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1322 lines
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---
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title: Introduction
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excerpt: Why do we do Condensed Matter theory at all?
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layout: none
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image:
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---
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<!DOCTYPE html>
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<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
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<meta name="description" content="Why do we do Condensed Matter theory at all?" />
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<title>Introduction</title>
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<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
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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="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<!-- Table of Contents -->
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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
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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="#bibliography" id="toc-bibliography">Bibliography</a></li>
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-->
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<!-- Main Page Body -->
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<div id="page-header">
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<p>1 Introduction</p>
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<hr />
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</div>
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<section id="interacting-quantum-many-body-systems" class="level1">
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<h1>Interacting Quantum Many Body Systems</h1>
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<p>When you take many objects and let them interact together, it is
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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
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natural description of this phenomena is couched in terms of the flock
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rather than of the individual birds.</p>
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<p>The behaviours of the flock are an <em>emergent phenomena</em>. The
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starlings are only interacting with their immediate six or seven
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neighbours <span class="citation"
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||
data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"> [<a
|
||
href="#ref-king2012murmurations" role="doc-biblioref">1</a>,<a
|
||
href="#ref-balleriniInteractionRulingAnimal2008"
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||
role="doc-biblioref">2</a>]</span>, what a physicist would call a
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||
<em>local interaction</em>. There is much philosophical debate about how
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||
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
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||
that emergence is the fact that the aggregate behaviour of many
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||
interacting objects may necessitate a description very different from
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||
that of the individual objects.</p>
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<div id="fig:Studland_Starlings" class="fignos">
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||
<figure>
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||
<img src="/assets/thesis/intro_chapter/Studland_Starlings.jpeg"
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||
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
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starlings. Dorset, UK. Credit <a
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||
href="https://twitter.com/arripay">Tanya Hart</a>, “Studland Starlings”,
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||
2017, <a
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||
href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA
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||
3.0</a></figcaption>
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</figure>
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</div>
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<p>To give an example closer to the topic at hand, our understanding of
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||
thermodynamics began with bulk properties like heat, energy, pressure
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||
and temperature <span class="citation"
|
||
data-cites="saslowHistoryThermodynamicsMissing2020"> [<a
|
||
href="#ref-saslowHistoryThermodynamicsMissing2020"
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||
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
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||
class="citation" data-cites="flammHistoryOutlookStatistical1998"> [<a
|
||
href="#ref-flammHistoryOutlookStatistical1998"
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||
role="doc-biblioref">6</a>]</span>.</p>
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||
<p>Condensed Matter is, at its heart, the study of what behaviours
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||
emerge from large numbers of interacting quantum objects at low energy.
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||
When these three properties are present together: a large number of
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||
objects, those objects being quantum and there are interaction between
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||
the objects, we call it an interacting quantum many body system. From
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||
these three ingredients nature builds all manner of weird and wonderful
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||
materials.</p>
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||
<p>Historically, we made initial headway in the study of many-body
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||
systems, ignoring interactions and quantum properties. The ideal gas law
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||
and the Drude classical electron gas <span class="citation"
|
||
data-cites="ashcroftSolidStatePhysics1976"> [<a
|
||
href="#ref-ashcroftSolidStatePhysics1976"
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||
role="doc-biblioref">7</a>]</span> are good examples. Including
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||
interactions into many-body physics leads to the Ising model <span
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||
class="citation" data-cites="isingBeitragZurTheorie1925"> [<a
|
||
href="#ref-isingBeitragZurTheorie1925"
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||
role="doc-biblioref">8</a>]</span>, Landau theory <span class="citation"
|
||
data-cites="landau2013fluid"> [<a href="#ref-landau2013fluid"
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||
role="doc-biblioref">9</a>]</span> and the classical theory of phase
|
||
transitions <span class="citation"
|
||
data-cites="jaegerEhrenfestClassificationPhase1998"> [<a
|
||
href="#ref-jaegerEhrenfestClassificationPhase1998"
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||
role="doc-biblioref">10</a>]</span>. In contrast, condensed matter
|
||
theory got it state in quantum many-body theory. Bloch’s theorem <span
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||
class="citation"
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||
data-cites="blochÜberQuantenmechanikElektronen1929"> [<a
|
||
href="#ref-blochÜberQuantenmechanikElektronen1929"
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||
role="doc-biblioref">11</a>]</span> predicted the properties of
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||
non-interacting electrons in crystal lattices, leading to band theory.
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||
In the same vein, advances were made in understanding the quantum
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||
origins of magnetism, including ferromagnetism and
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||
antiferromagnetism <span class="citation"
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||
data-cites="MagnetismCondensedMatter"> [<a
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||
href="#ref-MagnetismCondensedMatter"
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||
role="doc-biblioref">12</a>]</span>.</p>
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||
<p>The development of Landau-Fermi Liquid theory explained why band
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||
theory works so well even in cases where an analysis of the relevant
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||
energies suggests that it should not <span class="citation"
|
||
data-cites="wenQuantumFieldTheory2007"> [<a
|
||
href="#ref-wenQuantumFieldTheory2007"
|
||
role="doc-biblioref">13</a>]</span>. Landau Fermi Liquid theory
|
||
demonstrates that in many cases where electron-electron interactions are
|
||
significant, the system can still be described in terms on generalised
|
||
non-interacting quasiparticles.</p>
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||
<p>However there are systems where even Landau Fermi Liquid theory
|
||
fails. An effective theoretical description of these systems must
|
||
include electron-electron correlations and they are thus called Strongly
|
||
Correlated Materials <span class="citation"
|
||
data-cites="morosanStronglyCorrelatedMaterials2012"> [<a
|
||
href="#ref-morosanStronglyCorrelatedMaterials2012"
|
||
role="doc-biblioref">14</a>]</span>, Correlated Electron systems or
|
||
Quantum Materials. The canonical examples are superconductivity <span
|
||
class="citation" data-cites="MicroscopicTheorySuperconductivity"> [<a
|
||
href="#ref-MicroscopicTheorySuperconductivity"
|
||
role="doc-biblioref">15</a>]</span>, the fractional quantum hall
|
||
effect <span class="citation"
|
||
data-cites="feldmanFractionalChargeFractional2021"> [<a
|
||
href="#ref-feldmanFractionalChargeFractional2021"
|
||
role="doc-biblioref">16</a>]</span> and the Mott insulators <span
|
||
class="citation"
|
||
data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a
|
||
href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">17</a>,<a
|
||
href="#ref-fisherMottInsulatorsSpin1999"
|
||
role="doc-biblioref">18</a>]</span>. We’ll start by looking at the
|
||
latter but shall see that there are many links between three topics.</p>
|
||
</section>
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||
<section id="mott-insulators" class="level1">
|
||
<h1>Mott Insulators</h1>
|
||
<p>Mott Insulators are remarkable because their electrical insulator
|
||
properties come from electron-electron interactions. Electrical
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||
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
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||
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:100.0%"
|
||
alt="Figure 2: Three key adjectives. Many Body, the fact of describing systems in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials." />
|
||
<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">19</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">20</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">21</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">22</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">23</a>–<a
|
||
href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
|
||
role="doc-biblioref">25</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">26</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">27</a>]</span>. However, Mott insulators have been
|
||
found <span class="citation"
|
||
data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
|
||
href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">28</a>,<a
|
||
href="#ref-ribakGaplessExcitationsGround2017"
|
||
role="doc-biblioref">29</a>]</span> 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">30</a>]</span> and dynamical mean-field
|
||
theory <span class="citation"
|
||
data-cites="greinerQuantumPhaseTransition2002"> [<a
|
||
href="#ref-greinerQuantumPhaseTransition2002"
|
||
role="doc-biblioref">31</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">32</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">33</a>,<a
|
||
href="#ref-baskoMetalInsulatorTransition2006"
|
||
role="doc-biblioref">34</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">35</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">36</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">37</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">38</a>–<a
|
||
href="#ref-herrmannNonequilibriumDynamicalCluster2016"
|
||
role="doc-biblioref">41</a>]</span>.</p>
|
||
<p>In Chapter 3 I will introduce a generalized Falikov-Kimball model in
|
||
one dimension I call the Long-Range Falikov-Kimball model. With the
|
||
addition of long-range interactions in the background field, the model
|
||
shows a similarly rich phase diagram its higher dimensional cousins. 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>
|
||
</section>
|
||
<section id="quantum-spin-liquids" class="level1">
|
||
<h1>Quantum Spin Liquids</h1>
|
||
<p>To turn to the other key topic of this thesis, we have already
|
||
discussed the AFM ordering of local moments in the Mott insulating
|
||
state. Landau-Ginzburg-Wilson theory characterises phases of matter as
|
||
inextricably linked to the emergence of long range order via a
|
||
spontaneously broken symmetry. So within this paradigm we would not
|
||
expect any interesting phases of matter not associated with AFM or other
|
||
long-range order. However, Anderson first proposed in 1973 <span
|
||
class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a
|
||
href="#ref-andersonResonatingValenceBonds1973"
|
||
role="doc-biblioref">42</a>]</span> that if long range order is
|
||
suppressed by some mechanism, it might lead to a liquid-like state even
|
||
at zero temperature, the Quantum Spin Liquid (QSL).</p>
|
||
<p>This QSL state would exist at zero or very low temperatures, so we
|
||
would expect quantum effects to be very strong, which will turn out to
|
||
have far reaching consequences. It was the discovery of a different
|
||
phase, however that really kickstarted interest in the topic. The
|
||
fractional quantum Hall (FQH) state, discovered in the 1980s is an
|
||
explicit example of an interacting electron system that falls outside of
|
||
the Landau-Ginzburg-Wilson paradigm. It shares many phenomenological
|
||
properties with the QSL state. They both exhibit fractionalised
|
||
excitations, braiding statistics and non-trivial topological
|
||
properties <span class="citation"
|
||
data-cites="broholmQuantumSpinLiquids2020"> [<a
|
||
href="#ref-broholmQuantumSpinLiquids2020"
|
||
role="doc-biblioref">43</a>]</span>. The many-body ground state of such
|
||
systems acts as a complex and highly entangled vacuum. This vacuum can
|
||
support quasiparticle excitations with properties unbound from that of
|
||
the Dirac fermions of the standard model.</p>
|
||
<p>How do we actually make a QSL? Frustration is one mechanism that we
|
||
can use to suppress magnetic order in spin models <span class="citation"
|
||
data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022"
|
||
role="doc-biblioref">44</a>]</span>. Frustration can be geometric,
|
||
triangular lattices for instance cannot support AFM order. It can also
|
||
come about as a result of spin-orbit coupling or other physics. There
|
||
are also other routes to QSLs besides frustrated spin systems that we
|
||
will not discuss here <span class="citation"
|
||
data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a
|
||
href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">45</a>–<a
|
||
href="#ref-linExactSymmetryWeaklyinteracting1998"
|
||
role="doc-biblioref">47</a>]</span>.</p>
|
||
<!-- Experimentally, Mott insulating systems without magnetic order have been proposed as QSL systems\ [@law1TTaS2QuantumSpin2017; @ribakGaplessExcitationsGround2017]. -->
|
||
<!-- Other exampels: Quantum spin liquids are the analogous phase of matter for spin systems. Spin ice support deconfined magnetic monopoles. -->
|
||
<div id="fig:correlation_spin_orbit_PT" class="fignos">
|
||
<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 [44]." />
|
||
<figcaption aria-hidden="true"><span>Figure 3:</span> From <span
|
||
class="citation" data-cites="TrebstPhysRep2022"> [<a
|
||
href="#ref-TrebstPhysRep2022"
|
||
role="doc-biblioref">44</a>]</span>.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<p>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 fields
|
||
to which the electron spin couples. This effectively couples the spatial
|
||
and spin parts of the electron wavefunction, meaning that the lattice
|
||
structure can influence the form of the spin-spin interactions leading
|
||
to spatial anisotropy. This anisotropy will be how we frustrate the Mott
|
||
insulators <span class="citation"
|
||
data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a
|
||
href="#ref-jackeliMottInsulatorsStrong2009"
|
||
role="doc-biblioref">48</a>,<a
|
||
href="#ref-khaliullinOrbitalOrderFluctuations2005"
|
||
role="doc-biblioref">49</a>]</span>. As we saw with the Hubbard model,
|
||
interaction effects are only strong or weak in comparison to the
|
||
bandwidth or hopping integral <span class="math inline">\(t\)</span> so
|
||
what we need to see strong frustration is a material with strong
|
||
spin-orbit coupling <span class="math inline">\(\lambda\)</span>
|
||
relative to its bandwidth <span class="math inline">\(t\)</span>.</p>
|
||
<p>In certain transition metal based compounds, such as those based on
|
||
Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling
|
||
and narrow bandwidths lead to effective spin-<span
|
||
class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with
|
||
strongly anisotropic spin-spin couplings. These transition metal
|
||
compounds, known Kitaev Materials, draw their name from the celebrated
|
||
Kitaev Honeycomb Model which is expected to model their low temperature
|
||
behaviour <span class="citation"
|
||
data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a
|
||
href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>,<a
|
||
href="#ref-Jackeli2009" role="doc-biblioref">50</a>–<a
|
||
href="#ref-Takagi2019" role="doc-biblioref">53</a>]</span>.</p>
|
||
<p>At this point we can sketch out a phase diagram like that of fig. <a
|
||
href="#fig:correlation_spin_orbit_PT">3</a>. When both electron-electron
|
||
interactions <span class="math inline">\(U\)</span> and spin-orbit
|
||
couplings <span class="math inline">\(\lambda\)</span> are small
|
||
relative to the bandwidth <span class="math inline">\(t\)</span> we
|
||
recover standard band theory of band insulators and metals. In the upper
|
||
left we have the simple Mott insulating state as described by the
|
||
Hubbard model. In the lower right, strong spin-orbit coupling gives rise
|
||
to Topological insulators (TIs) characterised by symmetry protected edge
|
||
modes and non-zero Chern number. Kitaev materials occur in the region
|
||
where strong electron-electron interaction and spin-orbit coupling
|
||
interact. See <span class="citation"
|
||
data-cites="witczak-krempaCorrelatedQuantumPhenomena2014"> [<a
|
||
href="#ref-witczak-krempaCorrelatedQuantumPhenomena2014"
|
||
role="doc-biblioref">54</a>]</span> for a much more expansive version of
|
||
this diagram.</p>
|
||
<p>The Kitaev Honeycomb model <span class="citation"
|
||
data-cites="kitaevAnyonsExactlySolved2006"> [<a
|
||
href="#ref-kitaevAnyonsExactlySolved2006"
|
||
role="doc-biblioref">55</a>]</span> was the first concrete spin model
|
||
with a QSL ground state. It is defined on the two dimensional honeycomb
|
||
lattice and provides an exactly solvable model that can be reduced to a
|
||
free fermion problem via a mapping to Majorana fermions. This 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 but also for its
|
||
fractionalised excitations with non-trivial braiding statistics. It has
|
||
a rich phase diagram hosting gapless, Abelian and non-Abelian
|
||
phases <span class="citation"
|
||
data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a
|
||
href="#ref-knolleDynamicsFractionalizationQuantum2015"
|
||
role="doc-biblioref">56</a>]</span> and a finite temperature phase
|
||
transition to a thermal metal state <span class="citation"
|
||
data-cites="selfThermallyInducedMetallic2019"> [<a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">57</a>]</span>. It been proposed that its
|
||
non-Abelian excitations could be used to support robust topological
|
||
quantum computing <span class="citation"
|
||
data-cites="kitaev_fault-tolerant_2003 freedmanTopologicalQuantumComputation2003 nayakNonAbelianAnyonsTopological2008"> [<a
|
||
href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">58</a>–<a
|
||
href="#ref-nayakNonAbelianAnyonsTopological2008"
|
||
role="doc-biblioref">60</a>]</span>.</p>
|
||
<p>As Kitaev points out in his original paper, the model remains
|
||
solvable on any tri-coordinated <span class="math inline">\(z=3\)</span>
|
||
graph which can be 3-edge-coloured. Indeed many generalisations of the
|
||
model to <span class="citation"
|
||
data-cites="Baskaran2007 Baskaran2008 Nussinov2009 OBrienPRB2016 hermanns2015weyl"> [<a
|
||
href="#ref-Baskaran2007" role="doc-biblioref">61</a>–<a
|
||
href="#ref-hermanns2015weyl" role="doc-biblioref">65</a>]</span>.
|
||
Notably, the Yao-Kivelson model <span class="citation"
|
||
data-cites="yaoExactChiralSpin2007"> [<a
|
||
href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">66</a>]</span>
|
||
introduces triangular plaquettes to the honeycomb lattice leading to
|
||
spontaneous chiral symmetry breaking. These extensions all retain
|
||
translation symmetry, likely because edge-colouring and finding the
|
||
ground state become much harder without it. 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">67</a>,<a
|
||
href="#ref-Peri2020" role="doc-biblioref">68</a>]</span>. Undeterred,
|
||
this gap lead us to wonder what might happen if we remove translation
|
||
symmetry from the Kitaev Model. This might would be a model of a
|
||
tri-coordinated, highly bond anisotropic but otherwise amorphous
|
||
material.</p>
|
||
<p>Amorphous materials do no have long-range lattice regularities but
|
||
covalent compounds can induce short-range regularities in the lattice
|
||
structure such as fixed coordination number <span
|
||
class="math inline">\(z\)</span>. The best examples being amorphous
|
||
Silicon and Germanium with <span class="math inline">\(z=4\)</span>
|
||
which are used to make thin-film solar cells <span class="citation"
|
||
data-cites="Weaire1971 betteridge1973possible"> [<a
|
||
href="#ref-Weaire1971" role="doc-biblioref">69</a>,<a
|
||
href="#ref-betteridge1973possible" role="doc-biblioref">70</a>]</span>.
|
||
Recently is has been shown that topological insulating (TI) phases can
|
||
exist in amorphous systems. Amorphous TIs are characterized by similar
|
||
protected edge states to their translation invariant cousins and
|
||
generalised topological bulk invariants <span class="citation"
|
||
data-cites="mitchellAmorphousTopologicalInsulators2018 agarwala2019topological marsalTopologicalWeaireThorpeModels2020 costa2019toward agarwala2020higher spring2021amorphous corbae2019evidence"> [<a
|
||
href="#ref-mitchellAmorphousTopologicalInsulators2018"
|
||
role="doc-biblioref">71</a>–<a href="#ref-corbae2019evidence"
|
||
role="doc-biblioref">77</a>]</span>. However, research on amorphous
|
||
electronic systems has been mostly focused on non-interacting systems
|
||
with a few exceptions, for example, to account for the observation of
|
||
superconductivity <span class="citation"
|
||
data-cites="buckel1954einfluss mcmillan1981electron meisel1981eliashberg bergmann1976amorphous mannaNoncrystallineTopologicalSuperconductors2022"> [<a
|
||
href="#ref-buckel1954einfluss" role="doc-biblioref">78</a>–<a
|
||
href="#ref-mannaNoncrystallineTopologicalSuperconductors2022"
|
||
role="doc-biblioref">82</a>]</span> in amorphous materials or very
|
||
recently to understand the effect of strong electron repulsion in
|
||
TIs <span class="citation" data-cites="kim2022fractionalization"> [<a
|
||
href="#ref-kim2022fractionalization"
|
||
role="doc-biblioref">83</a>]</span>.</p>
|
||
<p>Amorphous <em>magnetic</em> systems has been investigated since the
|
||
1960s, mostly through the adaptation of theoretical tools developed for
|
||
disordered systems <span class="citation"
|
||
data-cites="aharony1975critical Petrakovski1981 kaneyoshi1992introduction Kaneyoshi2018"> [<a
|
||
href="#ref-aharony1975critical" role="doc-biblioref">84</a>–<a
|
||
href="#ref-Kaneyoshi2018" role="doc-biblioref">87</a>]</span> and with
|
||
numerical methods <span class="citation"
|
||
data-cites="fahnle1984monte plascak2000ising"> [<a
|
||
href="#ref-fahnle1984monte" role="doc-biblioref">88</a>,<a
|
||
href="#ref-plascak2000ising" role="doc-biblioref">89</a>]</span>.
|
||
Research on classical Heisenberg and Ising models has been shown to
|
||
account for observed behaviour of ferromagnetism, disordered
|
||
antiferromagnetism and widely observed spin glass behaviour <span
|
||
class="citation" data-cites="coey1978amorphous"> [<a
|
||
href="#ref-coey1978amorphous" role="doc-biblioref">90</a>]</span>.
|
||
However, the role of spin-anisotropic interactions and quantum effects
|
||
in amorphous magnets has not been addressed. It is an open question
|
||
whether frustrated magnetic interactions on amorphous lattices can give
|
||
rise genuine quantum phases, i.e. to long-range entangled quantum spin
|
||
liquids (QSL) <span class="citation"
|
||
data-cites="Anderson1973 Knolle2019 Savary2016 Lacroix2011"> [<a
|
||
href="#ref-Anderson1973" role="doc-biblioref">91</a>–<a
|
||
href="#ref-Lacroix2011" role="doc-biblioref">94</a>]</span>.</p>
|
||
<p>In Chapter 4 I will introduce the Amorphous Kitaev model, a
|
||
generalisation of the Kitaev honeycomb model to random lattices with
|
||
fixed coordination number three. We will show that this model is a
|
||
soluble chiral amorphous quantum spin liquid. The model retains its
|
||
exact solubility but, as with the Yao-Kivelson model <span
|
||
class="citation" data-cites="yaoExactChiralSpin2007"> [<a
|
||
href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">66</a>]</span>,
|
||
the presence of plaquettes with an odd number of sides leads to a
|
||
spontaneous breaking of time reversal symmetry. We will confirm prior
|
||
observations that the form of the ground state can be written in terms
|
||
of the number of sides of elementary plaquettes of the model <span
|
||
class="citation"
|
||
data-cites="OBrienPRB2016 eschmannThermodynamicClassificationThreedimensional2020"> [<a
|
||
href="#ref-OBrienPRB2016" role="doc-biblioref">64</a>,<a
|
||
href="#ref-eschmannThermodynamicClassificationThreedimensional2020"
|
||
role="doc-biblioref">95</a>]</span>. We unearth a rich phase diagram
|
||
displaying Abelian as well as a non-Abelian chiral spin liquid phases.
|
||
Furthermore, I show that the system undergoes a finite-temperature phase
|
||
transition to a conducting thermal metal state and discuss possible
|
||
experimental realisations.</p>
|
||
<p>The next chapter, Chapter 2, will introduce some necessary background
|
||
to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and
|
||
localisation. Then Chapter 3 introduces and studies the Long Range
|
||
Falikov-Kimball Model in one dimension while Chapter 4 focusses on the
|
||
Amorphous Kitaev Model.</p>
|
||
<p>Next Chapter: <a
|
||
href="../2_Background/2.1_FK_Model.html#the-falikov-kimball-model">2
|
||
Background</a></p>
|
||
</section>
|
||
<section id="bibliography" class="level1 unnumbered">
|
||
<h1 class="unnumbered">Bibliography</h1>
|
||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
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Ising, <em><a href="https://doi.org/10.1007/BF02980577">Beitrag zur
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||
D. Landau and E. M. Lifshitz, <em>Fluid Mechanics: Landau and Lifshitz:
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||
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||
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||
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|
||
Jaeger, <em><a href="https://doi.org/10.1007/s004070050021">The
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||
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||
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(1998).</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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|
||
<strong>52</strong>, 555 (1929).</div>
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||
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|
||
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||
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|
||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">S.
|
||
Blundell, <em>Magnetism in Condensed Matter</em> (OUP Oxford,
|
||
2001).</div>
|
||
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|
||
<div id="ref-wenQuantumFieldTheory2007" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
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