mirror of
https://github.com/TomHodson/tomhodson.github.com.git
synced 2025-06-26 10:01:18 +02:00
841 lines
39 KiB
HTML
841 lines
39 KiB
HTML
---
|
||
title: The Amorphous Kitaev Model - Introduction
|
||
excerpt: A short introduction to the weird and wonderful world of exactly solvable quantum models.
|
||
layout: none
|
||
image:
|
||
|
||
---
|
||
<!DOCTYPE html>
|
||
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
|
||
<head>
|
||
<meta charset="utf-8" />
|
||
<meta name="generator" content="pandoc" />
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||
<meta name="description" content="A short introduction to the weird and wonderful world of exactly solvable quantum models." />
|
||
<title>The Amorphous Kitaev Model - Introduction</title>
|
||
<!-- <style>
|
||
html {
|
||
line-height: 1.5;
|
||
font-family: Georgia, serif;
|
||
font-size: 20px;
|
||
color: #1a1a1a;
|
||
background-color: #fdfdfd;
|
||
}
|
||
body {
|
||
margin: 0 auto;
|
||
max-width: 36em;
|
||
padding-left: 50px;
|
||
padding-right: 50px;
|
||
padding-top: 50px;
|
||
padding-bottom: 50px;
|
||
hyphens: auto;
|
||
overflow-wrap: break-word;
|
||
text-rendering: optimizeLegibility;
|
||
font-kerning: normal;
|
||
}
|
||
@media (max-width: 600px) {
|
||
body {
|
||
font-size: 0.9em;
|
||
padding: 1em;
|
||
}
|
||
h1 {
|
||
font-size: 1.8em;
|
||
}
|
||
}
|
||
@media print {
|
||
body {
|
||
background-color: transparent;
|
||
color: black;
|
||
font-size: 12pt;
|
||
}
|
||
p, h2, h3 {
|
||
orphans: 3;
|
||
widows: 3;
|
||
}
|
||
h2, h3, h4 {
|
||
page-break-after: avoid;
|
||
}
|
||
}
|
||
p {
|
||
margin: 1em 0;
|
||
}
|
||
a {
|
||
color: #1a1a1a;
|
||
}
|
||
a:visited {
|
||
color: #1a1a1a;
|
||
}
|
||
img {
|
||
max-width: 100%;
|
||
}
|
||
h1, h2, h3, h4, h5, h6 {
|
||
margin-top: 1.4em;
|
||
}
|
||
h5, h6 {
|
||
font-size: 1em;
|
||
font-style: italic;
|
||
}
|
||
h6 {
|
||
font-weight: normal;
|
||
}
|
||
ol, ul {
|
||
padding-left: 1.7em;
|
||
margin-top: 1em;
|
||
}
|
||
li > ol, li > ul {
|
||
margin-top: 0;
|
||
}
|
||
blockquote {
|
||
margin: 1em 0 1em 1.7em;
|
||
padding-left: 1em;
|
||
border-left: 2px solid #e6e6e6;
|
||
color: #606060;
|
||
}
|
||
code {
|
||
font-family: Menlo, Monaco, 'Lucida Console', Consolas, monospace;
|
||
font-size: 85%;
|
||
margin: 0;
|
||
}
|
||
pre {
|
||
margin: 1em 0;
|
||
overflow: auto;
|
||
}
|
||
pre code {
|
||
padding: 0;
|
||
overflow: visible;
|
||
overflow-wrap: normal;
|
||
}
|
||
.sourceCode {
|
||
background-color: transparent;
|
||
overflow: visible;
|
||
}
|
||
hr {
|
||
background-color: #1a1a1a;
|
||
border: none;
|
||
height: 1px;
|
||
margin: 1em 0;
|
||
}
|
||
table {
|
||
margin: 1em 0;
|
||
border-collapse: collapse;
|
||
width: 100%;
|
||
overflow-x: auto;
|
||
display: block;
|
||
font-variant-numeric: lining-nums tabular-nums;
|
||
}
|
||
table caption {
|
||
margin-bottom: 0.75em;
|
||
}
|
||
tbody {
|
||
margin-top: 0.5em;
|
||
border-top: 1px solid #1a1a1a;
|
||
border-bottom: 1px solid #1a1a1a;
|
||
}
|
||
th {
|
||
border-top: 1px solid #1a1a1a;
|
||
padding: 0.25em 0.5em 0.25em 0.5em;
|
||
}
|
||
td {
|
||
padding: 0.125em 0.5em 0.25em 0.5em;
|
||
}
|
||
header {
|
||
margin-bottom: 4em;
|
||
text-align: center;
|
||
}
|
||
#TOC li {
|
||
list-style: none;
|
||
}
|
||
#TOC ul {
|
||
padding-left: 1.3em;
|
||
}
|
||
#TOC > ul {
|
||
padding-left: 0;
|
||
}
|
||
#TOC a:not(:hover) {
|
||
text-decoration: none;
|
||
}
|
||
code{white-space: pre-wrap;}
|
||
span.smallcaps{font-variant: small-caps;}
|
||
span.underline{text-decoration: underline;}
|
||
div.column{display: inline-block; vertical-align: top; width: 50%;}
|
||
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
|
||
ul.task-list{list-style: none;}
|
||
div.csl-bib-body { }
|
||
div.csl-entry {
|
||
clear: both;
|
||
}
|
||
.hanging div.csl-entry {
|
||
margin-left:2em;
|
||
text-indent:-2em;
|
||
}
|
||
div.csl-left-margin {
|
||
min-width:2em;
|
||
float:left;
|
||
}
|
||
div.csl-right-inline {
|
||
margin-left:2em;
|
||
padding-left:1em;
|
||
}
|
||
div.csl-indent {
|
||
margin-left: 2em;
|
||
}
|
||
</style> -->
|
||
|
||
<!-- <script
|
||
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js"
|
||
type="text/javascript"></script>
|
||
-->
|
||
|
||
<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
|
||
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
|
||
-->
|
||
|
||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||
|
||
|
||
<!--[if lt IE 9]>
|
||
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
|
||
<![endif]-->
|
||
<link rel="stylesheet" href="/assets/css/styles.css">
|
||
<script src="/assets/js/index.js"></script>
|
||
</head>
|
||
<body>
|
||
{% include header.html %}
|
||
|
||
<main>
|
||
<nav id="TOC" role="doc-toc">
|
||
<ul>
|
||
<li><a href="#introduction" id="toc-introduction">Introduction</a>
|
||
<ul>
|
||
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
|
||
Systems</a></li>
|
||
<li><a href="#chapter-outline" id="toc-chapter-outline">Chapter
|
||
outline</a></li>
|
||
</ul></li>
|
||
<li><a href="#an-in-depth-look-at-the-kitaev-model"
|
||
id="toc-an-in-depth-look-at-the-kitaev-model">An in-depth look at the
|
||
Kitaev Model</a>
|
||
<ul>
|
||
<li><a href="#commutation-relations"
|
||
id="toc-commutation-relations">Commutation relations</a>
|
||
<ul>
|
||
<li><a href="#spins" id="toc-spins">Spins</a></li>
|
||
<li><a href="#fermions-and-majoranas"
|
||
id="toc-fermions-and-majoranas">Fermions and Majoranas</a></li>
|
||
</ul></li>
|
||
<li><a href="#the-hamiltonian" id="toc-the-hamiltonian">The
|
||
Hamiltonian</a></li>
|
||
<li><a href="#from-spins-to-majorana-operators"
|
||
id="toc-from-spins-to-majorana-operators">From Spins to Majorana
|
||
operators</a>
|
||
<ul>
|
||
<li><a href="#for-a-single-spin" id="toc-for-a-single-spin">For a single
|
||
spin</a></li>
|
||
<li><a href="#for-multiple-spins" id="toc-for-multiple-spins">For
|
||
multiple spins</a></li>
|
||
</ul></li>
|
||
<li><a href="#partitioning-the-hilbert-space-into-bond-sectors"
|
||
id="toc-partitioning-the-hilbert-space-into-bond-sectors">Partitioning
|
||
the Hilbert Space into Bond sectors</a></li>
|
||
<li><a href="#the-majorana-hamiltonian"
|
||
id="toc-the-majorana-hamiltonian">The Majorana Hamiltonian</a></li>
|
||
<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace"
|
||
id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
|
||
back from Bond Sectors to the Physical Subspace</a></li>
|
||
</ul></li>
|
||
</ul>
|
||
</nav>
|
||
<h1 id="introduction">Introduction</h1>
|
||
<p>The Kitaev-Honeycomb model is remarkable because it was the first
|
||
such model that combined three key properties.</p>
|
||
<p>First, it is a plausible tight binding Hamiltonian. The form of the
|
||
Hamiltonian could be realised by a real material. Indeed candidate
|
||
materials such as <span
|
||
class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span> were quickly
|
||
found<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> that are expected to behave
|
||
according to the Kitaev with small corrections.</p>
|
||
<p>Second, the Kitaev Honeycomb 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, particles that can only exist in two dimensions
|
||
that break the normal fermion/boson dichotomy. Anyons have been the
|
||
subject of much attention because, among other reasons, there are
|
||
proposals to braid them through space and time 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>
|
||
<p>Third and perhaps most importantly, it a rare many body interacting
|
||
quantum system that can be treated analytically. It is exactly
|
||
solveable. 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
|
||
because the model has extensively many conserved degrees of freedom that
|
||
mediate the interactions between quantum degrees of freedom.</p>
|
||
<h2 id="amorphous-systems">Amorphous Systems</h2>
|
||
<p><strong>Insert discussion of why a generalisation to the amorphous
|
||
case is intersting</strong></p>
|
||
<h2 id="chapter-outline">Chapter outline</h2>
|
||
<p>In this chapter I will discuss the physics of the Kitaev Model on
|
||
amorphous lattices.</p>
|
||
<p>I’ll start by discussing the physics of the Kitaev model in much more
|
||
detail. Here I will look at the gauge symmetries of the model as well as
|
||
its solution via a transformation to a Majorana hamiltonian. From this
|
||
discusssion we will see that for the the model to be sovleable it need
|
||
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>
|
||
<p>In the methods section, I will discuss how to generate such lattices
|
||
and colour them as well as how to map back and forth between
|
||
configurations of the gauge field and configurations of the gauge
|
||
invariant quantities.</p>
|
||
<p>In results section, I will begin by looking at the zero temperature
|
||
physics. I’ll present numerical evidence that the ground state of the
|
||
model is given by a simple rule. I’ll make an assessment of the gapless,
|
||
abelian and non-abelian phases that are present as well as spontaneous
|
||
chiral symmetry breaking and topological edge states. We will also
|
||
compare the zero temperature phase diagram to that of the Kitaev
|
||
Honeycomb Model. Next I will take the model to finite temperature and
|
||
demonstrate that there is a phase transition to a thermal metal
|
||
state.</p>
|
||
<p>In the Discussion I will consider possible physical realisations of
|
||
this model as well the motivations for doing so. I will alao discuss how
|
||
a well known quantum error correcting code defined on the Kitaev
|
||
Honeycomb could be generalised to the amorphous case.</p>
|
||
<p>Various generalisations have been made, one mode replaces pairs of
|
||
hexagons with heptagons and pentagons and another that replaces vertices
|
||
of the hexagons with triangles . When we generalise this to the
|
||
amorphous case, the key property that will remain is that each vertex
|
||
interacts with exactly three others via an x, y and z edge. However the
|
||
lattice will no longer be bipartite, breaking chiral symmetry among
|
||
other things.</p>
|
||
<p>Kitaev-Heisenberg Model In real materials there will generally be an
|
||
addtional small Heisenberg term <span class="math display">\[H_{KH} = -
|
||
\sum_{\langle j,k\rangle_\alpha}
|
||
J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} +
|
||
\sigma_j\sigma_k\]</span></p>
|
||
<h1 id="an-in-depth-look-at-the-kitaev-model">An in-depth look at the
|
||
Kitaev Model</h1>
|
||
<h2 id="commutation-relations">Commutation relations</h2>
|
||
<p>Before diving into the Hamiltonian of the Kitaev Model, here is a
|
||
quick refresher of the key commutation relations of spins, fermions and
|
||
Majoranas.</p>
|
||
<h3 id="spins">Spins</h3>
|
||
<p>Skip this is you’re super familiar with the algebra of the Pauli
|
||
martrices. Scalars like <span class="math inline">\(\delta_{ij}\)</span>
|
||
should be understood to be multiplied by an implicit identity <span
|
||
class="math inline">\(\mathbb{1}\)</span> where necessary.</p>
|
||
<p>We can represent a single spin<span
|
||
class="math inline">\(-1/2\)</span> particle using the Pauli matrices
|
||
<span class="math inline">\((\sigma^x, \sigma^y, \sigma^z) =
|
||
\vec{\sigma}\)</span>, these matrices all square to the identity <span
|
||
class="math inline">\(\sigma^\alpha \sigma^\alpha = \mathbb{1}\)</span>
|
||
and obey nice commutation and exchange rules: <span
|
||
class="math display">\[\sigma^\alpha \sigma^\beta = \delta^{\alpha
|
||
\beta} + i \epsilon^{\alpha \beta \gamma} \sigma^\gamma\]</span> <span
|
||
class="math display">\[[\sigma^\alpha, \sigma^\beta] = 2 i
|
||
\epsilon^{\alpha \beta \gamma} \sigma^\gamma\]</span></p>
|
||
<p>Adding a sites indices <span class="math inline">\(ijk...\)</span>,
|
||
spins at different spatial sites commute always <span
|
||
class="math inline">\([\vec{\sigma}_i, \vec{\sigma}_j] = 0\)</span> so
|
||
when <span class="math inline">\(i \neq j\)</span> <span
|
||
class="math display">\[\sigma_i^\alpha \sigma_j^\beta = \sigma_j^\alpha
|
||
\sigma_i^\beta\]</span> <span class="math display">\[[\sigma_i^\alpha,
|
||
\sigma_j^\beta] = 0\]</span> while the previous equations hold for <span
|
||
class="math inline">\(i = j\)</span>.</p>
|
||
<p>Two extra relations that will be useful for the Kitaev model are the
|
||
value of <span class="math inline">\(\sigma^\alpha \sigma^\beta
|
||
\sigma^\gamma\)</span> and <span class="math inline">\([\sigma^\alpha
|
||
\sigma^\beta, \sigma^\gamma]\)</span> when <span
|
||
class="math inline">\(\alpha \neq \beta \neq \gamma\)</span> these can
|
||
be computed quite easily by appling the above relations yielding: <span
|
||
class="math display">\[\sigma^\alpha \sigma^\beta \sigma^\gamma = i
|
||
\epsilon^{\alpha\beta\gamma}\]</span> and <span
|
||
class="math display">\[[\sigma^\alpha \sigma^\beta, \sigma^\gamma] =
|
||
0\]</span></p>
|
||
<h3 id="fermions-and-majoranas">Fermions and Majoranas</h3>
|
||
<p>The fermionic creation and anhilation operators are defined by the
|
||
canonical anticommutation relations <span
|
||
class="math display">\[\begin{aligned}
|
||
\{f_i, f_j\} &= \{f^\dagger_i, f^\dagger_j\} = 0\\
|
||
\{f_i, f^\dagger_j\} &= \delta_{ij}
|
||
\end{aligned}\]</span> which give us the exchange statistics and Pauli
|
||
exclusion principle.</p>
|
||
<p>From fermionic operators, we can construct Majorana operators: <span
|
||
class="math display">\[\begin{aligned}
|
||
f_i &= 1/2 (a_i + ib_i)\\
|
||
f^\dagger_i &= 1/2(a_i - ib_i)\\
|
||
a_i &= f_i + f^\dagger_i = 2\mathbb{R}f\\
|
||
b_i &= 1/i(f_i - f^\dagger_i) = 2\mathbb{I} f
|
||
\end{aligned}\]</span></p>
|
||
<p>Majorana operators are the real and imaginary parts of the fermionic
|
||
operators, physically they correspond to the orthogonal superpositions
|
||
of the presence and absence of the fermion and are thus a kind of
|
||
quasiparticle.</p>
|
||
<p>Once we involve multiple fermions there is quite a bit of freedom in
|
||
how we can perform the transformation from <span
|
||
class="math inline">\(n\)</span> fermions <span
|
||
class="math inline">\(f_i\)</span> to <span
|
||
class="math inline">\(2n\)</span> Majoranas <span
|
||
class="math inline">\(c_i\)</span>. The property that must be preserved
|
||
however is that the Majoranas still anticommute:</p>
|
||
<p><span class="math display">\[ \{c_i, c_j\} =
|
||
2\delta_{ij}\]</span></p>
|
||
<h2 id="the-hamiltonian">The Hamiltonian</h2>
|
||
<p>To get down to brass tacks, the Kitaev Honeycomb model is a model of
|
||
interacting spin<span class="math inline">\(-1/2\)</span>s on the
|
||
vertices of a honeycomb lattice. Each bond in the lattice is assigned a
|
||
label <span class="math inline">\(\alpha \in \{ x, y, z\}\)</span> and
|
||
that bond couples its two spin neighbours along the <span
|
||
class="math inline">\(\alpha\)</span> axis. See fig. <a
|
||
href="#fig:visual_kitaev_1">1</a> for a diagram.</p>
|
||
<p>This gives us the Hamiltonian <span class="math display">\[H = -
|
||
\sum_{\langle j,k\rangle_\alpha}
|
||
J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha},\]</span> where <span
|
||
class="math inline">\(\sigma^\alpha_j\)</span> is a Pauli matrix acting
|
||
on site <span class="math inline">\(j\)</span> and <span
|
||
class="math inline">\(\langle j,k\rangle_\alpha\)</span> is a pair of
|
||
nearest-neighbour indices connected by an <span
|
||
class="math inline">\(\alpha\)</span>-bond with exchange coupling <span
|
||
class="math inline">\(J^\alpha\)</span><span class="citation"
|
||
data-cites="kitaevAnyonsExactlySolved2006"><sup><a
|
||
href="#ref-kitaevAnyonsExactlySolved2006"
|
||
role="doc-biblioref">4</a></sup></span>. For notational brevity is is
|
||
useful to introduce the bond operators <span
|
||
class="math inline">\(K_{ij} =
|
||
\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> where <span
|
||
class="math inline">\(\alpha\)</span> is a function of <span
|
||
class="math inline">\(i,j\)</span> that picks the correct bond type.</p>
|
||
<div id="fig:visual_kitaev_1" class="fignos">
|
||
<figure>
|
||
<img src="/assets/thesis/figure_code/amk_chapter/visual_kitaev_1.svg"
|
||
style="width:100.0%" alt="Figure 1: " />
|
||
<figcaption aria-hidden="true"><span>Figure 1:</span> </figcaption>
|
||
</figure>
|
||
</div>
|
||
<p>This Kitaev model has a set of conserved quantities that, in the spin
|
||
language, take the form of Wilson loop operators <span
|
||
class="math inline">\(W_p\)</span> winding around a closed path on the
|
||
lattice. The direction doesn’t matter, but I will stick to clockwise
|
||
here. I’ll use the term plaquette and the symbol <span
|
||
class="math inline">\(\phi\)</span> to refer to a Wilson loop operator
|
||
that does not enclose any other sites, such as a single hexagon in a
|
||
honeycomb lattice.</p>
|
||
<p><span class="math display">\[W_p = \prod_{\mathrm{i,j}\; \in\; p}
|
||
K_{ij} = \sigma_1^z \sigma_2^x \sigma_2^y \sigma_3^y .. \sigma_n^y
|
||
\sigma_n^y \sigma_1^z\]</span></p>
|
||
<p><strong>add a diagram of a single plaquette with labelled site and
|
||
bond types</strong></p>
|
||
<p>In closed loops, each site appears twice in the product with two of
|
||
the three bond types. Applying <span class="math inline">\(\sigma^\alpha
|
||
\sigma^\beta = \epsilon^{\alpha \beta \gamma} \sigma^\gamma, \alpha \neq
|
||
\beta\)</span> then gives us a product containing a single pauli matrix
|
||
associated with each site in the loop with the type of the
|
||
<em>outward</em> pointing bond. From this we see that the <span
|
||
class="math inline">\(W_p\)</span> associated with hexagons or shapes
|
||
with an even number of sides all square to 1 and hence have eigenvalues
|
||
<span class="math inline">\(\pm 1\)</span>.</p>
|
||
<p>A consequence of the fact that the honeycomb lattice is bipartite is
|
||
that there are no closed loops that contain an even number of edges<a
|
||
href="#fn1" class="footnote-ref" id="fnref1"
|
||
role="doc-noteref"><sup>1</sup></a> and hence all the <span
|
||
class="math inline">\(W_p\)</span> have eigenvalues <span
|
||
class="math inline">\(\pm 1\)</span> on bipartite lattices. Later we
|
||
will show that plaquettes with an odd number of sides (odd plaquettes
|
||
for short) will have eigenvalues <span class="math inline">\(\pm
|
||
i\)</span>.</p>
|
||
<div id="fig:regular_plaquettes" class="fignos">
|
||
<figure>
|
||
<img
|
||
src="/assets/thesis/figure_code/amk_chapter/regular_plaquettes/regular_plaquettes.svg"
|
||
style="width:86.0%"
|
||
alt="Figure 2: The eigenvalues of a loop or plaquette operators depend on how many bonds in its enclosing path." />
|
||
<figcaption aria-hidden="true"><span>Figure 2:</span> The eigenvalues of
|
||
a loop or plaquette operators depend on how many bonds in its enclosing
|
||
path.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<p>Remarkably, all of the spin bond operators <span
|
||
class="math inline">\(K_{ij}\)</span> commute with all the Wilson loop
|
||
operators <span class="math inline">\(W_p\)</span>. <span
|
||
class="math display">\[[W_p, J_{ij}] = 0\]</span> We can prove this by
|
||
considering the three cases: 1. neither <span
|
||
class="math inline">\(i\)</span> nor <span
|
||
class="math inline">\(j\)</span> is part of the loop 2. one of <span
|
||
class="math inline">\(i\)</span> or <span
|
||
class="math inline">\(j\)</span> are part of the loop 3. both are part
|
||
of the loop</p>
|
||
<p>The first case is trivial while the other two require a bit of
|
||
algebra, outlined in fig. <a href="#fig:visual_kitaev_2">3</a>.</p>
|
||
<div id="fig:visual_kitaev_2" class="fignos">
|
||
<figure>
|
||
<img src="/assets/thesis/figure_code/amk_chapter/visual_kitaev_2.svg"
|
||
style="width:143.0%" alt="Figure 3: " />
|
||
<figcaption aria-hidden="true"><span>Figure 3:</span> </figcaption>
|
||
</figure>
|
||
</div>
|
||
<p>Since the Hamiltonian is just a linear combination of bond operators,
|
||
it also commutes with the plaquette operators! This is great because it
|
||
means that the there’s a simultaneous eigenbasis for the Hamiltonian and
|
||
the plaquette operators. We can thus work in a basis in which the
|
||
eigenvalues of the plaquette operators take on a definite value and for
|
||
all intents and purposes act like classical degrees of freedom. These
|
||
are the extensively many conserved quantities that make the model
|
||
tractable.</p>
|
||
<p>Plaquette operators measure flux. We will find that the ground state
|
||
of the model corresponds to some particular choice of flux through each
|
||
plaquette. I will refer to excitations which flip the expectation value
|
||
of a plaqutte operator away from the ground state as
|
||
<strong>vortices</strong>.</p>
|
||
<p>Fixing a configuration of the vortices thus partitions the many-body
|
||
Hilbert space into a set of ‘vortex sectors’ labelled by that particular
|
||
flux configuration <span class="math inline">\(\phi_i = \pm 1,\pm
|
||
i\)</span>.</p>
|
||
<h2 id="from-spins-to-majorana-operators">From Spins to Majorana
|
||
operators</h2>
|
||
<h3 id="for-a-single-spin">For a single spin</h3>
|
||
<p>Let’s start by considering just one site and its <span
|
||
class="math inline">\(\sigma^x, \sigma^y\)</span> and <span
|
||
class="math inline">\(\sigma^z\)</span> operators which live in a two
|
||
dimensional Hilbert space <span
|
||
class="math inline">\(\mathcal{L}\)</span>.</p>
|
||
<p>We will introduce two fermionic modes <span
|
||
class="math inline">\(f\)</span> and <span
|
||
class="math inline">\(g\)</span> that satisy the canonical
|
||
anticommutation relations along with their number operators <span
|
||
class="math inline">\(n_f = f^\dagger f, n_g = g^\dagger g\)</span> and
|
||
the total fermionic parity operator <span class="math inline">\(F_p =
|
||
(2n_f - 1)(2n_g - 1)\)</span> which we can use to divide their Fock
|
||
space up into even and odd parity subspaces which are separated by the
|
||
addition or removal of one fermion.</p>
|
||
<p>From these two fermionic modes we can build four Majorana operators:
|
||
<span class="math display">\[\begin{aligned}
|
||
b^x &= f + f^\dagger\\
|
||
b^y &= -i(f - f^\dagger)\\
|
||
b^z &= g + g^\dagger\\
|
||
c &= -i(g - g^\dagger)
|
||
\end{aligned}\]</span></p>
|
||
<p>The Majoranas obey the usual commutation relations, squaring to one
|
||
and anticommuting with eachother. The fermions and Majorana live in a 4
|
||
dimenional Fock space <span
|
||
class="math inline">\(\mathcal{\tilde{L}}\)</span>. We can therefore
|
||
identify the two dimensional space <span
|
||
class="math inline">\(\mathcal{M}\)</span> with one of the partity
|
||
subspaces of <span class="math inline">\(\mathcal{\tilde{L}}\)</span>
|
||
which we will call the <em>physical subspace</em> <span
|
||
class="math inline">\(\mathcal{\tilde{L}}_p\)</span>. Kitaev defines the
|
||
operator <span class="math display">\[D = b^xb^yb^zc\]</span> which can
|
||
be expanded out to <span class="math display">\[D = -(2n_f - 1)(2n_g -
|
||
1) = -F_p\]</span> and labels the physical subspace as the space sanned
|
||
by states for which <span class="math display">\[ D|\phi\rangle =
|
||
|\phi\rangle\]</span></p>
|
||
<p>We can also think of the physical subspace as whatever is left after
|
||
applying the projector <span class="math display">\[P = \frac{1 -
|
||
D}{2}\]</span> to it. This formulation will be useful for taking states
|
||
that span the extended space <span
|
||
class="math inline">\(\mathcal{\tilde{M}}\)</span> and projecting them
|
||
into the physical subspace.</p>
|
||
<p>So now, with the caveat that we are working in the physical subspace,
|
||
we can define new pauli operators:</p>
|
||
<p><span class="math display">\[\tilde{\sigma}^x = i b^x c,\;
|
||
\tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^y = i b^y c\]</span></p>
|
||
<p>These extended space pauli operators satisfy all the usual
|
||
commutation relations, the only difference being that if we evaluate
|
||
<span class="math inline">\(\sigma^x \sigma^y \sigma^z = i\)</span> we
|
||
instead get <span class="math display">\[
|
||
\tilde{\sigma}^x\tilde{\sigma}^y\tilde{\sigma}^z = iD \]</span></p>
|
||
<p>Which indeed makes sense, as long as we promise to confine ourselves
|
||
to the physical subspace <span class="math inline">\(D = 1\)</span> and
|
||
this all makes sense.</p>
|
||
<div id="fig:majorana" class="fignos">
|
||
<figure>
|
||
<img src="/assets/thesis/figure_code/majorana.png" style="width:71.0%"
|
||
alt="Figure 4: " />
|
||
<figcaption aria-hidden="true"><span>Figure 4:</span> </figcaption>
|
||
</figure>
|
||
</div>
|
||
<h3 id="for-multiple-spins">For multiple spins</h3>
|
||
<p>This construction generalises easily to the case of multiple spins:
|
||
we get a set of 4 Majoranas <span class="math inline">\(b^x_j,\;
|
||
b^y_j,\;b^z_j,\; c_j\)</span> and a <span class="math inline">\(D_j =
|
||
b^x_jb^y_jb^z_jc_j\)</span> operator for every spin. For a state to be
|
||
physical we require that <span class="math inline">\(D_j |\psi\rangle =
|
||
|\psi\rangle\)</span> for all <span
|
||
class="math inline">\(j\)</span>.</p>
|
||
<p>From these each Pauli operator can be constructed: <span
|
||
class="math display">\[\tilde{\sigma}^\alpha_j = i b^\alpha_j
|
||
c_j\]</span></p>
|
||
<p>This is where the magic happens. We can promote the spin hamiltonian
|
||
from <span class="math inline">\(\mathcal{L}\)</span> into the extended
|
||
space <span class="math inline">\(\mathcal{\tilde{L}}\)</span>, safe in
|
||
the knowledge that nothing changes so long as we only actually work with
|
||
physical states. The Hamiltonian <span
|
||
class="math display">\[\begin{aligned}
|
||
\tilde{H} &= - \sum_{\langle j,k\rangle_\alpha}
|
||
J^{\alpha}\tilde{\sigma}_j^{\alpha}\tilde{\sigma}_k^{\alpha}\\
|
||
&= \frac{i}{4} \sum_{\langle j,k\rangle_\alpha}
|
||
2J^{\alpha} (ib^\alpha_i b^\alpha_j) c_i c_j\\
|
||
&= \frac{i}{4} \sum_{\langle i,j\rangle_\alpha}
|
||
2J^{\alpha} \hat{u}_{ij} \hat{c}_i \hat{c}_j
|
||
\end{aligned}\]</span></p>
|
||
<p>We can factor out the Majorana bond operators <span
|
||
class="math inline">\(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\)</span>.
|
||
Note that these bond operators are not equal to the spin bond operators
|
||
<span class="math inline">\(K_{ij} = \sigma^\alpha_i \sigma^\alpha_j = -
|
||
\hat{u}_{ij} c_i c_j\)</span>. In what follows we will work much more
|
||
frequently with the Majorana bond operators so when I refer to bond
|
||
operators without qualification, I am refering to the Majorana
|
||
variety.</p>
|
||
<p>Similar to the argument with the spin bond operators <span
|
||
class="math inline">\(K_{ij}\)</span> we can quickly verify by
|
||
considering three cases that the Majorana bond operators <span
|
||
class="math inline">\(u_{ij}\)</span> all commute with one another. They
|
||
square to one so have eigenvalues <span class="math inline">\(\pm
|
||
1\)</span> and they also commute with the <span
|
||
class="math inline">\(c_i\)</span> operators.</p>
|
||
<p>Another important point here is that the operators <span
|
||
class="math inline">\(D_i = b^x_i b^y_i b^z_i c_i\)</span> commute with
|
||
<span class="math inline">\(K_{ij}\)</span> and therefore with <span
|
||
class="math inline">\(\tilde{H}\)</span>. We will show later that the
|
||
action of <span class="math inline">\(D_i\)</span> on a state is to flip
|
||
the values of the three <span class="math inline">\(u_{ij}\)</span>
|
||
bonds that connect to site <span class="math inline">\(i\)</span>.
|
||
Physcially this is telling us that <span
|
||
class="math inline">\(u_{ij}\)</span> is a gauge field with a high
|
||
degree of degeneracy.</p>
|
||
<p>In summary Majorana bond operators <span
|
||
class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span
|
||
class="math inline">\(\mathbb{Z_2}\)</span> gauge field!</p>
|
||
<h2 id="partitioning-the-hilbert-space-into-bond-sectors">Partitioning
|
||
the Hilbert Space into Bond sectors</h2>
|
||
<p>Similar to the story with the plaquette operators from the spin
|
||
language, we can break the Hilbert space <span
|
||
class="math inline">\(\mathcal{L}\)</span> up into sectors labelled by
|
||
the a set of choices <span class="math inline">\(\{\pm 1\}\)</span> for
|
||
the value of each <span class="math inline">\(u_{ij}\)</span> operator
|
||
which I denote by <span class="math inline">\(\mathcal{L}_u\)</span>.
|
||
Since <span class="math inline">\(u_{ij} = -u_{ji}\)</span> we can
|
||
represent the <span class="math inline">\(u_{ij}\)</span> graphically
|
||
with an arrow that points along each bond in the direction in which
|
||
<span class="math inline">\(u_{ij} = 1\)</span>.</p>
|
||
<p>Once confined to a particular <span
|
||
class="math inline">\(\mathcal{L}_u\)</span>, we can ‘remove the hats’
|
||
from the <span class="math inline">\(\hat{u}_{ij}\)</span> and the
|
||
hamiltonian becomes a quadratic, free fermion problem <span
|
||
class="math display">\[\tilde{H_u} = \frac{i}{4} \sum_{\langle
|
||
i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\]</span> the ground state
|
||
of which, <span class="math inline">\(|\psi_u\rangle\)</span> can be
|
||
found easily via matrix diagonalisation. If you have been paying very
|
||
close attention, you may at this point ask whether the <span
|
||
class="math inline">\(\mathcal{L}_u\)</span> are confined entirely
|
||
within the physical subspace <span
|
||
class="math inline">\(\mathcal{L}_p\)</span> and indeed we will see that
|
||
they are not. However it will be helpful to first develop the theory of
|
||
the Majorana Hamiltonian a little more.</p>
|
||
<div id="fig:intro_figure_template" class="fignos">
|
||
<figure>
|
||
<img
|
||
src="/assets/thesis/figure_code/amk_chapter/honeycomb_zoom/intro_figure_template.svg"
|
||
style="width:100.0%"
|
||
alt="Figure 5: (a) The standard Kitaev Model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solveable it is that each vertex is joined by exactly three bonds i.e the lattice is trivalent. One of three labels is assigned to each (b) We represent the antisymmetric gauge degree of freedom u_{jk} = \pm 1 with arrows that point in the direction u_{jk} = +1 (c) The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \mathbb{Z}_2 gauge field u_{ij} leaving just a single Majorana c_i per site." />
|
||
<figcaption aria-hidden="true"><span>Figure 5:</span>
|
||
<strong>(a)</strong> The standard Kitaev Model is defined on a honeycomb
|
||
lattice. The special feature of the honeycomb lattice that makes the
|
||
model solveable it is that each vertex is joined by exactly three bonds
|
||
i.e the lattice is trivalent. One of three labels is assigned to each
|
||
<strong>(b)</strong> We represent the antisymmetric gauge degree of
|
||
freedom <span class="math inline">\(u_{jk} = \pm 1\)</span> with arrows
|
||
that point in the direction <span class="math inline">\(u_{jk} =
|
||
+1\)</span> <strong>(c)</strong> The Majorana transformation can be
|
||
visualised as breaking each spin into four Majoranas which then pair
|
||
along the bonds. The pairs of x,y and z Majoranas become part of the
|
||
classical <span class="math inline">\(\mathbb{Z}_2\)</span> gauge field
|
||
<span class="math inline">\(u_{ij}\)</span> leaving just a single
|
||
Majorana <span class="math inline">\(c_i\)</span> per site.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<h2 id="the-majorana-hamiltonian">The Majorana Hamiltonian</h2>
|
||
<p>We now have a quadtratic hamiltonian <span class="math display">\[
|
||
\tilde{H} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha}
|
||
u_{ij} c_i c_j\]</span> in which most of the Majorana degrees of freedom
|
||
have paired along bonds to become a classical gauge field <span
|
||
class="math inline">\(u_{ij}\)</span>. What follows is relatively
|
||
standard theory for quadratic Majorana Hamiltonians<span
|
||
class="citation" data-cites="BlaizotRipka1986"><sup><a
|
||
href="#ref-BlaizotRipka1986"
|
||
role="doc-biblioref">6</a></sup></span>.</p>
|
||
<p>As a consequence of the 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 pairs <span class="math inline">\(\pm \epsilon_m\)</span>. This
|
||
redundant information is a consequence of the doubling of the Hilbert
|
||
space which occured when we transformed to the Majorana
|
||
representation.</p>
|
||
<p>If we pair organise the eigenmodes of <span
|
||
class="math inline">\(H\)</span> into pairs such that <span
|
||
class="math inline">\(b_m\)</span> and <span
|
||
class="math inline">\(b_m'\)</span> have energies <span
|
||
class="math inline">\(\epsilon_m\)</span> and <span
|
||
class="math inline">\(-\epsilon_m\)</span> we can construct the
|
||
transformation <span class="math inline">\(Q\)</span> <span
|
||
class="math display">\[(c_1, c_2... c_{2N}) Q = (b_1, b_1', b_2,
|
||
b_2' ... b_{N}, b_{N}')\]</span> and put the Hamiltonian into
|
||
the form <span class="math display">\[\tilde{H}_u = \frac{i}{2} \sum_m
|
||
\epsilon_m b_m b_m'\]</span></p>
|
||
<p>The determinant of <span class="math inline">\(Q\)</span> will be
|
||
useful later when we consider the projector from <span
|
||
class="math inline">\(\mathcal{\tilde{L}}\)</span> to <span
|
||
class="math inline">\(\mathcal{L}\)</span> but otherwise the <span
|
||
class="math inline">\(b_m\)</span> are just an intermediate step. From
|
||
them we form fermionic operators <span class="math display">\[ f_i =
|
||
\tfrac{1}{2} (b_m + ib_m')\]</span> with their associated number
|
||
operators <span class="math inline">\(n_i = f^\dagger_i f_i\)</span>.
|
||
These let us write the Hamiltonian neatly as</p>
|
||
<p><span class="math display">\[ \tilde{H}_u = \sum_m \epsilon_m (n_m -
|
||
\tfrac{1}{2}).\]</span></p>
|
||
<p>The ground state <span class="math inline">\(|n_m = 0\rangle\)</span>
|
||
of the many body system at fixed <span class="math inline">\(u\)</span>
|
||
is then <span class="math display">\[E_{u,0} = -\frac{1}{2}\sum_m
|
||
\epsilon_m \]</span> and we can construct any state from a particular
|
||
choice of <span class="math inline">\(n_m = 0,1\)</span>.</p>
|
||
<p>In cases where all we care about it the value of <span
|
||
class="math inline">\(E_{u,0}\)</span> it is possible to skip forming
|
||
the fermionic operators. The eigenvalues obtained directly from
|
||
diagonalising <span class="math inline">\(J^{\alpha} u_{ij}\)</span>
|
||
come in <span class="math inline">\(\pm \epsilon_m\)</span> pairs. We
|
||
can take half the absolute value of the whole set to recover <span
|
||
class="math inline">\(\sum_m \epsilon_m\)</span> easily.</p>
|
||
<p><strong>The Majorana Hamiltonian is quadratic within a Bond
|
||
Sector.</strong></p>
|
||
<h2 id="mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
|
||
back from Bond Sectors to the Physical Subspace</h2>
|
||
<p>At this point, given a particular bond configuration <span
|
||
class="math inline">\(u_{ij} = \pm 1\)</span> we are able to construct a
|
||
quadratic Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span>
|
||
in the extended space and diagonalise it to find its ground state <span
|
||
class="math inline">\(|\vec{u}, \vec{n} = 0\rangle\)</span>. This is not
|
||
necessarily the ground state of the system as a whole, it just the
|
||
lowest energy state within the subspace <span
|
||
class="math inline">\(\mathcal{L}_u\)</span></p>
|
||
<p><strong>However, <span class="math inline">\(|u, n_m =
|
||
0\rangle\)</span> does not lie in the physical subspace</strong>. As an
|
||
example let’s take the lowest energy state associated with <span
|
||
class="math inline">\(u_{ij} = +1\)</span>, this state satisfies <span
|
||
class="math display">\[u_{ij} |\vec{u}=1, \vec{n} = 0\rangle =
|
||
|\vec{u}=1, \vec{n} = 0\rangle\]</span> for all bonds <span
|
||
class="math inline">\(i,j\)</span>.</p>
|
||
<p>If we act on it this state with one of the gauge operators <span
|
||
class="math inline">\(D_j = b_j^x b_j^y b_j^z c_j\)</span> we see that
|
||
<span class="math inline">\(D_j\)</span> flips the value of the three
|
||
bonds <span class="math inline">\(u_{ij}\)</span> that surround site
|
||
<span class="math inline">\(k\)</span>:</p>
|
||
<p><span class="math display">\[ |u'\rangle = D_j |u=1, n_m =
|
||
0\rangle\]</span></p>
|
||
<p><span class="math display">\[ \begin{aligned}
|
||
\langle u'|u_{ij}|u'\rangle &= \langle u| b_j^x b_j^y b_j^z
|
||
c_j \;ib^x_i b^x_j\; b_j^x b_j^y b_j^z c_j|u\rangle\\
|
||
&= -1
|
||
\end{aligned}\]</span></p>
|
||
<p>Since <span class="math inline">\(D_j\)</span> commutes with the
|
||
hamiltonian in the extended space <span
|
||
class="math inline">\(\tilde{H}\)</span>, the fact that <span
|
||
class="math inline">\(D_j\)</span> flips the value of bond operators is
|
||
telling us that there is a gauge degeneracy between the ground state of
|
||
<span class="math inline">\(\tilde{H}_u\)</span> and the set of <span
|
||
class="math inline">\(\tilde{H}_{u'}\)</span> related to it by gauge
|
||
transformations <span class="math inline">\(D_j\)</span>. I.e we can
|
||
flip any three bonds around a vertex and the physics will stay the
|
||
same.</p>
|
||
<p>We can turn this into a symmetrisation procedure by taking a
|
||
superposition of every possible gauge transformation. Every possible
|
||
gauge transformation is just every possible subset of <span
|
||
class="math inline">\({D_0, D_1 ... D_n}\)</span> which can be neatly
|
||
expressed as <span class="math display">\[|\phi_w\rangle = \prod_i
|
||
\left( \frac{1 + D_i}{2}\right) |\tilde{\phi}_u\rangle\]</span> this is
|
||
nice because the quantity <span class="math inline">\(\frac{1 +
|
||
D_i}{2}\)</span> is also the local projector onto the physical subspace.
|
||
Here <span class="math inline">\(|\phi_w\rangle\)</span> is a gauge
|
||
invariant state that lives in <span
|
||
class="math inline">\(\mathcal{L}_p\)</span> which has been constructed
|
||
from a set of states in different <span
|
||
class="math inline">\(\mathcal{L}_u\)</span>.</p>
|
||
<p>This gauge degeneracy leads nicely onto the next topic which is how
|
||
to construct a set of gauge invariant quantities out of the <span
|
||
class="math inline">\(u_{ij}\)</span>, these will turn out to just be
|
||
the plaquette operators.</p>
|
||
<p><strong>The Bond Sectors overlap with the physical subspace but are
|
||
not contained within it.</strong></p>
|
||
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
|
||
role="doc-bibliography">
|
||
<div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">1. </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>.
|
||
<em>Nature Mater</em> <strong>15</strong>, 733–740 (2016).</div>
|
||
</div>
|
||
<div id="ref-trebstKitaevMaterials2022" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">2. </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
|
||
(2022).</div>
|
||
</div>
|
||
<div id="ref-freedmanTopologicalQuantumComputation2003"
|
||
class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">3. </div><div
|
||
class="csl-right-inline">Freedman, M., Kitaev, A., Larsen, M. &
|
||
Wang, Z. <a
|
||
href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological quantum
|
||
computation</a>. <em>Bull. Amer. Math. Soc.</em> <strong>40</strong>,
|
||
31–38 (2003).</div>
|
||
</div>
|
||
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">4. </div><div
|
||
class="csl-right-inline">Kitaev, A. <a
|
||
href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons in an exactly
|
||
solved model and beyond</a>. <em>Annals of Physics</em>
|
||
<strong>321</strong>, 2–111 (2006-01-01, 2006).</div>
|
||
</div>
|
||
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">5. </div><div
|
||
class="csl-right-inline">Nussinov, Z. & Ortiz, G. <a
|
||
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond algebras and
|
||
exact solvability of <span>Hamiltonians</span>: Spin
|
||
<span>S</span>=<span><span
|
||
class="math inline">\(\frac{1}{2}\)</span></span> multilayer
|
||
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
|
||
class="csl-right-inline">Blaizot, J.-P. & Ripka, G. <em>Quantum
|
||
theory of finite systems</em>. (<span>The MIT Press</span>, 1986).</div>
|
||
</div>
|
||
</div>
|
||
<section class="footnotes footnotes-end-of-document"
|
||
role="doc-endnotes">
|
||
<hr />
|
||
<ol>
|
||
<li id="fn1" role="doc-endnote"><p>A bipartite lattice is composed of A
|
||
and B sublattices with no intra-sublattice edges i.e no A-A or B-B
|
||
edges. Any closed loop must begin and at the same site, let’s say it’s
|
||
an A site. The loop must go A-B-A-B… until it returns to the original
|
||
site and must therefore must contain an even number of edges in order to
|
||
end on the same sublattice that it started on.<a href="#fnref1"
|
||
class="footnote-back" role="doc-backlink">↩︎</a></p></li>
|
||
</ol>
|
||
</section>
|
||
</main>
|
||
</body>
|
||
</html>
|