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_thesis/0.1_Intro.html
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@ -250,6 +250,15 @@ image:
<nav id="TOC" role="doc-toc">
<ul>
<li><a href="#themes" id="toc-themes">Themes</a></li>
<li><a href="#condsened-matter-systems"
id="toc-condsened-matter-systems">Condsened Matter Systems</a>
<ul>
<li><a href="#spin-orbit-coupling"
id="toc-spin-orbit-coupling">Spin-Orbit Coupling</a></li>
<li><a href="#electronic-correlations-the-hubbard-model"
id="toc-electronic-correlations-the-hubbard-model">Electronic
correlations: The Hubbard Model</a></li>
</ul></li>
</ul>
</nav>
<div class="sourceCode" id="cb1"><pre
@ -283,8 +292,72 @@ properties depending on the system parameters</p>
<li><p>localisation</p></li>
<li><p>lengthscales</p></li>
</ul>
<h2 id="condsened-matter-systems">Condsened Matter Systems</h2>
<h3 id="spin-orbit-coupling">Spin-Orbit Coupling</h3>
<p>Electronic wavefunctions can be understood as quantum extensions
of</p>
<p>This can be loosely understood as a consequence of that fact that
electrons are ‘orbiting’ their host nucleus and in doing so they are
moving with respect to an electric field generated by the positive
charge of the nucleus. The electric field looks like a magnetic field in
the rest frame of the electron and this magnetic field couples to the
magnetic spin moment of the electron.</p>
<p>This analogy is wrong on many levels but it suffices to understand
that there should be such an effect.</p>
<p>Going one level deeper we can estimate the scale of the effect by
combining the non-relativistic quantum theory of a spin in a magnetic
field with the classical relativistic electromagnetism prediction for
how the electric field turns into a magnetic field in the rest frame of
the electron. This gets us within a factor to two of the correct answer
but it fails to account for an extra relativistic effect called Thomas
Precession <strong>cite</strong>.</p>
<p>The next level would be to compute this effect within relativistic QM
using the Dirac equation. And finally, we could do the full calculation
within Quantum Electrodynamics where we would find tiny corrections that
come about from virtual processes involving particle-antiparticle pairs
that spring form from the vacuum.</p>
<h3 id="electronic-correlations-the-hubbard-model">Electronic
correlations: The Hubbard Model</h3>
<figure>
<img
src="/assets/thesis/figure_code/5d575ef5-9414-4f30-a2cc-9a2b8cd44cc0.png"
alt="image.png" />
<figcaption aria-hidden="true">image.png</figcaption>
</figure>
<p>These are easiest to understand within the context of the Hubbard
model, if we take spin <span class="math inline">\(1/2\)</span> fermions
hopping on the lattice with hopping parameter <span
class="math inline">\(t\)</span> and interaction strength <span
class="math inline">\(U\)</span> <span class="math display">\[ H = -t
\sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} +
\sum_i c^\dagger_{i\uparrow} c_{i\downarrow}\]</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>. Pauli exclusion prevents
two electrons with the same spin being at the same site so which is why
the interaction term only couples opposite spin electrons. The only
physically relevant parameter here is <span
class="math inline">\(U/t\)</span> which compared the interaction
strength <span class="math inline">\(U\)</span> to the importance of
kinetic energy <span class="math inline">\(t\)</span>.</p>
<p>In the free fermion limit <span class="math inline">\(U/t =
0\)</span>, we can just find the single particle eigenstates and fill
them up to the fermi level. The many body ground state has no particular
electron-electron correlations.</p>
<p>In the interacting limit, <span class="math inline">\(t/U =
0\)</span>, there’s no hopping so electrons just site wherever we put
them. We can fill the system up until there is one electron per site
without any energy penalty at all. The maximum we can fill the system up
to</p>
<figure>
<img
src="/assets/thesis/figure_code/f25fb28d-4239-4184-9a9e-b6704189019d.png"
alt="Stolen from https://arxiv.org/pdf/1701.07056.pdf" />
<figcaption aria-hidden="true">Stolen from
https://arxiv.org/pdf/1701.07056.pdf</figcaption>
</figure>
<div class="sourceCode" id="cb2"><pre
class="sourceCode python"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>__ Connection between </span></code></pre></div>
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
</main>
</body>
</html>

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---
title: 1.2_FK_Results
excerpt:
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@ -244,7 +244,19 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
<p>which was a joint project of the first three authors with advice and
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.</p>
Johannes’ expertise on the Kitaev model. The idea to use voronoi
partitions came from<span class="citation"
data-cites="marsalTopologicalWeaireThorpe2020"><sup><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
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
rest of the programming for Koala while pair programming and
’whiteboard’ing, this included the phase diagram, edge mode and finite
temperature analyses as well as the derivation of the projector in the
amorphous case.</p>
<h1 id="introduction">Introduction</h1>
<p>The Kitaev Honeycomb model is remarkable because it combines three
key properties.</p>
@ -256,8 +268,8 @@ class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span><span
class="citation"
data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"><sup><a
href="#ref-banerjeeProximateKitaevQuantum2016"
role="doc-biblioref">1</a>,<a href="#ref-trebstKitaevMaterials2022"
role="doc-biblioref">2</a></sup></span>.</p>
role="doc-biblioref">2</a>,<a href="#ref-trebstKitaevMaterials2022"
role="doc-biblioref">3</a></sup></span>.</p>
<p>Second, this model is deeply interesting to modern condensed matter
theory. Its ground state is almost the canonical example of the long
sought after quantum spin liquid state. Its excitations are anyons,
@ -267,14 +279,14 @@ because, among other reasons, they can be braided through spacetime to
achieve noise tolerant quantum computations<span class="citation"
data-cites="freedmanTopologicalQuantumComputation2003"><sup><a
href="#ref-freedmanTopologicalQuantumComputation2003"
role="doc-biblioref">3</a></sup></span>.</p>
role="doc-biblioref">4</a></sup></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
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a></sup></span>. Its solubility comes about
role="doc-biblioref">5</a></sup></span>. Its solubility comes about
because the model has many conserved degrees of freedom that mediate the
interactions between quantum degrees of freedom.</p>
<h2 id="amorphous-systems">Amorphous Systems</h2>
@ -288,7 +300,7 @@ 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>
role="doc-biblioref">6</a></sup></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
@ -478,7 +490,7 @@ class="math inline">\(\alpha\)</span>-bond with exchange coupling <span
class="math inline">\(J^\alpha\)</span><span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"><sup><a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a></sup></span>. For notational brevity, it is
role="doc-biblioref">5</a></sup></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
@ -731,7 +743,7 @@ 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>
role="doc-biblioref">7</a></sup></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
@ -851,9 +863,18 @@ 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="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry"
<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
href="https://doi.org/10.1073/pnas.2007384117">Topological
<span>Weaire</span> models of amorphous matter</a>. <em>Proceedings of
the National Academy of Sciences</em> <strong>117</strong>, 30260–30265
(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
href="https://doi.org/10.1038/nmat4604">Proximate <span>Kitaev Quantum
Spin Liquid Behaviour</span> in {\alpha}-<span>RuCl</span>$_3$</a>.
@ -861,7 +882,7 @@ Spin Liquid Behaviour</span> in {\alpha}-<span>RuCl</span>$_3$</a>.
</div>
<div id="ref-trebstKitaevMaterials2022" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div
<div class="csl-left-margin">3. </div><div
class="csl-right-inline">Trebst, S. &amp; Hickey, C. <a
href="https://doi.org/10.1016/j.physrep.2021.11.003">Kitaev
materials</a>. <em>Physics Reports</em> <strong>950</strong>, 1–37
@ -869,7 +890,7 @@ materials</a>. <em>Physics Reports</em> <strong>950</strong>, 1–37
</div>
<div id="ref-freedmanTopologicalQuantumComputation2003"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div
<div class="csl-left-margin">4. </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
@ -878,14 +899,14 @@ computation</a>. <em>Bull. Amer. Math. Soc.</em> <strong>40</strong>,
</div>
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div
<div class="csl-left-margin">5. </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>
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div
<div class="csl-left-margin">6. </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 <span>Hamiltonians</span>: Spin
@ -895,7 +916,7 @@ systems</a>. <em>Physical Review B</em> <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
<div class="csl-left-margin">7. </div><div
class="csl-right-inline">Blaizot, J.-P. &amp; Ripka, G. <em>Quantum
theory of finite systems</em>. (<span>The MIT Press</span>, 1986).</div>
</div>

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<li>Introduction</li>
<ul><ul>
<li><a href="./0.1_Intro.html#themes">Themes</a></li>
<li><a href="./0.1_Intro.html#condsened-matter-systems">Condsened Matter Systems</a></li>
</ul></ul>
<li>Chapter 1: The Long Range Falikov-Kimball Model</li>
<ul>
@ -13,7 +14,17 @@
<li><a href="./1.1_FK_Intro.html#localisation">Localisation</a></li>
<li><a href="./1.1_FK_Intro.html#numerical-methods">Numerical Methods</a></li>
<li><a href="./1.1_FK_Intro.html#markov-chain-monte-carlo-in-practice}">Markov Chain Monte-Carlo in Practice}</a></li>
</ul></ul>
</ul>
<li><a href="./1.1_FK_Intro.html#introduction">Introduction</a></li>
<li><a href="./1.1_FK_Intro.html#the-long-ranged-falikov-kimball-model">The Long-Ranged Falikov-Kimball Model</a></li>
<li><a href="./1.1_FK_Intro.html#the-phase-diagram">The Phase Diagram</a></li>
<li><a href="./1.1_FK_Intro.html#markov-chain-monte-carlo-and-emergent-disorder">Markov Chain Monte Carlo and Emergent Disorder</a></li>
<li><a href="./1.1_FK_Intro.html#localisation-properties">Localisation Properties</a></li>
<li><a href="./1.1_FK_Intro.html#discussion-&-conclusion">Discussion & Conclusion</a></li>
<li><a href="./1.1_FK_Intro.html#acknowledgments">Acknowledgments</a></li>
<li><a href="./1.1_FK_Intro.html#[]{#app:balance-label="app:balance"}-detailed-balance">[]{#app:balance label="app:balance"} DETAILED BALANCE</a></li>
<li><a href="./1.1_FK_Intro.html#[]{#app:disorder_model-label="app:disorder_model"}-uncorrelated-disorder-model">[]{#app:disorder_model label="app:disorder_model"} UNCORRELATED DISORDER MODEL</a></li>
</ul>
<li>Chapter 2: The Amorphous Kitaev Model</li>
<ul>
<li><a href="./2.1_AMK_Intro.html#contributions">Contributions</a></li>