mirror of
https://github.com/TomHodson/tomhodson.github.com.git
synced 2025-06-26 10:01:18 +02:00
364 lines
13 KiB
HTML
364 lines
13 KiB
HTML
---
|
||
title: Introduction
|
||
excerpt: Why do we do Condensed Matter theory at all?
|
||
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="Why do we do Condensed Matter theory at all?" />
|
||
<title>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;}
|
||
pre > code.sourceCode { white-space: pre; position: relative; }
|
||
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
|
||
pre > code.sourceCode > span:empty { height: 1.2em; }
|
||
.sourceCode { overflow: visible; }
|
||
code.sourceCode > span { color: inherit; text-decoration: inherit; }
|
||
div.sourceCode { margin: 1em 0; }
|
||
pre.sourceCode { margin: 0; }
|
||
@media screen {
|
||
div.sourceCode { overflow: auto; }
|
||
}
|
||
@media print {
|
||
pre > code.sourceCode { white-space: pre-wrap; }
|
||
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
|
||
}
|
||
pre.numberSource code
|
||
{ counter-reset: source-line 0; }
|
||
pre.numberSource code > span
|
||
{ position: relative; left: -4em; counter-increment: source-line; }
|
||
pre.numberSource code > span > a:first-child::before
|
||
{ content: counter(source-line);
|
||
position: relative; left: -1em; text-align: right; vertical-align: baseline;
|
||
border: none; display: inline-block;
|
||
-webkit-touch-callout: none; -webkit-user-select: none;
|
||
-khtml-user-select: none; -moz-user-select: none;
|
||
-ms-user-select: none; user-select: none;
|
||
padding: 0 4px; width: 4em;
|
||
color: #aaaaaa;
|
||
}
|
||
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
|
||
div.sourceCode
|
||
{ }
|
||
@media screen {
|
||
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
|
||
}
|
||
code span.al { color: #ff0000; font-weight: bold; } /* Alert */
|
||
code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
|
||
code span.at { color: #7d9029; } /* Attribute */
|
||
code span.bn { color: #40a070; } /* BaseN */
|
||
code span.bu { } /* BuiltIn */
|
||
code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
|
||
code span.ch { color: #4070a0; } /* Char */
|
||
code span.cn { color: #880000; } /* Constant */
|
||
code span.co { color: #60a0b0; font-style: italic; } /* Comment */
|
||
code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */
|
||
code span.do { color: #ba2121; font-style: italic; } /* Documentation */
|
||
code span.dt { color: #902000; } /* DataType */
|
||
code span.dv { color: #40a070; } /* DecVal */
|
||
code span.er { color: #ff0000; font-weight: bold; } /* Error */
|
||
code span.ex { } /* Extension */
|
||
code span.fl { color: #40a070; } /* Float */
|
||
code span.fu { color: #06287e; } /* Function */
|
||
code span.im { } /* Import */
|
||
code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
|
||
code span.kw { color: #007020; font-weight: bold; } /* Keyword */
|
||
code span.op { color: #666666; } /* Operator */
|
||
code span.ot { color: #007020; } /* Other */
|
||
code span.pp { color: #bc7a00; } /* Preprocessor */
|
||
code span.sc { color: #4070a0; } /* SpecialChar */
|
||
code span.ss { color: #bb6688; } /* SpecialString */
|
||
code span.st { color: #4070a0; } /* String */
|
||
code span.va { color: #19177c; } /* Variable */
|
||
code span.vs { color: #4070a0; } /* VerbatimString */
|
||
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
|
||
</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="#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
|
||
class="sourceCode python"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="op">%%</span>html</span></code></pre></div>
|
||
<p>One of the most interesting and perhaps surprising features of many
|
||
body physics is the existence of distinct phases of matter.</p>
|
||
<p>Why does liquid turn to ice? Well there are two key ingredients.
|
||
First we need a system composed of a large number of objects and second
|
||
we need those objects to interact with eachother.</p>
|
||
<p>It turns out that the more objects there are, the greater the effect
|
||
that their interactions has on the whole. A hundred <span
|
||
class="math inline">\(H_2O\)</span> molecules can’t actually form the
|
||
nice regular structure that characterises ice, instead you’d get more of
|
||
a blob. However, any human scale amount of water contains an
|
||
unimaginable huge number of molecules.</p>
|
||
<p>Phases come about when the interactions between individuals
|
||
components serve the reinforce</p>
|
||
<p>When a many body, interacting system can display radically different
|
||
properties depending on the system parameters</p>
|
||
<h2 id="themes">Themes</h2>
|
||
<ul>
|
||
<li><p>many body</p></li>
|
||
<li><p>interactions</p></li>
|
||
<li><p>quantum</p></li>
|
||
<li><p>topology</p></li>
|
||
<li><p>disorder</p></li>
|
||
<li><p>quasiparticles</p></li>
|
||
<li><p>topological order</p></li>
|
||
<li><p>protected edge states</p></li>
|
||
<li><p>abelian and non-abelian anyons</p></li>
|
||
<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"></code></pre></div>
|
||
</main>
|
||
</body>
|
||
</html>
|