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updates for FK
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@ -250,6 +250,15 @@ image:
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<nav id="TOC" role="doc-toc">
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<ul>
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<li><a href="#themes" id="toc-themes">Themes</a></li>
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<li><a href="#condsened-matter-systems"
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id="toc-condsened-matter-systems">Condsened Matter Systems</a>
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<ul>
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<li><a href="#spin-orbit-coupling"
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id="toc-spin-orbit-coupling">Spin-Orbit Coupling</a></li>
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<li><a href="#electronic-correlations-the-hubbard-model"
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id="toc-electronic-correlations-the-hubbard-model">Electronic
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correlations: The Hubbard Model</a></li>
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</ul></li>
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</ul>
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</nav>
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<div class="sourceCode" id="cb1"><pre
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@ -283,8 +292,72 @@ properties depending on the system parameters</p>
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<li><p>localisation</p></li>
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<li><p>lengthscales</p></li>
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</ul>
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<h2 id="condsened-matter-systems">Condsened Matter Systems</h2>
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<h3 id="spin-orbit-coupling">Spin-Orbit Coupling</h3>
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<p>Electronic wavefunctions can be understood as quantum extensions
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of</p>
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<p>This can be loosely understood as a consequence of that fact that
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electrons are ‘orbiting’ their host nucleus and in doing so they are
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moving with respect to an electric field generated by the positive
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charge of the nucleus. The electric field looks like a magnetic field in
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the rest frame of the electron and this magnetic field couples to the
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magnetic spin moment of the electron.</p>
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<p>This analogy is wrong on many levels but it suffices to understand
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that there should be such an effect.</p>
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<p>Going one level deeper we can estimate the scale of the effect by
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combining the non-relativistic quantum theory of a spin in a magnetic
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field with the classical relativistic electromagnetism prediction for
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how the electric field turns into a magnetic field in the rest frame of
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the electron. This gets us within a factor to two of the correct answer
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but it fails to account for an extra relativistic effect called Thomas
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Precession <strong>cite</strong>.</p>
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<p>The next level would be to compute this effect within relativistic QM
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using the Dirac equation. And finally, we could do the full calculation
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within Quantum Electrodynamics where we would find tiny corrections that
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come about from virtual processes involving particle-antiparticle pairs
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that spring form from the vacuum.</p>
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<h3 id="electronic-correlations-the-hubbard-model">Electronic
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correlations: The Hubbard Model</h3>
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<figure>
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<img
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src="/assets/thesis/figure_code/5d575ef5-9414-4f30-a2cc-9a2b8cd44cc0.png"
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alt="image.png" />
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<figcaption aria-hidden="true">image.png</figcaption>
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</figure>
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<p>These are easiest to understand within the context of the Hubbard
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model, if we take spin <span class="math inline">\(1/2\)</span> fermions
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hopping on the lattice with hopping parameter <span
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class="math inline">\(t\)</span> and interaction strength <span
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class="math inline">\(U\)</span> <span class="math display">\[ H = -t
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\sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} +
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\sum_i c^\dagger_{i\uparrow} c_{i\downarrow}\]</span></p>
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<p>where <span class="math inline">\(c^\dagger_{i\alpha}\)</span>
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creates a spin <span class="math inline">\(\alpha\)</span> electron at
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site <span class="math inline">\(i\)</span>. Pauli exclusion prevents
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two electrons with the same spin being at the same site so which is why
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the interaction term only couples opposite spin electrons. The only
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physically relevant parameter here is <span
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class="math inline">\(U/t\)</span> which compared the interaction
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strength <span class="math inline">\(U\)</span> to the importance of
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kinetic energy <span class="math inline">\(t\)</span>.</p>
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<p>In the free fermion limit <span class="math inline">\(U/t =
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0\)</span>, we can just find the single particle eigenstates and fill
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them up to the fermi level. The many body ground state has no particular
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electron-electron correlations.</p>
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<p>In the interacting limit, <span class="math inline">\(t/U =
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0\)</span>, there’s no hopping so electrons just site wherever we put
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them. We can fill the system up until there is one electron per site
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without any energy penalty at all. The maximum we can fill the system up
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to</p>
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<figure>
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<img
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src="/assets/thesis/figure_code/f25fb28d-4239-4184-9a9e-b6704189019d.png"
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alt="Stolen from https://arxiv.org/pdf/1701.07056.pdf" />
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<figcaption aria-hidden="true">Stolen from
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https://arxiv.org/pdf/1701.07056.pdf</figcaption>
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</figure>
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<div class="sourceCode" id="cb2"><pre
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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>
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class="sourceCode python"><code class="sourceCode python"></code></pre></div>
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</main>
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</body>
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</html>
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_thesis/1.2_FK_Methods.html
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_thesis/1.2_FK_Methods.html
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---
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title: 1.2_FK_Methods
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excerpt:
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layout: none
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image:
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---
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</style> -->
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<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
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<script src="/assets/js/index.js"></script>
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<body>
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{% include header.html %}
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<main>
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</main>
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</body>
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</html>
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186
_thesis/1.2_FK_Results.html
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186
_thesis/1.2_FK_Results.html
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@ -0,0 +1,186 @@
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---
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title: 1.2_FK_Results
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excerpt:
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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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<head>
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<meta charset="utf-8" />
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<meta name="generator" content="pandoc" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
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<title>1.2_FK_Results</title>
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<!-- <style>
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<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
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<script src="/assets/js/index.js"></script>
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<body>
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</main>
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</html>
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@ -244,7 +244,19 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
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<p>which was a joint project of the first three authors with advice and
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guidance from Willian and Johannes. The project grew out of an interest
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Gino, Peru and I had in studying amorphous systems, coupled with
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Johannes’ expertise on the Kitaev model.</p>
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Johannes’ expertise on the Kitaev model. The idea to use voronoi
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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.
|
||||
</i,j></i,j></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. & 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. & 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. &
|
||||
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. & 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. & Ripka, G. <em>Quantum
|
||||
theory of finite systems</em>. (<span>The MIT Press</span>, 1986).</div>
|
||||
</div>
|
||||
|
||||
@ -2,6 +2,7 @@
|
||||
<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>
|
||||
|
||||
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