M. H. &amp; Knolle, J. <a
<div class="csl-left-margin">[42] </div><div class="csl-right-inline">W.
M. H. Natori and J. Knolle, <em><a
href="https://doi.org/10.1103/PhysRevLett.125.067201">Dynamics of a
two-dimensional quantum spin-orbital liquid: Spectroscopic signatures of
fermionic magnons</a>. <em>Phys. Rev. Lett.</em> <strong>125</strong>,
Two-Dimensional Quantum Spin-Orbital Liquid: Spectroscopic Signatures of
Fermionic Magnons</a></em>, Phys. Rev. Lett. <strong>125</strong>,
067201 (2020).</div>
</div>
<div id="ref-Chulliparambil2020" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">43. </div><div
class="csl-right-inline">Chulliparambil, S., Seifert, U. F. P., Vojta,
M., Janssen, L. &amp; Tu, H.-H. <a
href="https://doi.org/10.1103/PhysRevB.102.201111">Microscopic models
for Kitaev’s sixteenfold way of anyon theories</a>. <em>Phys. Rev.
B</em> <strong>102</strong>, 201111 (2020).</div>
<div class="csl-left-margin">[43] </div><div class="csl-right-inline">S.
Chulliparambil, U. F. P. Seifert, M. Vojta, L. Janssen, and H.-H. Tu,
<em><a href="https://doi.org/10.1103/PhysRevB.102.201111">Microscopic
Models for Kitaev’s Sixteenfold Way of Anyon Theories</a></em>, Phys.
Rev. B <strong>102</strong>, 201111 (2020).</div>
</div>
<div id="ref-Chulliparambil2021" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">44. </div><div
class="csl-right-inline">Chulliparambil, S., Janssen, L., Vojta, M., Tu,
H.-H. &amp; Seifert, U. F. P. <a
href="https://doi.org/10.1103/PhysRevB.103.075144">Flux crystals,
Majorana metals, and flat bands in exactly solvable spin-orbital
liquids</a>. <em>Phys. Rev. B</em> <strong>103</strong>, 075144
<div class="csl-left-margin">[44] </div><div class="csl-right-inline">S.
Chulliparambil, L. Janssen, M. Vojta, H.-H. Tu, and U. F. P. Seifert,
<em><a href="https://doi.org/10.1103/PhysRevB.103.075144">Flux Crystals,
Majorana Metals, and Flat Bands in Exactly Solvable Spin-Orbital
Liquids</a></em>, Phys. Rev. B <strong>103</strong>, 075144
(2021).</div>
</div>
<div id="ref-Seifert2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">45. </div><div
class="csl-right-inline">Seifert, U. F. P. <em>et al.</em> <a
<div class="csl-left-margin">[45] </div><div class="csl-right-inline">U.
F. P. Seifert, X.-Y. Dong, S. Chulliparambil, M. Vojta, H.-H. Tu, and L.
Janssen, <em><a
href="https://doi.org/10.1103/PhysRevLett.125.257202">Fractionalized
fermionic quantum criticality in spin-orbital mott insulators</a>.
<em>Phys. Rev. Lett.</em> <strong>125</strong>, 257202 (2020).</div>
Fermionic Quantum Criticality in Spin-Orbital Mott Insulators</a></em>,
Phys. Rev. Lett. <strong>125</strong>, 257202 (2020).</div>
</div>
<div id="ref-WangHaoranPRB2021" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">46. </div><div
class="csl-right-inline">Wang, H. &amp; Principi, A. <a
href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana edge and
corner states in square and kagome quantum spin-3/2 liquids</a>.
<em>Phys. Rev. B</em> <strong>104</strong>, 214422 (2021).</div>
<div class="csl-left-margin">[46] </div><div class="csl-right-inline">H.
Wang and A. Principi, <em><a
href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana Edge and
Corner States in Square and Kagome Quantum Spin-3/2 Liquids</a></em>,
Phys. Rev. B <strong>104</strong>, 214422 (2021).</div>
</div>
<div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">47. </div><div class="csl-right-inline">Wu,
C., Arovas, D. &amp; Hung, H.-H. Γ-matrix generalization of the Kitaev
model. <em>Physical Review B</em> <strong>79</strong>, 134427
(2009).</div>
<div class="csl-left-margin">[47] </div><div class="csl-right-inline">C.
Wu, D. Arovas, and H.-H. Hung, <em>Γ-Matrix Generalization of the Kitaev
Model</em>, Physical Review B <strong>79</strong>, 134427 (2009).</div>
</div>
</div>
</main>

BIN
assets/thesis/fk_chapter/FK_phase_diagram_2d.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 57 KiB