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1114 lines
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HTML
1114 lines
52 KiB
HTML
---
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title: 1_Intro
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<!--Capture the table of contents from pandoc as a jekyll variable -->
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{% capture tableOfContents %}
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<br>
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Contents:
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<ul>
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<li><a href="#interacting-quantum-many-body-systems"
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id="toc-interacting-quantum-many-body-systems">Interacting Quantum Many
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Body Systems</a></li>
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<li><a href="#mott-insulators" id="toc-mott-insulators">Mott
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Insulators</a></li>
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<li><a href="#quantum-spin-liquids"
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id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
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<li><a href="#outline" id="toc-outline">Outline</a></li>
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</ul>
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{% endcapture %}
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<main>
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<!-- <nav id="TOC" role="doc-toc">
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<ul>
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<li><a href="#interacting-quantum-many-body-systems"
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id="toc-interacting-quantum-many-body-systems">Interacting Quantum Many
|
||
Body Systems</a></li>
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||
<li><a href="#mott-insulators" id="toc-mott-insulators">Mott
|
||
Insulators</a></li>
|
||
<li><a href="#quantum-spin-liquids"
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||
id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
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<li><a href="#outline" id="toc-outline">Outline</a></li>
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</ul>
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</nav>
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-->
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<h1 id="interacting-quantum-many-body-systems">Interacting Quantum Many
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Body Systems</h1>
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<p>When you take many objects and let them interact together, it is
|
||
often simpler to describe the behaviour of the group differently from
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||
the way one would describe the individual objects. Consider a flock of
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||
starlings like that of fig. <a href="#fig:Studland_Starlings">1</a>.
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||
Watching the flock you’ll see that it has a distinct outline, that waves
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of density will sometimes propagate through the closely packed birds and
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||
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
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||
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>]</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>]</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"
|
||
data-short-caption="A murmuration of Starlings" style="width:100.0%"
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||
alt="Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0" />
|
||
<figcaption aria-hidden="true"><span>Figure 1:</span> A murmuration of
|
||
starlings. Dorset, UK. Credit <a href="twitter.com/arripay">Tanya
|
||
Hart</a>, “Studland Starlings”, 2017, <a
|
||
href="creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA
|
||
3.0</a></figcaption>
|
||
</figure>
|
||
</div>
|
||
<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>]</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>]</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
|
||
objects, those objects being quantum and there are interaction between
|
||
the objects, we call it an interacting quantum many body system. From
|
||
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"> [<a
|
||
href="#ref-ashcroftSolidStatePhysics1976"
|
||
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>]</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>]</span>. In contrast, condensed matter
|
||
theory got it state in quantum many-body theory. Bloch’s theorem <span
|
||
class="citation"
|
||
data-cites="blochÜberQuantenmechanikElektronen1929"> [<a
|
||
href="#ref-blochÜberQuantenmechanikElektronen1929"
|
||
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"> [<a
|
||
href="#ref-MagnetismCondensedMatter"
|
||
role="doc-biblioref">12</a>]</span>.</p>
|
||
<p>However, at some point we had to start on the interacting quantum
|
||
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>]</span>, the fractional quantum hall
|
||
effect <span class="citation"
|
||
data-cites="feldmanFractionalChargeFractional2021"> [<a
|
||
href="#ref-feldmanFractionalChargeFractional2021"
|
||
role="doc-biblioref">14</a>]</span> and the Mott insulators <span
|
||
class="citation"
|
||
data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a
|
||
href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">15</a>,<a
|
||
href="#ref-fisherMottInsulatorsSpin1999"
|
||
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>
|
||
<h1 id="mott-insulators">Mott Insulators</h1>
|
||
<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, 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"
|
||
data-short-caption="Interacting Quantum Many Body Systems Venn Diagram"
|
||
style="width:57.0%"
|
||
alt="Figure 2: Three key adjectives. Many Body, the fact of describing systems in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials." />
|
||
<figcaption aria-hidden="true"><span>Figure 2:</span> Three key
|
||
adjectives. Many Body, the fact of describing systems in the limit of
|
||
large numbers of particles. Quantum, objects whose behaviour requires
|
||
quantum mechanics to describe accurately. Interacting, the constituent
|
||
particles of the system affect one another via forces, either directly
|
||
or indirectly. When taken together, these three properties can give rise
|
||
to what are called strongly correlated materials.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<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"> [<a
|
||
href="#ref-boerSemiconductorsPartiallyCompletely1937"
|
||
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>]</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>]</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>]</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>]</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">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j
|
||
\rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow}
|
||
n_{i\downarrow} - \mu \sum_{i,\alpha} n_{i\alpha}\]</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>. 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">\(U <<
|
||
t\)</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">\(U >>
|
||
t\)</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"> [<a
|
||
href="#ref-hubbardElectronCorrelationsNarrow1964"
|
||
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. 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>]</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"> [<a
|
||
href="#ref-slaterMagneticEffectsHartreeFock1951"
|
||
role="doc-biblioref">28</a>]</span> and dynamical mean-field
|
||
theory <span class="citation"
|
||
data-cites="greinerQuantumPhaseTransition2002"> [<a
|
||
href="#ref-greinerQuantumPhaseTransition2002"
|
||
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"> [<a
|
||
href="#ref-mazurenkoColdatomFermiHubbard2017"
|
||
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 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">\[\begin{aligned}
|
||
H_{\mathrm{FK}} = & -\;t \sum_{\langle i,j \rangle}
|
||
c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
|
||
\tfrac{1}{2}). \\
|
||
\end{aligned}\]</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">\(T > 0\)</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>
|
||
<h1 id="quantum-spin-liquids">Quantum Spin Liquids</h1>
|
||
<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">27</a>]</span>, giving rise to what is known as a
|
||
quantum spin liquid (QSL) state.</p>
|
||
<p>Landau theory characterises phases of matter as inextricably linked
|
||
to the emergence of long range order via a spontaneously broken
|
||
symmetry. The fractional quantum Hall (FQH) state, discovered in the
|
||
1980s is an explicit example of an electronic system that falls outside
|
||
of the Landau paradigm. FQH systems exhibit fractionalised excitations
|
||
linked to their ground state having long range entanglement and
|
||
non-trivial topological properties <span class="citation"
|
||
data-cites="broholmQuantumSpinLiquids2020"> [<a
|
||
href="#ref-broholmQuantumSpinLiquids2020"
|
||
role="doc-biblioref">40</a>]</span>. Quantum spin liquids are the
|
||
analogous phase of matter for spin systems. Remarkably the existence of
|
||
QSLs was first suggested by Anderson in 1973 <span class="citation"
|
||
data-cites="andersonResonatingValenceBonds1973"> [<a
|
||
href="#ref-andersonResonatingValenceBonds1973"
|
||
role="doc-biblioref">41</a>]</span>.</p>
|
||
<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: From [42]." />
|
||
<figcaption aria-hidden="true"><span>Figure 3:</span> From <span
|
||
class="citation" data-cites="TrebstPhysRep2022"> [<a
|
||
href="#ref-TrebstPhysRep2022"
|
||
role="doc-biblioref">42</a>]</span>.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<p>The main route to QSLs, though there are others <span
|
||
class="citation"
|
||
data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a
|
||
href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">43</a>–<a
|
||
href="#ref-linExactSymmetryWeaklyinteracting1998"
|
||
role="doc-biblioref">45</a>]</span>, is via frustration of spin models
|
||
that would otherwise order have AFM order. This frustration can come
|
||
geometrically, triangular lattices for instance cannot support AFM
|
||
order. It can also come about as a result of spin-orbit coupling.</p>
|
||
<p>Electron spin naturally couples to magnetic fields. Spin-orbit
|
||
coupling is a relativistic effect, that very roughly corresponds to the
|
||
fact that in the frame of reference of a moving electron, the electric
|
||
field of nearby nuclei look like magnetic field to which the electron
|
||
spin couples. In certain transition metal based compounds, such as those
|
||
based on Iridium and Rutheniun, crystal field effects, strong spin-orbit
|
||
coupling and narrow bandwidths lead to effective spin-<span
|
||
class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with
|
||
strongly anisotropic spin-spin couplings known as Kitaev Materials <span
|
||
class="citation"
|
||
data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a
|
||
href="#ref-TrebstPhysRep2022" role="doc-biblioref">42</a>,<a
|
||
href="#ref-Jackeli2009" role="doc-biblioref">46</a>–<a
|
||
href="#ref-Takagi2019" role="doc-biblioref">49</a>]</span>. Kitaev
|
||
materials draw their name from the celebrated Kitaev Honeycomb Model as
|
||
it is believed they will realise the QSL state via the mechanisms of the
|
||
Kitaev Model.</p>
|
||
<p>The Kitaev Honeycomb model <span class="citation"
|
||
data-cites="kitaevAnyonsExactlySolved2006"> [<a
|
||
href="#ref-kitaevAnyonsExactlySolved2006"
|
||
role="doc-biblioref">50</a>]</span> was the first concrete model with a
|
||
QSL ground state. It is defined on the honeycomb lattice and provides an
|
||
exactly solvable model whose ground state is a QSL characterized by a
|
||
static <span class="math inline">\(\mathbb Z_2\)</span> gauge field and
|
||
Majorana fermion excitations. It can be reduced to a free fermion
|
||
problem via a mapping to Majorana fermions which yields an extensive
|
||
number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes
|
||
tied to an emergent gauge field. The model is remarkable not only for
|
||
its QSL ground state, it supports a rich phase diagram hosting gapless,
|
||
Abelian and non-Abelian phases and a finite temperature phase transition
|
||
to a thermal metal state <span class="citation"
|
||
data-cites="selfThermallyInducedMetallic2019"> [<a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">51</a>]</span>. It has also been proposed that it
|
||
could be used to support topological quantum computing <span
|
||
class="citation"
|
||
data-cites="freedmanTopologicalQuantumComputation2003"> [<a
|
||
href="#ref-freedmanTopologicalQuantumComputation2003"
|
||
role="doc-biblioref">52</a>]</span>.</p>
|
||
<p>It is by now understood that the Kitaev model on any tri-coordinated
|
||
<span class="math inline">\(z=3\)</span> graph has conserved plaquette
|
||
operators and local symmetries <span class="citation"
|
||
data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007"
|
||
role="doc-biblioref">53</a>,<a href="#ref-Baskaran2008"
|
||
role="doc-biblioref">54</a>]</span> which allow a mapping onto effective
|
||
free Majorana fermion problems in a background of static <span
|
||
class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation"
|
||
data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a
|
||
href="#ref-Nussinov2009" role="doc-biblioref">55</a>–<a
|
||
href="#ref-hermanns2015weyl" role="doc-biblioref">58</a>]</span>.
|
||
However, depending on lattice symmetries, finding the ground state flux
|
||
sector and understanding the QSL properties can still be
|
||
challenging <span class="citation"
|
||
data-cites="eschmann2019thermodynamics Peri2020"> [<a
|
||
href="#ref-eschmann2019thermodynamics" role="doc-biblioref">59</a>,<a
|
||
href="#ref-Peri2020" role="doc-biblioref">60</a>]</span>.</p>
|
||
<p><strong>paragraph about amorphous lattices</strong></p>
|
||
<p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin
|
||
liquid by extending the Kitaev honeycomb model to random lattices with
|
||
fixed coordination number three. The model retains its exact solubility
|
||
but the presence of plaquettes with an odd number of sides leads to a
|
||
spontaneous breaking of time reversal symmetry. I unearth a rich phase
|
||
diagram displaying Abelian as well as a non-Abelian quantum spin liquid
|
||
phases with a remarkably simple ground state flux pattern. Furthermore,
|
||
I show that the system undergoes a finite-temperature phase transition
|
||
to a conducting thermal metal state and discuss possible experimental
|
||
realisations.</p>
|
||
<h1 id="outline">Outline</h1>
|
||
<p>The next chapter, Chapter 2, will introduce some necessary background
|
||
to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and
|
||
localisation.</p>
|
||
<p>In Chapter 3 I introduce the Long Range Falikov-Kimball Model in
|
||
greater detail. I will present results that. Chapter 4 focusses on the
|
||
Amorphous Kitaev Model.</p>
|
||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||
<div id="ref-king2012murmurations" class="csl-entry"
|
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J. King and D. J. Sumpter, <em>Murmurations</em>, Current Biology
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|
||
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|
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Collective Behavior Depends on Topological Rather Than Metric Distance:
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Evidence from a Field Study</a></em>, Proceedings of the National
|
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Academy of Sciences <strong>105</strong>, 1232 (2008).</div>
|
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</div>
|
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<div id="ref-andersonMoreDifferent1972" class="csl-entry"
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<div class="csl-left-margin">[3] </div><div class="csl-right-inline">P.
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W. Anderson, <em><a
|
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href="https://doi.org/10.1126/science.177.4047.393">More Is
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Different</a></em>, Science <strong>177</strong>, 393 (1972).</div>
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Kivelson and S. A. Kivelson, <em><a
|
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M. Saslow, <em><a href="https://doi.org/10.3390/e22010077">A History of
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Thermodynamics: The Missing Manual</a></em>, Entropy (Basel)
|
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<strong>22</strong>, 77 (2020).</div>
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||
Flamm, <em><a
|
||
href="https://doi.org/10.48550/arXiv.physics/9803005">History and
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||
Outlook of Statistical Physics</a></em>, arXiv:physics/9803005.</div>
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<div id="ref-ashcroftSolidStatePhysics1976" class="csl-entry"
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<div class="csl-left-margin">[7] </div><div class="csl-right-inline">N.
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||
W. Ashcroft and N. D. Mermin, <em>Solid State Physics</em> (Holt,
|
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Rinehart and Winston, 1976).</div>
|
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</div>
|
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<div id="ref-isingBeitragZurTheorie1925" class="csl-entry"
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<div class="csl-left-margin">[8] </div><div class="csl-right-inline">E.
|
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Ising, <em><a href="https://doi.org/10.1007/BF02980577">Beitrag zur
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Theorie des Ferromagnetismus</a></em>, Z. Physik <strong>31</strong>,
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||
253 (1925).</div>
|
||
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|
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<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,
|
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2013).</div>
|
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</div>
|
||
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<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>
|
||
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|
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|
||
Bloch, <em><a href="https://doi.org/10.1007/BF01339455">Über die
|
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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">S.
|
||
Blundell, <em>Magnetism in Condensed Matter</em> (OUP Oxford,
|
||
2001).</div>
|
||
</div>
|
||
<div id="ref-MicroscopicTheorySuperconductivity" class="csl-entry"
|
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|
||
<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
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(1957).</div>
|
||
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|
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|
||
<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">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
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||
(1949).</div>
|
||
</div>
|
||
<div id="ref-fisherMottInsulatorsSpin1999" class="csl-entry"
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||
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|
||
<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
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||
En Basse Dimension. Topological Aspects of Low Dimensional Systems</em>,
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||
edited by A. Comtet, T. Jolicœur, S. Ouvry, and F. David, Vol. 69
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||
(Springer Berlin Heidelberg, Berlin, Heidelberg, 1999), pp.
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||
575–641.</div>
|
||
</div>
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