tom/personal_site
1
0
Fork 0
mirror of https://github.com/TomHodson/tomhodson.github.com.git synced 2025-06-26 10:01:18 +02:00
Code Issues Packages Projects Releases Wiki Activity

updates

Browse Source
This commit is contained in:
Tom Hodson 2022-10-07 17:55:04 +01:00
parent 5bc4942765
commit a37e39ea5a
23 changed files with 1029 additions and 529 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

60
_thesis/0_Preface/0.1_Abstract.html Normal file
View File

@ -0,0 +1,60 @@
---
title: Abstract
excerpt:
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" />
<title>Abstract</title>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
"HTML-CSS": {
linebreaks: { automatic: true, width: "container" }
}
});
</script>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<script src="/assets/js/thesis_scrollspy.js"></script>
<script src="https://d3js.org/d3.v5.min.js" defer></script>
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
<nav aria-label="Table of Contents" class="page-table-of-contents">
</nav>
{% endcapture %}
<!-- Give the table of contents to header as a variable so it can be put into the sidebar-->
{% include header.html extra=tableOfContents %}
<main>
<!-- Table of Contents -->
<!-- -->
<!-- Main Page Body -->
<p>Large systems of interacting objects can give rise to a rich array of emergent behaviours. Make those objects quantum and the possibilities only expand. Interacting quantum many-body systems, as such systems are called, include essentially all physical systems. Luckily, we don’t usually need to consider this full quantum many-body description. The world at the human scale is essentially classical (not quantum), while at the microscopic scale of condensed matter physics we can often get by without interactions. Some systems, however, do require the full description. These are known as strongly correlated materials. Some of the most exciting topics in modern condensed matter fall under this umbrella: the spin liquids, the fractional quantum Hall effect, high temperature superconductivity and much more. Unfortunately, strongly correlated materials are notoriously difficult to study, defying many of the established theoretical techniques within the field. Enter <em>exactly solvable models</em>, these are interacting quantum many-body systems with extensively many local symmetries. The symmetries give rise to conserved charges which essentially break the model up into many non-interacting quantum systems which are more amenable to standard theoretical techniques. This thesis will focus on two such exactly solvable models.</p>
<p>The first, the Falicov-Kimball (FK) model is an exactly solvable limit of the famous Hubbard model which describes itinerant fermions interacting with a classical Ising background field. Originally introduced to explain metal-insulator transitions, it has a rich set of ground state and thermodynamic phases. Disorder or interactions can turn metals into insulators and the FK model features both transitions. We will define a generalised FK model in 1D with long-range interactions. This model which shows a similarly rich phase diagram to its higher dimensional cousins. We use an exact Markov Chain Monte Carlo method to map the phase diagram and compute the energy resolved localisation properties of the fermions. This allows us to look at how the move to 1D affects the physics of the model. We show that the model can be understood by comparison to a simpler model of fermions coupled to binary disorder.</p>
<p>The second, the Kitaev Honeycomb (KH) model, was the first solvable 2D model with a Quantum Spin Liquid (QSL) ground state. QSLs are generally expected to arise from Mott insulators when frustration prevents magnetic ordering all the way to zero temperature. The QSL state defies the traditional Landau-Ginzburg-Wilson paradigm of phases being defined by local order parameters. It is instead a topologically ordered phase. Recent work generalising non-interacting topological insulator phases to amorphous lattices raises the question of whether interacting phases like the QSLs can be similarly generalised. We extend the KH model to random lattices with fixed coordination number three generated by Voronoi partitions of the plane. We show that this model remains solvable and hosts a chiral amorphous QSL ground state. The presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian QSL phases with a remarkably simple ground state flux pattern. Furthermore, we show that the system undergoes a phase transition to a conducting thermal metal state and discuss possible experimental realisations.</p>
</main>
</body>
</html>

81
_thesis/0_Preface/0.1_Declarations.html Normal file
View File

@ -0,0 +1,81 @@
---
title: Declarations
excerpt:
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" />
<title>Declarations</title>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
"HTML-CSS": {
linebreaks: { automatic: true, width: "container" }
}
});
</script>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<script src="/assets/js/thesis_scrollspy.js"></script>
<script src="https://d3js.org/d3.v5.min.js" defer></script>
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
<nav aria-label="Table of Contents" class="page-table-of-contents">
<ul>
<li><a href="#versions-of-this-document" id="toc-versions-of-this-document">Versions of this document</a></li>
<li><a href="#statement-of-originality" id="toc-statement-of-originality">Statement of Originality</a></li>
<li><a href="#copyright-declaration" id="toc-copyright-declaration">Copyright Declaration</a></li>
</ul>
</nav>
{% endcapture %}
<!-- Give the table of contents to header as a variable so it can be put into the sidebar-->
{% include header.html extra=tableOfContents %}
<main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#versions-of-this-document" id="toc-versions-of-this-document">Versions of this document</a></li>
<li><a href="#statement-of-originality" id="toc-statement-of-originality">Statement of Originality</a></li>
<li><a href="#copyright-declaration" id="toc-copyright-declaration">Copyright Declaration</a></li>
</ul>
</nav>
-->
<!-- Main Page Body -->
<section id="versions-of-this-document" class="level3">
<h3>Versions of this document</h3>
<p>A PDF version of document is available online with either <a href="https://github.com/TomHodson/Thesis/raw/main/thesis.pdf">normal</a> or <a href="https://github.com/TomHodson/Thesis/raw/main/double_line_spaced.pdf">double</a> line spacing. An <a href="http://thomashodson.com/thesis/">HTML version</a> is also available with some figures animated.</p>
<p>https://doi.org/10.5281/zenodo.7143205</p>
</section>
<section id="statement-of-originality" class="level3">
<h3>Statement of Originality</h3>
<p>I declare that the following work is entirely my own except where stated otherwise. Contributions from collaborators and others have been acknowledged by standard referencing practices. I have permissions to reproduce any third party copyrighted material which are included at the end of this thesis.</p>
</section>
<section id="copyright-declaration" class="level3">
<h3>Copyright Declaration</h3>
<p>The copyright of this thesis rests with the author. Unless otherwise indicated, its contents are licensed under a Creative Commons Attribution-Non Commercial 4.0 International Licence (CC BY-NC). Under this licence, you may copy and redistribute the material in any medium or format. You may also create and distribute modified versions of the work. This is on the condition that: you credit the author and do not use it, or any derivative works, for a commercial purpose. When reusing or sharing this work, ensure you make the licence terms clear to others by naming the licence and linking to the licence text. Where a work has been adapted, you should indicate that the work has been changed and describe those changes. Please seek permission from the copyright holder for uses of this work that are not included in this licence or permitted under UK Copyright Law.</p>
</section>
</main>
</body>
</html>

72
_thesis/0_Preface/0.2_Aknowledgements.html Normal file
View File

@ -0,0 +1,72 @@
---
title: Aknowledgements
excerpt:
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" />
<title>Aknowledgements</title>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
"HTML-CSS": {
linebreaks: { automatic: true, width: "container" }
}
});
</script>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<script src="/assets/js/thesis_scrollspy.js"></script>
<script src="https://d3js.org/d3.v5.min.js" defer></script>
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
<nav aria-label="Table of Contents" class="page-table-of-contents">
</nav>
{% endcapture %}
<!-- Give the table of contents to header as a variable so it can be put into the sidebar-->
{% include header.html extra=tableOfContents %}
<main>
<!-- Table of Contents -->
<!-- -->
<!-- Main Page Body -->
<p>I would like to thank my supervisor, Professor Johannes Knolle and co-supervisor Professor Derek Lee for guidance and support during this long process.</p>
<p>Dan Hdidouan for being an example of how to weather the stress of a PhD with grace and kindness.</p>
<p>Arnaud for help and guidance…</p>
<p>Carolyn, Juraci, Ievgeniia and Loli for their patience and support.</p>
<p>Nina del Ser</p>
<p>Brian Tam for his endless energy on our many many calls while we served as joint Postgraduate reps for the department.</p>
<p>All the students in CMTH, Halvard, Tom, Chris, Krishnan, David, Tonny, Emanuele … and particularly to Thank you to the CMT group at TUM in Munich, Alex and Rohit.</p>
<p>Gino, Peru and Willian for their collaboration on the Amorphous Kitaev Model.</p>
<p>Mr Jeffries who encouraged me to pursue physics</p>
<p>All the gang from Munich, Toni, Mine, Mike, Claudi.</p>
<p>Dan Simpson, the poet in residence at Imperial and one of my favourite collaborators during my time at Imperial.</p>
<p>Lou Khalfaoui for keeping me sane during the lockdown of March 2022. Sophie Nadel, Julie Ketcher and Kim ??? for their graphic design expertise and patience.</p>
<p>All the I-Stemm team, Katerina, Jeremey, John, ….</p>
<p>And finally, I’d like the thank the staff of the Camberwell Public Library where the majority of this thesis was written.</p>
<p>We thank Angus MacKinnon for helpful discussions, Sophie Nadel for input when preparing the figures.</p>
</main>
</body>
</html>

80
_thesis/0_Preface/0.2_Declarations.html Normal file
View File

@ -0,0 +1,80 @@
---
title: Declarations
excerpt:
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" />
<title>Declarations</title>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
"HTML-CSS": {
linebreaks: { automatic: true, width: "container" }
}
});
</script>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<script src="/assets/js/thesis_scrollspy.js"></script>
<script src="https://d3js.org/d3.v5.min.js" defer></script>
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
<nav aria-label="Table of Contents" class="page-table-of-contents">
<ul>
<li><a href="#versions-of-this-document" id="toc-versions-of-this-document">Versions of this document</a></li>
<li><a href="#statement-of-originality" id="toc-statement-of-originality">Statement of Originality</a></li>
<li><a href="#copyright-declaration" id="toc-copyright-declaration">Copyright Declaration</a></li>
</ul>
</nav>
{% endcapture %}
<!-- Give the table of contents to header as a variable so it can be put into the sidebar-->
{% include header.html extra=tableOfContents %}
<main>
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#versions-of-this-document" id="toc-versions-of-this-document">Versions of this document</a></li>
<li><a href="#statement-of-originality" id="toc-statement-of-originality">Statement of Originality</a></li>
<li><a href="#copyright-declaration" id="toc-copyright-declaration">Copyright Declaration</a></li>
</ul>
</nav>
-->
<!-- Main Page Body -->
<section id="versions-of-this-document" class="level3">
<h3>Versions of this document</h3>
<p>A PDF version of document is available online with either <a href="https://github.com/TomHodson/Thesis/raw/main/thesis.pdf">normal</a> or <a href="https://github.com/TomHodson/Thesis/raw/main/double_line_spaced.pdf">double</a> line spacing. An <a href="http://thomashodson.com/thesis/">HTML version</a> is also available with some figures animated.</p>
</section>
<section id="statement-of-originality" class="level3">
<h3>Statement of Originality</h3>
<p>I declare that the following work is entirely my own except where stated otherwise. Contributions from collaborators and others have been acknowledged by standard referencing practices. I have permissions to reproduce any third party copyrighted material which are included at the end of this thesis.</p>
</section>
<section id="copyright-declaration" class="level3">
<h3>Copyright Declaration</h3>
<p>The copyright of this thesis rests with the author. Unless otherwise indicated, its contents are licensed under a Creative Commons Attribution-Non Commercial 4.0 International Licence (CC BY-NC). Under this licence, you may copy and redistribute the material in any medium or format. You may also create and distribute modified versions of the work. This is on the condition that: you credit the author and do not use it, or any derivative works, for a commercial purpose. When reusing or sharing this work, ensure you make the licence terms clear to others by naming the licence and linking to the licence text. Where a work has been adapted, you should indicate that the work has been changed and describe those changes. Please seek permission from the copyright holder for uses of this work that are not included in this licence or permitted under UK Copyright Law. \ https://doi.org/10.5281/zenodo.7143205</p>
</section>
</main>
</body>
</html>

86
_thesis/0_Preface/0.3_Aknowledgements.html Normal file
View File

@ -0,0 +1,86 @@
---
title: Aknowledgements
excerpt:
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" />
<title>Aknowledgements</title>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
"HTML-CSS": {
linebreaks: { automatic: true, width: "container" }
}
});
</script>
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
<script src="/assets/js/thesis_scrollspy.js"></script>
<script src="https://d3js.org/d3.v5.min.js" defer></script>
<link rel="stylesheet" href="/assets/css/styles.css">
<script src="/assets/js/index.js"></script>
</head>
<body>
<!--Capture the table of contents from pandoc as a jekyll variable -->
{% capture tableOfContents %}
<br>
<nav aria-label="Table of Contents" class="page-table-of-contents">
</nav>
{% endcapture %}
<!-- Give the table of contents to header as a variable so it can be put into the sidebar-->
{% include header.html extra=tableOfContents %}
<main>
<!-- Table of Contents -->
<!-- -->
<!-- Main Page Body -->
<p>Acknowledgements here.</p>
<!-- I would like to thank my supervisor, Professor Johannes Knolle and co-supervisor Professor Derek Lee for guidance and support during this long process. Thanks also to Professors Kim Christensen, Angus MacKinnon and Matthew Foulkes for helpful discussions.
Dan Hdidouan for being an example of how to weather the stress of a PhD with grace and kindness.
Arnaud for help and guidance...
Carolyn, Juraci, Ievgeniia and Loli for their patience and support.
Nina del Ser
Brian Tam for his endless energy on our many many calls while we served as joint Postgraduate reps for the department.
All the students in CMTH, Halvard, Tom, Chris, Krishnan, David, Tonny, Emanuele ... and particularly to Thank you to the CMT group at TUM in Munich, Alex and Rohit.
Gino, Peru and Willian for their collaboration on the Amorphous Kitaev model.
Mr Jeffries who encouraged me to pursue physics
All the gang from Munich, Toni, Mine, Mike, Claudi.
Dan Simpson, the poet in residence at Imperial and one of my favourite collaborators during my time at Imperial.
Lou Khalfaoui for keeping me sane during the lockdown of March 2022. Sophie Nadel, Julie Ketcher and Kim ??? for their graphic design expertise and patience.
All the I-Stemm team, Katerina, Jeremey, John, ....
And finally, I'd like the thank the staff of the Camberwell Public Library where the majority of this thesis was written. -->
</main>
</body>
</html>

53
_thesis/1_Introduction/1_Intro.html
View File

@ -41,7 +41,10 @@ image:
<ul>
<li><a href="#chap:1-introduction" id="toc-chap:1-introduction">1 Introduction</a></li>
<li><a href="#interacting-quantum-many-body-systems" id="toc-interacting-quantum-many-body-systems">Interacting Quantum Many Body Systems</a></li>
<li><a href="#mott-insulators" id="toc-mott-insulators">Mott Insulators</a></li>
<li><a href="#mott-insulators" id="toc-mott-insulators">Mott Insulators</a>
<ul>
<li><a href="#intro-the-fk-model" id="toc-intro-the-fk-model">The Falicov-Kimball Model</a></li>
</ul></li>
<li><a href="#quantum-spin-liquids" id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
@ -58,7 +61,10 @@ image:
<ul>
<li><a href="#chap:1-introduction" id="toc-chap:1-introduction">1 Introduction</a></li>
<li><a href="#interacting-quantum-many-body-systems" id="toc-interacting-quantum-many-body-systems">Interacting Quantum Many Body Systems</a></li>
<li><a href="#mott-insulators" id="toc-mott-insulators">Mott Insulators</a></li>
<li><a href="#mott-insulators" id="toc-mott-insulators">Mott Insulators</a>
<ul>
<li><a href="#intro-the-fk-model" id="toc-intro-the-fk-model">The Falicov-Kimball Model</a></li>
</ul></li>
<li><a href="#quantum-spin-liquids" id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
@ -75,8 +81,8 @@ image:
</section>
<section id="interacting-quantum-many-body-systems" class="level1">
<h1>Interacting Quantum Many Body Systems</h1>
<p>When you take many objects and let them interact together, it is often easier to describe the behaviour of the group as a whole rather than the behaviour of the individual objects. Consider a flock of starlings like that of fig. <a href="#fig:Studland_Starlings">1</a>. Watching the flock you’ll see that it has a distinct outline, that waves of density will sometimes propagate through the closely packed birds and that the flock seems to respond to predators as a distinct object. The natural description of this phenomenon is in terms of the flock, not the individual birds.</p>
<p>The behaviours of the flock are an <em>emergent phenomenon</em>. The starlings are only interacting with their immediate six or seven neighbours <span class="citation" data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"> [<a href="#ref-king2012murmurations" role="doc-biblioref">1</a>,<a href="#ref-balleriniInteractionRulingAnimal2008" role="doc-biblioref">2</a>]</span>, what a physicist would call a <em>local interaction</em>. There is much philosophical debate about how exactly to define emergence <span class="citation" data-cites="andersonMoreDifferent1972 kivelsonDefiningEmergencePhysics2016"> [<a href="#ref-andersonMoreDifferent1972" role="doc-biblioref">3</a>,<a href="#ref-kivelsonDefiningEmergencePhysics2016" role="doc-biblioref">4</a>]</span>. For our purposes, it is enough to say that emergence is the fact that the aggregate behaviour of many interacting objects may necessitate a radically different description from that of the individual objects.</p>
<p>When you take many objects and let them interact together, when we describe the behaviours that arise it is often easier to talks in terms of the group rather than the behaviour of the individual objects. A flock of starlings like that of fig. <a href="#fig:Studland_Starlings">1</a> is a good example. If you were to sit and watch a flock like this, you’d see that it has a distinct outline, that waves of density will sometimes propagate through the closely packed birds and that the flock seems to respond to predators as a distinct object. The natural description of this phenomenon is in terms of the flock, not the individual birds.</p>
<p>A flock is an <em>emergent phenomenon</em>. The starlings are only interacting with their immediate six or seven neighbours <span class="citation" data-cites="king2012murmurations balleriniInteractionRulingAnimal2008"> [<a href="#ref-king2012murmurations" role="doc-biblioref">1</a>,<a href="#ref-balleriniInteractionRulingAnimal2008" role="doc-biblioref">2</a>]</span>, what a physicist would call a <em>local interaction</em>. There is much philosophical debate about how exactly to define emergence <span class="citation" data-cites="andersonMoreDifferent1972 kivelsonDefiningEmergencePhysics2016"> [<a href="#ref-andersonMoreDifferent1972" role="doc-biblioref">3</a>,<a href="#ref-kivelsonDefiningEmergencePhysics2016" role="doc-biblioref">4</a>]</span>. For our purposes, it is enough to say that emergence is the fact that the aggregate behaviour of many interacting objects may necessitate a radically different description from that of the individual objects.</p>
<figure>
<img src="/assets/thesis/intro_chapter/Studland_Starlings.jpeg" id="fig-Studland_Starlings" data-short-caption="A murmuration of Starlings" style="width:100.0%" alt="Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0" />
<figcaption aria-hidden="true">Figure 1: A murmuration of starlings. Dorset, UK. Credit <a href="https://twitter.com/arripay">Tanya Hart</a>, “Studland Starlings”, 2017, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a></figcaption>
@ -84,12 +90,12 @@ image:
<p>To give an example closer to the topic at hand, our understanding of thermodynamics began with bulk properties like heat, energy, pressure and temperature <span class="citation" data-cites="saslowHistoryThermodynamicsMissing2020"> [<a href="#ref-saslowHistoryThermodynamicsMissing2020" role="doc-biblioref">5</a>]</span>. It was only later that we gained an understanding of how these properties emerge from microscopic interactions between very large numbers of particles <span class="citation" data-cites="flammHistoryOutlookStatistical1998"> [<a href="#ref-flammHistoryOutlookStatistical1998" role="doc-biblioref">6</a>]</span>.</p>
<p>At its heart, condensed matter is the study of the behaviours that can emerge from large numbers of interacting quantum objects at low energy. When these three properties are present together (a large number of objects, those objects being quantum and the presence interactions between the objects), we call it an interacting quantum many body system. From these three ingredients, nature builds all manner of weird and wonderful materials.</p>
<p>Historically, we first made headway by ignoring interactions and quantum properties and looking at purely many-body systems. The ideal gas law and the Drude classical electron gas <span class="citation" data-cites="ashcroftSolidStatePhysics1976"> [<a href="#ref-ashcroftSolidStatePhysics1976" role="doc-biblioref">7</a>]</span> are good examples. Including interactions leads to the Ising model <span class="citation" data-cites="isingBeitragZurTheorie1925"> [<a href="#ref-isingBeitragZurTheorie1925" role="doc-biblioref">8</a>]</span>, the Landau theory <span class="citation" data-cites="landau2013fluid"> [<a href="#ref-landau2013fluid" role="doc-biblioref">9</a>]</span> and the classical theory of phase transitions <span class="citation" data-cites="jaegerEhrenfestClassificationPhase1998"> [<a href="#ref-jaegerEhrenfestClassificationPhase1998" role="doc-biblioref">10</a>]</span>. In contrast, condensed matter theory got its start in quantum many-body theory where the only electron-electron interaction considered is the Pauli exclusion principle. Bloch’s theorem <span class="citation" data-cites="blochÜberQuantenmechanikElektronen1929"> [<a href="#ref-blochÜberQuantenmechanikElektronen1929" role="doc-biblioref">11</a>]</span>, the core result of band theory, predicted the properties of non-interacting electrons in crystal lattices. It predicted, in particular, that band insulators arise when the electrons bands are filled, leaving the fermi level in a bandgap <span class="citation" data-cites="ashcroftSolidStatePhysics1976"> [<a href="#ref-ashcroftSolidStatePhysics1976" role="doc-biblioref">7</a>]</span>. In the same vein, advances were made in understanding the quantum origins of magnetism, including ferromagnetism and antiferromagnetism <span class="citation" data-cites="MagnetismCondensedMatter"> [<a href="#ref-MagnetismCondensedMatter" role="doc-biblioref">12</a>]</span>.</p>
<p>The development of Landau-Fermi Liquid theory explained why band theory works so well even where an analysis of the relevant energies suggests that it should not <span class="citation" data-cites="wenQuantumFieldTheory2007"> [<a href="#ref-wenQuantumFieldTheory2007" role="doc-biblioref">13</a>]</span>. Landau Fermi Liquid theory demonstrates that, in many cases where electron-electron interactions are significant, the system can still be described in terms of generalised non-interacting quasiparticles. This happens when the properties of the quasiparticles in the interacting system can be smoothly connected to the free fermions of the non-interacting system.</p>
<p>However, there are systems where even Landau Fermi Liquid theory fails. An effective theoretical description of these systems must include electron-electron correlations. They are thus called strongly correlated materials <span class="citation" data-cites="morosanStronglyCorrelatedMaterials2012"> [<a href="#ref-morosanStronglyCorrelatedMaterials2012" role="doc-biblioref">14</a>]</span>. The canonical examples are superconductivity <span class="citation" data-cites="MicroscopicTheorySuperconductivity"> [<a href="#ref-MicroscopicTheorySuperconductivity" role="doc-biblioref">15</a>]</span>, the fractional quantum hall effect <span class="citation" data-cites="feldmanFractionalChargeFractional2021"> [<a href="#ref-feldmanFractionalChargeFractional2021" role="doc-biblioref">16</a>]</span> and the Mott insulators <span class="citation" data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">17</a>,<a href="#ref-fisherMottInsulatorsSpin1999" role="doc-biblioref">18</a>]</span>. We’ll start by looking at the latter but shall see that there are many links between the three topics.</p>
<p>The development of Landau-Fermi liquid theory explained why band theory works so well even where an analysis of the relevant energies suggests that it should not <span class="citation" data-cites="wenQuantumFieldTheory2007"> [<a href="#ref-wenQuantumFieldTheory2007" role="doc-biblioref">13</a>]</span>. Landau Fermi Liquid theory demonstrates that, in many cases where electron-electron interactions are significant, the system can still be described in terms of generalised non-interacting quasiparticles. This happens when the properties of the quasiparticles in the interacting system can be smoothly connected to the free fermions of the non-interacting system.</p>
<p>However, there are systems where even Landau-Fermi liquid theory fails. An effective theoretical description of these systems must include electron-electron correlations. They are thus called strongly correlated materials <span class="citation" data-cites="morosanStronglyCorrelatedMaterials2012"> [<a href="#ref-morosanStronglyCorrelatedMaterials2012" role="doc-biblioref">14</a>]</span>. The canonical examples are superconductivity <span class="citation" data-cites="MicroscopicTheorySuperconductivity"> [<a href="#ref-MicroscopicTheorySuperconductivity" role="doc-biblioref">15</a>]</span>, the fractional quantum hall effect <span class="citation" data-cites="feldmanFractionalChargeFractional2021"> [<a href="#ref-feldmanFractionalChargeFractional2021" role="doc-biblioref">16</a>]</span> and the Mott insulators <span class="citation" data-cites="mottBasisElectronTheory1949 fisherMottInsulatorsSpin1999"> [<a href="#ref-mottBasisElectronTheory1949" role="doc-biblioref">17</a>,<a href="#ref-fisherMottInsulatorsSpin1999" role="doc-biblioref">18</a>]</span>. We’ll start by looking at the latter but shall see that there are many links between the three topics.</p>
</section>
<section id="mott-insulators" class="level1">
<h1>Mott Insulators</h1>
<p>Mott insulators (MIs) are remarkable because their electrical insulator properties come not from having filled bands but from electron-electron interactions other than Pauli exclusion. Electrical conductivity, the bulk movement of electrons, requires both that there are electronic states very close in energy to the ground state and that those states are delocalised so that they can contribute to macroscopic transport. Band insulators are systems whose Fermi level falls within a gap in the density of states and thus fail the first criteria. Band insulators derive their character from the characteristics of the underlying lattice. A third kind of insulator, the Anderson insulators, have only localised electronic states near the fermi level and therefore fail the second criteria. In a later section, I will discuss Anderson insulators and the disorder that drives them.</p>
<p>Mott Insulators (MIs) are remarkable because their electrical insulator properties come not from having filled bands but from electron-electron interactions other than Pauli exclusion. Electrical conductivity, the bulk movement of electrons, requires both that there are electronic states very close in energy to the ground state and that those states are delocalised so that they can contribute to macroscopic transport. Band insulators are systems whose Fermi level falls within a gap in the density of states and thus fail the first criteria. Band insulators derive their character from the characteristics of the underlying lattice. A third kind of insulator, the Anderson insulators, have only localised electronic states near the fermi level and therefore fail the second criteria. In a later section, I will discuss Anderson insulators and the disorder that drives them.</p>
<p>Both band and Anderson insulators occur without electron-electron interactions. MIs, by contrast, require a many body picture to understand and thus elude band theory and single-particle methods.</p>
<figure>
<img src="/assets/thesis/intro_chapter/venn_diagram.svg" id="fig-venn_diagram" data-short-caption="Interacting Quantum Many Body Systems Venn Diagram" style="width:100.0%" alt="Figure 2: Three key adjectives. Many Body: systems considered in the limit of large numbers of particles. Quantum: objects whose behaviour requires quantum mechanics to describe accurately. Interacting: the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to strongly correlated materials." />
@ -100,11 +106,13 @@ image:
<p><span class="math display">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow} n_{i\downarrow} - \mu \sum_{i,\alpha} n_{i\alpha},\]</span></p>
<p>where <span class="math inline">\(c^\dagger_{i\alpha}\)</span> creates a spin <span class="math inline">\(\alpha\)</span> electron at site <span class="math inline">\(i\)</span> and the number operator <span class="math inline">\(n_{i\alpha}\)</span> measures the number of electrons with spin <span class="math inline">\(\alpha\)</span> at site <span class="math inline">\(i\)</span>. The sum runs over lattice neighbours <span class="math inline">\(\langle i,j \rangle\)</span> including both <span class="math inline">\(\langle i,j \rangle\)</span> and <span class="math inline">\(\langle j,i \rangle\)</span> so that the model is Hermitian.</p>
<p>In the non-interacting limit <span class="math inline">\(U &lt;&lt; t\)</span>, the model reduces to free fermions and the many-body ground state is a separable product of Bloch waves filled up to the Fermi level. On the other hand, the ground state in the interacting limit <span class="math inline">\(U &gt;&gt; t\)</span> is a direct product of the local Hilbert spaces <span class="math inline">\(|0\rangle, |\uparrow\rangle, |\downarrow\rangle, |\uparrow\downarrow\rangle\)</span>. At half filling, one electron per site, each site becomes a <em>local moment</em> in the reduced Hilbert space <span class="math inline">\(|\uparrow\rangle, |\downarrow\rangle\)</span> and thus acts like a spin-<span class="math inline">\(1/2\)</span> <span class="citation" data-cites="hubbardElectronCorrelationsNarrow1964"> [<a href="#ref-hubbardElectronCorrelationsNarrow1964" role="doc-biblioref">26</a>]</span>.</p>
<p>The Mott insulating phase occurs at half filling <span class="math inline">\(\mu = \tfrac{U}{2}\)</span>. Here the model can be rewritten in a symmetric form <span class="math display">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2}).\]</span></p>
<p>The basic reason that the half filled state is insulating seems trivial. Any excitation must include states of double occupancy that cost energy <span class="math inline">\(U\)</span>. Hence, the system has a finite bandgap and is an interaction-driven MI. Depending on the lattice, the local moments may then order antiferromagnetically. Originally it was proposed that this antiferromagnetic (AFM) order was actually the reason for the insulating behaviour. This would make sense since AFM order doubles the unit cell and can turn a system into a band insulator with an even number of electrons per unit cell <span class="citation" data-cites="mottMetalInsulatorTransitions1990"> [<a href="#ref-mottMetalInsulatorTransitions1990" role="doc-biblioref">27</a>]</span>. However, MIs have been found without magnetic order <span class="citation" data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">28</a>,<a href="#ref-ribakGaplessExcitationsGround2017" role="doc-biblioref">29</a>]</span>. Instead, the local moments may form a highly entangled state known as a quantum spin liquid, which will be discussed shortly.</p>
<p>Various theoretical treatments of the Hubbard model have been made, including those based on Fermi liquid theory, mean field treatments, the local density approximation (LDA) <span class="citation" data-cites="slaterMagneticEffectsHartreeFock1951"> [<a href="#ref-slaterMagneticEffectsHartreeFock1951" role="doc-biblioref">30</a>]</span>, dynamical mean-field theory <span class="citation" data-cites="greinerQuantumPhaseTransition2002"> [<a href="#ref-greinerQuantumPhaseTransition2002" role="doc-biblioref">31</a>]</span>, density matrix renormalisation group methods <span class="citation" data-cites="hallbergNewTrendsDensity2006 schollwöckDensitymatrixRenormalizationGroup2005 whiteDensityMatrixFormulation1992"> [<a href="#ref-hallbergNewTrendsDensity2006" role="doc-biblioref">32</a>–<a href="#ref-whiteDensityMatrixFormulation1992" role="doc-biblioref">34</a>]</span> and Markov chain Monte Carlo <span class="citation" data-cites="blankenbeclerMonteCarloCalculations1981 hirschDiscreteHubbardStratonovichTransformation1983 whiteNumericalStudyTwodimensional1989"> [<a href="#ref-blankenbeclerMonteCarloCalculations1981" role="doc-biblioref">35</a>–<a href="#ref-whiteNumericalStudyTwodimensional1989" role="doc-biblioref">37</a>]</span>. None of these approaches are perfect. Strong correlations are poorly described by Fermi liquid theory and LDA approaches while mean field approximations do poorly in low dimensional systems. This theoretical difficulty has made the Hubbard model a target for cold atom simulations <span class="citation" data-cites="mazurenkoColdatomFermiHubbard2017"> [<a href="#ref-mazurenkoColdatomFermiHubbard2017" role="doc-biblioref">38</a>]</span>.</p>
<p>From here, the discussion will branch in two directions. First, I will discuss a limit of the Hubbard model called the Falicov-Kimball model. Second, I will look at quantum spin liquids and the Kitaev honeycomb model.</p>
<p><strong>The Falicov-Kimball Model</strong></p>
<p>The Mott insulating phase occurs at half filling <span class="math inline">\(\mu = \tfrac{U}{2}\)</span>. Here the model can be rewritten in a symmetric form</p>
<p><span class="math display">\[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2}).\]</span></p>
<p>The basic reason that the half filled state is insulating seems trivial. Any excitation must include states of double occupancy that cost energy <span class="math inline">\(U\)</span>. Hence, the system has a finite bandgap and is an interaction-driven MI. Depending on the lattice, the local moments may then order antiferromagnetically. Originally it was proposed that this Antiferromagnetic (AFM) order was actually the reason for the insulating behaviour. This would make sense since AFM order doubles the unit cell and can turn a system into a band insulator with an even number of electrons per unit cell <span class="citation" data-cites="mottMetalInsulatorTransitions1990"> [<a href="#ref-mottMetalInsulatorTransitions1990" role="doc-biblioref">27</a>]</span>. However, MIs have been found without magnetic order <span class="citation" data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">28</a>,<a href="#ref-ribakGaplessExcitationsGround2017" role="doc-biblioref">29</a>]</span>. Instead, the local moments may form a highly entangled state known as a Quantum Spin Liquid (QSL), which will be discussed shortly.</p>
<p>Various theoretical treatments of the Hubbard model have been made, including those based on Fermi liquid theory, mean field treatments, the local density approximation <span class="citation" data-cites="slaterMagneticEffectsHartreeFock1951"> [<a href="#ref-slaterMagneticEffectsHartreeFock1951" role="doc-biblioref">30</a>]</span>, dynamical mean-field theory <span class="citation" data-cites="greinerQuantumPhaseTransition2002"> [<a href="#ref-greinerQuantumPhaseTransition2002" role="doc-biblioref">31</a>]</span>, density matrix renormalisation group methods <span class="citation" data-cites="hallbergNewTrendsDensity2006 schollwöckDensitymatrixRenormalizationGroup2005 whiteDensityMatrixFormulation1992"> [<a href="#ref-hallbergNewTrendsDensity2006" role="doc-biblioref">32</a>–<a href="#ref-whiteDensityMatrixFormulation1992" role="doc-biblioref">34</a>]</span> and Markov chain Monte Carlo <span class="citation" data-cites="blankenbeclerMonteCarloCalculations1981 hirschDiscreteHubbardStratonovichTransformation1983 whiteNumericalStudyTwodimensional1989"> [<a href="#ref-blankenbeclerMonteCarloCalculations1981" role="doc-biblioref">35</a>–<a href="#ref-whiteNumericalStudyTwodimensional1989" role="doc-biblioref">37</a>]</span>. None of these approaches are perfect. Strong correlations are poorly described by Fermi liquid theory and LDA approaches while mean field approximations do poorly in low dimensional systems. This theoretical difficulty has made the Hubbard model a target for cold atom simulations <span class="citation" data-cites="mazurenkoColdatomFermiHubbard2017"> [<a href="#ref-mazurenkoColdatomFermiHubbard2017" role="doc-biblioref">38</a>]</span>.</p>
<p>From here, the discussion will branch in two directions. First, I will discuss a limit of the Hubbard model called the Falicov-Kimball model. Second, I will look at QSLs and the Kitaev honeycomb model.</p>
<section id="intro-the-fk-model" class="level3">
<h3>The Falicov-Kimball Model</h3>
<figure>
<img src="/assets/thesis/intro_chapter/fk_schematic.svg" id="fig-fk_schematic" data-short-caption="Falicov-Kimball Model Diagram" style="width:100.0%" alt="Figure 3: The Falicov-Kimball model can be viewed as a model of classical spins S_i coupled to spinless fermions \hat{c}_i where the fermions are mobile with hopping t and the fermions are coupled to the spins by an Ising type interaction with strength U." />
<figcaption aria-hidden="true">Figure 3: The Falicov-Kimball model can be viewed as a model of classical spins <span class="math inline">\(S_i\)</span> coupled to spinless fermions <span class="math inline">\(\hat{c}_i\)</span> where the fermions are mobile with hopping <span class="math inline">\(t\)</span> and the fermions are coupled to the spins by an Ising type interaction with strength <span class="math inline">\(U\)</span>.</figcaption>
@ -113,12 +121,13 @@ image:
<p><span class="math display">\[\begin{aligned}
H_{\mathrm{FK}} = &amp; -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}). \\
\end{aligned}\]</span></p>
<p>The physics of states near the metal-insulator transition is still poorly understood <span class="citation" data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a href="#ref-belitzAndersonMottTransition1994" role="doc-biblioref">40</a>,<a href="#ref-baskoMetalInsulatorTransition2006" role="doc-biblioref">41</a>]</span>. As a result, the FK model provides a rich test bed to explore interaction-driven metal-insulator transition physics. Despite its simplicity, the model has a rich phase diagram in <span class="math inline">\(D \geq 2\)</span> dimensions. It shows a Mott insulator transition even at high temperature, similar to the corresponding Hubbard Model <span class="citation" data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a href="#ref-brandtThermodynamicsCorrelationFunctions1989" role="doc-biblioref">42</a>]</span>. In one dimension, the ground state phenomenology as a function of filling can be rich <span class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">43</a>]</span>, but the system is disordered for all <span class="math inline">\(T &gt; 0\)</span> <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">44</a>]</span>. The model has also been a test-bed for many-body methods. Interest took off when an exact dynamical mean-field theory solution in the infinite dimensional case was found <span class="citation" data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">45</a>–<a href="#ref-herrmannNonequilibriumDynamicalCluster2016" role="doc-biblioref">48</a>]</span>.</p>
<p>In <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">chapter 3</a>, I will introduce a generalized Falicov-Kimball model in one dimension I call the Long-Range Falicov-Kimball model. With the addition of long-range interactions in the background field, the model shows a similarly rich phase diagram like its higher dimensional cousins. Our goal is to understand the Mott transition in more detail, the phase transition into a charge density wave state and how the localisation properties of the fermionic sector behave in one dimension. We were particularly interested to see if correlations in the disorder potential are enough to bring about localisation effects, such as mobility edges, that are normally only seen in higher dimensions. I use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localisation properties of the fermions. We observe what appears to be a hint of coexisting localised and delocalised states. However, after careful comparison to an Anderson model of uncorrelated binary disorder about a background charge density wave field, we confirm that the fermionic sector does fully localise at larger system sizes as expected for one dimensional systems.</p>
<p>The physics of states near the metal-insulator transition is still poorly understood <span class="citation" data-cites="belitzAndersonMottTransition1994 baskoMetalInsulatorTransition2006"> [<a href="#ref-belitzAndersonMottTransition1994" role="doc-biblioref">40</a>,<a href="#ref-baskoMetalInsulatorTransition2006" role="doc-biblioref">41</a>]</span>. As a result, the FK model provides a rich test bed to explore interaction-driven metal-insulator transition physics. Despite its simplicity, the model has a rich phase diagram in <span class="math inline">\(D \geq 2\)</span> dimensions. It shows a Mott insulator transition even at high temperature, similar to the corresponding Hubbard model <span class="citation" data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a href="#ref-brandtThermodynamicsCorrelationFunctions1989" role="doc-biblioref">42</a>]</span>. In 1D, the ground state phenomenology as a function of filling can be rich <span class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">43</a>]</span>, but the system is disordered for all <span class="math inline">\(T &gt; 0\)</span> <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">44</a>]</span>. The model has also been a test-bed for many-body methods. Interest took off when an exact dynamical mean-field theory solution in the infinite dimensional case was found <span class="citation" data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">45</a>–<a href="#ref-herrmannNonequilibriumDynamicalCluster2016" role="doc-biblioref">48</a>]</span>.</p>
<p>In <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">chapter 3</a>, I will introduce a generalised Falicov-Kimball model in 1D I call the Long-Range Falicov-Kimball model. With the addition of long-range interactions in the background field, the model shows a similarly rich phase diagram like its higher dimensional cousins. Our goal is to understand the Mott transition in more detail, the phase transition into a charge density wave state and how the localisation properties of the fermionic sector behave in 1D. We were particularly interested to see if correlations in the disorder potential are enough to bring about localisation effects, such as mobility edges, that are normally only seen in higher dimensions. I use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localisation properties of the fermions. We observe what appears to be a hint of coexisting localised and delocalised states. However, after careful comparison to an Anderson model of uncorrelated binary disorder about a background charge density wave field, we confirm that the fermionic sector does fully localise at larger system sizes as expected for 1D systems.</p>
</section>
</section>
<section id="quantum-spin-liquids" class="level1">
<h1>Quantum Spin Liquids</h1>
<p>Turning to the other key topic of this thesis, we have already discussed the AFM ordering of local moments in the Mott insulating state. Landau-Ginzburg-Wilson theory characterises phases of matter as inextricably linked to the emergence of long range order via a spontaneously broken symmetry. Within this paradigm, we would not expect any interesting phases of matter not associated with AFM or other long-range order. However, Anderson first proposed in 1973 <span class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a href="#ref-andersonResonatingValenceBonds1973" role="doc-biblioref">49</a>]</span> that, if long range order is suppressed by some mechanism, it might lead to a liquid-like state even at zero temperature: a Quantum Spin Liquid (QSL).</p>
<p>Turning to the other key topic of this thesis, we have already discussed the AFM ordering of local moments in the Mott insulating state. Landau-Ginzburg-Wilson theory characterises phases of matter as inextricably linked to the emergence of long-range order via a spontaneously broken symmetry. Within this paradigm, we would not expect any interesting phases of matter not associated with AFM or other long-range order. However, Anderson first proposed in 1973 <span class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a href="#ref-andersonResonatingValenceBonds1973" role="doc-biblioref">49</a>]</span> that, if long-range order is suppressed by some mechanism, it might lead to a liquid-like state even at zero temperature: a QSL.</p>
<p>This QSL state would exist at zero or very low temperatures. Therefore, we would expect quantum effects to be very strong, which will have far reaching consequences. It was the discovery of a different phase, however, that really kickstarted interest in the topic. The fractional quantum Hall state, discovered in the 1980s <span class="citation" data-cites="laughlinAnomalousQuantumHall1983"> [<a href="#ref-laughlinAnomalousQuantumHall1983" role="doc-biblioref">50</a>]</span> is an explicit example of an interacting electron system that falls outside of the Landau-Ginzburg-Wilson paradigm<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a>. It shares many phenomenological properties with the QSL state. They both exhibit fractionalised excitations, braiding statistics and non-trivial topological properties <span class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a href="#ref-broholmQuantumSpinLiquids2020" role="doc-biblioref">55</a>]</span>. The many-body ground state of such systems acts as a complex and highly entangled vacuum. This vacuum can support quasiparticle excitations with properties unbound from that of the Dirac fermions of the standard model.</p>
<p>How do we actually make a QSL? Frustration is one mechanism that we can use to suppress magnetic order in spin models <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">56</a>]</span>. Frustration can be geometric. Triangular lattices, for instance, cannot support AFM order. It can also come about as a result of spin-orbit coupling or other physics. There are also other routes to QSLs besides frustrated spin systems that we will not discuss here <span class="citation" data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">57</a>–<a href="#ref-linExactSymmetryWeaklyinteracting1998" role="doc-biblioref">59</a>]</span>.</p>
<!-- Experimentally, Mott insulating systems without magnetic order have been proposed as QSL systems\ [@law1TTaS2QuantumSpin2017; @ribakGaplessExcitationsGround2017]. -->
@ -128,15 +137,15 @@ H_{\mathrm{FK}} = &amp; -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U
<figcaption aria-hidden="true">Figure 4: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev model. Reproduced from <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">56</a>]</span>.</figcaption>
</figure>
<p>Spin-orbit coupling is a relativistic effect that, very roughly, corresponds to the fact that in the frame of reference of a moving electron the electric field of nearby nuclei looks like a magnetic field to which the electron spin couples. This couples the spatial and spin parts of the electron wavefunction. The lattice structure can therefore influence the form of the spin-spin interactions, leading to spatial anisotropy in the effective interactions. This spatial anisotropy can frustrate an MI state <span class="citation" data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a href="#ref-jackeliMottInsulatorsStrong2009" role="doc-biblioref">60</a>,<a href="#ref-khaliullinOrbitalOrderFluctuations2005" role="doc-biblioref">61</a>]</span> leading to more exotic ground states than the AFM order we have seen so far. As with the Hubbard model, interaction effects are only strong or weak in comparison to the bandwidth or hopping integral <span class="math inline">\(t\)</span>. Hence, we will see strong frustration in materials with strong spin-orbit coupling <span class="math inline">\(\lambda\)</span> relative to their bandwidth <span class="math inline">\(t\)</span>.</p>
<p>In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-<span class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known as Kitaev materials, draw their name from the celebrated Kitaev Honeycomb model which is expected to model their low temperature behaviour <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">56</a>,<a href="#ref-Jackeli2009" role="doc-biblioref">62</a>–<a href="#ref-Takagi2019" role="doc-biblioref">65</a>]</span>.</p>
<p>In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-<span class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known as Kitaev materials, draw their name from the celebrated Kitaev Honeycomb (KH) model which is expected to model their low temperature behaviour <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">56</a>,<a href="#ref-Jackeli2009" role="doc-biblioref">62</a>–<a href="#ref-Takagi2019" role="doc-biblioref">65</a>]</span>.</p>
<p>At this point, we can sketch out a phase diagram like that of fig. <a href="#fig:kitaev-material-phase-diagram">4</a>. When both electron-electron interactions <span class="math inline">\(U\)</span> and spin-orbit couplings <span class="math inline">\(\lambda\)</span> are small relative to the bandwidth <span class="math inline">\(t\)</span>, we recover standard band theory of band insulators and metals. In the upper left, we have the simple Mott insulating state as described by the Hubbard model. In the lower right, strong spin-orbit coupling gives rise to topological insulators characterised by symmetry protected edge modes and non-zero Chern number. Kitaev materials occur in the region where strong electron-electron interaction and spin-orbit coupling interact. See <span class="citation" data-cites="witczak-krempaCorrelatedQuantumPhenomena2014"> [<a href="#ref-witczak-krempaCorrelatedQuantumPhenomena2014" role="doc-biblioref">66</a>]</span> for a more expansive version of this diagram.</p>
<p>The Kitaev Honeycomb model <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">67</a>]</span> was one of the first exactly solvable spin models with a QSL ground state. It is defined on the two dimensional honeycomb lattice and provides an exactly solvable model that can be reduced to a free fermion problem via a mapping to Majorana fermions. This yields an extensive number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes tied to an emergent gauge field. The model is remarkable not only for its QSL ground state, but also for its fractionalised excitations with non-trivial braiding statistics. It has a rich phase diagram hosting gapless, Abelian and non-Abelian phases <span class="citation" data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a href="#ref-knolleDynamicsFractionalizationQuantum2015" role="doc-biblioref">68</a>]</span> and a finite temperature phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">69</a>]</span>. It has been proposed that its non-Abelian excitations could be used to support robust topological quantum computing <span class="citation" data-cites="kitaev_fault-tolerant_2003 freedmanTopologicalQuantumComputation2003 nayakNonAbelianAnyonsTopological2008"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">70</a>–<a href="#ref-nayakNonAbelianAnyonsTopological2008" role="doc-biblioref">72</a>]</span>.</p>
<p>The Kitaev model and FK model have quite a bit of conceptual overlap. They are both effectively models of spinless fermions coupled to a classical Ising background field. This is what makes them exactly solvable. At finite temperatures, fluctuations in their background fields provide an effective disorder potential for the fermionic sector, so both models can be studied at finite temperature with Markov chain Monte Carlo methods <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">69</a>,<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">73</a>]</span>.</p>
<p>As Kitaev points out in his original paper, the model remains solvable on any tri-coordinated <span class="math inline">\(z=3\)</span> graph which can be 3-edge-coloured. Indeed many generalisations of the model exist <span class="citation" data-cites="Baskaran2007 Baskaran2008 Nussinov2009 OBrienPRB2016 hermanns2015weyl"> [<a href="#ref-Baskaran2007" role="doc-biblioref">74</a>–<a href="#ref-hermanns2015weyl" role="doc-biblioref">78</a>]</span>. Notably, the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">79</a>]</span> introduces triangular plaquettes to the honeycomb lattice leading to spontaneous chiral symmetry breaking. These extensions all retain translation symmetry. This is likely because edge-colouring, finding the ground state and understanding the QSL properties are much harder without it <span class="citation" data-cites="eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmann2019thermodynamics" role="doc-biblioref">80</a>,<a href="#ref-Peri2020" role="doc-biblioref">81</a>]</span>. Undeterred, this gap lead us to wonder what might happen if we remove translation symmetry from the Kitaev model. This would be a model of a tri-coordinated, highly bond anisotropic but otherwise amorphous material.</p>
<p>Amorphous materials do not have long-range lattice regularities but may have short-range regularities in the lattice structure, such as fixed coordination number <span class="math inline">\(z\)</span> as in some covalent compounds. The best examples are amorphous Silicon and Germanium with <span class="math inline">\(z=4\)</span> which are used to make thin-film solar cells <span class="citation" data-cites="Weaire1971 betteridge1973possible"> [<a href="#ref-Weaire1971" role="doc-biblioref">82</a>,<a href="#ref-betteridge1973possible" role="doc-biblioref">83</a>]</span>. Recently, it has been shown that topological insulating phases can exist in amorphous systems. Amorphous topological insulators are characterised by similar protected edge states to their translation invariant cousins and generalised topological bulk invariants <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 agarwala2019topological marsalTopologicalWeaireThorpeModels2020 costa2019toward agarwala2020higher spring2021amorphous corbae2019evidence"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">84</a>–<a href="#ref-corbae2019evidence" role="doc-biblioref">90</a>]</span>. However, research on amorphous electronic systems has mostly focused on non-interacting systems with a few exceptions, for example, to account for the observation of superconductivity <span class="citation" data-cites="buckel1954einfluss mcmillan1981electron meisel1981eliashberg bergmann1976amorphous mannaNoncrystallineTopologicalSuperconductors2022"> [<a href="#ref-buckel1954einfluss" role="doc-biblioref">91</a>–<a href="#ref-mannaNoncrystallineTopologicalSuperconductors2022" role="doc-biblioref">95</a>]</span> in amorphous materials or very recently to understand the effect of strong electron repulsion in TIs <span class="citation" data-cites="kim2022fractionalization"> [<a href="#ref-kim2022fractionalization" role="doc-biblioref">96</a>]</span>.</p>
<p>The KH model <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">67</a>]</span> was one of the first exactly solvable spin models with a QSL ground state. It is defined on the 2D honeycomb lattice and provides an exactly solvable model that can be reduced to a free fermion problem via a mapping to Majorana fermions. This yields an extensive number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes tied to an emergent gauge field. The model is remarkable not only for its QSL ground state, but also for its fractionalised excitations with non-trivial braiding statistics. It has a rich phase diagram hosting gapless, Abelian and non-Abelian phases <span class="citation" data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a href="#ref-knolleDynamicsFractionalizationQuantum2015" role="doc-biblioref">68</a>]</span> and a finite temperature phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">69</a>]</span>. It has been proposed that its non-Abelian excitations could be used to support robust topological quantum computing <span class="citation" data-cites="kitaev_fault-tolerant_2003 freedmanTopologicalQuantumComputation2003 nayakNonAbelianAnyonsTopological2008"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">70</a>–<a href="#ref-nayakNonAbelianAnyonsTopological2008" role="doc-biblioref">72</a>]</span>.</p>
<p>The KH and FK models have quite a bit of conceptual overlap. They can both be seen as models of spinless fermions coupled to a classical Ising background field. This is what makes them exactly solvable. At finite temperatures, fluctuations in their background fields provide an effective disorder potential for the fermionic sector, so both models can be studied at finite temperature with Markov chain Monte Carlo methods <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">69</a>,<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">73</a>]</span>.</p>
<p>As Kitaev points out in his original paper, the KH model remains solvable on any trivalent <span class="math inline">\(z=3\)</span> graph which can be three-edge-coloured. Indeed many generalisations of the model exist <span class="citation" data-cites="Baskaran2007 Baskaran2008 Nussinov2009 OBrienPRB2016 hermanns2015weyl"> [<a href="#ref-Baskaran2007" role="doc-biblioref">74</a>–<a href="#ref-hermanns2015weyl" role="doc-biblioref">78</a>]</span>. Notably, the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">79</a>]</span> introduces triangular plaquettes to the honeycomb lattice leading to spontaneous chiral symmetry breaking. These extensions all retain translation symmetry. This is likely because edge-colouring, finding the ground state and understanding the QSL properties are much harder without it <span class="citation" data-cites="eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmann2019thermodynamics" role="doc-biblioref">80</a>,<a href="#ref-Peri2020" role="doc-biblioref">81</a>]</span>. Undeterred, this gap lead us to wonder what might happen if we remove translation symmetry from the Kitaev model. This would be a model of a trivalent, highly bond anisotropic but otherwise amorphous material.</p>
<p>Amorphous materials do not have long-range lattice regularities but may have short-range regularities in the lattice structure, such as fixed coordination number <span class="math inline">\(z\)</span> as in some covalent compounds. The best examples are amorphous Silicon and Germanium with <span class="math inline">\(z=4\)</span> which are used to make thin-film solar cells <span class="citation" data-cites="Weaire1971 betteridge1973possible"> [<a href="#ref-Weaire1971" role="doc-biblioref">82</a>,<a href="#ref-betteridge1973possible" role="doc-biblioref">83</a>]</span>. Recently, it has been shown that topological insulating (TI) phases can exist in amorphous systems. Amorphous TIs are characterised by similar protected edge states to their translation invariant cousins and generalised topological bulk invariants <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 agarwala2019topological marsalTopologicalWeaireThorpeModels2020 costa2019toward agarwala2020higher spring2021amorphous corbae2019evidence"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">84</a>–<a href="#ref-corbae2019evidence" role="doc-biblioref">90</a>]</span>. However, research on amorphous electronic systems has mostly focused on non-interacting systems with a few exceptions, for example, to account for the observation of superconductivity <span class="citation" data-cites="buckel1954einfluss mcmillan1981electron meisel1981eliashberg bergmann1976amorphous mannaNoncrystallineTopologicalSuperconductors2022"> [<a href="#ref-buckel1954einfluss" role="doc-biblioref">91</a>–<a href="#ref-mannaNoncrystallineTopologicalSuperconductors2022" role="doc-biblioref">95</a>]</span> in amorphous materials or very recently to understand the effect of strong electron repulsion in TIs <span class="citation" data-cites="kim2022fractionalization"> [<a href="#ref-kim2022fractionalization" role="doc-biblioref">96</a>]</span>.</p>
<p>Amorphous <em>magnetic</em> systems have been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems <span class="citation" data-cites="aharony1975critical Petrakovski1981 kaneyoshi1992introduction Kaneyoshi2018"> [<a href="#ref-aharony1975critical" role="doc-biblioref">97</a>–<a href="#ref-Kaneyoshi2018" role="doc-biblioref">100</a>]</span> and with numerical methods <span class="citation" data-cites="fahnle1984monte plascak2000ising"> [<a href="#ref-fahnle1984monte" role="doc-biblioref">101</a>,<a href="#ref-plascak2000ising" role="doc-biblioref">102</a>]</span>. Research on classical Heisenberg and Ising models accounts for the observed behaviour of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour <span class="citation" data-cites="coey1978amorphous"> [<a href="#ref-coey1978amorphous" role="doc-biblioref">103</a>]</span>. However, the role of spin-anisotropic interactions and quantum effects in amorphous magnets has not been addressed.</p>
<p>In <a href="../4_Amorphous_Kitaev_Model/4.1_AMK_Model.html">chapter 4</a>, I will address the question of whether frustrated magnetic interactions on amorphous lattices can give rise to genuine quantum phases, i.e. to long-range entangled quantum spin liquids (QSL) <span class="citation" data-cites="Anderson1973 Knolle2019 Savary2016 Lacroix2011"> [<a href="#ref-Anderson1973" role="doc-biblioref">104</a>–<a href="#ref-Lacroix2011" role="doc-biblioref">107</a>]</span>. We will find that the answer is yes. I will introduce the Amorphous Kitaev (AK) model, a generalisation of the Kitaev honeycomb model to random lattices with fixed coordination number three. I will show that this model is a solvable, amorphous, chiral spin liquid. As with the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">79</a>]</span>, the AK model retains its exact solubility but the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. I will confirm prior observations that the form of the ground state is relatively simple <span class="citation" data-cites="OBrienPRB2016 eschmannThermodynamicClassificationThreedimensional2020"> [<a href="#ref-OBrienPRB2016" role="doc-biblioref">77</a>,<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">108</a>]</span> and unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I will show that the system undergoes a finite-temperature phase transition to a thermal metal state and discuss possible experimental realisations.</p>
<p>The next chapter, <a href="../2_Background/2.1_FK_Model.html">Chapter 2</a>, will introduce some necessary background to the Falicov-Kimball model, the Kitaev Honeycomb model, and disorder and localisation.</p>
<p>In <a href="../4_Amorphous_Kitaev_Model/4.1_AMK_Model.html">chapter 4</a>, I will address the question of whether frustrated magnetic interactions on amorphous lattices can give rise to genuine quantum phases, i.e. to long-range entangled QSL <span class="citation" data-cites="Anderson1973 Knolle2019 Savary2016 Lacroix2011"> [<a href="#ref-Anderson1973" role="doc-biblioref">104</a>–<a href="#ref-Lacroix2011" role="doc-biblioref">107</a>]</span>. We will find that the answer is yes. I will introduce the amorphous Kitaev model, a generalisation of the KH model to random lattices with fixed coordination number three. I will show that this model is a solvable, amorphous, chiral spin liquid. As with the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">79</a>]</span>, the amorphous Kitaev model retains its exact solubility but the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. I will confirm prior observations that the form of the ground state is relatively simple <span class="citation" data-cites="OBrienPRB2016 eschmannThermodynamicClassificationThreedimensional2020"> [<a href="#ref-OBrienPRB2016" role="doc-biblioref">77</a>,<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">108</a>]</span> and unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I will show that the system undergoes a finite-temperature phase transition to a thermal metal state and discuss possible experimental realisations.</p>
<p>The next chapter, <a href="../2_Background/2.1_FK_Model.html">Chapter 2</a>, will introduce some necessary background to the FK model, the KH model, and disorder and localisation.</p>
<p>Next Chapter: <a href="../2_Background/2.1_FK_Model.html">2 Background</a></p>
</section>
<section id="bibliography" class="level1 unnumbered">

50
_thesis/2_Background/2.1_FK_Model.html
View File

@ -43,7 +43,7 @@ image:
<ul>
<li><a href="#the-model" id="toc-the-model">The Model</a></li>
<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase Diagrams</a></li>
<li><a href="#long-ranged-ising-model" id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
<li><a href="#long-ranged-ising-model" id="toc-long-ranged-ising-model">Long-Ranged Ising model</a></li>
</ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
@ -63,7 +63,7 @@ image:
<ul>
<li><a href="#the-model" id="toc-the-model">The Model</a></li>
<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase Diagrams</a></li>
<li><a href="#long-ranged-ising-model" id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
<li><a href="#long-ranged-ising-model" id="toc-long-ranged-ising-model">Long-Ranged Ising model</a></li>
</ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
@ -82,19 +82,19 @@ image:
<h1>The Falicov Kimball Model</h1>
<section id="the-model" class="level2">
<h2>The Model</h2>
<p>The Falicov-Kimball (FK) model is one of the simplest models of the correlated electron problem. It captures the essence of the interaction between itinerant and localized electrons. It was originally introduced to explain the metal-insulator transition in f-electron systems but in its long history it has been interpreted variously as a model of electrons and ions, binary alloys or of crystal formation <span class="citation" data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicovSimpleModelSemiconductorMetal1969 gruberFalicovKimballModelReview1996 gruberFalicovKimballModel2006"> [<a href="#ref-hubbardj.ElectronCorrelationsNarrow1963" role="doc-biblioref">1</a>–<a href="#ref-gruberFalicovKimballModel2006" role="doc-biblioref">4</a>]</span>. In terms of immobile fermions <span class="math inline">\(d_i\)</span> and light fermions <span class="math inline">\(c_i\)</span> and with chemical potential fixed at half-filling, the model reads</p>
<p>The Falicov-Kimball (FK) model is one of the simplest models of the correlated electron problem. It captures the essence of the interaction between itinerant and localised electrons. It was originally introduced to explain the metal-insulator transition in f-electron systems but in its long history it has been interpreted variously as a model of electrons and ions, binary alloys or of crystal formation <span class="citation" data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicovSimpleModelSemiconductorMetal1969 gruberFalicovKimballModelReview1996 gruberFalicovKimballModel2006"> [<a href="#ref-hubbardj.ElectronCorrelationsNarrow1963" role="doc-biblioref">1</a>–<a href="#ref-gruberFalicovKimballModel2006" role="doc-biblioref">4</a>]</span>. In terms of immobile fermions <span class="math inline">\(d_i\)</span> and light fermions <span class="math inline">\(c_i\)</span> and with chemical potential fixed at half-filling, the model reads</p>
<p><span class="math display">\[\begin{aligned}
H_{\mathrm{FK}} = &amp; \;U \sum_{i} (d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
\end{aligned}\]</span></p>
<p>Here we will only discuss the hypercubic lattices, i.e the chain, the square lattice, the cubic lattice and so on. The connection to the Hubbard model is that we have relabelled the up and down spin electron states and removed the hopping term for one spin state, the equivalent of taking the limit of infinite mass ratio <span class="citation" data-cites="devriesSimplifiedHubbardModel1993"> [<a href="#ref-devriesSimplifiedHubbardModel1993" role="doc-biblioref">5</a>]</span>.</p>
<p>Like other exactly solvable models <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">6</a>]</span>, the FK model possesses extensively many conserved degrees of freedom <span class="math inline">\([d^\dagger_{i}d_{i}, H] = 0\)</span>. Similarly, the Kitaev Model model contains an extensive number of conserved fluxes. So in both models, the Hilbert space breaks up into a set of sectors in which these operators take a definite value. Crucially, this reduces the interaction terms in the model from being quartic in fermion operators to quadratic. This is what makes the two models exactly solvable, in contrast to the Hubbard model. For the FK model the interaction term <span class="math inline">\((d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\)</span> becomes quadratic when <span class="math inline">\(d^\dagger_{i}d_{i}\)</span> is replaced with on of its eigenvalues <span class="math inline">\(\{0,1\}\)</span>. The same thing happens in the Kitaev model, though after first applying a clever transformation which we will discuss later.</p>
<p>Like other exactly solvable models <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">6</a>]</span>, the FK model possesses extensively many conserved degrees of freedom <span class="math inline">\([d^\dagger_{i}d_{i}, H] = 0\)</span>. Similarly, the Kitaev model contains an extensive number of conserved fluxes. So in both models, the Hilbert space breaks up into a set of sectors in which these operators take a definite value. Crucially, this reduces the interaction terms in the model from being quartic in fermion operators to quadratic. This is what makes the two models exactly solvable, in contrast to the Hubbard model. For the FK model the interaction term <span class="math inline">\((d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\)</span> becomes quadratic when <span class="math inline">\(d^\dagger_{i}d_{i}\)</span> is replaced with on of its eigenvalues <span class="math inline">\(\{0,1\}\)</span>. The same thing happens in the Kitaev model, though after first applying a clever transformation which we will discuss later.</p>
<p>Due to Pauli exclusion, maximum filling occurs when each lattice site is fully occupied, <span class="math inline">\(\langle n_c + n_d \rangle = 2\)</span>. Here we will focus on the half filled case <span class="math inline">\(\langle n_c + n_d \rangle = 1\)</span>. The ground state phenomenology as the model is doped away from the half-filled state can be rich <span class="citation" data-cites="jedrzejewskiFalicovKimballModels2001 gruberGroundStatesSpinless1990"> [<a href="#ref-jedrzejewskiFalicovKimballModels2001" role="doc-biblioref">7</a>,<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">8</a>]</span> but the half-filled point has symmetries that make it particularly interesting. From this point on we will only consider the half-filled point.</p>
<p>At half-filling and on bipartite lattices, the FK the model is particle-hole (PH) symmetric. That is, the Hamiltonian anticommutes with the particle hole operator <span class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a consequence, the energy spectrum is symmetric about <span class="math inline">\(E = 0\)</span>, which is the Fermi energy. The particle hole operator corresponds to the substitution <span class="math inline">\(c^\dagger_i \rightarrow \epsilon_i c_i, d^\dagger_i \rightarrow d_i\)</span> where <span class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and <span class="math inline">\(-1\)</span> for the B sublattice <span class="citation" data-cites="gruberFalicovKimballModel2005"> [<a href="#ref-gruberFalicovKimballModel2005" role="doc-biblioref">9</a>]</span>. The absence of a hopping term for the heavy electrons means they do not need the factor of <span class="math inline">\(\epsilon_i\)</span> but they would need it in the corresponding Hubbard model. See appendix <a href="../6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">A.1</a> for a full derivation of the PH symmetry.</p>
<p>At half-filling and on bipartite lattices, the FK the model is particle-hole symmetric. That is, the Hamiltonian anticommutes with the particle hole operator <span class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a consequence, the energy spectrum is symmetric about <span class="math inline">\(E = 0\)</span>, which is the Fermi energy. The particle hole operator corresponds to the substitution <span class="math inline">\(c^\dagger_i \rightarrow \epsilon_i c_i, d^\dagger_i \rightarrow d_i\)</span> where <span class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and <span class="math inline">\(-1\)</span> for the B sublattice <span class="citation" data-cites="gruberFalicovKimballModel2005"> [<a href="#ref-gruberFalicovKimballModel2005" role="doc-biblioref">9</a>]</span>. The absence of a hopping term for the heavy electrons means they do not need the factor of <span class="math inline">\(\epsilon_i\)</span> but they would need it in the corresponding Hubbard model. See appendix <a href="../6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">A.1</a> for a full derivation of the particle-hole symmetry.</p>
<figure>
<img src="/assets/thesis/background_chapter/simple_DOS.svg" id="fig-simple_DOS" data-short-caption="Cubic Lattice dispersion with disorder" style="width:100.0%" alt="Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in one dimension. (a) With no external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder. The top rows shows the analytic dispersion in orange compared with the integral of the DOS in dotted black." />
<figcaption aria-hidden="true">Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model <span class="math inline">\(H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}\)</span> in one dimension. (a) With no external potential. (b) With a static charge density wave background <span class="math inline">\(V_i = (-1)^i\)</span> (c) A static charge density wave background with 2% binary disorder. The top rows shows the analytic dispersion in orange compared with the integral of the DOS in dotted black.</figcaption>
<img src="/assets/thesis/background_chapter/simple_DOS.svg" id="fig-simple_DOS" data-short-caption="Cubic Lattice dispersion with disorder" style="width:100.0%" alt="Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in 1D. (a) With no external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder. The top rows shows the analytic dispersion in orange compared with the integral of the DOS in dotted black." />
<figcaption aria-hidden="true">Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model <span class="math inline">\(H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}\)</span> in 1D. (a) With no external potential. (b) With a static charge density wave background <span class="math inline">\(V_i = (-1)^i\)</span> (c) A static charge density wave background with 2% binary disorder. The top rows shows the analytic dispersion in orange compared with the integral of the DOS in dotted black.</figcaption>
</figure>
<p>We will later add a long range interaction between the localised electrons at which point we will replace the immobile fermions with a classical Ising field <span class="math inline">\(S_i = 1 - 2d^\dagger_id_i = \pm1\)</span> which I will refer to as the spins.</p>
<p>We will later add a long-range interaction between the localised electrons at which point we will replace the immobile fermions with a classical Ising field <span class="math inline">\(S_i = 1 - 2d^\dagger_id_i = \pm1\)</span> which I will refer to as the spins.</p>
<p><span class="math display">\[\begin{aligned}
H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
\end{aligned}\]</span></p>
@ -106,27 +106,27 @@ H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\
<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg" id="fig-fk_phase_diagram" data-short-caption="Falicov-Kimball Temperatue-Interaction Phase Diagrams" style="width:100.0%" alt="Figure 2: Schematic Phase diagram of the Falicov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from  [10,14]" />
<figcaption aria-hidden="true">Figure 2: Schematic Phase diagram of the Falicov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">10</a>,<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">14</a>]</span></figcaption>
</figure>
<p>In dimensions greater than one, the FK model exhibits a phase transition at some <span class="math inline">\(U\)</span> dependent critical temperature <span class="math inline">\(T_c(U)\)</span> to a low temperature ordered phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">15</a>]</span>. In terms of the heavy electrons this corresponds to them occupying only one of the two sublattices A and B, known as a charge density wave (CDW) phase. In terms of spins this is an AFM phase.</p>
<p>In dimensions greater than one, the FK model exhibits a phase transition at some <span class="math inline">\(U\)</span> dependent critical temperature <span class="math inline">\(T_c(U)\)</span> to a low temperature ordered phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">15</a>]</span>. In terms of the heavy electrons this corresponds to them occupying only one of the two sublattices A and B, known as a Charge Density Wave (CDW) phase. In terms of spins this is an antiferromagnetic phase.</p>
<p>In the disordered region above <span class="math inline">\(T_c(U)\)</span> there are two insulating phases. For weak interactions <span class="math inline">\(U &lt;&lt; t\)</span>, thermal fluctuations in the spins act as an effective disorder potential for the fermions, causing them to localise and giving rise to an Anderson insulating (AI) phase <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">16</a>]</span> which we will discuss more in section <a href="../2_Background/2.4_Disorder.html#bg-disorder-and-localisation">2.3</a>. For strong interactions <span class="math inline">\(U &gt;&gt; t\)</span>, the spins are not ordered but nevertheless their interaction with the electrons opens a gap, leading to a Mott insulator analogous to that of the Hubbard model <span class="citation" data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a href="#ref-brandtThermodynamicsCorrelationFunctions1989" role="doc-biblioref">17</a>]</span>. The presence of an interaction driven phase like the Mott insulator in an exactly solvable model is part of what makes the FK model such an interesting system.</p>
<p>By contrast, in the one dimensional FK model there is no finite-temperature phase transition (FTPT) to an ordered CDW phase <span class="citation" data-cites="liebAbsenceMottTransition1968"> [<a href="#ref-liebAbsenceMottTransition1968" role="doc-biblioref">18</a>]</span>. Indeed dimensionality is crucial for the physics of both localisation and FTPTs. In one dimension, disorder generally dominates: even the weakest disorder exponentially localises <em>all</em> single particle eigenstates. In the one dimensional FK model this means the whole spectrum is localised at all finite temperatures <span class="citation" data-cites="goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">19</a>–<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">21</a>]</span>. Though at low temperatures the localisation length may be so large that the states appear extended in finite sized systems <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">14</a>]</span>. Only longer-range correlations of the disorder potential can potentially induce localisation-delocalisation transitions in one dimension <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">22</a>–<a href="#ref-dunlapAbsenceLocalizationRandomdimer1990" role="doc-biblioref">24</a>]</span></p>
<p>The absence of finite temperature ordered phases in one dimensional systems is a general feature. It can be understood as a consequence of the fact that domain walls are energetically cheap in one dimension. Thermodynamically, short-range interactions just cannot overcome the entropy of thermal defects in one dimension. However, the addition of longer range interactions can overcome this <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">25</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">26</a>]</span>.</p>
<p>However, the absence of an FTPT in the short ranged FK chain is far from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the fermions <span class="citation" data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">27</a>–<a href="#ref-yosidaMagneticPropertiesCuMn1957" role="doc-biblioref">30</a>]</span> decays as <span class="math inline">\(r^{-1}\)</span> in one dimension <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">31</a>]</span>. This could in principle induce the necessary long-range interactions for the classical Ising background to order at low temperatures <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">25</a>,<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">32</a>]</span>. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at <span class="math inline">\(T = 0\)</span> for the one dimensional FK model <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">26</a>]</span>.</p>
<p>The one dimensional FK model has been studied numerically, perturbatively in interaction strength <span class="math inline">\(U\)</span> and in the continuum limit <span class="citation" data-cites="bursillOneDimensionalContinuum1994"> [<a href="#ref-bursillOneDimensionalContinuum1994" role="doc-biblioref">33</a>]</span>. The main results are that for attractive <span class="math inline">\(U &gt; U_c\)</span> the system forms electron spin bound state ‘atoms’ which repel on another <span class="citation" data-cites="gruberGroundStateEnergyLowTemperature1993"> [<a href="#ref-gruberGroundStateEnergyLowTemperature1993" role="doc-biblioref">34</a>]</span> and that the ground state phase diagram has a has a fractal structure as a function of electron filling, a devil’s staircase <span class="citation" data-cites="freericksTwostateOnedimensionalSpinless1990 michelettiCompleteDevilStaircase1997"> [<a href="#ref-freericksTwostateOnedimensionalSpinless1990" role="doc-biblioref">35</a>,<a href="#ref-michelettiCompleteDevilStaircase1997" role="doc-biblioref">36</a>]</span>.</p>
<p>Based on this primacy of dimensionality, we will go digging into the one dimensional case. In chapter <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3</a> we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long Range Ising Model which is the subject of the next section.</p>
<p>By contrast, in the 1D FK model there is no Finite-Temperature Phase Transition (FTPT) to an ordered CDW phase <span class="citation" data-cites="liebAbsenceMottTransition1968"> [<a href="#ref-liebAbsenceMottTransition1968" role="doc-biblioref">18</a>]</span>. Indeed dimensionality is crucial for the physics of both localisation and FTPTs. In 1D, disorder generally dominates: even the weakest disorder exponentially localises <em>all</em> single particle eigenstates. In the 1D FK model this means the whole spectrum is localised at all finite temperatures <span class="citation" data-cites="goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">19</a>–<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">21</a>]</span>. Though at low temperatures the localisation length may be so large that the states appear extended in finite sized systems <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">14</a>]</span>. Only longer-range correlations of the disorder potential can potentially induce localisation-delocalisation transitions in 1D <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">22</a>–<a href="#ref-dunlapAbsenceLocalizationRandomdimer1990" role="doc-biblioref">24</a>]</span></p>
<p>The absence of finite temperature ordered phases in 1D systems is a general feature. It can be understood as a consequence of the fact that domain walls are energetically cheap in 1D. Thermodynamically, short-range interactions just cannot overcome the entropy of thermal defects in 1D. However, the addition of longer range interactions can overcome this <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">25</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">26</a>]</span>.</p>
<p>However, the absence of an FTPT in the short ranged FK chain is far from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the fermions <span class="citation" data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">27</a>–<a href="#ref-yosidaMagneticPropertiesCuMn1957" role="doc-biblioref">30</a>]</span> decays as <span class="math inline">\(r^{-1}\)</span> in 1D <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">31</a>]</span>. This could in principle induce the necessary long-range interactions for the classical Ising background to order at low temperatures <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">25</a>,<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">32</a>]</span>. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at <span class="math inline">\(T = 0\)</span> for the 1D FK model <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">26</a>]</span>.</p>
<p>The 1D FK model has been studied numerically, perturbatively in interaction strength <span class="math inline">\(U\)</span> and in the continuum limit <span class="citation" data-cites="bursillOneDimensionalContinuum1994"> [<a href="#ref-bursillOneDimensionalContinuum1994" role="doc-biblioref">33</a>]</span>. The main results are that for attractive <span class="math inline">\(U &gt; U_c\)</span> the system forms electron spin bound state ‘atoms’ which repel on another <span class="citation" data-cites="gruberGroundStateEnergyLowTemperature1993"> [<a href="#ref-gruberGroundStateEnergyLowTemperature1993" role="doc-biblioref">34</a>]</span> and that the ground state phase diagram has a has a fractal structure as a function of electron filling, a devil’s staircase <span class="citation" data-cites="freericksTwostateOnedimensionalSpinless1990 michelettiCompleteDevilStaircase1997"> [<a href="#ref-freericksTwostateOnedimensionalSpinless1990" role="doc-biblioref">35</a>,<a href="#ref-michelettiCompleteDevilStaircase1997" role="doc-biblioref">36</a>]</span>.</p>
<p>Based on this primacy of dimensionality, we will go digging into the 1D case. In chapter <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3</a> we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long-Range Ising (LRI) model which is the subject of the next section.</p>
</section>
<section id="long-ranged-ising-model" class="level2">
<h2>Long Ranged Ising model</h2>
<p>The suppression of phase transitions is a common phenomena in one dimensional systems and the Ising model serves as the canonical illustration of this. In terms of classical spins <span class="math inline">\(S_i = \pm 1\)</span> the standard Ising model reads</p>
<h2>Long-Ranged Ising model</h2>
<p>The suppression of phase transitions is a common phenomena in 1D systems and the Ising model serves as the canonical illustration of this. In terms of classical spins <span class="math inline">\(S_i = \pm 1\)</span> the standard Ising model reads</p>
<p><span class="math display">\[H_{\mathrm{I}} = \sum_{\langle ij \rangle} S_i S_j\]</span></p>
<p>Like the FK model, the Ising model shows an FTPT to an ordered state only in two dimensions and above. This can be understood via Peierls’ argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">25</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">26</a>]</span> to be a consequence of the low energy penalty for domain walls in one dimensional systems.</p>
<p>Like the FK model, the Ising model shows an FTPT to an ordered state only in 2D and above. This can be understood via Peierls’ argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">25</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">26</a>]</span> to be a consequence of the low energy penalty for domain walls in 1D systems.</p>
<figure>
<img src="/assets/thesis/intro_chapter/ising_model_domain_wall.svg" id="fig-ising_model_domain_wall" data-short-caption="Domain walls in the long range Ising Model" style="width:100.0%" alt="Figure 3: Domain walls in the one dimensional Ising model cost finite energy because they affect only one interaction while in the long range Ising model it depends on how the interactions decay with distance." />
<figcaption aria-hidden="true">Figure 3: Domain walls in the one dimensional Ising model cost finite energy because they affect only one interaction while in the long range Ising model it depends on how the interactions decay with distance.</figcaption>
<img src="/assets/thesis/intro_chapter/ising_model_domain_wall.svg" id="fig-ising_model_domain_wall" data-short-caption="Domain walls in the long-range Ising Model" style="width:100.0%" alt="Figure 3: Domain walls in the 1D Ising model cost finite energy because they affect only one interaction while in the Long-Range Ising (LRI) model it depends on how the interactions decay with distance." />
<figcaption aria-hidden="true">Figure 3: Domain walls in the 1D Ising model cost finite energy because they affect only one interaction while in the Long-Range Ising (LRI) model it depends on how the interactions decay with distance.</figcaption>
</figure>
<p>Following Peierls’ argument, consider the difference in free energy <span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span> between an ordered state and a state with single domain wall as in fig. <a href="#fig:ising_model_domain_wall">3</a>. If this value is negative it implies that the ordered state is unstable with respect to domain wall defects, and they will thus proliferate, destroying the ordered phase. If we consider the scaling of the two terms with system size <span class="math inline">\(L\)</span> we see that short range interactions produce a constant energy penalty <span class="math inline">\(\Delta E\)</span> for a domain wall. In contrast, the number of such single domain wall states scales linearly with system size so the entropy is <span class="math inline">\(\propto \ln L\)</span>. Thus the entropic contribution dominates (eventually) in the thermodynamic limit and no finite temperature order is possible. In two dimensions and above, the energy penalty of a large domain wall scales like <span class="math inline">\(L^{d-1}\)</span> which is why they can support ordered phases. This argument does not quite apply to the FK model because of the aforementioned RKKY interaction. Instead this argument will give us insight into how to recover an ordered phase in the one dimensional FK model.</p>
<p>In contrast the long range Ising (LRI) model <span class="math inline">\(H_{\mathrm{LRI}}\)</span> can have an FTPT in one dimension.</p>
<p>Following Peierls’ argument, consider the difference in free energy <span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span> between an ordered state and a state with single domain wall as in fig. <a href="#fig:ising_model_domain_wall">3</a>. If this value is negative it implies that the ordered state is unstable with respect to domain wall defects, and they will thus proliferate, destroying the ordered phase. If we consider the scaling of the two terms with system size <span class="math inline">\(L\)</span> we see that short range interactions produce a constant energy penalty <span class="math inline">\(\Delta E\)</span> for a domain wall. In contrast, the number of such single domain wall states scales linearly with system size so the entropy is <span class="math inline">\(\propto \ln L\)</span>. Thus the entropic contribution dominates (eventually) in the thermodynamic limit and no finite temperature order is possible. In 2D and above, the energy penalty of a large domain wall scales like <span class="math inline">\(L^{d-1}\)</span> which is why they can support ordered phases. This argument does not quite apply to the FK model because of the aforementioned RKKY interaction. Instead this argument will give us insight into how to recover an ordered phase in the 1D FK model.</p>
<p>In contrast the LRI model <span class="math inline">\(H_{\mathrm{LRI}}\)</span> can have an FTPT in 1D.</p>
<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|) S_i S_j = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\]</span></p>
<p>Renormalisation group analyses show that the LRI model has an ordered phase in 1D for <span class="math inline">\(1 &lt; \alpha &lt; 2\)</span>  <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">37</a>]</span>. Peierls’ argument can be extended <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">32</a>]</span> to long range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state <span class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span> and a domain wall state <span class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\)</span>. In the case of the LRI model, careful counting shows that this energy penalty is <span id="eq:bg-dw-penalty"><span class="math display">\[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\qquad{(1)}\]</span></span></p>
<p>Renormalisation group analyses show that the LRI model has an ordered phase in 1D for <span class="math inline">\(1 &lt; \alpha &lt; 2\)</span>  <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">37</a>]</span>. Peierls’ argument can be extended <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">32</a>]</span> to long-range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state <span class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span> and a domain wall state <span class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\)</span>. In the case of the LRI model, careful counting shows that this energy penalty is <span id="eq:bg-dw-penalty"><span class="math display">\[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\qquad{(1)}\]</span></span></p>
<p>because each interaction between spins separated across the domain by a bond length <span class="math inline">\(n\)</span> can be drawn between <span class="math inline">\(n\)</span> equivalent pairs of sites. The behaviour then depends crucially on how eq. <a href="#eq:bg-dw-penalty">1</a> scales with system size. Ruelle proved rigorously for a very general class of 1D systems that if <span class="math inline">\(\Delta E\)</span> or its many-body generalisation converges to a constant in the thermodynamic limit then the free energy is analytic <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">38</a>]</span>. This rules out a finite order phase transition, though not one of the Kosterlitz-Thouless type. Dyson also proves this though with a slightly different condition on <span class="math inline">\(J(n)\)</span> <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">37</a>]</span>.</p>
<p>With a power law form for <span class="math inline">\(J(n)\)</span>, there are a few cases to consider:</p>
<p>For <span class="math inline">\(\alpha = 0\)</span> i.e infinite range interactions, the Ising model is exactly solvable and mean field theory is exact <span class="citation" data-cites="lipkinValidityManybodyApproximation1965"> [<a href="#ref-lipkinValidityManybodyApproximation1965" role="doc-biblioref">39</a>]</span>. This limit is the same as the infinite dimensional limit.</p>
@ -135,10 +135,10 @@ H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\
<p>For <span class="math inline">\(\alpha = 2\)</span>, the energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transition <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">32</a>]</span>.</p>
<p>Finally, for <span class="math inline">\(2 &lt; \alpha\)</span> we have very quickly decaying interactions and domain walls again have a finite energy penalty, hence Peirels’ argument holds and there is no phase transition.</p>
<p>One final complexity is that for <span class="math inline">\(\tfrac{3}{2} &lt; \alpha &lt; 2\)</span> renormalisation group methods show that the critical point has non-universal critical exponents that depend on <span class="math inline">\(\alpha\)</span>  <span class="citation" data-cites="fisherCriticalExponentsLongRange1972"> [<a href="#ref-fisherCriticalExponentsLongRange1972" role="doc-biblioref">43</a>]</span>. To avoid this potential confounding factors we will park ourselves at <span class="math inline">\(\alpha = 1.25\)</span> when we apply these ideas to the FK model.</p>
<p>Were we to extend this to arbitrary dimension <span class="math inline">\(d\)</span> we would find that thermodynamics properties generally both <span class="math inline">\(d\)</span> and <span class="math inline">\(\alpha\)</span>, long range interactions can modify the ‘effective dimension’ of thermodynamic systems <span class="citation" data-cites="angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">44</a>]</span>.</p>
<p>Were we to extend this to arbitrary dimension <span class="math inline">\(d\)</span> we would find that thermodynamics properties generally both <span class="math inline">\(d\)</span> and <span class="math inline">\(\alpha\)</span>, long-range interactions can modify the ‘effective dimension’ of thermodynamic systems <span class="citation" data-cites="angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">44</a>]</span>.</p>
<figure>
<img src="/assets/thesis/background_chapter/alpha_diagram.svg" id="fig-alpha_diagram" data-short-caption="Long Range Ising Model Behaviour" style="width:100.0%" alt="Figure 4: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied. In my simulations I stick to a value of \alpha = \tfrac{5}{4} the complexity of non-universal critical exponents." />
<figcaption aria-hidden="true">Figure 4: The thermodynamic behaviour of the long range Ising model <span class="math inline">\(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span class="math inline">\(\alpha\)</span> is varied. In my simulations I stick to a value of <span class="math inline">\(\alpha = \tfrac{5}{4}\)</span> the complexity of non-universal critical exponents.</figcaption>
<img src="/assets/thesis/background_chapter/alpha_diagram.svg" id="fig-alpha_diagram" data-short-caption="Long-Range Ising Model Behaviour" style="width:100.0%" alt="Figure 4: The thermodynamic behaviour of the long-range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied. In my simulations I stick to a value of \alpha = \tfrac{5}{4} the complexity of non-universal critical exponents." />
<figcaption aria-hidden="true">Figure 4: The thermodynamic behaviour of the long-range Ising model <span class="math inline">\(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span class="math inline">\(\alpha\)</span> is varied. In my simulations I stick to a value of <span class="math inline">\(\alpha = \tfrac{5}{4}\)</span> the complexity of non-universal critical exponents.</figcaption>
</figure>
<p>Next Section: <a href="../2_Background/2.2_HKM_Model.html">The Kitaev Honeycomb Model</a></p>
</section>

50
_thesis/2_Background/2.2_HKM_Model.html
View File

@ -89,31 +89,31 @@ image:
</figure>
<section id="the-spin-hamiltonian" class="level2">
<h2>The Spin Hamiltonian</h2>
<p>This section introduces the Kitaev honeycomb (KH) model. The KH model is an exactly solvable model of interacting spin<span class="math inline">\(-1/2\)</span> spins 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 couple two spins along the <span class="math inline">\(\alpha\)</span> axis. See fig. <a href="#fig:intro_figure_by_hand">1</a> for a diagram of the setup.</p>
<p>The Kitaev Honeycomb (KH) model is an exactly solvable model of interacting spin<span class="math inline">\(-1/2\)</span> spins 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 couples two spins along the <span class="math inline">\(\alpha\)</span> axis. See fig. <a href="#fig:intro_figure_by_hand">1</a> for a diagram of the setup.</p>
<p>This gives us the Hamiltonian <span id="eq:bg-kh-model"><span class="math display">\[ H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha}, \qquad{(1)}\]</span></span> where <span class="math inline">\(\sigma^\alpha_j\)</span> is the <span class="math inline">\(\alpha\)</span> component of 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>. Kitaev introduced this model in his seminal 2006 paper <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>.</p>
<p>The KH can arise as the result of strong spin-orbit couplings in, for example, the transition metal based compounds <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-Jackeli2009" role="doc-biblioref">2</a>–<a href="#ref-Takagi2019" role="doc-biblioref">6</a>]</span>. The model is highly frustrated: each spin would like to align along a different direction with each of its three neighbours but this cannot be achieved even classically <span class="citation" data-cites="chandraClassicalHeisenbergSpins2010 selaOrderbydisorderSpinorbitalLiquids2014"> [<a href="#ref-chandraClassicalHeisenbergSpins2010" role="doc-biblioref">7</a>,<a href="#ref-selaOrderbydisorderSpinorbitalLiquids2014" role="doc-biblioref">8</a>]</span>. This frustration leads the model to have a quantum spin liquid (QSL) ground state, a complex many body state with a high degree of entanglement but no long range magnetic order even at zero temperature. While the possibility of a QSL ground state was suggested much earlier <span class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a href="#ref-andersonResonatingValenceBonds1973" role="doc-biblioref">9</a>]</span>, the KH model was the first exactly solvable models of the QSL state. The KH model has a rich ground state phase diagram with gapless and gapped phases, the latter supporting fractionalised quasiparticles with both Abelian and non-Abelian quasiparticle excitations. Anyons have been the subject of much attention because, among other reasons, they can be braided through spacetime to achieve noise tolerant quantum computations <span class="citation" data-cites="freedmanTopologicalQuantumComputation2003"> [<a href="#ref-freedmanTopologicalQuantumComputation2003" role="doc-biblioref">10</a>]</span>. At finite temperature the KH model undergoes a phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">11</a>]</span>. The KH model can be solved exactly via a mapping to Majorana fermions. This mapping yields an extensive number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes tied to an emergent gauge field with the remaining fermions are governed by a free fermion hamiltonian.</p>
<p>The KH can arise as the result of strong spin-orbit couplings in, for example, the transition metal based compounds <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-Jackeli2009" role="doc-biblioref">2</a>–<a href="#ref-Takagi2019" role="doc-biblioref">6</a>]</span>. The model is highly frustrated: each spin would like to align along a different direction with each of its three neighbours but this cannot be achieved even classically <span class="citation" data-cites="chandraClassicalHeisenbergSpins2010 selaOrderbydisorderSpinorbitalLiquids2014"> [<a href="#ref-chandraClassicalHeisenbergSpins2010" role="doc-biblioref">7</a>,<a href="#ref-selaOrderbydisorderSpinorbitalLiquids2014" role="doc-biblioref">8</a>]</span>. This frustration leads the model to have a Quantum Sping Liquid (QSL) ground state, a complex many-body state with a high degree of entanglement but no long-range magnetic order even at zero temperature. While the possibility of a QSL ground state was suggested much earlier <span class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a href="#ref-andersonResonatingValenceBonds1973" role="doc-biblioref">9</a>]</span>, the KH model was the first exactly solvable models of the QSL state. The KH model has a rich ground state phase diagram with gapless and gapped phases, the latter supporting fractionalised quasiparticles with both Abelian and non-Abelian quasiparticle excitations. Anyons have been the subject of much attention because, among other reasons, they can be braided through spacetime to achieve noise tolerant quantum computations <span class="citation" data-cites="freedmanTopologicalQuantumComputation2003"> [<a href="#ref-freedmanTopologicalQuantumComputation2003" role="doc-biblioref">10</a>]</span>. At finite temperature the KH model undergoes a phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">11</a>]</span>. The KH model can be solved exactly via a mapping to Majorana fermions. This mapping yields an extensive number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes tied to an emergent gauge field with the remaining fermions are governed by a free fermion hamiltonian.</p>
<p>This section will go over the standard model in detail, first discussing <a href="../2_Background/2.2_HKM_Model.html#the-spin-model">the spin model</a>, then detailing the transformation to a <a href="../2_Background/2.2_HKM_Model.html#the-majorana-model">Majorana hamiltonian</a> that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the <a href="../2_Background/2.2_HKM_Model.html#an-emergent-gauge-field">emergent gauge fields</a> and the projector. The <a href="../2_Background/2.2_HKM_Model.html#sec:anyons">next section</a> will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics. Finally will then discuss the ground state found via Lieb’s theorem as well as work on generalisations of the ground state to other lattices. Finally we will look at the <a href="../2_Background/2.2_HKM_Model.html#ground-state-phases">phase diagram</a>.</p>
</section>
<section id="the-spin-model" class="level2">
<h2>The Spin Model</h2>
<figure>
<img src="/assets/thesis/amk_chapter/visual_kitaev_1.svg" id="fig-visual_kitaev_1" data-short-caption="A Visual Intro to the Kitaev Model" style="width:100.0%" alt="Figure 2: A visual introduction to the Kitaev Model." />
<figcaption aria-hidden="true">Figure 2: A visual introduction to the Kitaev Model.</figcaption>
<img src="/assets/thesis/amk_chapter/visual_kitaev_1.svg" id="fig-visual_kitaev_1" data-short-caption="A Visual Intro to the Kitaev Model" style="width:100.0%" alt="Figure 2: A visual introduction to the Kitaev honeycomb model." />
<figcaption aria-hidden="true">Figure 2: A visual introduction to the Kitaev honeycomb model.</figcaption>
</figure>
<p>As discussed in the introduction, spin hamiltonians like that of the Kitaev model arise in electronic systems as the result the balance of multiple effects <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span>. For instance, in certain transition metal systems with <span class="math inline">\(d^5\)</span> valence electrons, crystal field and spin-orbit couplings conspire to shift and split the <span class="math inline">\(d\)</span> orbitals into moments with spin <span class="math inline">\(j = 1/2\)</span> and <span class="math inline">\(j = 3/2\)</span>. Of these, the bandwidth <span class="math inline">\(t\)</span> of the <span class="math inline">\(j= 1/2\)</span> band is small, meaning that even relatively meagre electron correlations (such those induced by the <span class="math inline">\(U\)</span> term in the Hubbard model) can lead to the opening of a Mott gap. From there we have a <span class="math inline">\(j = 1/2\)</span> Mott insulator whose effective spin-spin interactions are again shaped by the lattice geometry and spin-orbit coupling leading some materials to have strong bond-directional Ising-type interactions <span class="citation" data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a href="#ref-jackeliMottInsulatorsStrong2009" role="doc-biblioref">12</a>,<a href="#ref-khaliullinOrbitalOrderFluctuations2005" role="doc-biblioref">13</a>]</span>. In the Kitaev Model the bond directionality refers to the fact that the coupling axis <span class="math inline">\(\alpha\)</span> in terms like <span class="math inline">\(\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> is strongly bond dependent.</p>
<p>As discussed in the introduction, spin hamiltonians like that of the KH model arise in electronic systems as the result the balance of multiple effects <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span>. For instance, in certain transition metal systems with <span class="math inline">\(d^5\)</span> valence electrons, crystal field and spin-orbit couplings conspire to shift and split the <span class="math inline">\(d\)</span> orbitals into moments with spin <span class="math inline">\(j = 1/2\)</span> and <span class="math inline">\(j = 3/2\)</span>. Of these, the bandwidth <span class="math inline">\(t\)</span> of the <span class="math inline">\(j= 1/2\)</span> band is small, meaning that even relatively meagre electron correlations (such those induced by the <span class="math inline">\(U\)</span> term in the Hubbard model) can lead to the opening of a Mott gap. From there we have a <span class="math inline">\(j = 1/2\)</span> Mott insulator whose effective spin-spin interactions are again shaped by the lattice geometry and spin-orbit coupling leading some materials to have strong bond-directional Ising-type interactions <span class="citation" data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a href="#ref-jackeliMottInsulatorsStrong2009" role="doc-biblioref">12</a>,<a href="#ref-khaliullinOrbitalOrderFluctuations2005" role="doc-biblioref">13</a>]</span>. In the KH model the bond directionality refers to the fact that the coupling axis <span class="math inline">\(\alpha\)</span> in terms like <span class="math inline">\(\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> is strongly bond dependent.</p>
<p>In the spin hamiltonian eq. <a href="#eq:bg-kh-model">1</a> we can already tease out a set of conserved fluxes that will be key to the model’s solution. These fluxes are the expectations of Wilson loop operators</p>
<p><span class="math display">\[\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha},\]</span></p>
<p>the products of bonds winding around a closed path <span class="math inline">\(p\)</span> on the lattice. These operators commute with the Hamiltonian and so have no time dynamics. The winding direction does not matter so long as it is fixed. By convention we will always use clockwise. Each closed path on the lattice is associated with a flux. The number of conserved quantities grows linearly with system size and is thus extensive, this is a common property for exactly solvable systems and can be compared to the heavy electrons present in the Falicov-Kimball model. The square of two loop operators is one so any contractible loop can be expressed as a product of loops around plaquettes of the lattice, as in fig. <a href="#fig:stokes_theorem">3</a>. For the honeycomb lattice the plaquettes are the hexagons. The expectations of <span class="math inline">\(\hat{W}_p\)</span> through each plaquette, the fluxes, are therefore enough to describe the whole flux sector. We will focus on these fluxes, denoting them by <span class="math inline">\(\phi_i\)</span>. Once we have made the mapping to the Majorana Hamiltonian I will explain how these fluxes can be connected to an emergent <span class="math inline">\(B\)</span> field which makes their interpretation as fluxes clear.</p>
<figure>
<img src="/assets/thesis/amk_chapter/stokes_theorem/stokes_theorem.svg" id="fig-stokes_theorem" data-short-caption="We can construct arbitrary loops from plaquette fluxes." style="width:71.0%" alt="Figure 3: In the Kitaev Honeycomb model, Wilson loop operators \hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha} can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette \phi_i. This relationship between the u_{ij} around a region and fluxes with one is evocative of Stokes’ theorem for classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later." />
<figcaption aria-hidden="true">Figure 3: In the Kitaev Honeycomb model, Wilson loop operators <span class="math inline">\(\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha}\)</span> can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette <span class="math inline">\(\phi_i\)</span>. This relationship between the <span class="math inline">\(u_{ij}\)</span> around a region and fluxes with one is evocative of Stokes’ theorem for classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later.</figcaption>
<img src="/assets/thesis/amk_chapter/stokes_theorem/stokes_theorem.svg" id="fig-stokes_theorem" data-short-caption="We can construct arbitrary loops from plaquette fluxes." style="width:71.0%" alt="Figure 3: In the Kitaev honeycomb model, Wilson loop operators \hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha} can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette \phi_i. This relationship between the u_{ij} around a region and fluxes with one is evocative of Stokes’ theorem from classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later." />
<figcaption aria-hidden="true">Figure 3: In the Kitaev honeycomb model, Wilson loop operators <span class="math inline">\(\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha}\)</span> can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette <span class="math inline">\(\phi_i\)</span>. This relationship between the <span class="math inline">\(u_{ij}\)</span> around a region and fluxes with one is evocative of Stokes’ theorem from classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later.</figcaption>
</figure>
<p>It is worth noting in passing that the effective Hamiltonian for many Kitaev materials incorporates a contribution from an isotropic Heisenberg term <span class="math inline">\(\sum_{i,j} \vec{\sigma}_i\cdot\vec{\sigma}_j\)</span>, this is referred to as the Heisenberg-Kitaev Model <span class="citation" data-cites="Chaloupka2010"> [<a href="#ref-Chaloupka2010" role="doc-biblioref">14</a>]</span>. Materials for which the Kitaev term dominates are generally known as Kitaev Materials. See <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span> for a full discussion of Kitaev Materials.</p>
<p>As with the Falicov-Kimball model, the KH model has a extensive number of conserved quantities, the fluxes. As with the FK model it will make sense to work in the simultaneous eigenbasis of the fluxes and the Hamiltonian so that we can treat the fluxes like a classical degree of freedom. This is part of what makes the model tractable. We will find that the ground state of the model corresponds to some particular choice of fluxes. We will refer to local excitations away from the flux ground state as <em>vortices</em>. In order to fully solve the model however, we must first move to a Majorana picture.</p>
<p>It is worth noting in passing that the effective Hamiltonian for many Kitaev materials incorporates a contribution from an isotropic Heisenberg term <span class="math inline">\(\sum_{i,j} \vec{\sigma}_i\cdot\vec{\sigma}_j\)</span>, this is referred to as the Heisenberg-Kitaev model <span class="citation" data-cites="Chaloupka2010"> [<a href="#ref-Chaloupka2010" role="doc-biblioref">14</a>]</span>. Materials for which the Kitaev term dominates are generally known as Kitaev Materials. See <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span> for a full discussion of Kitaev Materials.</p>
<p>As with the Falicov-Kimball model, the KH model has a extensive number of conserved quantities, the fluxes. So again we will work in the simultaneous eigenbasis of the fluxes and the Hamiltonian so that we can treat the fluxes like a classical degree of freedom. This is part of what makes the model tractable. We will find that the ground state of the model corresponds to some particular choice of fluxes. We will refer to local excitations away from the flux ground state as <em>vortices</em>. In order to fully solve the model however, we must first move to a Majorana picture.</p>
</section>
<section id="the-majorana-model" class="level2">
<h2>The Majorana Model</h2>
<p>Majorana fermions are something like ‘half of a complex fermion’ and are their own antiparticle. From a set of <span class="math inline">\(N\)</span> fermionic creation <span class="math inline">\(f_i^\dagger\)</span> and anhilation <span class="math inline">\(f_i\)</span> operators we can construct <span class="math inline">\(2N\)</span> Majorana operators <span class="math inline">\(c_m\)</span>. We can do this construction in multiple ways subject to only mild constraints required to keep the overall commutations relations correct <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. Majorana operators square to one but otherwise have standard fermionic commutation relations.</p>
<p>Majorana fermions are something like ‘half of a complex fermion’ and are their own antiparticle. From a set of <span class="math inline">\(N\)</span> fermionic creation <span class="math inline">\(f_i^\dagger\)</span> and anhilation <span class="math inline">\(f_i\)</span> operators we can construct <span class="math inline">\(2N\)</span> Majorana operators <span class="math inline">\(c_m\)</span>. We can do this construction in multiple ways subject to only mild constraints required to keep the overall commutations relations correct <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. Majorana operators square to one but otherwise have standard fermionic anti-commutation relations.</p>
<p><span class="math inline">\(N\)</span> spins can be mapped to <span class="math inline">\(N\)</span> fermions with the well known Jordan-Wigner transformation and indeed this approach can be used to solve the Kitaev model <span class="citation" data-cites="chenExactResultsKitaev2008"> [<a href="#ref-chenExactResultsKitaev2008" role="doc-biblioref">15</a>]</span>. Here I will introduce the method Kitaev used in the original paper as this forms the basis for the results that will be presented in this thesis. Rather than mapping to <span class="math inline">\(N\)</span> fermions, Kitaev maps to <span class="math inline">\(4N\)</span> Majoranas, effectively <span class="math inline">\(2N\)</span> fermions. In contrast to the Jordan-Wigner approach which makes fermions out of strings of spin operators in order to correctly produce fermionic commutation relations, the Kitaev transformation maps each spin locally to four Majoranas. The downside is that this enlarges the Hilbert space from <span class="math inline">\(2^N\)</span> to <span class="math inline">\(4^N\)</span>. We will have to employ a projector <span class="math inline">\(\hat{P}\)</span> to come back down to the physical Hilbert space later. As everything is local, I will drop the site indices <span class="math inline">\(ijk\)</span> in expressions that refer to only a single site.</p>
<p>The mapping is defined in terms of four Majoranas per site <span class="math inline">\(b_i^x,\;b_i^y,\;b_i^z,\;c_i\)</span> such that</p>
<p><span id="eq:bg-kh-mapping"><span class="math display">\[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^z = i b^z c\qquad{(2)}\]</span></span></p>
@ -126,7 +126,7 @@ b^x = (f + f^\dagger),\;\;&amp; b^y = -i(f - f^\dagger),\\
b^z = (g + g^\dagger),\;\;&amp; c = -i(g - g^\dagger),
\end{aligned}\]</span></p>
<p>Working through the algebra we see that the operator <span class="math inline">\(D = b^x b^y b^z c\)</span> is equal to the fermion parity <span class="math inline">\(D = -(2n_f - 1)(2n_g - 1) = \pm1\)</span> where <span class="math inline">\(n_f,\; n_g\)</span> are the number operators. So setting <span class="math inline">\(D = 1\)</span> everywhere is equivalent to restricting to the <span class="math inline">\(\{|01\rangle,|10\rangle\}\)</span> though we could equally well have used the other one.</p>
<p>Expanding the product <span class="math inline">\(\prod_i P_i\)</span> out, we find that the projector corresponds to a symmetrisation over <span class="math inline">\(\{u_{ij}\}\)</span> states within a flux sector and and overall fermion parity <span class="math inline">\(\prod_i D_i\)</span>. The significance of this is that an arbitrary many body state can be made to have non-zero overlap with the physical subspace via the addition or removal of just a single fermion. This implies that in the thermodynamic limit the projection step is not generally necessary to extract physical results, see <span class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a href="#ref-pedrocchiPhysicalSolutionsKitaev2011" role="doc-biblioref">17</a>]</span> or <a href="../6_Appendices/A.5_The_Projector.html#app-the-projector">appendix A.5</a> for more details.</p>
<p>Expanding the product <span class="math inline">\(\prod_i P_i\)</span> out, we find that the projector corresponds to a symmetrisation over <span class="math inline">\(\{u_{ij}\}\)</span> states within a flux sector and and overall fermion parity <span class="math inline">\(\prod_i D_i\)</span>, see <span class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a href="#ref-pedrocchiPhysicalSolutionsKitaev2011" role="doc-biblioref">17</a>]</span> or <a href="../6_Appendices/A.5_The_Projector.html#app-the-projector">appendix A.5</a> for the full derivation. The significance of this is that an arbitrary many-body state can be made to have non-zero overlap with the physical subspace via the addition or removal of just a single fermion. This implies that in the thermodynamic limit the projection step is not generally necessary to extract physical results</p>
<p>We can now rewrite the spin hamiltonian in Majorana form with the caveat that they are only strictly equivalent after projection. The Ising interactions <span class="math inline">\(\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> decouple into the form <span class="math inline">\(-i (i b^\alpha_i b^\alpha_j) c_i c_j\)</span>. We factor out the <em>bond operators</em> <span class="math inline">\(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\)</span> which are Hermitian and, remarkably, commute with the Hamiltonian and each other.</p>
<p><span class="math display">\[\begin{aligned}
\tilde{H} &amp;= - \sum_{\langle i,j\rangle_\alpha} J^{\alpha}\tilde{\sigma}_i^{\alpha}\tilde{\sigma}_j^{\alpha}\\
@ -148,7 +148,7 @@ H &amp;= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}
<p><span class="math display">\[ f_i = \tfrac{1}{2} (b_m + ib_m&#39;)\]</span></p>
<p>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">\[ H = \sum_m \epsilon_m (n_m - \tfrac{1}{2}).\]</span></p>
<p>The energy of the ground state <span class="math inline">\(|n_m = 0\rangle\)</span> of the many body system at fixed <span class="math inline">\(\{u_{ij}\}\)</span> is</p>
<p>The energy of the ground state <span class="math inline">\(|n_m = 0\rangle\)</span> of the many-body system at fixed <span class="math inline">\(\{u_{ij}\}\)</span> is</p>
<p><span class="math display">\[E_{0} = -\frac{1}{2}\sum_m \epsilon_m \]</span></p>
<p>and we can construct any state from a particular choice of <span class="math inline">\(n_m = 0,1\)</span>. If we only care about the ground state energy <span class="math inline">\(E_{0}\)</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 set to recover <span class="math inline">\(\sum_m \epsilon_m\)</span> directly.</p>
</section>
@ -157,11 +157,11 @@ H &amp;= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}
<p>We have transformed the spin Hamiltonian into a Majorana hamiltonian <span class="math inline">\(H = i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\)</span> describing the dynamics of a classical field <span class="math inline">\(u_{ij}\)</span> and Majoranas <span class="math inline">\(c_i\)</span>. It is natural to ask how the classical field <span class="math inline">\(u_{ij}\)</span> relates to the fluxes of the original spin model. We can evaluate the fluxes <span class="math inline">\(\phi_i\)</span> in terms of the bond operators</p>
<p><span id="eq:flux-majorana"><span class="math display">\[\phi_i = \prod_{\langle j,k\rangle \in \mathcal{P}_i} i u_{jk}.\qquad{(4)}\]</span></span></p>
<figure>
<img src="/assets/thesis/amk_chapter/intro/gauge_symmetries/gauge_symmetries.svg" id="fig-gauge_symmetries" data-short-caption="Gauge Symmetries" style="width:100.0%" alt="Figure 4: A honeycomb lattice with edges in light grey, along with its dual, the triangle lattice in light blue. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator D_j at a vertex is to flip the value of the three u_{jk} variables (black lines) surrounding site j. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent D_j operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing D_j operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of u_{ij}s flipped on its edge. Therefore, all retain the same flux \phi_i." />
<figcaption aria-hidden="true">Figure 4: A honeycomb lattice with edges in light grey, along with its dual, the triangle lattice in light blue. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator <span class="math inline">\(D_j\)</span> at a vertex is to flip the value of the three <span class="math inline">\(u_{jk}\)</span> variables (black lines) surrounding site <span class="math inline">\(j\)</span>. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent <span class="math inline">\(D_j\)</span> operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing <span class="math inline">\(D_j\)</span> operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of <span class="math inline">\(u_{ij}\)</span>s flipped on its edge. Therefore, all retain the same flux <span class="math inline">\(\phi_i\)</span>.</figcaption>
<img src="/assets/thesis/amk_chapter/intro/gauge_symmetries/gauge_symmetries.svg" id="fig-gauge_symmetries" data-short-caption="Gauge Symmetries" style="width:100.0%" alt="Figure 4: A honeycomb lattice with edges in grey, along with its dual, the triangle lattice in red. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator D_j at a vertex is to flip the value of the three u_{jk} variables (black lines) surrounding site j. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent D_j operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing D_j operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of u_{ij}s flipped on its edge. Therefore, all retain the same flux \phi_i." />
<figcaption aria-hidden="true">Figure 4: A honeycomb lattice with edges in grey, along with its dual, the triangle lattice in red. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator <span class="math inline">\(D_j\)</span> at a vertex is to flip the value of the three <span class="math inline">\(u_{jk}\)</span> variables (black lines) surrounding site <span class="math inline">\(j\)</span>. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent <span class="math inline">\(D_j\)</span> operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing <span class="math inline">\(D_j\)</span> operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of <span class="math inline">\(u_{ij}\)</span>s flipped on its edge. Therefore, all retain the same flux <span class="math inline">\(\phi_i\)</span>.</figcaption>
</figure>
<p>In addition, the bond operators form a highly degenerate description of the system. 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">\(H\)</span> forming a set of local symmetries. 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>. This changes the bond configuration <span class="math inline">\(\{u_{ij}\}\)</span> but leaves the flux configuration <span class="math inline">\(\{\phi_i\}\)</span> unchanged. Physically, we interpret <span class="math inline">\(u_{ij}\)</span> as a gauge field with a high degree of degeneracy and <span class="math inline">\(\{D_i\}\)</span> as the set of gauge symmetries. The Majorana bond operators <span class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span class="math inline">\(\mathbb{Z}_2\)</span> gauge field! The flux configuration <span class="math inline">\(\{\phi_i\}\)</span> is what encodes physical information about the system without all the gauge degeneracy.</p>
<p>The ground state of the Kitaev Honeycomb model is the all one flux configuration <span class="math inline">\(\{\phi_i = +1\; \forall \; i\}\)</span>. This can be proven via Lieb’s theorem <span class="citation" data-cites="lieb_flux_1994"> [<a href="#ref-lieb_flux_1994" role="doc-biblioref">19</a>]</span> which gives the lowest energy magnetic flux configuration for a system of electrons hopping in a magnetic field. Kitaev remarks in his original paper that he was not initially aware of the relevance of Lieb’s 1994 result. This is not surprising because at first glance the two models seem quite different but the connection is quite instructive for understanding the Kitaev Model and its generalisations.</p>
<p>The ground state of the KH model is the flux configuration where all fluxes are one <span class="math inline">\(\{\phi_i = +1\; \forall \; i\}\)</span>. This can be proven via Lieb’s theorem <span class="citation" data-cites="lieb_flux_1994"> [<a href="#ref-lieb_flux_1994" role="doc-biblioref">19</a>]</span> which gives the lowest energy magnetic flux configuration for a system of electrons hopping in a magnetic field. Kitaev remarks in his original paper that he was not initially aware of the relevance of Lieb’s 1994 result. This is not surprising because at first glance the two models seem quite different but the connection is quite instructive for understanding the KH and its generalisations.</p>
<p>Lieb discusses a model of mobile electrons</p>
<p><span class="math display">\[H = \sum_{ij} t_{ij} c^\dagger_i c_j\]</span></p>
<p>where the hopping terms <span class="math inline">\(t_{ij} = |t_{ij}|\exp(i\theta_{ij})\)</span> incorporate Aharanhov-Bohm (AB) phases <span class="citation" data-cites="aharonovSignificanceElectromagneticPotentials1959"> [<a href="#ref-aharonovSignificanceElectromagneticPotentials1959" role="doc-biblioref">20</a>]</span> <span class="math inline">\(\theta_{ij}\)</span>. The AB phases model the effect of a slowly varying magnetic field on the electrons through the integral of the magnetic vector potential <span class="math inline">\(\theta_{ij} = \int_i^j \vec{A} \cdot d\vec{l}\)</span>, a Peierls substitution <span class="citation" data-cites="peierlsZurTheorieDiamagnetismus1933"> [<a href="#ref-peierlsZurTheorieDiamagnetismus1933" role="doc-biblioref">21</a>]</span>. If we map the Majorana form of the Kitaev model to Lieb’s model we see that our <span class="math inline">\(t_{ij} = i J^\alpha u_{ij}\)</span>. The <span class="math inline">\(i u_{ij} = \pm i\)</span> correspond to AB phases <span class="math inline">\(\theta_{ij} = \pi/2\)</span> or <span class="math inline">\(3\pi/2\)</span> along each bond.</p>
@ -179,7 +179,7 @@ H &amp;= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}
\end{aligned}\qquad{(5)}\]</span></span></p>
<p>Thus we can interpret the fluxes <span class="math inline">\(\phi_i\)</span> as the exponential of magnetic fluxes <span class="math inline">\(Q_m\)</span> of some fictitious gauge field <span class="math inline">\(\vec{A}\)</span> and the bond operators as <span class="math inline">\(i u_{ij} = \exp i \int_i^j \vec{A} \cdot d\vec{l}\)</span>. In this analogy to classical electromagnetism, the sets <span class="math inline">\(\{u_{ij}\}\)</span> that correspond to the same <span class="math inline">\(\{\phi_i\}\)</span> are all gauge equivalent as we have already seen via other means. The fact that fluxes can be written as products of bond operators and composed is a consequence of eq. <a href="#eq:flux-magnetic">5</a>. If the lattice contains odd plaquettes, as in the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">26</a>]</span>, the complex fluxes that appear are a sign that chiral symmetry has been broken.</p>
<p>In full, Lieb’s theorem states that the ground state has magnetic flux <span class="math inline">\(Q_i = \sum_{\mathcal{P}_i}\theta_{ij} = \pi \; (\mathrm{mod} \;2\pi)\)</span> for plaquettes with <span class="math inline">\(0 \; (\mathrm{mod}\;4)\)</span> sides and <span class="math inline">\(0 \; (\mathrm{mod}\;2\pi)\)</span> for plaquettes with <span class="math inline">\(2 \; (\mathrm{mod}\;4)\)</span> sides. In terms of our fluxes, this means <span class="math inline">\(\phi = -1\)</span> for squares, <span class="math inline">\(\phi = 1\)</span> for hexagons and so on.</p>
<p>While Lieb’s theorem is restricted to bipartite lattices with translational symmetry, other works has shown numerically that it tends to hold for more general lattices too <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">22</a>–<a href="#ref-Peri2020" role="doc-biblioref">25</a>]</span>. From this we find that the generalisation to odd sided plaquettes is similar but with an additional chiral symmetry, so <span class="math inline">\(\phi = \pm i\)</span> for plaquettes with <span class="math inline">\(1 \; (\mathrm{mod}\;4)\)</span> sides and <span class="math inline">\(\mp i\)</span> for those with <span class="math inline">\(3 \; (\mathrm{mod}\;4)\)</span> sides. Overall we can write <span class="math inline">\(\phi = -(\pm i)^{n_{\mathrm{sides}}}\)</span>. Later I will present numerical evidence that this rule continues to hold for general amorphous lattices.</p>
<p>While Lieb’s theorem is restricted to bipartite lattices with translational symmetry, other works have shown numerically that it tends to hold for more general lattices too <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">22</a>–<a href="#ref-Peri2020" role="doc-biblioref">25</a>]</span>. From this we find that the generalisation to odd sided plaquettes is similar but with an additional chiral symmetry, so <span class="math inline">\(\phi = \pm i\)</span> for plaquettes with <span class="math inline">\(1 \; (\mathrm{mod}\;4)\)</span> sides and <span class="math inline">\(\mp i\)</span> for those with <span class="math inline">\(3 \; (\mathrm{mod}\;4)\)</span> sides. Overall we can write <span class="math inline">\(\phi = -(\pm i)^{n_{\mathrm{sides}}}\)</span>. Later I will present numerical evidence that this rule continues to hold for general amorphous lattices.</p>
<p>Understanding <span class="math inline">\(u_{ij}\)</span> as a gauge field provides another way to understand the action of the projector. The local projector <span class="math inline">\(P_i = \frac{1 + D_i}{2}\)</span> applied to a state constructs a superposition of the original state and the gauge equivalent state linked to it by flipping the three <span class="math inline">\(u_{ij}\)</span> around site <span class="math inline">\(i\)</span>. The overall projector <span class="math inline">\(P = \prod_i P_i\)</span> can thus be understood as a symmetrisation over all gauge equivalent states, removing the gauge degeneracy introduced by the mapping from spins to Majoranas.</p>
<!-- <figure>
<img src="../../figure_code/amk_chapter/intro/flood_fill/flood_fill.gif" style="max-width:700px;" title="Gauge Operators">
@ -196,12 +196,12 @@ A honeycomb lattice (in black) along with its dual (in red). (Left) The product
<section id="sec:anyons" class="level2">
<h2>Anyons, Topology and the Chern number</h2>
<figure>
<img src="/assets/thesis/amk_chapter/braiding.png" id="fig-braiding" data-short-caption="Braiding in Two Dimensions" style="width:71.0%" alt="Figure 7: Worldlines of particles in two dimensions can become tangled or braided with one another." />
<figcaption aria-hidden="true">Figure 7: Worldlines of particles in two dimensions can become tangled or <em>braided</em> with one another.</figcaption>
<img src="/assets/thesis/amk_chapter/braiding.png" id="fig-braiding" data-short-caption="Braiding in Two Dimensions" style="width:71.0%" alt="Figure 7: Worldlines of particles in 2D can become tangled or braided with one another." />
<figcaption aria-hidden="true">Figure 7: Worldlines of particles in 2D can become tangled or <em>braided</em> with one another.</figcaption>
</figure>
<p>To discuss different ground state phases of the KH model we must first review the topic of anyons and topology. The standard argument for the existence of Fermions and Bosons goes like this: the quantum state of a system must pick up a factor of <span class="math inline">\(\pm1\)</span> if two identical particles are swapped. Only <span class="math inline">\(\pm1\)</span> are allowed since swapping twice must correspond to the identity. This argument works in three dimensions for states without topological degeneracy, which seems to be true of the real world, but condensed matter systems are subject to no such constraints.</p>
<p>To discuss different ground state phases of the KH model we must first review the topic of anyons and topology. The standard argument for the existence of Fermions and Bosons goes like this: the quantum state of a system must pick up a factor of <span class="math inline">\(\pm1\)</span> if two identical particles are swapped. Only <span class="math inline">\(\pm1\)</span> are allowed since swapping twice must correspond to the identity. This argument works in 3D for states without topological degeneracy, which seems to be true of the real world, but condensed matter systems are subject to no such constraints.</p>
<p>In gapped condensed matter systems, all equal time correlators decay exponentially with distance <span class="citation" data-cites="hastingsLiebSchultzMattisHigherDimensions2004"> [<a href="#ref-hastingsLiebSchultzMattisHigherDimensions2004" role="doc-biblioref">28</a>]</span>. Put another way, gapped systems support quasiparticles with a definite location in space and finite extent. As such it is meaningful to consider what would happen to the overall quantum state if we were to adiabatically carry out a series of swaps as described above. This is known as braiding. Recently, braiding in topological systems has attracted interest because of proposals to use ground state degeneracy to implement both passively fault tolerant and actively stabilised quantum computations <span class="citation" data-cites="kitaev_fault-tolerant_2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">29</a>–<a href="#ref-hastingsDynamicallyGeneratedLogical2021" role="doc-biblioref">31</a>]</span>.</p>
<p>First we realise that in two dimensions, swapping identical particles twice is not topologically equivalent to the identity, see fig. <a href="#fig:braiding">7</a>. Instead it corresponds to encircling one particle around the other. This means we can in general pick up any complex phase <span class="math inline">\(e^{i\theta}\)</span>, hence the name <strong>any</strong>-ons upon exchange. These are known as Abelian anyons because complex multiplication commutes and hence the group of braiding operations forms an Abelian group.</p>
<p>First we realise that in 2D, swapping identical particles twice is not topologically equivalent to the identity, see fig. <a href="#fig:braiding">7</a>. Instead it corresponds to encircling one particle around the other. This means we can in general pick up any complex phase <span class="math inline">\(e^{i\theta}\)</span> upon exchange, hence the name <strong>any</strong>-ons. These are known as Abelian anyons because complex multiplication commutes and hence the group of braiding operations forms an Abelian group.</p>
<p>The KH model has a topologically degenerate ground state with sectors labelled by the values of the topological fluxes <span class="math inline">\((\Phi_x\)</span>, <span class="math inline">\(\Phi_y)\)</span>. Consider the operation in which a quasiparticle pair is created from the ground state, transported around one of the non-contractible loops and then annihilated together, call them <span class="math inline">\(\mathcal{T}_{x}\)</span> and <span class="math inline">\(\mathcal{T}_{y}\)</span>. These operations move the system around within the ground state manifold and they need not commute. This leads to non-Abelian anyons. As Kitaev points out, these operations are not specific to the torus: the operation <span class="math inline">\(\mathcal{T}_{x}\mathcal{T}_{y}\mathcal{T}_{x}^{-1}\mathcal{T}_{y}^{-1}\)</span> corresponds to an operation in which none of the particles crosses the torus, rather one simply winds around the other, hence these effects are relevant even for the planar case.</p>
<!-- <figure>
<img src="../../figure_code/amk_chapter/intro/types_of_dual_loops_animated/types_of_dual_loops_animated.gif" style="max-width:700px;" title="Dual Loops and Vortex Pairs">
@ -216,16 +216,16 @@ This all works the same way for the amorphous lattice but the diagram is a lot m
<section id="ground-state-phases" class="level2">
<h2>Ground State Phases</h2>
<figure>
<img src="/assets/thesis/background_chapter/KH_phase_diagram.svg" id="fig-KH_phase_diagram" data-short-caption="Kitaev Honeycomb Model Phase Diagram" style="width:100.0%" alt="Figure 8: Setting the energy scale of the Kitaev Model with the constraint that J_x + J_y + J_z = 1 yields a triangular phase diagram where each of the corners represents J_\alpha = 1. For each corner \alpha the region |J_\alpha &gt; |J_\beta| + |J_\gamma| supports a gapped non-Abelian phase equivalent to that of the Toric code  [29,37]. The point around equal coupling J_x = J_y = J_z, the B phase, is gapless. The B phase is known a Majorana metal and on the honeycomb lattice it has a Dirac cone dispersion similar to that of graphene." />
<figcaption aria-hidden="true">Figure 8: Setting the energy scale of the Kitaev Model with the constraint that <span class="math inline">\(J_x + J_y + J_z = 1\)</span> yields a triangular phase diagram where each of the corners represents <span class="math inline">\(J_\alpha = 1\)</span>. For each corner <span class="math inline">\(\alpha\)</span> the region <span class="math inline">\(|J_\alpha &gt; |J_\beta| + |J_\gamma|\)</span> supports a gapped non-Abelian phase equivalent to that of the Toric code <span class="citation" data-cites="kitaev1997quantum kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">29</a>,<a href="#ref-kitaev1997quantum" role="doc-biblioref">37</a>]</span>. The point around equal coupling <span class="math inline">\(J_x = J_y = J_z\)</span>, the B phase, is gapless. The B phase is known a Majorana metal and on the honeycomb lattice it has a Dirac cone dispersion similar to that of graphene.</figcaption>
<img src="/assets/thesis/background_chapter/KH_phase_diagram.svg" id="fig-KH_phase_diagram" data-short-caption="Kitaev Honeycomb Model Phase Diagram" style="width:100.0%" alt="Figure 8: Setting the energy scale of the KH model with the constraint that J_x + J_y + J_z = 1 yields a triangular phase diagram where each of the corners represents J_\alpha = 1. For each corner \alpha the region |J_\alpha &gt; |J_\beta| + |J_\gamma| supports a gapped non-Abelian phase equivalent to that of the Toric code  [29,37]. The point around equal coupling J_x = J_y = J_z, the B phase, is gapless. The B phase is known a Majorana metal and on the honeycomb lattice it has a Dirac cone dispersion similar to that of graphene." />
<figcaption aria-hidden="true">Figure 8: Setting the energy scale of the KH model with the constraint that <span class="math inline">\(J_x + J_y + J_z = 1\)</span> yields a triangular phase diagram where each of the corners represents <span class="math inline">\(J_\alpha = 1\)</span>. For each corner <span class="math inline">\(\alpha\)</span> the region <span class="math inline">\(|J_\alpha &gt; |J_\beta| + |J_\gamma|\)</span> supports a gapped non-Abelian phase equivalent to that of the Toric code <span class="citation" data-cites="kitaev1997quantum kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">29</a>,<a href="#ref-kitaev1997quantum" role="doc-biblioref">37</a>]</span>. The point around equal coupling <span class="math inline">\(J_x = J_y = J_z\)</span>, the B phase, is gapless. The B phase is known a Majorana metal and on the honeycomb lattice it has a Dirac cone dispersion similar to that of graphene.</figcaption>
</figure>
<p>Setting the overall energy scale with the constraint <span class="math inline">\(J_x + J_y + J_z = 1\)</span> yields a triangular phase diagram. In each of the corners one of the spin-coupling directions dominates, <span class="math inline">\(|J_\alpha &gt; |J_\beta| + |J_\gamma|\)</span>, yielding three equivalent <span class="math inline">\(A_\alpha\)</span> phases while the central triangle around <span class="math inline">\(J_x = J_y = J_z\)</span> is called the B phase. Both phases support two kinds of quasiparticles, fermions and <span class="math inline">\(\mathbb{Z}_2\)</span>-vortices. In the A phases, the vortices have bosonic statistics with respect to themselves but act like fermions with respect to the fermions, hence they are Abelian anyons, This phase has the same anyonic structure as the Toric code <span class="citation" data-cites="kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">29</a>]</span>. The B phase can be described as a semi-metal of the Majorana fermions <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span>. Since the B phase is gapless, the quasiparticles aren’t localised and so don’t have braiding statistics.</p>
<p>An external magnetic can be used to break chiral symmetry. The lowest order term that breaks chiral symmetry but retains the solvability of the model is the three spin term <span class="math display">\[
\sum_{(i,j,k)} \sigma_i^{\alpha} \sigma_j^{\beta} \sigma_k^{\gamma}
\]</span> where the sum <span class="math inline">\((i,j,k)\)</span> runs over consecutive indices around plaquettes. The addition of this to the spin model leads to two bond terms in the corresponding Majorana model. The effect of breaking chiral symmetry is to open a gap in the B phase. The vortices of the gapped B phase are non-Abelian anyons. This phase has the same anyonic exchange statistics as <span class="math inline">\(p_x + ip_y\)</span> superconductor <span class="citation" data-cites="readPairedStatesFermions2000"> [<a href="#ref-readPairedStatesFermions2000" role="doc-biblioref">38</a>]</span>, the Moore-Read state for the <span class="math inline">\(\nu = 5/2\)</span> fractional quantum Hall state <span class="citation" data-cites="mooreNonabelionsFractionalQuantum1991"> [<a href="#ref-mooreNonabelionsFractionalQuantum1991" role="doc-biblioref">39</a>]</span> and many other systems <span class="citation" data-cites="aliceaNonAbelianStatisticsTopological2011 fuSuperconductingProximityEffect2008 lutchynMajoranaFermionsTopological2010 oregHelicalLiquidsMajorana2010 sauGenericNewPlatform2010"> [<a href="#ref-aliceaNonAbelianStatisticsTopological2011" role="doc-biblioref">40</a>–<a href="#ref-sauGenericNewPlatform2010" role="doc-biblioref">44</a>]</span>. Collectively these systems have attracted interest as possible physical realisations for braiding based quantum computers.</p>
<p>At finite temperatures, recent work has shown that the KH model undergoes a transition to a thermal metal phase.</p>
<p>To surmise, the Kitaev Honeycomb model is remarkable because it combines three key properties. First, the form of the Hamiltonian plausibly be realised by a real material. Candidate materials, such as <span class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have sufficiently strong spin-orbit coupling and the correct lattice structure to behave according to the Kitaev Honeycomb model with small corrections <span class="citation" data-cites="banerjeeProximateKitaevQuantum2016 TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>,<a href="#ref-banerjeeProximateKitaevQuantum2016" role="doc-biblioref">45</a>]</span>. Second, its ground state is the canonical example of the long sought after quantum spin liquid state, its dynamical spin-spin correlation functions are zero beyond nearest neighbour separation <span class="citation" data-cites="baskaranExactResultsSpin2007"> [<a href="#ref-baskaranExactResultsSpin2007" role="doc-biblioref">46</a>]</span>. Its excitations are anyons, particles that can only exist in two dimensions that break the normal fermion/boson dichotomy.</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"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. The solubility of the Kitaev Honeycomb Model, like the Falicov-Kimball model of chapter 1, comes about because the model has extensively many conserved degrees of freedom. These conserved quantities can be factored out as classical degrees of freedom, leaving behind a non-interacting quantum model that is easy to solve.</p>
<p>At finite temperatures, recent work has shown that the KH model undergoes a transition to a thermal metal phase. Vortex disorder causes the fermion gap to fill up and the DOS has a characteristic logarithmic divergence at zero energy which can be understood from random matrix theory <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">11</a>]</span>.</p>
<p>To surmise, the KH model is remarkable because it combines three key properties. First, the form of the Hamiltonian can plausibly be realised by a real material. Candidate materials, such as <span class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have sufficiently strong spin-orbit coupling and the correct lattice structure to behave according to the KH model with small corrections <span class="citation" data-cites="banerjeeProximateKitaevQuantum2016 TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>,<a href="#ref-banerjeeProximateKitaevQuantum2016" role="doc-biblioref">45</a>]</span>. Second, its ground state is the canonical example of the long sought after QSL state, its dynamical spin-spin correlation functions are zero beyond nearest neighbour separation <span class="citation" data-cites="baskaranExactResultsSpin2007"> [<a href="#ref-baskaranExactResultsSpin2007" role="doc-biblioref">46</a>]</span>. Its excitations are anyons, particles that can only exist in 2D that break the normal fermion/boson dichotomy.</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"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. The solubility of the KH model, like the Falicov-Kimball model of chapter 1, comes about because the model has extensively many conserved degrees of freedom. These conserved quantities can be factored out as classical degrees of freedom, leaving behind a non-interacting quantum model that is easy to solve.</p>
<p>Next Section: <a href="../2_Background/2.4_Disorder.html">Disorder and Localisation</a></p>
</section>
</section>

36
_thesis/2_Background/2.4_Disorder.html
View File

@ -41,7 +41,10 @@ image:
<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
<ul>
<li><a href="#topological-disorder" id="toc-topological-disorder">Topological Disorder</a></li>
<li><a href="#diagnosing-localisation-in-practice" id="toc-diagnosing-localisation-in-practice">Diagnosing Localisation in practice</a></li>
<li><a href="#diagnosing-localisation-in-practice" id="toc-diagnosing-localisation-in-practice">Diagnosing Localisation in practice</a>
<ul>
<li><a href="#chapter-summary" id="toc-chapter-summary">Chapter Summary</a></li>
</ul></li>
</ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
@ -59,7 +62,10 @@ image:
<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
<ul>
<li><a href="#topological-disorder" id="toc-topological-disorder">Topological Disorder</a></li>
<li><a href="#diagnosing-localisation-in-practice" id="toc-diagnosing-localisation-in-practice">Diagnosing Localisation in practice</a></li>
<li><a href="#diagnosing-localisation-in-practice" id="toc-diagnosing-localisation-in-practice">Diagnosing Localisation in practice</a>
<ul>
<li><a href="#chapter-summary" id="toc-chapter-summary">Chapter Summary</a></li>
</ul></li>
</ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
@ -79,21 +85,21 @@ image:
H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j.
\qquad{(1)}\]</span></span></p>
<p>It is one of non-interacting fermions subject to a disorder potential <span class="math inline">\(V_j\)</span> drawn uniformly from the interval <span class="math inline">\([-W,W]\)</span>. The discovery of localisation in quantum systems was surprising at the time given the seeming ubiquity of extended Bloch states. Within the Anderson model, all the states localise at the same disorder strength <span class="math inline">\(W\)</span>. Later Mott showed that in other contexts extended Bloch states and localised states can coexist at the same disorder strength but at different energies. The transition in energy between localised and extended states is known as a mobility edge <span class="citation" data-cites="mottMetalInsulatorTransitions1978"> [<a href="#ref-mottMetalInsulatorTransitions1978" role="doc-biblioref">4</a>]</span>.</p>
<p>Localisation phenomena are strongly dimension dependent. In three dimensions the scaling theory of localisation <span class="citation" data-cites="edwardsNumericalStudiesLocalization1972 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-edwardsNumericalStudiesLocalization1972" role="doc-biblioref">5</a>,<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span> shows that Anderson localisation is a critical phenomenon with critical exponents both for how the conductivity vanishes with energy when approaching the mobility edge and for how the localisation length increases below it. By contrast, in one dimension disorder generally dominates. Even the weakest disorder exponentially localises <em>all</em> single particle eigenstates in the one dimensional Anderson model. Only long-range spatial correlations of the disorder potential can induce delocalisation <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990 izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">7</a>–<a href="#ref-izrailevAnomalousLocalizationLowDimensional2012" role="doc-biblioref">12</a>]</span>.</p>
<p>Localisation phenomena are strongly dimension dependent. In 3D the scaling theory of localisation <span class="citation" data-cites="edwardsNumericalStudiesLocalization1972 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-edwardsNumericalStudiesLocalization1972" role="doc-biblioref">5</a>,<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span> shows that Anderson localisation is a critical phenomenon with critical exponents both for how the conductivity vanishes with energy when approaching the mobility edge and for how the localisation length increases below it. By contrast, in 1D disorder generally dominates. Even the weakest disorder exponentially localises <em>all</em> single particle eigenstates in the 1D Anderson model. Only long-range spatial correlations of the disorder potential can induce delocalisation <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990 izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">7</a>–<a href="#ref-izrailevAnomalousLocalizationLowDimensional2012" role="doc-biblioref">12</a>]</span>.</p>
<p>Later localisation was found in disordered interacting many-body systems:</p>
<p><span class="math display">\[
H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U\sum_{jk} n_j n_k
\]</span> Here, in contrast to the Anderson model, localisation phenomena are robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL) <span class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016 gogolinEquilibrationThermalisationEmergence2016"> [<a href="#ref-imbrieManyBodyLocalizationQuantum2016" role="doc-biblioref">13</a>,<a href="#ref-gogolinEquilibrationThermalisationEmergence2016" role="doc-biblioref">14</a>]</span>.</p>
<p>Both MBL and Anderson localisation depend crucially on the presence of <em>quenched</em> disorder. Quenched disorder takes the form a static background field drawn from an arbitrary probability distribution to which the model is coupled. Disorder may also be introduced into the initial state of the system rather than the Hamiltonian. This has led to ongoing interest in the possibility of disorder-free localisation where the disorder is instead <em>annealed</em>. In this scenario the disorder necessary to generate localisation is generated entirely from the thermal fluctuations of the model.</p>
\]</span> Here, in contrast to the Anderson model, localisation phenomena are robust to weak perturbations of the Hamiltonian. This is called many-body localisation <span class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016 gogolinEquilibrationThermalisationEmergence2016"> [<a href="#ref-imbrieManyBodyLocalizationQuantum2016" role="doc-biblioref">13</a>,<a href="#ref-gogolinEquilibrationThermalisationEmergence2016" role="doc-biblioref">14</a>]</span>.</p>
<p>Both many-body localisation and Anderson localisation depend crucially on the presence of <em>quenched</em> disorder. Quenched disorder takes the form a static background field drawn from an arbitrary probability distribution to which the model is coupled. Disorder may also be introduced into the initial state of the system rather than the Hamiltonian. This has led to ongoing interest in the possibility of disorder-free localisation where the disorder is instead <em>annealed</em>. In this scenario the disorder necessary to generate localisation is generated entirely from the thermal fluctuations of the model.</p>
<p>The concept of disorder-free localisation was first proposed in the context of Helium mixtures <span class="citation" data-cites="kagan1984localization"> [<a href="#ref-kagan1984localization" role="doc-biblioref">15</a>]</span> and then extended to heavy-light mixtures in which multiple species with large mass ratios interact. The idea is that the heavier particles act as an effective disorder potential for the lighter ones, inducing localisation. Two such models <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">16</a>,<a href="#ref-schiulazDynamicsManybodyLocalized2015" role="doc-biblioref">17</a>]</span> instead find that the models thermalise exponentially slowly in system size, which Ref. <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">16</a>]</span> dubs Quasi-MBL.</p>
<p>True disorder-free localisation does occur in exactly solvable models with extensively many conserved quantities <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">18</a>]</span>. As conserved quantities have no time dynamics this can be thought of as taking the separation of timescales to the infinite limit. The localisation phenomena present in the Falicov-Kimball model are instead the result of annealed disorder. A strong separation of timescales means that the heavy species is approximated as immobile with respect to the lighter itinerant species. At finite temperature the heavy species acts as a disorder potential for the lighter one. However, in contrast to quenched disorder, the probability distribution of annealed disorder is entirely determined by the thermodynamics of the Hamiltonian. In the two dimensional FK model this leads to multiple phases where localisation effects are relevant. At low temperatures the heavy species orders to a symmetry broken CDW phase, leading to a traditional band gap insulator. At higher temperatures however thermal disorder causes the light species to localise. At weak coupling, the localisation length can be very large, so finite sized systems may still conduct, an effect known as weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">19</a>]</span>.</p>
<p>In Chapter 3 we will consider a generalised FK model in one dimension and study how the disorder generated near a one dimensional thermodynamic phase transition interacts with localisation physics.</p>
<p>True disorder-free localisation does occur in exactly solvable models with extensively many conserved quantities <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">18</a>]</span>. As conserved quantities have no time dynamics this can be thought of as taking the separation of timescales to the infinite limit. The localisation phenomena present in the Falicov-Kimball model are instead the result of annealed disorder. A strong separation of timescales means that the heavy species is approximated as immobile with respect to the lighter itinerant species. At finite temperature the heavy species acts as a disorder potential for the lighter one. However, in contrast to quenched disorder, the probability distribution of annealed disorder is entirely determined by the thermodynamics of the Hamiltonian. In the 2D FK model this leads to multiple phases where localisation effects are relevant. At low temperatures the heavy species orders to a symmetry broken CDW phase, leading to a traditional band gap insulator. At higher temperatures however thermal disorder causes the light species to localise. At weak coupling, the localisation length can be very large, so finite sized systems may still conduct, an effect known as weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">19</a>]</span>.</p>
<p>In Chapter 3 we will consider a generalised FK model in 1D and study how the disorder generated near a 1D thermodynamic phase transition interacts with localisation physics.</p>
<section id="topological-disorder" class="level2">
<h2>Topological Disorder</h2>
<p>So far we have considered disorder as a static or dynamic field coupled to a model defined on a translation invariant lattice. Another kind of disordered system that worthy of study are amorphous systems. Amorphous systems have disordered bond connectivity, so called <em>topological disorder</em>. As discussed in the introduction these include amorphous semiconductors such as amorphous Germanium and Silicon  <span class="citation" data-cites="Yonezawa1983 zallen2008physics Weaire1971 betteridge1973possible"> [<a href="#ref-Yonezawa1983" role="doc-biblioref">20</a>–<a href="#ref-betteridge1973possible" role="doc-biblioref">23</a>]</span>. While materials do not have long range lattice structure they can enforce local constraints such as the approximate coordination number <span class="math inline">\(z = 4\)</span> of silicon.</p>
<p>Topological disorder can be qualitatively different from other disordered systems. Disordered graphs are constrained by fixed coordination number and the Euler equation. A standard method for generating such graphs with coordination number <span class="math inline">\(d+1\)</span> is Voronoi tessellation <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">24</a>,<a href="#ref-marsalTopologicalWeaireThorpeModels2020" role="doc-biblioref">25</a>]</span>. The Harris <span class="citation" data-cites="harrisEffectRandomDefects1974"> [<a href="#ref-harrisEffectRandomDefects1974" role="doc-biblioref">26</a>]</span> and the Imry-Mar <span class="citation" data-cites="imryRandomFieldInstabilityOrdered1975"> [<a href="#ref-imryRandomFieldInstabilityOrdered1975" role="doc-biblioref">27</a>]</span> criteria are key results on the effect of disorder on thermodynamic phase transitions. The Harris criterion signals when disorder will affect the universality of a thermodynamic critical point while the Imry-Ma criterion simply forbids the formation of long range ordered states in <span class="math inline">\(d \leq 2\)</span> dimensions in the presence of disorder. Both these criteria are modified for the case of topological disorder. This is because the Euler equation and vertex degree constraints lead to strong anti-correlations which mean that topological disorder is effectively weaker than standard disorder in two dimensions <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">28</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">29</a>]</span>. This does not apply to three dimensional Voronoi lattices where the Euler equation contains an extra volume term and so is effectively a weaker constraint.</p>
<p>Lastly it is worth exploring how quantum spin liquids and disorder interact. The KH model has been studied subject to both flux <span class="citation" data-cites="Nasu_Thermal_2015"> [<a href="#ref-Nasu_Thermal_2015" role="doc-biblioref">30</a>]</span> and bond <span class="citation" data-cites="knolle_dynamics_2016"> [<a href="#ref-knolle_dynamics_2016" role="doc-biblioref">31</a>]</span> disorder. In some instances it seems that disorder can even promote the formation of a QSL ground state <span class="citation" data-cites="wenDisorderedRouteCoulomb2017"> [<a href="#ref-wenDisorderedRouteCoulomb2017" role="doc-biblioref">32</a>]</span>. It has also been shown that the KH model exhibits disorder free localisation after a quantum quench <span class="citation" data-cites="zhuSubdiffusiveDynamicsCritical2021"> [<a href="#ref-zhuSubdiffusiveDynamicsCritical2021" role="doc-biblioref">33</a>]</span>.</p>
<p>In chapter 4 we will put the Kitaev model onto two dimensional Voronoi lattices and show that much of the rich character of the model is preserved despite the lack of long range order.</p>
<p>So far we have considered disorder as a static or dynamic field coupled to a model defined on a translation invariant lattice. Another kind of disordered system that worthy of study are amorphous systems. Amorphous systems have disordered bond connectivity, so called <em>topological disorder</em>. As discussed in the introduction these include amorphous semiconductors such as amorphous Germanium and Silicon  <span class="citation" data-cites="Yonezawa1983 zallen2008physics Weaire1971 betteridge1973possible"> [<a href="#ref-Yonezawa1983" role="doc-biblioref">20</a>–<a href="#ref-betteridge1973possible" role="doc-biblioref">23</a>]</span>. While materials do not have long-range lattice structure they can enforce local constraints such as the approximate coordination number <span class="math inline">\(z = 4\)</span> of silicon.</p>
<p>Topological disorder can be qualitatively different from other disordered systems. Disordered graphs are constrained by fixed coordination number and the Euler equation. A standard method for generating such graphs with coordination number <span class="math inline">\(d+1\)</span> is Voronoi tessellation <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">24</a>,<a href="#ref-marsalTopologicalWeaireThorpeModels2020" role="doc-biblioref">25</a>]</span>. The Harris <span class="citation" data-cites="harrisEffectRandomDefects1974"> [<a href="#ref-harrisEffectRandomDefects1974" role="doc-biblioref">26</a>]</span> and the Imry-Mar <span class="citation" data-cites="imryRandomFieldInstabilityOrdered1975"> [<a href="#ref-imryRandomFieldInstabilityOrdered1975" role="doc-biblioref">27</a>]</span> criteria are key results on the effect of disorder on thermodynamic phase transitions. The Harris criterion signals when disorder will affect the universality of a thermodynamic critical point while the Imry-Ma criterion simply forbids the formation of long-range ordered states in <span class="math inline">\(d \leq 2\)</span> dimensions in the presence of disorder. Both these criteria are modified for the case of topological disorder. This is because the Euler equation and vertex degree constraints lead to strong anti-correlations which mean that topological disorder is effectively weaker than standard disorder in 2D <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">28</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">29</a>]</span>. This does not apply to 3D Voronoi lattices where the Euler equation contains an extra volume term and so is effectively a weaker constraint.</p>
<p>Lastly it is worth exploring how QSLs and disorder interact. The KH model has been studied subject to both flux <span class="citation" data-cites="Nasu_Thermal_2015"> [<a href="#ref-Nasu_Thermal_2015" role="doc-biblioref">30</a>]</span> and bond <span class="citation" data-cites="knolle_dynamics_2016"> [<a href="#ref-knolle_dynamics_2016" role="doc-biblioref">31</a>]</span> disorder. In some instances it seems that disorder can even promote the formation of a QSL ground state <span class="citation" data-cites="wenDisorderedRouteCoulomb2017"> [<a href="#ref-wenDisorderedRouteCoulomb2017" role="doc-biblioref">32</a>]</span>. It has also been shown that the KH model exhibits disorder-free localisation after a quantum quench <span class="citation" data-cites="zhuSubdiffusiveDynamicsCritical2021"> [<a href="#ref-zhuSubdiffusiveDynamicsCritical2021" role="doc-biblioref">33</a>]</span>.</p>
<p>In chapter 4 we will put the Kitaev model onto 2D Voronoi lattices and show that much of the rich character of the model is preserved despite the lack of long-range order.</p>
</section>
<section id="diagnosing-localisation-in-practice" class="level2">
<h2>Diagnosing Localisation in practice</h2>
@ -103,7 +109,7 @@ H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U
</figure>
<p>Looking at practical tools for diagnosing localisation, there are a few standard methods <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span>.</p>
<p>The most direct method would be to fit a function of the form <span class="math inline">\(\psi(x) = f(x) e^{-|x-x_0|/\lambda}\)</span> to each single particle wavefunction to extract the localisation length <span class="math inline">\(\lambda\)</span>. This method is little used in practice since it requires storing and processing full wavefunctions which quickly becomes expensive for large systems.</p>
<p>For low dimensional systems with quenched disorder, transfer matrix methods can be used to directly extract the localisation length. These work by turning the time independent Schrödinger equation <span class="math inline">\(\hat{H}|\psi\rangle = E|\psi\rangle\)</span> into a matrix equation linking the amplitude of <span class="math inline">\(\psi\)</span> on each <span class="math inline">\(d-1\)</span> dimensional slice of the system to the next and looking at average properties of this transmission matrix. This method is less useful for systems like the FK model where the disorder as a whole must be sampled from the thermodynamic ensemble. It is also problematic for the Kitaev Model on an amorphous lattice as the slicing procedure is complex to define in the absence of a regular lattice.</p>
<p>For low dimensional systems with quenched disorder, transfer matrix methods can be used to directly extract the localisation length. These work by turning the time independent Schrödinger equation <span class="math inline">\(\hat{H}|\psi\rangle = E|\psi\rangle\)</span> into a matrix equation linking the amplitude of <span class="math inline">\(\psi\)</span> on each <span class="math inline">\(d-1\)</span> dimensional slice of the system to the next and looking at average properties of this transmission matrix. This method is less useful for systems like the FK model where the disorder as a whole must be sampled from the thermodynamic ensemble.</p>
<p>A more versatile method is based on the inverse participation ratio (IPR). The IPR is defined for a normalised wave function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i |\psi_i|^2 = 1\)</span> as its fourth moment <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">6</a>]</span>:</p>
<p><span class="math display">\[
P^{-1} = \sum_i |\psi_i|^4
@ -120,11 +126,13 @@ DOS(\omega) &amp;= \sum_n \delta(\omega - \epsilon_n)\\
IPR(\omega) &amp;= DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) |\psi_{n,i}|^4
\end{aligned}
\]</span> Where <span class="math inline">\(\psi_{n,i}\)</span> is the wavefunction corresponding to the energy <span class="math inline">\(\epsilon_n\)</span> at the ith site. In practice I bin the energies and IPRs into a fine energy grid and use the mean within each bin.</p>
<p><strong>Chapter Summary</strong></p>
<p>In this chapter we have covered the Falicov-Kimball model, the Kitaev Honeycomb model and the theory of disorder and localisation. We saw that the FK model is one of immobile species (spins) interacting with an itinerant quantum species (electrons). While the KH model is specified in terms of spins on a honeycomb lattice interacting via a highly anisotropic Ising coupling, it can be transformed into one of Majorana fermions interacting with a classical gauge field that supports immobile flux excitations. In each case it is the immobile species that makes each model exactly solvable. Both models have rich ground state and thermodynamic phase diagrams. The last part of this chapter dealt with disorder and how it almost inevitably leads to localisation. Both the FK and KH models are effectively disordered at finite temperatures by their immobile species. In the next chapter we will look at a version of the FK model in one dimension augmented with long range interactions in order to retain its ordered phase. The model is translation invariant but we will see that it exhibits disorder free localisation. After that we will look at the KH model defined on an amorphous lattice with vertex degree <span class="math inline">\(z=3\)</span>.</p>
<section id="chapter-summary" class="level3">
<h3>Chapter Summary</h3>
<p>In this chapter we have covered the Falicov-Kimball model, the Kitaev Honeycomb model and the theory of disorder and localisation. We saw that the FK model is one of immobile species (spins) interacting with an itinerant quantum species (electrons). While the KH model is specified in terms of spins on a honeycomb lattice interacting via a highly anisotropic Ising coupling, it can be transformed into one of Majorana fermions interacting with a classical gauge field that supports immobile flux excitations. In each case it is the immobile species that makes each model exactly solvable. Both models have rich ground state and thermodynamic phase diagrams. The last part of this chapter dealt with disorder and how it almost inevitably leads to localisation. Both the FK and KH models are effectively disordered at finite temperatures by their immobile species. In the next chapter we will look at a version of the FK model in 1D augmented with long-range interactions in order to retain its ordered phase. The model is translation invariant but we will see that it exhibits disorder-free localisation. After that we will look at the KH model defined on an amorphous lattice with vertex degree <span class="math inline">\(z=3\)</span>.</p>
<p>Next Chapter: <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falicov-Kimball Model</a></p>
</section>
</section>
</section>
<section id="bibliography" class="level1 unnumbered">
<h1 class="unnumbered">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography">

44
_thesis/3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html
View File

@ -1,5 +1,5 @@
---
title: The Long Range Falicov-Kimball Model - The Model
title: The Long-Range Falicov-Kimball Model - The Model
excerpt:
layout: none
image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>The Long Range Falicov-Kimball Model - The Model</title>
<title>The Long-Range Falicov-Kimball Model - The Model</title>
<script type="text/x-mathjax-config">
@ -38,7 +38,11 @@ image:
<br>
<nav aria-label="Table of Contents" class="page-table-of-contents">
<ul>
<li><a href="#chap:3-the-long-range-falicov-kimball-model" id="toc-chap:3-the-long-range-falicov-kimball-model">3 The Long Range Falicov-Kimball Model</a></li>
<li><a href="#chap:3-the-long-range-falicov-kimball-model" id="toc-chap:3-the-long-range-falicov-kimball-model">3 The Long Range Falicov-Kimball Model</a>
<ul>
<li><a href="#lrfk-contributions" id="toc-lrfk-contributions">Contributions</a></li>
<li><a href="#lrfk-chapter-summary" id="toc-lrfk-chapter-summary">Chapter Summary</a></li>
</ul></li>
<li><a href="#sec:lrfk-model" id="toc-sec:lrfk-model">The Model</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
@ -53,7 +57,11 @@ image:
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#chap:3-the-long-range-falicov-kimball-model" id="toc-chap:3-the-long-range-falicov-kimball-model">3 The Long Range Falicov-Kimball Model</a></li>
<li><a href="#chap:3-the-long-range-falicov-kimball-model" id="toc-chap:3-the-long-range-falicov-kimball-model">3 The Long Range Falicov-Kimball Model</a>
<ul>
<li><a href="#lrfk-contributions" id="toc-lrfk-contributions">Contributions</a></li>
<li><a href="#lrfk-chapter-summary" id="toc-lrfk-chapter-summary">Chapter Summary</a></li>
</ul></li>
<li><a href="#sec:lrfk-model" id="toc-sec:lrfk-model">The Model</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
@ -67,33 +75,37 @@ image:
</div>
<section id="chap:3-the-long-range-falicov-kimball-model" class="level1">
<h1>3 The Long Range Falicov-Kimball Model</h1>
<p><strong>Contributions</strong></p>
<section id="lrfk-contributions" class="level3">
<h3>Contributions</h3>
<p>This chapter expands on work presented in</p>
<p> <span class="citation" data-cites="hodsonOnedimensionalLongrangeFalikovKimball2021"> [<a href="#ref-hodsonOnedimensionalLongrangeFalikovKimball2021" role="doc-biblioref">1</a>]</span> <a href="https://link.aps.org/doi/10.1103/PhysRevB.104.045116">One-dimensional long-range Falikov-Kimball model: Thermal phase transition and disorder-free localization</a>, Hodson, T. and Willsher, J. and Knolle, J., Phys. Rev. B, <strong>104</strong>, 4, 2021,</p>
<p> <span class="citation" data-cites="hodsonOnedimensionalLongrangeFalikovKimball2021"> [<a href="#ref-hodsonOnedimensionalLongrangeFalikovKimball2021" role="doc-biblioref">1</a>]</span> <a href="https://link.aps.org/doi/10.1103/PhysRevB.104.045116">One-dimensional long-range Falicov-Kimball model: Thermal phase transition and disorder-free localization</a>, Hodson, T. and Willsher, J. and Knolle, J., Phys. Rev. B, <strong>104</strong>, 4, 2021,</p>
<p>The code is available online <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">2</a>]</span>.</p>
<p>Johannes had the initial idea to use a long range Ising term to stabilise order in a one dimension Falicov-Kimball model. Josef developed a proof of concept during a summer project at Imperial along with Alexander Belcik. I wrote the simulation code and performed all the analysis presented here.</p>
<p><strong>Chapter Summary</strong></p>
<p>The paper is organised as follows. First, I will introduce the long range Falicov-Kimball (LRFK) model and motivate its definition. Second, I will present the <a href="../3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html#sec:lrfk-methods">methods</a> used to solve it numerically, including Markov chain Monte Carlo and finite size scaling. I will then present and interpret the <a href="../3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html#sec:lrfk-results">results</a> obtained.</p>
<p>Johannes had the initial idea to use a long-range Ising term to stabilise order in a 1D Falicov-Kimball model. Josef developed a proof of concept during a summer project at Imperial along with Alexander Belcik. I wrote the simulation code and performed all the analysis presented here.</p>
</section>
<section id="lrfk-chapter-summary" class="level3">
<h3>Chapter Summary</h3>
<p>The paper is organised as follows. First, I will introduce the Long-Range Falicov-Kimball (LRFK) model and motivate its definition. Second, I will present the <a href="../3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html#sec:lrfk-methods">methods</a> used to solve it numerically, including Markov chain Monte Carlo and finite size scaling. I will then present and interpret the <a href="../3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html#sec:lrfk-results">results</a> obtained.</p>
</section>
</section>
<section id="sec:lrfk-model" class="level1">
<h1>The Model</h1>
<p>Dimensionality is crucial for the physics of both localisation and phase transitions. We have already seen that the one dimensional standard FK model cannot support an ordered phase at finite temperatures and therefore has no finite temperature phase transition (FTPT).</p>
<p>On bipartite lattices in dimensions 2 and above the FK model exhibits a finite temperature phase transition to an ordered charge density wave (CDW) phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">3</a>]</span>. In this phase, the spins order anti-ferromagnetically, breaking the <span class="math inline">\(\mathbb{Z}_2\)</span> symmetry. In 1D, however, Peierls’s argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">4</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">5</a>]</span> states that domain walls only introduce a constant energy penalty into the free energy while bringing a entropic contribution logarithmic in system size. Hence the 1D model does not have a finite temperature phase transition. However 1D systems are much easier to study numerically and admit simpler realisations experimentally. We therefore introduce a long-range coupling between the ions in order to stabilise a CDW phase in 1D.</p>
<p>Dimensionality is crucial for the physics of both localisation and phase transitions. We have already seen that the 1D standard Falicov-Kimball (FK) model cannot support an ordered phase at finite temperatures and therefore has no Finite-Temperature Phase Transition (FTPT).</p>
<p>On bipartite lattices in dimensions 2 and above the FK model exhibits a finite temperature phase transition to an ordered Charge Density Wave (CDW) phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">3</a>]</span>. In this phase, the spins order anti-ferromagnetically, breaking the <span class="math inline">\(\mathbb{Z}_2\)</span> symmetry. In 1D, however, Peierls’s argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">4</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">5</a>]</span> states that domain walls only introduce a constant energy penalty into the free energy while bringing a entropic contribution logarithmic in system size. Hence the 1D model does not have a finite temperature phase transition. However 1D systems are much easier to study numerically and admit simpler realisations experimentally. We therefore introduce a long-range coupling between the ions in order to stabilise a CDW phase in 1D.</p>
<figure>
<img src="/assets/thesis/intro_chapter/lrfk_schematic.svg" id="fig-lrfk_schematic" data-short-caption="Falicov-Kimball Model Diagram" style="width:100.0%" alt="Figure 1: The Long Range Falicov-Kimball (LRFK) Model is a model of classical spins S_i coupled to spinless fermions \hat{c}_i where the fermions are mobile with hopping t and the fermions are coupled to the spins by an Ising type interaction with strength U. The difference from the standard FK model is the presence of a long range interaction between the spins J_{ij}S_i S_j." />
<figcaption aria-hidden="true">Figure 1: The Long Range Falicov-Kimball (LRFK) Model is a model of classical spins <span class="math inline">\(S_i\)</span> coupled to spinless fermions <span class="math inline">\(\hat{c}_i\)</span> where the fermions are mobile with hopping <span class="math inline">\(t\)</span> and the fermions are coupled to the spins by an Ising type interaction with strength <span class="math inline">\(U\)</span>. The difference from the standard FK model is the presence of a long range interaction between the spins <span class="math inline">\(J_{ij}S_i S_j\)</span>.</figcaption>
<img src="/assets/thesis/intro_chapter/lrfk_schematic.svg" id="fig-lrfk_schematic" data-short-caption="Falicov-Kimball Model Diagram" style="width:100.0%" alt="Figure 1: The Long-Range Falicov-Kimball (LRFK) Model is a model of classical spins S_i coupled to spinless fermions \hat{c}_i where the fermions are mobile with hopping t and the fermions are coupled to the spins by an Ising type interaction with strength U. The difference from the standard FK model is the presence of a long-range interaction between the spins J_{ij}S_i S_j." />
<figcaption aria-hidden="true">Figure 1: The Long-Range Falicov-Kimball (LRFK) Model is a model of classical spins <span class="math inline">\(S_i\)</span> coupled to spinless fermions <span class="math inline">\(\hat{c}_i\)</span> where the fermions are mobile with hopping <span class="math inline">\(t\)</span> and the fermions are coupled to the spins by an Ising type interaction with strength <span class="math inline">\(U\)</span>. The difference from the standard FK model is the presence of a long-range interaction between the spins <span class="math inline">\(J_{ij}S_i S_j\)</span>.</figcaption>
</figure>
<p>We interpret the FK model as a model of spinless fermions, <span class="math inline">\(c^\dagger_{i}\)</span>, hopping on a 1D lattice against a classical Ising spin background, <span class="math inline">\(S_i \in {\pm \frac{1}{2}}\)</span>. The fermions couple to the spins via an onsite interaction with strength <span class="math inline">\(U\)</span> which we supplement by a long-range interaction, <span class="math display">\[
J_{ij} = 4\kappa J\; (-1)^{|i-j|} |i-j|^{-\alpha},
\]</span></p>
<p>between the spins. The additional coupling is very similar to that of the long range Ising model, it stabilises the Antiferromagnetic (AFM) order of the Ising spins which promotes the finite temperature CDW phase of the fermionic sector.</p>
<p>between the spins. The additional coupling is very similar to that of the long-range Ising (LRI) model, it stabilises the Antiferromagnetic (AFM) order of the Ising spins which promotes the finite temperature CDW phase of the fermionic sector.</p>
<p>The hopping strength of the electrons, <span class="math inline">\(t = 1\)</span>, sets the overall energy scale and we concentrate throughout on the particle-hole symmetric point at zero chemical potential and half filling <span class="citation" data-cites="gruberFalicovKimballModelReview1996"> [<a href="#ref-gruberFalicovKimballModelReview1996" role="doc-biblioref">6</a>]</span>.</p>
<p><span class="math display">\[\begin{aligned}
H_{\mathrm{FK}} = &amp; \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{i} (c^\dagger_{i}c_{i+1} + \textit{h.c.)}\\
&amp; + \sum_{i, j}^{N} J_{ij} S_i S_j
\label{eq:HFK}\end{aligned}\]</span></p>
<p>Without proper normalisation, the long range coupling would render the critical temperature strongly system size dependent for small system sizes. Within a mean field approximation the critical temperature scales with the effective coupling to all the neighbours of each site, which for a system with <span class="math inline">\(N\)</span> sites is <span class="math inline">\(\sum_{i=1}^{N} i^{-\alpha}\)</span>. Hence, the normalisation <span class="math inline">\(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\)</span>, renders the critical temperature independent of system size in the mean field approximation. This greatly improves the finite size behaviour of the model.</p>
<p>Taking the limit <span class="math inline">\(U = 0\)</span> decouples the spins from the fermions, which gives a spin sector governed by a classical LRI model. Note, the transformation of the spins <span class="math inline">\(S_i \to (-1)^{i} S_i\)</span> maps the AFM model to the FM one. As discussed in the background section, Peierls’ classic argument can be extended to long range couplings to show that, for the 1D LRI model, a power law decay of <span class="math inline">\(\alpha &lt; 2\)</span> is required for a FTPT as the energy of defect domain then scales with the system size and can overcome the entropic contribution. A renormalisation group analysis supports this finding and shows that the critical exponents are only universal for <span class="math inline">\(\alpha \leq 3/2\)</span> <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968 thoulessLongRangeOrderOneDimensional1969 angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">7</a>–<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">9</a>]</span>. In the following, we choose <span class="math inline">\(\alpha = 5/4\)</span> to avoid the additional complexity of non-universal critical points.</p>
<p>Without proper normalisation, the long-range coupling would render the critical temperature strongly system size dependent for small system sizes. Within a mean field approximation the critical temperature scales with the effective coupling to all the neighbours of each site, which for a system with <span class="math inline">\(N\)</span> sites is <span class="math inline">\(\sum_{i=1}^{N} i^{-\alpha}\)</span>. Hence, the normalisation <span class="math inline">\(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\)</span>, renders the critical temperature independent of system size in the mean field approximation. This greatly improves the finite size behaviour of the model.</p>
<p>Taking the limit <span class="math inline">\(U = 0\)</span> decouples the spins from the fermions, which gives a spin sector governed by a classical long-range Ising model. Note, the transformation of the spins <span class="math inline">\(S_i \to (-1)^{i} S_i\)</span> maps the AFM model to the FM one. As discussed in the background section, Peierls’ classic argument can be extended to long-range couplings to show that, for the 1D LRI model, a power law decay of <span class="math inline">\(\alpha &lt; 2\)</span> is required for a FTPT as the energy of defect domain then scales with the system size and can overcome the entropic contribution. A renormalisation group analysis supports this finding and shows that the critical exponents are only universal for <span class="math inline">\(\alpha \leq 3/2\)</span> <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968 thoulessLongRangeOrderOneDimensional1969 angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">7</a>–<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">9</a>]</span>. In the following, we choose <span class="math inline">\(\alpha = 5/4\)</span> to avoid the additional complexity of non-universal critical points.</p>
<p>Next Section: <a href="../3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html">Methods</a></p>
</section>
<section id="bibliography" class="level1 unnumbered">

16
_thesis/3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html
View File

@ -1,5 +1,5 @@
---
title: The Long Range Falicov-Kimball Model - Methods
title: The Long-Range Falicov-Kimball Model - Methods
excerpt:
layout: none
image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>The Long Range Falicov-Kimball Model - Methods</title>
<title>The Long-Range Falicov-Kimball Model - Methods</title>
<script type="text/x-mathjax-config">
@ -75,14 +75,14 @@ image:
</div>
<section id="sec:lrfk-methods" class="level1">
<h1>Methods</h1>
<p>To evaluate thermodynamic averages I perform classical Markov Chain Monte Carlo random walks over the space of spin configurations of the LRFK model, at each step diagonalising the effective electronic Hamiltonian <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">1</a>]</span>. Using a Binder-cumulant method <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">2</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">3</a>]</span>, I demonstrate the model has a finite temperature phase transition when the interaction is sufficiently long ranged. In this section I will discuss the thermodynamics of the model and how they are amenable to an exact Markov Chain Monte Carlo method.</p>
<p>To evaluate thermodynamic averages I perform classical Markov Chain Monte Carlo random walks over the space of spin configurations of the LRFK model, at each step diagonalising the effective electronic Hamiltonian <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">1</a>]</span>. Using a Binder-cumulant method <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">2</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">3</a>]</span>, I demonstrate the model has a finite temperature phase transition when the interaction is sufficiently long-ranged. In this section I will discuss the thermodynamics of the model and how they are amenable to an exact Markov Chain Monte Carlo method.</p>
<section id="thermodynamics-of-the-lrfk-model" class="level2">
<h2>Thermodynamics of the LRFK Model</h2>
<figure>
<img src="/assets/thesis/fk_chapter/lsr/pdf_figs/raw_steps_single_flip.svg" id="fig-raw_steps_single_flip" data-short-caption="Comparison of different proposal distributions" style="width:100.0%" alt="Figure 1: Two MCMC walks starting from the CDW state for a system with N = 100 sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation m = N^{-1} \sum_i (-1)^i \; S_i order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5" />
<figcaption aria-hidden="true">Figure 1: Two MCMC walks starting from the CDW state for a system with <span class="math inline">\(N = 100\)</span> sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. <span class="math inline">\(t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5\)</span></figcaption>
<img src="/assets/thesis/fk_chapter/lsr/pdf_figs/raw_steps_single_flip.svg" id="fig-raw_steps_single_flip" data-short-caption="Comparison of different proposal distributions" style="width:100.0%" alt="Figure 1: Two Markov Chain Monte Carlo (MCMC) walks starting from the CDW state for a system with N = 100 sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation m = N^{-1} \sum_i (-1)^i \; S_i order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5" />
<figcaption aria-hidden="true">Figure 1: Two Markov Chain Monte Carlo (MCMC) walks starting from the CDW state for a system with <span class="math inline">\(N = 100\)</span> sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. <span class="math inline">\(t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5\)</span></figcaption>
</figure>
<p>The classical Markov Chain Monte Carlo (MCMC) method which we discuss in the following allows us to solve our long-range FK model efficiently, yielding unbiased estimates of thermal expectation values.</p>
<p>The classical Markov Chain Monte Carlo (MCMC) method which we discuss in the following allows us to solve our LRFK model efficiently, yielding unbiased estimates of thermal expectation values.</p>
<p>Since the spin configurations are classical, the LRFK Hamiltonian can be split into a classical spin part <span class="math inline">\(H_s\)</span> and an operator valued part <span class="math inline">\(H_c\)</span>.</p>
<p><span class="math display">\[\begin{aligned}
H_s&amp; = - \frac{U}{2}S_i + \sum_{i, j}^{N} J_{ij} S_i S_j \\
@ -118,8 +118,8 @@ H_c&amp; = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{
<section id="scaling" class="level2">
<h2>Scaling</h2>
<figure>
<img src="/assets/thesis/fk_chapter/binder_cumulants/binder_cumulants.svg" id="fig-binder_cumulants" data-short-caption="Binder Cumulants" style="width:100.0%" alt="Figure 2: (Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. ¿fig:phase-diagram-lrfk?. All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly mentioned." />
<figcaption aria-hidden="true">Figure 2: (Left) The order parameters, <span class="math inline">\(\langle m^2 \rangle\)</span>(solid) and <span class="math inline">\(1 - f\)</span> (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle\)</span> . (Right) The crossing of the Binder cumulant, <span class="math inline">\(B = \langle m^4 \rangle / \langle m^2 \rangle^2\)</span>, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. <strong>¿fig:phase-diagram-lrfk?</strong>. All plots use system sizes <span class="math inline">\(N = [10,20,30,50,70,110,160,250]\)</span> and lines are coloured from <span class="math inline">\(N = 10\)</span> in dark blue to <span class="math inline">\(N = 250\)</span> in yellow. The parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span> except where explicitly mentioned.</figcaption>
<img src="/assets/thesis/fk_chapter/binder_cumulants/binder_cumulants.svg" id="fig-binder_cumulants" data-short-caption="Binder Cumulants" style="width:100.0%" alt="Figure 2: (Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it is used to estimate the critical lines shown in the phase diagram later. All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly mentioned." />
<figcaption aria-hidden="true">Figure 2: (Left) The order parameters, <span class="math inline">\(\langle m^2 \rangle\)</span>(solid) and <span class="math inline">\(1 - f\)</span> (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle\)</span> . (Right) The crossing of the Binder cumulant, <span class="math inline">\(B = \langle m^4 \rangle / \langle m^2 \rangle^2\)</span>, with system size provides a diagnostic that the phase transition is not a finite size effect, it is used to estimate the critical lines shown in the phase diagram later. All plots use system sizes <span class="math inline">\(N = [10,20,30,50,70,110,160,250]\)</span> and lines are coloured from <span class="math inline">\(N = 10\)</span> in dark blue to <span class="math inline">\(N = 250\)</span> in yellow. The parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span> except where explicitly mentioned.</figcaption>
</figure>
<p>To improve the scaling of finite size effects, we make the replacement <span class="math inline">\(|i - j|^{-\alpha} \rightarrow |f(i - j)|^{-\alpha}\)</span>, in both <span class="math inline">\(J_{ij}\)</span> and <span class="math inline">\(\kappa\)</span>, where <span class="math inline">\(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\)</span>. <span class="math inline">\(f\)</span> is smooth across the circular boundary and its effect effect diminished for larger systems <span class="citation" data-cites="fukuiOrderNClusterMonte2009"> [<a href="#ref-fukuiOrderNClusterMonte2009" role="doc-biblioref">12</a>]</span>. We only consider even system sizes given that odd system sizes are not commensurate with a CDW state.</p>
<p>To identify critical points we use the the Binder cumulant <span class="math inline">\(U_B\)</span> defined by</p>

75
_thesis/3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html
View File

@ -1,5 +1,5 @@
---
title: The Long Range Falicov-Kimball Model - Results
title: The Long-Range Falicov-Kimball Model - Results
excerpt:
layout: none
image:
@ -11,7 +11,7 @@ image:
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>The Long Range Falicov-Kimball Model - Results</title>
<title>The Long-Range Falicov-Kimball Model - Results</title>
<script type="text/x-mathjax-config">
@ -75,17 +75,17 @@ image:
</div>
<section id="sec:lrfk-results" class="level1">
<h1>Results</h1>
<p>Looking at the results of our MCMC simulations, we find a rich phase diagram with a CDW FTPT and interaction-tuned Anderson versus Mott localized phases similar to the 2D FK model <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>. We explore the localization properties of the fermionic sector and find that the localisation lengths vary dramatically across the phases and for different energies. Although moderate system sizes indicate the coexistence of localized and delocalized states within the CDW phase, we find quantitatively similar behaviour in a model of uncorrelated binary disorder on a CDW background. For large system sizes, i.e. for our 1D disorder model we can treat linear sizes of several thousand sites, we find that all states are eventually localized with a localization length which diverges towards zero temperature.</p>
<p>Looking at the results of our MCMC simulations, we find a rich phase diagram with a CDW FTPT and interaction-tuned Anderson versus Mott localised phases similar to the 2D FK model <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>. We explore the localisation properties of the fermionic sector and find that the localisation lengths vary dramatically across the phases and for different energies. Although moderate system sizes indicate the coexistence of localised and delocalised states within the CDW phase, we find quantitatively similar behaviour in a model of uncorrelated binary disorder on a CDW background. For large system sizes, i.e. for our 1D disorder model we can treat linear sizes of several thousand sites, we find that all states are eventually localised with a localisation length which diverges towards zero temperature.</p>
<figure>
<img src="/assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg" id="fig-phase-diagram-lrfk" data-short-caption="Long Range Falicov Kimball Model Phase Diagram" style="width:100.0%" alt="Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature T_c, linear in J. (Right) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas." />
<figcaption aria-hidden="true">Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at <span class="math inline">\(U = 5\)</span>: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (Right) The TU plane at <span class="math inline">\(J = 5\)</span>: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at <span class="math inline">\(E=0\)</span> in the single particle energy spectrum. <span class="math inline">\(U_c\)</span> is independent of temperature. At <span class="math inline">\(U = 0\)</span> the fermions are decoupled from the spins forming a simple Fermi gas.</figcaption>
<img src="/assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg" id="fig-phase-diagram-lrfk" data-short-caption="Long-Range Falicov Kimball Model Phase Diagram" style="width:100.0%" alt="Figure 1: Phase diagrams of the 1D long-range Falicov-Kimball model. (Left) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature T_c, linear in J. (Right) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a Mott insulator/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas." />
<figcaption aria-hidden="true">Figure 1: Phase diagrams of the 1D long-range Falicov-Kimball model. (Left) The TJ plane at <span class="math inline">\(U = 5\)</span>: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (Right) The TU plane at <span class="math inline">\(J = 5\)</span>: the disordered phase is split into two: at large/small U there’s a Mott insulator/Anderson phase characterised by the presence/absence of a gap at <span class="math inline">\(E=0\)</span> in the single particle energy spectrum. <span class="math inline">\(U_c\)</span> is independent of temperature. At <span class="math inline">\(U = 0\)</span> the fermions are decoupled from the spins forming a simple Fermi gas.</figcaption>
</figure>
<section id="lrfk-results-phase-diagram" class="level2">
<h2>Phase Diagram</h2>
<p>Using the MCMC methods described in the previous section I will now discuss the results of extensive MCMC simulations of the model, starting with the phase diagram in the fermion spin coupling <span class="math inline">\(U\)</span>, the strength of the long range spin-spin coupling <span class="math inline">\(J\)</span> and the temperature <span class="math inline">\(T\)</span>.</p>
<p>Using the MCMC methods described in the previous section I will now discuss the results of extensive MCMC simulations of the model, starting with the phase diagram in the fermion spin coupling <span class="math inline">\(U\)</span>, the strength of the long-range spin-spin coupling <span class="math inline">\(J\)</span> and the temperature <span class="math inline">\(T\)</span>.</p>
<p>Fig. <a href="#fig:phase-diagram-lrfk">1</a> shows the phase diagram for constant <span class="math inline">\(U=5\)</span> and constant <span class="math inline">\(J=5\)</span>, respectively. The transition temperatures were determined from the crossings of the Binder cumulants <span class="math inline">\(B_4 = \langle m^4 \rangle /\langle m^2 \rangle^2\)</span> <span class="citation" data-cites="binderFiniteSizeScaling1981"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">2</a>]</span>.</p>
<p>The CDW transition temperature is largely independent from the strength of the interaction <span class="math inline">\(U\)</span>. This demonstrates that the phase transition is driven by the long-range term <span class="math inline">\(J\)</span> with little effect from the coupling to the fermions <span class="math inline">\(U\)</span>. The physics of the spin sector in the long-range FK model mimics that of the long range Ising (LRI) model and is not significantly altered by the presence of the fermions. In two dimensions the transition to the CDW phase is mediated by an RKYY-like interaction <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">3</a>]</span> but this is insufficient to stabilise long range order in one dimension. That the critical temperature <span class="math inline">\(T_c\)</span> does not depend on <span class="math inline">\(U\)</span> in our model further confirms this.</p>
<p>The main order parameters for this model is the staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> that signals the onset of a charge density wave (CDW) phase at low temperature. However, my main interest concerns the additional structure of the fermionic sector in the high temperature phase. Following Ref. <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>, we can distinguish between the Mott and Anderson insulating phases. The Mott insulator is characterised by a gapped DOS in the absence of a CDW, instead the gap is driven entirely by interaction effect. Thus, the opening of a gap for large <span class="math inline">\(U\)</span> is distinct from the gap-opening induced by the translational symmetry breaking in the CDW state below <span class="math inline">\(T_c\)</span>. The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates hence it also has insulating character.</p>
<p>The CDW transition temperature is largely independent from the strength of the interaction <span class="math inline">\(U\)</span>. This demonstrates that the phase transition is driven by the long-range term <span class="math inline">\(J\)</span> with little effect from the coupling to the fermions <span class="math inline">\(U\)</span>. The physics of the spin sector in the LRFK model mimics that of the LRI model and is not significantly altered by the presence of the fermions. In 2D the transition to the CDW phase is mediated by an RKYY-like interaction <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">3</a>]</span> but this is insufficient to stabilise long-range order in 1D. That the critical temperature <span class="math inline">\(T_c\)</span> does not depend on <span class="math inline">\(U\)</span> in our model further confirms this.</p>
<p>The main order parameters for this model is the staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> that signals the onset of a CDW phase at low temperature. However, my main interest concerns the additional structure of the fermionic sector in the high temperature phase. Following Ref. <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>, we can distinguish between the Mott and Anderson insulating phases. The Mott insulator is characterised by a gapped DOS in the absence of a CDW, instead the gap is driven entirely by interaction effect. Thus, the opening of a gap for large <span class="math inline">\(U\)</span> is distinct from the gap-opening induced by the translational symmetry breaking in the CDW state below <span class="math inline">\(T_c\)</span>. The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates hence it also has insulating character.</p>
</section>
<section id="localisation-properties" class="level2">
<h2>Localisation Properties</h2>
@ -105,9 +105,9 @@ image:
<img src="/assets/thesis/fk_chapter/DOS/IPR_scaling.svg" id="fig-IPR_scaling" data-short-caption="Scaling of IPR($\omega$) against system size $N$." style="width:100.0%" alt="Figure 3: The IPR(\omega) scaling with N at fixed energy for each phase and for points both in the gap (\omega_0) and in a band (\omega_1). The slope of the line yields the scaling exponent \tau defined by \mathrm{IPR} \propto N^{-\tau}). \tau close to zero implies that the states at that energy are localised while \tau = -d corresponds to extended states where d is the system dimension. All but the bands of the charge density wave phase are approximately localised with \tau is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots J = 5,\;\alpha = 1.25. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states." />
<figcaption aria-hidden="true">Figure 3: The IPR(<span class="math inline">\(\omega\)</span>) scaling with <span class="math inline">\(N\)</span> at fixed energy for each phase and for points both in the gap (<span class="math inline">\(\omega_0\)</span>) and in a band (<span class="math inline">\(\omega_1\)</span>). The slope of the line yields the scaling exponent <span class="math inline">\(\tau\)</span> defined by <span class="math inline">\(\mathrm{IPR} \propto N^{-\tau}\)</span>). <span class="math inline">\(\tau\)</span> close to zero implies that the states at that energy are localised while <span class="math inline">\(\tau = -d\)</span> corresponds to extended states where <span class="math inline">\(d\)</span> is the system dimension. All but the bands of the charge density wave phase are approximately localised with <span class="math inline">\(\tau\)</span> is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span>. The measured <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\((0.06\pm0.01, 0.02\pm0.01\)</span> (b) <span class="math inline">\(0.04\pm0.02, 0.00\pm0.01\)</span> (c) <span class="math inline">\(0.05\pm0.03, 0.30\pm0.03\)</span> (d) <span class="math inline">\(0.06\pm0.04, 0.15\pm0.05\)</span> We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states.</figcaption>
</figure>
<p>The scaling of the IPR with system size <span class="math inline">\(\mathrm{IPR} \propto N^{-\tau}\)</span> depends on the localisation properties of states at that energy. For delocalised states, e.g. Bloch waves, <span class="math inline">\(\tau\)</span> is the physical dimension. For fully localised states <span class="math inline">\(\tau\)</span> goes to zero in the thermodynamic limit. However, for special types of disorder such as binary disorder, the localisation lengths can be large comparable to the system size at hand, which can make it difficult to extract the correct scaling. An additional complication arises from the fact that the scaling exponent may display intermediate behaviours for correlated disorder and in the vicinity of a localisation-delocalisation transition <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">5</a>,<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">6</a>]</span>. The thermal defects of the CDW phase lead to a binary disorder potential with a finite correlation length, which in principle could result in delocalized eigenstates.</p>
<p>The key question for our system is then: How is the <span class="math inline">\(T=0\)</span> CDW phase with fully delocalized fermionic states connected to the fully localized phase at high temperatures?</p>
<p>For a representative set of parameters covering all three phases fig. <a href="#fig:DOS">2</a> shows the density of states as function of energy while fig. <a href="#fig:IPR_scaling">3</a> shows <span class="math inline">\(\tau\)</span>, the scaling exponent of the IPR with system size, The DOS is symmetric about <span class="math inline">\(0\)</span> because of the particle hole symmetry of the model. At high temperatures, all of the eigenstates are localised in both the Mott and Anderson phases (with <span class="math inline">\(\tau \leq 0.07\)</span> for our system sizes). We also checked that the states are localised by direct inspection. Note that there are in-gap states for instance at <span class="math inline">\(\omega_0\)</span>, below the upper band which are localized and smoothly connected across the phase transition.</p>
<p>The scaling of the IPR with system size <span class="math inline">\(\mathrm{IPR} \propto N^{-\tau}\)</span> depends on the localisation properties of states at that energy. For delocalised states, e.g. Bloch waves, <span class="math inline">\(\tau\)</span> is the physical dimension. For fully localised states <span class="math inline">\(\tau\)</span> goes to zero in the thermodynamic limit. However, for special types of disorder such as binary disorder, the localisation lengths can be large comparable to the system size at hand, which can make it difficult to extract the correct scaling. An additional complication arises from the fact that the scaling exponent may display intermediate behaviours for correlated disorder and in the vicinity of a localisation-delocalisation transition <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">5</a>,<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">6</a>]</span>. The thermal defects of the CDW phase lead to a binary disorder potential with a finite correlation length, which in principle could result in delocalised eigenstates.</p>
<p>The key question for our system is then: How is the <span class="math inline">\(T=0\)</span> CDW phase with fully delocalised fermionic states connected to the fully localised phase at high temperatures?</p>
<p>For a representative set of parameters covering all three phases fig. <a href="#fig:DOS">2</a> shows the density of states as function of energy while fig. <a href="#fig:IPR_scaling">3</a> shows <span class="math inline">\(\tau\)</span>, the scaling exponent of the IPR with system size, The DOS is symmetric about <span class="math inline">\(0\)</span> because of the particle hole symmetry of the model. At high temperatures, all of the eigenstates are localised in both the Mott and Anderson phases (with <span class="math inline">\(\tau \leq 0.07\)</span> for our system sizes). We also checked that the states are localised by direct inspection. Note that there are in-gap states for instance at <span class="math inline">\(\omega_0\)</span>, below the upper band which are localised and smoothly connected across the phase transition.</p>
<figure>
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U5.svg" id="fig-gap_opening_U5" data-short-caption="The transition from CDW to the Mott phase" style="width:100.0%" alt="Figure 4: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
<figcaption aria-hidden="true">Figure 4: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW to gapped Mott phase transition at <span class="math inline">\(U=5\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
@ -118,29 +118,30 @@ image:
<img src="/assets/thesis/fk_chapter/gap_opening/gap_opening_U2.svg" id="fig-gap_opening_U2" data-short-caption="The transition from CDW to the Anderson Phase" style="width:100.0%" alt="Figure 5: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at U=2. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
<figcaption aria-hidden="true">Figure 5: The DOS (a) and scaling exponent <span class="math inline">\(\tau\)</span> (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at <span class="math inline">\(U=2\)</span>. Regions where the DOS is close to zero are shown in white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span></figcaption>
</figure>
<p>In order to understand the localization properties we can compare the behaviour of our model with that of a simpler Anderson disorder model (DM) in which the spins are replaced by a CDW background with uncorrelated binary defect potentials. This is defined by replacing the spin degree of freedom in the FK model <span class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder potential <span class="math inline">\(d_i = \pm \tfrac{1}{2}\)</span> controlled by a defect density <span class="math inline">\(\rho\)</span> such that <span class="math inline">\(d_i = -\tfrac{1}{2}\)</span> with probability <span class="math inline">\(\rho/2\)</span> and <span class="math inline">\(d_i = \tfrac{1}{2}\)</span> otherwise. <span class="math inline">\(\rho/2\)</span> is used rather than <span class="math inline">\(\rho\)</span> so that the disorder potential takes on the zero temperature CDW ground state at <span class="math inline">\(\rho = 0\)</span> and becomes a random choice over spin states at <span class="math inline">\(\rho = 1\)</span> i.e the infinite temperature limit.</p>
<p>In order to understand the localisation properties we can compare the behaviour of our model with that of a simpler Anderson disorder model in which the spins are replaced by a CDW background with uncorrelated binary defect potentials. This is defined by replacing the spin degree of freedom in the FK model <span class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder potential <span class="math inline">\(d_i = \pm \tfrac{1}{2}\)</span> controlled by a defect density <span class="math inline">\(\rho\)</span> such that <span class="math inline">\(d_i = -\tfrac{1}{2}\)</span> with probability <span class="math inline">\(\rho/2\)</span> and <span class="math inline">\(d_i = \tfrac{1}{2}\)</span> otherwise. <span class="math inline">\(\rho/2\)</span> is used rather than <span class="math inline">\(\rho\)</span> so that the disorder potential takes on the zero temperature CDW ground state at <span class="math inline">\(\rho = 0\)</span> and becomes a random choice over spin states at <span class="math inline">\(\rho = 1\)</span> i.e the infinite temperature limit.</p>
<p><span class="math display">\[\begin{aligned}
H_{\mathrm{DM}} = &amp; \;U \sum_{i} (-1)^i \; d_i \;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) \\
&amp; -\;t \sum_{i} c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i}
\end{aligned}\]</span></p>
<p>fig. <a href="#fig:DM_DOS">6</a> and fig. <a href="#fig:DM_IPR_scaling">7</a> compare the FK model to the disorder model at different system sizes, matching the defect densities of the disorder model to the FK model at <span class="math inline">\(N = 270\)</span> above and below the CDW transition. We find very good, even quantitative, agreement between the FK and disorder models, which suggests that correlations in the spin sector do not play a significant role.</p>
<figure>
<img src="/assets/thesis/fk_chapter/disorder_model/DM_DOS.svg" id="fig-DM_DOS" data-short-caption="FK model compared to binary disorder model: DOS" style="width:100.0%" alt="Figure 6: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 &lt; \rho &lt; 1 matched to the \rho for the largest corresponding FK model. As in fig. 2, the Energy resolved DOS(\omega) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters." />
<figcaption aria-hidden="true">Figure 6: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 &lt; \rho &lt; 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:DOS">2</a>, the Energy resolved DOS(<span class="math inline">\(\omega\)</span>) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters.</figcaption>
<img src="/assets/thesis/fk_chapter/disorder_model/DM_DOS.svg" id="fig-DM_DOS" data-short-caption="FK model compared to binary disorder model: DOS" style="width:100.0%" alt="Figure 6: A comparison of the full FK model to a simple binary disorder model with a CDW wave background perturbed by uncorrelated defects at density 0 &lt; \rho &lt; 1 matched to the \rho for the largest corresponding FK model. As in fig. 2, the Energy resolved DOS(\omega) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters." />
<figcaption aria-hidden="true">Figure 6: A comparison of the full FK model to a simple binary disorder model with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 &lt; \rho &lt; 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:DOS">2</a>, the Energy resolved DOS(<span class="math inline">\(\omega\)</span>) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters.</figcaption>
</figure>
<p>As we can sample directly from the disorder model, rather than through MCMC, the samples are uncorrelated. Hence we can evaluate much larger system sizes with the disorder model which enables us to pin down the correct localisation effects. In particular, what appear to be delocalized states for small system sizes eventually turn out to be states with large localization length. The localization length diverges towards the ordered zero temperature CDW state. The interplay of interactions, which here produce as peculiar binary potential, and localization can be very intricate and the added advantage of a 1D model is that we can explore very large system sizes.</p>
<p>As we can sample directly from the disorder model, rather than through MCMC, the samples are uncorrelated. Hence we can evaluate much larger system sizes with the disorder model which enables us to pin down the correct localisation effects. In particular, what appear to be delocalised states for small system sizes eventually turn out to be states with large localisation length. The localisation length diverges towards the ordered zero temperature CDW state. The interplay of interactions, which here produce as peculiar binary potential, and localisation can be very intricate and the added advantage of a 1D model is that we can explore very large system sizes.</p>
<figure>
<img src="/assets/thesis/fk_chapter/disorder_model/DM_IPR_scaling.svg" id="fig-DM_IPR_scaling" data-short-caption="FK model compared to binary disorder model: IPR Scaling" style="width:100.0%" alt="Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 &lt; \rho &lt; 1 matched to the \rho for the largest corresponding FK model. As in fig. 3 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation  [1], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N &gt; 400" />
<figcaption aria-hidden="true">Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 &lt; \rho &lt; 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:IPR_scaling">3</a> <span class="math inline">\(\tau(\omega)\)</span> the scaling of IPR(<span class="math inline">\(\omega\)</span>) with system size, , is shown both in gap (<span class="math inline">\(\omega_0\)</span>) and in the band (<span class="math inline">\(\omega_1\)</span>). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>, hence all the states are indeed localised as one would expect in 1D. The disorder model <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span class="math inline">\(0.01\pm0.04, -0.01\pm0.04\)</span> (c) <span class="math inline">\(0.05\pm0.06, 0.04\pm0.06\)</span> (d) <span class="math inline">\(-0.03\pm0.06, 0.01\pm0.06\)</span>. The lines are fit on system sizes <span class="math inline">\(N &gt; 400\)</span></figcaption>
<img src="/assets/thesis/fk_chapter/disorder_model/DM_IPR_scaling.svg" id="fig-DM_IPR_scaling" data-short-caption="FK model compared to binary disorder model: IPR Scaling" style="width:100.0%" alt="Figure 7: A comparison of the full FK model to a simple binary disorder model with a CDW wave background perturbed by uncorrelated defects at density 0 &lt; \rho &lt; 1 matched to the \rho for the largest corresponding FK model. As in fig. 3 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation  [1], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N &gt; 400" />
<figcaption aria-hidden="true">Figure 7: A comparison of the full FK model to a simple binary disorder model with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 &lt; \rho &lt; 1\)</span> matched to the <span class="math inline">\(\rho\)</span> for the largest corresponding FK model. As in fig. <a href="#fig:IPR_scaling">3</a> <span class="math inline">\(\tau(\omega)\)</span> the scaling of IPR(<span class="math inline">\(\omega\)</span>) with system size, , is shown both in gap (<span class="math inline">\(\omega_0\)</span>) and in the band (<span class="math inline">\(\omega_1\)</span>). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">1</a>]</span>, hence all the states are indeed localised as one would expect in 1D. The disorder model <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span class="math inline">\(0.01\pm0.04, -0.01\pm0.04\)</span> (c) <span class="math inline">\(0.05\pm0.06, 0.04\pm0.06\)</span> (d) <span class="math inline">\(-0.03\pm0.06, 0.01\pm0.06\)</span>. The lines are fit on system sizes <span class="math inline">\(N &gt; 400\)</span></figcaption>
</figure>
</section>
</section>
<section id="fk-conclusion" class="level1">
<h1>Discussion and Conclusion</h1>
<p>The FK model is one of the simplest non-trivial models of interacting fermions. We studied its thermodynamic and localisation properties brought down in dimensionality to one dimension by adding a novel long-ranged coupling designed to stabilise the CDW phase present in dimension two and above.</p>
<p>Our MCMC approach emphasises the presence of a disorder-free localization mechanism within our translationally invariant system. Further, it gives a significant speed up over the naive method. We show that our LRFK model retains much of the rich phase diagram of its higher dimensional cousins. Careful scaling analysis indicates that all the single particle eigenstates eventually localise at non-zero temperature albeit only for very large system sizes of several thousand.</p>
<p>Our work raises a number of interesting questions for future research. A straightforward but numerically challenging problem is to pin down the model’s behaviour closer to the critical point where correlations in the spin sector would become significant. Would this modify the localisation behaviour? Similar to other soluble models of disorder-free localisation, we expect intriguing out-of equilibrium physics, for example slow entanglement dynamics akin to more generic interacting systems <span class="citation" data-cites="hartLogarithmicEntanglementGrowth2020"> [<a href="#ref-hartLogarithmicEntanglementGrowth2020" role="doc-biblioref">11</a>]</span>. One could also investigate whether the rich ground state phenomenology of the FK model as a function of filling <span class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">12</a>]</span> such as the devil’s staircase <span class="citation" data-cites="michelettiCompleteDevilTextquotesingles1997"> [<a href="#ref-michelettiCompleteDevilTextquotesingles1997" role="doc-biblioref">13</a>]</span> as well as superconductor like states <span class="citation" data-cites="caiVisualizingEvolutionMott2016"> [<a href="#ref-caiVisualizingEvolutionMott2016" role="doc-biblioref">14</a>]</span> could be stabilised at finite temperature.</p>
<p>In a broader context, we envisage that long-range interactions can also be used to gain a deeper understanding of the temperature evolution of topological phases. One example would be a long-ranged FK version of the celebrated Su-Schrieffer-Heeger model where one could explore the interplay of topological bound states and thermal domain wall defects. Finally, the rich physics of our model should be realizable in systems with long-range interactions, such as trapped ion quantum simulators, where one can also explore the fully interacting regime with a dynamical background field.</p>
<p>The FK model is one of the simplest non-trivial models of interacting fermions. We studied its thermodynamic and localisation properties brought down in dimensionality to 1D by adding a novel long-ranged coupling designed to stabilise the CDW phase present in dimension two and above.</p>
<p>Our MCMC approach emphasises the presence of a disorder-free localisation mechanism within the translationally invariant system. Further, it gives a significant speed up over the naive method. We show that the LRFK model retains much of the rich phase diagram of its higher dimensional cousins. Careful scaling analysis indicates that all the single particle eigenstates eventually localise at non-zero temperature albeit only for very large system sizes of several thousand.</p>
<p>This work raises a number of interesting questions for future research. A straightforward but numerically challenging problem is to pin down the model’s behaviour closer to the critical point where correlations in the spin sector would become significant. Would this modify the localisation behaviour? It has been shown that Anderson models with long-range correlated disorder potentials can have complex localisation behaviour not normally seen in 1D <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 shimasakiAnomalousLocalizationMultifractality2022"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">11</a>,<a href="#ref-shimasakiAnomalousLocalizationMultifractality2022" role="doc-biblioref">12</a>]</span>. MCMC methods with local spin updates tend to experience critical slowing down near classical critical points <span class="citation" data-cites="geyerPracticalMarkovChain1992 levinMarkovChainsMixing2017 vatsMultivariateOutputAnalysis2015"> [<a href="#ref-geyerPracticalMarkovChain1992" role="doc-biblioref">13</a>–<a href="#ref-vatsMultivariateOutputAnalysis2015" role="doc-biblioref">15</a>]</span>. Cluster updates methods can help to alleviate this but it is not clear if they can be adapted to incorporate the energetic contributions from the fermion sector <span class="citation" data-cites="evertzClusterAlgorithmVertex1993 fukuiOrderNClusterMonte2009 wolffCollectiveMonteCarlo1989"> [<a href="#ref-evertzClusterAlgorithmVertex1993" role="doc-biblioref">16</a>–<a href="#ref-wolffCollectiveMonteCarlo1989" role="doc-biblioref">18</a>]</span>.</p>
<p>Similar to other solvable models of disorder-free localisation, we expect intriguing out-of equilibrium physics, for example slow entanglement dynamics akin to more generic interacting systems <span class="citation" data-cites="hartLogarithmicEntanglementGrowth2020"> [<a href="#ref-hartLogarithmicEntanglementGrowth2020" role="doc-biblioref">19</a>]</span>. One could also investigate whether the rich ground state phenomenology of the FK model as a function of filling <span class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">20</a>]</span> such as the devil’s staircase <span class="citation" data-cites="michelettiCompleteDevilTextquotesingles1997"> [<a href="#ref-michelettiCompleteDevilTextquotesingles1997" role="doc-biblioref">21</a>]</span> as well as superconductor like states <span class="citation" data-cites="caiVisualizingEvolutionMott2016"> [<a href="#ref-caiVisualizingEvolutionMott2016" role="doc-biblioref">22</a>]</span> could be stabilised at finite temperature.</p>
<p>In a broader context, we envisage that long-range interactions can also be used to gain a deeper understanding of the temperature evolution of topological phases. One example would be a LRFK version of the celebrated Su-Schrieffer-Heeger (SSH) model where one could explore the interplay of topological bound states and thermal domain wall defects. Finally, the rich physics of our model should be realisable in systems with long-range interactions, such as trapped ion quantum simulators, where one can also explore the fully interacting regime with a dynamical background field.</p>
<p>Next Chapter: <a href="../4_Amorphous_Kitaev_Model/4.1_AMK_Model.html">4 The Amorphous Kitaev Model</a></p>
</section>
<section id="bibliography" class="level1 unnumbered">
@ -176,17 +177,41 @@ H_{\mathrm{DM}} = &amp; \;U \sum_{i} (-1)^i \; d_i \;(c^\dagger_{i}c_{i} - \tfra
<div id="ref-hassanSpectralPropertiesChargedensitywave2007" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">S. R. Hassan and H. R. Krishnamurthy, <em><a href="https://doi.org/10.1103/PhysRevB.76.205109">Spectral Properties in the Charge-Density-Wave Phase of the Half-Filled Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>76</strong>, 205109 (2007).</div>
</div>
<div id="ref-aubryAnalyticityBreakingAnderson1980" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">S. Aubry and G. André, <em>Analyticity Breaking and Anderson Localization in Incommensurate Lattices</em>, Proceedings, VIII International Colloquium on Group-Theoretical Methods in Physics <strong>3</strong>, 18 (1980).</div>
</div>
<div id="ref-shimasakiAnomalousLocalizationMultifractality2022" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">T. Shimasaki, M. Prichard, H. E. Kondakci, J. Pagett, Y. Bai, P. Dotti, A. Cao, T.-C. Lu, T. Grover, and D. M. Weld, <em><a href="https://doi.org/10.48550/arXiv.2203.09442">Anomalous Localization and Multifractality in a Kicked Quasicrystal</a></em>, arXiv:2203.09442.</div>
</div>
<div id="ref-geyerPracticalMarkovChain1992" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">C. J. Geyer, <em><a href="https://doi.org/10.1214/ss/1177011137">Practical Markov Chain Monte Carlo</a></em>, Statist. Sci. <strong>7</strong>, 473 (1992).</div>
</div>
<div id="ref-levinMarkovChainsMixing2017" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">D. A. Levin and Y. Peres, <em>Markov Chains and Mixing Times</em> (American Mathematical Soc., 2017).</div>
</div>
<div id="ref-vatsMultivariateOutputAnalysis2015" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">D. Vats, J. M. Flegal, and G. L. Jones, <em><a href="http://arxiv.org/abs/1512.07713">Multivariate Output Analysis for Markov Chain Monte Carlo</a></em>, arXiv:1512.07713 [Math, Stat] (2015).</div>
</div>
<div id="ref-evertzClusterAlgorithmVertex1993" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">H. G. Evertz, G. Lana, and M. Marcu, <em><a href="https://doi.org/10.1103/PhysRevLett.70.875">Cluster Algorithm for Vertex Models</a></em>, Phys. Rev. Lett. <strong>70</strong>, 875 (1993).</div>
</div>
<div id="ref-fukuiOrderNClusterMonte2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">K. Fukui and S. Todo, <em><a href="https://doi.org/10.1016/j.jcp.2008.12.022">Order-N Cluster Monte Carlo Method for Spin Systems with Long-Range Interactions</a></em>, Journal of Computational Physics <strong>228</strong>, 2629 (2009).</div>
</div>
<div id="ref-wolffCollectiveMonteCarlo1989" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">U. Wolff, <em><a href="https://doi.org/10.1103/PhysRevLett.62.361">Collective Monte Carlo Updating for Spin Systems</a></em>, Phys. Rev. Lett. <strong>62</strong>, 361 (1989).</div>
</div>
<div id="ref-hartLogarithmicEntanglementGrowth2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">O. Hart, S. Gopalakrishnan, and C. Castelnovo, <em><a href="http://arxiv.org/abs/2009.00618">Logarithmic Entanglement Growth from Disorder-Free Localisation in the Two-Leg Compass Ladder</a></em>, arXiv:2009.00618 [Cond-Mat] (2020).</div>
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">O. Hart, S. Gopalakrishnan, and C. Castelnovo, <em><a href="http://arxiv.org/abs/2009.00618">Logarithmic Entanglement Growth from Disorder-Free Localisation in the Two-Leg Compass Ladder</a></em>, arXiv:2009.00618 [Cond-Mat] (2020).</div>
</div>
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">C. Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>41</strong>, 2198 (1990).</div>
<div class="csl-left-margin">[20] </div><div class="csl-right-inline">C. Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>41</strong>, 2198 (1990).</div>
</div>
<div id="ref-michelettiCompleteDevilTextquotesingles1997" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">C. Micheletti, A. B. Harris, and J. M. Yeomans, <em><a href="https://doi.org/10.1088/0305-4470/30/21/002">A Complete Devil\textquotesingles Staircase in the Falicov - Kimball Model</a></em>, J. Phys. A: Math. Gen. <strong>30</strong>, L711 (1997).</div>
<div class="csl-left-margin">[21] </div><div class="csl-right-inline">C. Micheletti, A. B. Harris, and J. M. Yeomans, <em><a href="https://doi.org/10.1088/0305-4470/30/21/002">A Complete Devil\textquotesingles Staircase in the Falicov - Kimball Model</a></em>, J. Phys. A: Math. Gen. <strong>30</strong>, L711 (1997).</div>
</div>
<div id="ref-caiVisualizingEvolutionMott2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">P. Cai, W. Ruan, Y. Peng, C. Ye, X. Li, Z. Hao, X. Zhou, D.-H. Lee, and Y. Wang, <em><a href="https://doi.org/10.1038/nphys3840">Visualizing the Evolution from the Mott Insulator to a Charge-Ordered Insulator in Lightly Doped Cuprates</a></em>, Nature Phys <strong>12</strong>, 11 (2016).</div>
<div class="csl-left-margin">[22] </div><div class="csl-right-inline">P. Cai, W. Ruan, Y. Peng, C. Ye, X. Li, Z. Hao, X. Zhou, D.-H. Lee, and Y. Wang, <em><a href="https://doi.org/10.1038/nphys3840">Visualizing the Evolution from the Mott Insulator to a Charge-Ordered Insulator in Lightly Doped Cuprates</a></em>, Nature Phys <strong>12</strong>, 11 (2016).</div>
</div>
</div>
</section>

57
_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html
View File

@ -39,7 +39,11 @@ image:
<br>
<nav aria-label="Table of Contents" class="page-table-of-contents">
<ul>
<li><a href="#chap:4-the-amorphous-kitaev-model" id="toc-chap:4-the-amorphous-kitaev-model">4 The Amorphous Kitaev Model</a></li>
<li><a href="#chap:4-the-amorphous-kitaev-model" id="toc-chap:4-the-amorphous-kitaev-model">4 The Amorphous Kitaev Model</a>
<ul>
<li><a href="#ak-contributions" id="toc-ak-contributions">Contributions</a></li>
<li><a href="#ak-summary" id="toc-ak-summary">Chapter Summary</a></li>
</ul></li>
<li><a href="#amk-Model" id="toc-amk-Model">The Model</a>
<ul>
<li><a href="#the-euler-equation" id="toc-the-euler-equation">The Euler Equation</a></li>
@ -57,7 +61,11 @@ image:
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#chap:4-the-amorphous-kitaev-model" id="toc-chap:4-the-amorphous-kitaev-model">4 The Amorphous Kitaev Model</a></li>
<li><a href="#chap:4-the-amorphous-kitaev-model" id="toc-chap:4-the-amorphous-kitaev-model">4 The Amorphous Kitaev Model</a>
<ul>
<li><a href="#ak-contributions" id="toc-ak-contributions">Contributions</a></li>
<li><a href="#ak-summary" id="toc-ak-summary">Chapter Summary</a></li>
</ul></li>
<li><a href="#amk-Model" id="toc-amk-Model">The Model</a>
<ul>
<li><a href="#the-euler-equation" id="toc-the-euler-equation">The Euler Equation</a></li>
@ -74,38 +82,45 @@ image:
</div>
<section id="chap:4-the-amorphous-kitaev-model" class="level1">
<h1>4 The Amorphous Kitaev Model</h1>
<p><strong>Contributions</strong></p>
<section id="ak-contributions" class="level3">
<h3>Contributions</h3>
<p>The material in this chapter expands on work presented in</p>
<p> <span class="citation" data-cites="cassellaExactChiralAmorphous2022"> [<a href="#ref-cassellaExactChiralAmorphous2022" role="doc-biblioref">1</a>]</span> Cassella, G., D’Ornellas, P., Hodson, T., Natori, W. M., &amp; Knolle, J. (2022). An exact chiral amorphous spin liquid. <em>arXiv preprint arXiv:2208.08246.</em></p>
<p>All the code is available online <span class="citation" data-cites="hodsonKoalaKitaevAmorphous2022"> [<a href="#ref-hodsonKoalaKitaevAmorphous2022" role="doc-biblioref">2</a>]</span>.</p>
<p>This was a joint project of Gino, Peru and myself with advice and guidance from Willian and Johannes (all authors of the above). The project grew out of an interest the three of us had in studying amorphous systems, coupled with Johannes’ expertise on the Kitaev model. The idea to use Voronoi partitions came from <span class="citation" data-cites="marsalTopologicalWeaireThorpe2020"> [<a href="#ref-marsalTopologicalWeaireThorpe2020" role="doc-biblioref">3</a>]</span> and Gino did the implementation of this. The idea and implementation of the edge colouring using SAT solvers and the mapping from flux sector to bond sector using A* search were both entirely my work. Peru produced the numerical evidence for the ground state and implemented the local markers. Gino and I did much of the rest of the programming for Koala collaboratively, often pair programming, this included the phase diagram, edge mode and finite temperature analyses as well as the derivation of the projector in the amorphous case.</p>
<p>All the code is available online as a Python package called Koala <span class="citation" data-cites="hodsonKoalaKitaevAmorphous2022"> [<a href="#ref-hodsonKoalaKitaevAmorphous2022" role="doc-biblioref">2</a>]</span>.</p>
<p>This was a joint project of Gino, Peru and myself with advice and guidance from Willian and Johannes, all authors of the above. The project grew out of an interest the three of us had in studying amorphous systems, coupled with Johannes’ expertise on the Kitaev model. The idea to use Voronoi partitions came from <span class="citation" data-cites="marsalTopologicalWeaireThorpe2020"> [<a href="#ref-marsalTopologicalWeaireThorpe2020" role="doc-biblioref">3</a>]</span> and Gino did the implementation of this. The idea and implementation of the edge colouring using SAT solvers and the mapping from flux sector to bond sector using A* search were both entirely my work. Peru produced the numerical evidence for the ground state and implemented the local markers. Gino and I did much of the rest of the programming for Koala collaboratively, often pair programming, this included the phase diagram, edge mode and finite temperature analyses as well as the derivation of the projector in the amorphous case.</p>
</section>
<section id="ak-summary" class="level3">
<h3>Chapter Summary</h3>
<p>In this chapter I will, first, define the amorphous Kitaev (AK) model and discuss the construction of amorphous lattices. Second, I will present some more details of the the methods, including Voronisation and graph colouring in the <a href="../4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html#amk-methods">methods</a> section. Finally I will then present and interpret the <a href="../4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#amk-results">results</a> obtained.</p>
<p>From its introduction it was known that Kitaev Honeycomb (KH) model would remain solvable on any trivalent graph. This has been used to generalise the model to many lattices <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">4</a>–<a href="#ref-Peri2020" role="doc-biblioref">7</a>]</span> but so far none have fully broken the translation symmetry of the model.</p>
<p>Amorphous lattices are characterised by local constraints but no long-range order. They arise, for instance, in amorphous semiconductors like Silicon and Germanium <span class="citation" data-cites="Yonezawa1983 zallen2008physics"> [<a href="#ref-Yonezawa1983" role="doc-biblioref">8</a>,<a href="#ref-zallen2008physics" role="doc-biblioref">9</a>]</span>. Recent work has shown that topological insulating (TI) phases, characterised by protected edge states and topological bulk invariants, can exist in amorphous systems <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 agarwala2019topological marsalTopologicalWeaireThorpeModels2020 costa2019toward agarwala2020higher spring2021amorphous corbae2019evidence"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">10</a>–<a href="#ref-corbae2019evidence" role="doc-biblioref">16</a>]</span>. TI phases, however, arise in non-interacting systems. In this context, we might ask whether Quantum Spin Liquid (QSL) systems and the Kitaev Honeycomb (KH) model, in particular, could be realised on amorphous lattices. The phases of the KH model have many similarities with TIs but differ in that the KH model is an interacting system. In general, research on amorphous electronic systems has been focused mainly on non-interacting systems with the exception of amorphous superconductivity <span class="citation" data-cites="buckel1954einfluss mcmillan1981electron meisel1981eliashberg bergmann1976amorphous mannaNoncrystallineTopologicalSuperconductors2022"> [<a href="#ref-buckel1954einfluss" role="doc-biblioref">17</a>–<a href="#ref-mannaNoncrystallineTopologicalSuperconductors2022" role="doc-biblioref">21</a>]</span> or very recent work looking to understand the effect of strong electron repulsion in TIs <span class="citation" data-cites="kim2022fractionalization"> [<a href="#ref-kim2022fractionalization" role="doc-biblioref">22</a>]</span>.</p>
<p>The KH model is a magnetic system. Magnetism in amorphous systems has been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems <span class="citation" data-cites="aharony1975critical Petrakovski1981 kaneyoshi1992introduction Kaneyoshi2018"> [<a href="#ref-aharony1975critical" role="doc-biblioref">23</a>–<a href="#ref-Kaneyoshi2018" role="doc-biblioref">26</a>]</span>. This is not always ideal, we have already seen that the topological disorder of amorphous lattices can be qualitatively different from standard bond or site disorder, especially in 2D <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">27</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">28</a>]</span>. Research focused on classical Heisenberg and Ising models has accounted for the observed behaviour of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour <span class="citation" data-cites="coey1978amorphous"> [<a href="#ref-coey1978amorphous" role="doc-biblioref">29</a>]</span>. However, the role of the spin-anisotropic interactions and quantum effects that we see in the KH model has not been addressed in amorphous magnets. It is an open question whether frustrated magnetic interactions on amorphous lattices can give rise to genuine quantum phases such as QSLs <span class="citation" data-cites="Anderson1973 Knolle2019 Savary2016 Lacroix2011"> [<a href="#ref-Anderson1973" role="doc-biblioref">30</a>–<a href="#ref-Lacroix2011" role="doc-biblioref">33</a>]</span>. This chapter will answer that question by demonstrating that the Kitaev model on amorphous lattices leads to a kind of QSL called a chiral spin liquid.</p>
<p>In this section I will discuss how to generalise the KH to an amorphous lattice. The <a href="../4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html#amk-methods">methods section</a> discusses how to generate such lattices using Voronoi partitions of the plane <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">10</a>,<a href="#ref-marsalTopologicalWeaireThorpeModels2020" role="doc-biblioref">12</a>]</span>, colour them using a SAT solver and how to map back and forth between gauge field configurations and flux configurations. In the <a href="../4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#amk-results">results section</a>, I will show extensive numerical evidence that the model follows the simple generalisation to Lieb’s theorem <span class="citation" data-cites="lieb_flux_1994"> [<a href="#ref-lieb_flux_1994" role="doc-biblioref">34</a>]</span> found by other works <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">4</a>–<a href="#ref-Peri2020" role="doc-biblioref">7</a>]</span>. I then map out the phase diagram of the model and show that the chiral phase around the symmetric point (<span class="math inline">\(J_x = J_y = J_z\)</span>) is gapped and non-Abelian. We use a quantised local Chern number <span class="math inline">\(\nu\)</span> <span class="citation" data-cites="peru_preprint mitchellAmorphousTopologicalInsulators2018"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">10</a>,<a href="#ref-peru_preprint" role="doc-biblioref">35</a>]</span> as well as the presence of protected chiral Majorana edge modes to determine this. Finally, I look at the role of finite temperature fluctuations and show that the proliferation of flux excitations leads to an Anderson transition, similar to that of the Falicov-Kimball model, to a thermal metal phase <span class="citation" data-cites="Laumann2012 lahtinenTopologicalLiquidNucleation2012 selfThermallyInducedMetallic2019"> [<a href="#ref-Laumann2012" role="doc-biblioref">36</a>–<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">38</a>]</span>. Finally, I consider possible physical realisations of the model and other generalisations.</p>
</section>
</section>
<section id="amk-Model" class="level1">
<h1>The Model</h1>
<p>Already at its introduction it was known that Kitaev honeycomb model would remain solvable on any tri-coordinated graph. This has been used to generalise the model to many lattices <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">4</a>–<a href="#ref-Peri2020" role="doc-biblioref">7</a>]</span> but so far none have fully broken the translation symmetry of the model by putting it onto a amorphous lattice.</p>
<p>Amorphous lattices are characterised by local constraints but no long range order. These arise, for instance, in amorphous semiconductors like Silicon and Germanium <span class="citation" data-cites="Yonezawa1983 zallen2008physics"> [<a href="#ref-Yonezawa1983" role="doc-biblioref">8</a>,<a href="#ref-zallen2008physics" role="doc-biblioref">9</a>]</span>. Recent work has shown that topological insulating (TI) phases, characterized by protected edge states and topological bulk invariants, can exist in amorphous systems <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 agarwala2019topological marsalTopologicalWeaireThorpeModels2020 costa2019toward agarwala2020higher spring2021amorphous corbae2019evidence"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">10</a>–<a href="#ref-corbae2019evidence" role="doc-biblioref">16</a>]</span>. TI phases, however, arise in non-interacting systems. In this context, we might ask whether QSL systems and the KH model in particular could be realised on amorphous lattices. The phases of the KH model have many similarities with TIs but differ in that the KH model is an interacting system. In general, research on amorphous electronic systems has been focused mainly on non-interacting systems with the exception of amorphous superconductivity <span class="citation" data-cites="buckel1954einfluss mcmillan1981electron meisel1981eliashberg bergmann1976amorphous mannaNoncrystallineTopologicalSuperconductors2022"> [<a href="#ref-buckel1954einfluss" role="doc-biblioref">17</a>–<a href="#ref-mannaNoncrystallineTopologicalSuperconductors2022" role="doc-biblioref">21</a>]</span> or very recent work looking to understand the effect of strong electron repulsion in TIs <span class="citation" data-cites="kim2022fractionalization"> [<a href="#ref-kim2022fractionalization" role="doc-biblioref">22</a>]</span>.</p>
<p>Looking more towards the KH model, magnetism in amorphous systems has been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems <span class="citation" data-cites="aharony1975critical Petrakovski1981 kaneyoshi1992introduction Kaneyoshi2018"> [<a href="#ref-aharony1975critical" role="doc-biblioref">23</a>–<a href="#ref-Kaneyoshi2018" role="doc-biblioref">26</a>]</span>. We have already seen that the topological disorder of amorphous lattices can be qualitatively different from standard bond or site disorder, especially in two dimensions <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">27</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">28</a>]</span>. Research focused on classical Heisenberg and Ising models has accounted for the observed behaviour of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour <span class="citation" data-cites="coey1978amorphous"> [<a href="#ref-coey1978amorphous" role="doc-biblioref">29</a>]</span>. However, the role of the spin-anisotropic interactions and quantum effects that we see in the KH model has not been addressed in amorphous magnets. It is an open question whether frustrated magnetic interactions on amorphous lattices can give rise genuine quantum phases such as QSLs <span class="citation" data-cites="Anderson1973 Knolle2019 Savary2016 Lacroix2011"> [<a href="#ref-Anderson1973" role="doc-biblioref">30</a>–<a href="#ref-Lacroix2011" role="doc-biblioref">33</a>]</span>. This chapter will answer that question by demonstrating that the Kitaev model on amorphous lattices leads to a kind of QSL called a chiral spin liquid (CSL).</p>
<p>In this section I will discuss how to generalise the Kitaev model to an amorphous lattice. The methods section discusses how to generate such lattices using Voronoi partitions of the plane <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">10</a>,<a href="#ref-marsalTopologicalWeaireThorpeModels2020" role="doc-biblioref">12</a>]</span>, colour them using a SAT solver and how to map back and forth between gauge field configurations and flux configurations. In the results section, I will show extensive numerical evidence that the model follows the simple generalisation to Lieb’s theorem <span class="citation" data-cites="lieb_flux_1994"> [<a href="#ref-lieb_flux_1994" role="doc-biblioref">34</a>]</span> found by other works <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">4</a>–<a href="#ref-Peri2020" role="doc-biblioref">7</a>]</span>. I then map out the phase diagram of the model and show that the chiral phase around the symmetric point (<span class="math inline">\(J_x = J_y = J_z\)</span>) is gapped and non-Abelian as characterized by a quantized local Chern number <span class="math inline">\(\nu\)</span> <span class="citation" data-cites="peru_preprint mitchellAmorphousTopologicalInsulators2018"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">10</a>,<a href="#ref-peru_preprint" role="doc-biblioref">35</a>]</span> as well as protected chiral Majorana edge modes. Finally, I look at the role of finite temperature fluctuations and show that the proliferation of flux excitations leads to an Anderson transition (similar to that of the FK model) to a thermal metal phase <span class="citation" data-cites="Laumann2012 lahtinenTopologicalLiquidNucleation2012 selfThermallyInducedMetallic2019"> [<a href="#ref-Laumann2012" role="doc-biblioref">36</a>–<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">38</a>]</span>. Finally I consider possible physical realisations of the model and other generalisations.</p>
<figure>
<img src="/assets/thesis/amk_chapter/intro/amk_zoom/amk_zoom_by_hand.svg" id="fig-amk-zoom" data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%" alt="Figure 1: (a) The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable 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}. This leaves a single Majorana c_i per site." />
<figcaption aria-hidden="true">Figure 1: <strong>(a)</strong> The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable 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>. This leaves a single Majorana <span class="math inline">\(c_i\)</span> per site.</figcaption>
</figure>
<p>The KH model must remain solvable on any lattice which satisfies two properties. First the lattice must first be trivalent, every vertex must three edges attached to it <span class="citation" data-cites="kitaevAnyonsExactlySolved2006 Nussinov2009"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">39</a>,<a href="#ref-Nussinov2009" role="doc-biblioref">40</a>]</span>. A well studied class of amorphous trivalent lattices are the 2D Voronoi lattices <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 florescu_designer_2009 marsalTopologicalWeaireThorpeModels2020"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">10</a>,<a href="#ref-marsalTopologicalWeaireThorpeModels2020" role="doc-biblioref">12</a>,<a href="#ref-florescu_designer_2009" role="doc-biblioref">41</a>]</span>. These arise from Voronoi partitions of the plane. Given a set of seed points, the Voronoi partition divides the plane into regions based on which seed point is closest by some metric, usually the euclidean metric. The ‘spheres of influence’ of each seed point form the plaquettes of the resulting lattices, while the boundaries become the edges. The Voronoi partition exists in arbitrary dimensions and produces lattices with coordination number <span class="math inline">\(d+1\)</span> except for degenerate cases with measure zero <span class="citation" data-cites="voronoiNouvellesApplicationsParamètres1908 watsonComputingNdimensionalDelaunay1981"> [<a href="#ref-voronoiNouvellesApplicationsParamètres1908" role="doc-biblioref">42</a>,<a href="#ref-watsonComputingNdimensionalDelaunay1981" role="doc-biblioref">43</a>]</span>. Hence Voronoi lattices in two dimensions lends themselves naturally to the Kitaev model.</p>
<p>Other methods of lattice generation are possible, one can connect randomly placed sites based on proximity <span class="citation" data-cites="agarwala2019topological"> [<a href="#ref-agarwala2019topological" role="doc-biblioref">11</a>]</span> or create simplices from random sites <span class="citation" data-cites="christRandomLatticeField1982"> [<a href="#ref-christRandomLatticeField1982" role="doc-biblioref">44</a>]</span>. However these methods do not present a natural way to restrict the vertex degree to a constant. Perhaps ideally, we would sample uniformly from the space of possible trivalent graphs, there has been some work on how to do this using a Markov Chain Monte Carlo approach <span class="citation" data-cites="alyamiUniformSamplingDirected2016"> [<a href="#ref-alyamiUniformSamplingDirected2016" role="doc-biblioref">45</a>]</span>. However, it does not guarantee that the resulting graph is planar, which is necessary to be able to 3-edge-colour the lattice, our second constraint.</p>
<p>The second constraint required for the Kitaev model to remain solvable is that we must be able to assign labels to each bond <span class="math inline">\(\{x,y,z\}\)</span> such that each no two edges of the same label meet at a vertex. Such an assignment is is known as a 3-edge-colouring. For translation invariant models we need only find a solution for the unit cell which is usually small enough that this can be done by hand. For amorphous lattices the difficulty is compounded by the fact that, to the best of my knowledge, the problem of edge-colouring is in NP. To find colourings in practice, we will employ a standard method from the computer science literature for finding solutions of NP problems called a SAT solver, this is discussed in more detail in the <a href="../4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html#amk-methods">methods secton</a>.</p>
<p>We find that for larger lattices there are many valid colourings. In the isotropic case <span class="math inline">\(J^\alpha = 1\)</span> the colouring has no physical significance. As the definition of the four Majoranas at a site is arbitrary, we can define a local operator that transforms the colouring of any particular site to another permutation and show that these operators commute with the Hamiltonian, see <a href="../6_Appendices/A.4_Lattice_Colouring.html#lattice-colouring">appendix A.4</a>. We cannot do this in the anisotropic case but we nevertheless expect the lattices to exhibit a self averaging behaviour in larger systems when the colouring is chosen arbitrarily.</p>
<p>The KH model is solvable on any lattice which satisfies two properties. First the lattice must first be trivalent, every vertex must have three edges attached to it <span class="citation" data-cites="kitaevAnyonsExactlySolved2006 Nussinov2009"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">39</a>,<a href="#ref-Nussinov2009" role="doc-biblioref">40</a>]</span>. A well studied class of amorphous trivalent lattices are the 2D Voronoi lattices <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 florescu_designer_2009 marsalTopologicalWeaireThorpeModels2020"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">10</a>,<a href="#ref-marsalTopologicalWeaireThorpeModels2020" role="doc-biblioref">12</a>,<a href="#ref-florescu_designer_2009" role="doc-biblioref">41</a>]</span>. These arise from Voronoi partitions of the plane. Given a set of seed points, the Voronoi partition divides the plane into basins, based on which seed point is closest by some metric, usually the euclidean metric. The basins of each seed point form the plaquettes of the resulting lattices, while the boundaries become the edges. The Voronoi partition exists in arbitrary dimension <span class="math inline">\(d\)</span> and produces lattices with degree <span class="math inline">\(d+1\)</span> except for degenerate cases with measure zero <span class="citation" data-cites="voronoiNouvellesApplicationsParamètres1908 watsonComputingNdimensionalDelaunay1981"> [<a href="#ref-voronoiNouvellesApplicationsParamètres1908" role="doc-biblioref">42</a>,<a href="#ref-watsonComputingNdimensionalDelaunay1981" role="doc-biblioref">43</a>]</span>. Hence Voronoi lattices in 2D lends themselves naturally to the Kitaev model.</p>
<p>Other methods of lattice generation are possible. One can connect randomly placed sites based on proximity <span class="citation" data-cites="agarwala2019topological"> [<a href="#ref-agarwala2019topological" role="doc-biblioref">11</a>]</span> or create simplices from random sites <span class="citation" data-cites="christRandomLatticeField1982"> [<a href="#ref-christRandomLatticeField1982" role="doc-biblioref">44</a>]</span>. However these methods do not present a natural way to restrict the vertex degree to a constant. Perhaps ideally, we would sample uniformly from the space of possible trivalent graphs, there has been some work on how to do this using a Markov Chain Monte Carlo approach <span class="citation" data-cites="alyamiUniformSamplingDirected2016"> [<a href="#ref-alyamiUniformSamplingDirected2016" role="doc-biblioref">45</a>]</span>. However, it does not guarantee that the resulting graph is planar, which is necessary to be able to three-edge-colour the lattice, our second constraint.</p>
<p>The second constraint required for the Kitaev model to remain solvable is that we must be able to assign labels to each bond <span class="math inline">\(\{x,y,z\}\)</span> such that each no two edges with the same label meet at a vertex. Such an assignment is is known as a three-edge-colouring. For translation invariant models we need only find a solution for the unit cell. This problem is usually small enough that this can be done by hand or using symmetry. For amorphous lattices, the difficulty is that, to the best of my knowledge, the problem of edge-colouring these lattices in general is in NP. To find colourings in practice, we will employ a standard method from the computer science literature for finding solutions of NP problems called a SAT solver, this is discussed in more detail in the <a href="../4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html#amk-methods">methods secton</a>.</p>
<p>We find that for large lattices there are many valid colourings. In the isotropic case <span class="math inline">\(J^\alpha = 1\)</span> the colouring has no physical significance as the definition of the four Majoranas at a site is arbitrary. In the anisotropic case this symmetry is broken at the local level but we nevertheless expect the lattices to exhibit a self averaging behaviour in larger systems such that the choice of colouring doesn’t matter.</p>
<figure>
<img src="/assets/thesis/amk_chapter/intro/state_decomposition_animated/state_decomposition_animated.gif" id="fig-state_decomposition_animated" data-short-caption="State Decomposition" style="width:100.0%" alt="Figure 2: (Bond Sector) A state in the bond sector is specified by assigning \pm 1 to each edge of the lattice. However, this description has a substantial gauge degeneracy. We can simplify things by decomposing each state into the product of three kinds of objects: (Vortex Sector) Only a small number of bonds need to be flipped (compared to some arbitrary reference) to reconstruct the vortex sector. Here, the edges are chosen from a spanning tree of the dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of the bond sector can be factored out. This gives a network of loops that can always be written as a product of the gauge operators D_j. (Topological Sector) Finally, there are two loops that have no effect on the vortex sector, nor can they be constructed from gauge symmetries. These can be thought of as two fluxes \Phi_{x/y} that thread through the major and minor axes of the torus. Measuring \Phi_{x/y} amounts to constructing Wilson loops around the axes of the torus. We can flip the value of \Phi_{x} by transporting a vortex pair around the torus in the y direction, as shown here. In each of the three figures on the right, black bonds correspond to those that must be flipped. Composing the three together gives back the original bond sector on the left." />
<figcaption aria-hidden="true">Figure 2: (Bond Sector) A state in the bond sector is specified by assigning <span class="math inline">\(\pm 1\)</span> to each edge of the lattice. However, this description has a substantial gauge degeneracy. We can simplify things by decomposing each state into the product of three kinds of objects: (Vortex Sector) Only a small number of bonds need to be flipped (compared to some arbitrary reference) to reconstruct the vortex sector. Here, the edges are chosen from a spanning tree of the dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of the bond sector can be factored out. This gives a network of loops that can always be written as a product of the gauge operators <span class="math inline">\(D_j\)</span>. (Topological Sector) Finally, there are two loops that have no effect on the vortex sector, nor can they be constructed from gauge symmetries. These can be thought of as two fluxes <span class="math inline">\(\Phi_{x/y}\)</span> that thread through the major and minor axes of the torus. Measuring <span class="math inline">\(\Phi_{x/y}\)</span> amounts to constructing Wilson loops around the axes of the torus. We can flip the value of <span class="math inline">\(\Phi_{x}\)</span> by transporting a vortex pair around the torus in the <span class="math inline">\(y\)</span> direction, as shown here. In each of the three figures on the right, black bonds correspond to those that must be flipped. Composing the three together gives back the original bond sector on the left.</figcaption>
<img src="/assets/thesis/amk_chapter/intro/state_decomposition_animated/state_decomposition_animated.gif" id="fig-state_decomposition_animated" data-short-caption="State Decomposition" style="width:100.0%" alt="Figure 2: (Bond Sector) A state in the bond sector is specified by assigning \pm 1 to each edge of the lattice. However, this description has a substantial gauge degeneracy. To remove it, we decompose each state into the product of three kinds of objects: (Flux Sector) The main physically relevant quantities. Only a small number of bonds need to be flipped (compared to some arbitrary fixed reference) to reconstruct the flux sector. Here, the edges are chosen from a spanning tree of the dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of the bond sector is in this part. This is a network of loops that can always be written as a product of the gauge operators D_j. (Topological Sector) Finally, there are two loops that have no effect on the vortex sector, nor can they be constructed from gauge symmetries D_j. These can be thought of as two fluxes \Phi_{x/y} that thread through the major and minor axes of the torus. Measuring \Phi_{x/y} amounts to constructing Wilson loops around the axes of the torus. We can flip the value of \Phi_{x} by transporting a vortex pair around the torus in the y direction, as shown here. In each of the three figures on the right, black bonds correspond to those that must be flipped, while red line are those same edges on the dual lattice. Composing the three objects together gives back the original bond sector on the left." />
<figcaption aria-hidden="true">Figure 2: (Bond Sector) A state in the bond sector is specified by assigning <span class="math inline">\(\pm 1\)</span> to each edge of the lattice. However, this description has a substantial gauge degeneracy. To remove it, we decompose each state into the product of three kinds of objects: (Flux Sector) The main physically relevant quantities. Only a small number of bonds need to be flipped (compared to some arbitrary fixed reference) to reconstruct the flux sector. Here, the edges are chosen from a spanning tree of the dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of the bond sector is in this part. This is a network of loops that can always be written as a product of the gauge operators <span class="math inline">\(D_j\)</span>. (Topological Sector) Finally, there are two loops that have no effect on the vortex sector, nor can they be constructed from gauge symmetries <span class="math inline">\(D_j\)</span>. These can be thought of as two fluxes <span class="math inline">\(\Phi_{x/y}\)</span> that thread through the major and minor axes of the torus. Measuring <span class="math inline">\(\Phi_{x/y}\)</span> amounts to constructing Wilson loops around the axes of the torus. We can flip the value of <span class="math inline">\(\Phi_{x}\)</span> by transporting a vortex pair around the torus in the <span class="math inline">\(y\)</span> direction, as shown here. In each of the three figures on the right, black bonds correspond to those that must be flipped, while red line are those same edges on the dual lattice. Composing the three objects together gives back the original bond sector on the left.</figcaption>
</figure>
<p>On a lattice with the above properties, the solution laid out in <a href="../2_Background/2.2_HKM_Model.html#bg-hkm-model">the kitaev model</a> remains applicable to our Amorphous Kitaev (AK) model. See fig. <a href="#fig:amk-zoom">1</a> for an example lattice generated by our method. The main differences are twofold. Firstly, the lattices are no longer bipartite in general and therefore contain plaquettes with an odd number of sides which have flux <span class="math inline">\(\pm i\)</span>. This leads the AK model to have a ground state with spontaneously broken chiral symmetry <span class="citation" data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"> [<a href="#ref-Peri2020" role="doc-biblioref">7</a>,<a href="#ref-Chua2011" role="doc-biblioref">46</a>–<a href="#ref-WangHaoranPRB2021" role="doc-biblioref">52</a>]</span>. This is also similar to the behaviour of the original Kitaev model in response to a magnetic field. One ground state is related to the other by globally inverting the imaginary <span class="math inline">\(\phi_i\)</span> fluxes <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">47</a>]</span>.</p>
<p>Secondly, as the model is no longer translationally invariant, Lieb’s theorem for the ground state flux sector no longer applies. However as discussed in the background, a simple generalisation of Lieb’s theorem has been shown numerically to be applicable to many generalised Kitaev models <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">4</a>–<a href="#ref-Peri2020" role="doc-biblioref">7</a>]</span>. This generalisation states that the ground state flux configuration depends on on the number of sides of each plaquette <span class="math inline">\(\phi = -(\pm i)^{n_{\mathrm{sides}}}\)</span> with a twofold global chiral degeneracy.</p>
<p>The obvious approach would be to verify that this generalises to the AK model numerically via exhaustive checking of flux configurations. However this is problematic because the number of states to check scales exponentially with system size. We side step this by gluing together two methods, we first work with lattices small enough that we can fully enumerate their flux sectors but tile them to reduce finite size effects. We then show that the effect of tiling scales away with system size.</p>
<p>On a lattice with the above properties, the solution for the KH model in <a href="../2_Background/2.2_HKM_Model.html#bg-hkm-model">section 2.2</a> remains applicable to our AK model. See fig. <a href="#fig:amk-zoom">1</a> for an example lattice generated by our method. The main differences are twofold. Firstly, the lattices are no longer bipartite in general and therefore contain plaquettes with an odd number of sides which have flux <span class="math inline">\(\pm i\)</span>. This leads the AK model to have a ground state with spontaneously broken chiral symmetry <span class="citation" data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"> [<a href="#ref-Peri2020" role="doc-biblioref">7</a>,<a href="#ref-Chua2011" role="doc-biblioref">46</a>–<a href="#ref-WangHaoranPRB2021" role="doc-biblioref">52</a>]</span>. This is similar to the behaviour of the original Kitaev model in response to a magnetic field. One ground state is related to the other by globally inverting the imaginary <span class="math inline">\(\phi_i\)</span> fluxes <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">47</a>]</span>. Secondly, as the model is no longer translationally invariant, Lieb’s theorem for the ground state flux sector no longer applies. However as discussed in the background, a simple generalisation of Lieb’s theorem has been shown numerically to be applicable to many generalised Kitaev models <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">4</a>–<a href="#ref-Peri2020" role="doc-biblioref">7</a>]</span>. This generalisation states that the ground state flux configuration depends on on the number of sides of each plaquette as</p>
<p><span id="eq:gs-flux-sector"><span class="math display">\[\phi = -(\pm i)^{n_{\mathrm{sides}}}\qquad{(1)}\]</span></span></p>
<p>with a twofold global chiral degeneracy (picking either <span class="math inline">\(+i\)</span> or <span class="math inline">\(-i\)</span> in eq. <a href="#eq:gs-flux-sector">1</a>).</p>
<p>To verify that this generalises to the AK model numerically, the obvious approach would be via exhaustive checking of flux configurations. However this is problematic because the number of states to check scales exponentially with system size. We side-step this by gluing together two methods, we first work with lattices small enough that we can fully enumerate their flux sectors but tile them to reduce finite size effects. We then show that the effect of tiling scales away with system size.</p>
<figure>
<img src="/assets/thesis/amk_chapter/majorana_bound_states/majorana_bound_states.svg" id="fig-majorana_bound_states" data-short-caption="Majorana Bound States" style="width:100.0%" alt="Figure 3: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object." />
<figcaption aria-hidden="true">Figure 3: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object.</figcaption>
</figure>
<p>In order to evaluate the Chern marker later, we need a way to evaluate the model on open boundary conditions. Simply removing bonds from the lattice leaves behind unpaired <span class="math inline">\(b^\alpha\)</span> operators that must be paired in some way to arrive at fermionic modes. To fix a pairing, we always start from a lattice defined on the torus and generate a lattice with open boundary conditions by defining the bond coupling <span class="math inline">\(J^{\alpha}_{ij} = 0\)</span> for sites joined by bonds <span class="math inline">\((i,j)\)</span> that we want to remove. This creates fermionic zero modes <span class="math inline">\(u_{ij}\)</span> associated with these cut bonds which we set to 1 when calculating the projector. Alternatively, since all the fermionic zero modes are degenerate anyway, an arbitrary pairing of the unpaired <span class="math inline">\(b^\alpha\)</span> operators could be performed.</p>
<p>In order to evaluate the Chern marker later, we need a way to evaluate the model on open boundary conditions. Simply removing bonds from the lattice leaves behind unpaired <span class="math inline">\(b^\alpha\)</span> operators that must be paired in some way to arrive at fermionic modes. To fix a pairing, we always start from a lattice defined on the torus and generate a lattice with open boundary conditions by defining the bond coupling <span class="math inline">\(J^{\alpha}_{ij} = 0\)</span> for sites joined by bonds <span class="math inline">\((i,j)\)</span> that we want to remove. This creates fermionic zero modes <span class="math inline">\(u_{ij}\)</span> associated with these cut bonds which we set to 1 when calculating the projector. Alternatively, since all the fermionic zero modes are degenerate anyway, an arbitrary pairing of the unpaired <span class="math inline">\(b^\alpha\)</span> operators can be performed.</p>
<!-- <figure>
<img src="../../figure_code/amk_chapter/loops_and_dual_loops/loops_and_dual_loops.svg" style="max-width:700px;" title="Topological Loops and Dual Loops">
<figcaption>
@ -120,12 +135,12 @@ The same as @fig:flood_fill but for the amorphous lattice.
</figure> -->
<section id="the-euler-equation" class="level2">
<h2>The Euler Equation</h2>
<p>Euler’s equation provides a convenient way to understand how the states of the AK model factorise into flux sectors, gauge sectors and topological sectors. The Euler equation states if we embed a lattice with <span class="math inline">\(B\)</span> bonds, <span class="math inline">\(P\)</span> plaquettes and <span class="math inline">\(V\)</span> vertices onto a closed surface of genus <span class="math inline">\(g\)</span>, (<span class="math inline">\(0\)</span> for the sphere, <span class="math inline">\(1\)</span> for the torus) then</p>
<p>Euler’s equation provides a convenient way to understand how the states of the AK model factorise into flux sectors, gauge sectors and topological sectors as in fig. <a href="#fig:state_decomposition_animated">2</a>. The Euler equation states that if we embed a lattice with <span class="math inline">\(B\)</span> bonds, <span class="math inline">\(P\)</span> plaquettes and <span class="math inline">\(V\)</span> vertices onto a closed surface of genus <span class="math inline">\(g\)</span>, (<span class="math inline">\(0\)</span> for the sphere, <span class="math inline">\(1\)</span> for the torus) then</p>
<p><span class="math display">\[B = P + V + 2 - 2g\]</span></p>
<p>For the case of the torus where <span class="math inline">\(g = 1\)</span>, we can rearrange this and exponentiate it to read:</p>
<p><span class="math display">\[2^B = 2^{P-1}\cdot 2^{V-1} \cdot 2^2\]</span></p>
<p>There are <span class="math inline">\(2^B\)</span> configurations of the bond variables <span class="math inline">\(\{u_{ij}\}\)</span>. Each of these configurations can be uniquely decomposed into a flux sector, a gauge sector and a topological sector, see fig. <a href="#fig:state_decomposition_animated">2</a>. Each of the <span class="math inline">\(P\)</span> plaquette operators <span class="math inline">\(\phi_i\)</span> takes two values but vortices are created in pairs so there are <span class="math inline">\(2^{P-1}\)</span> vortex sectors in total. There are <span class="math inline">\(2^{V-1}\)</span> gauge symmetries formed from the <span class="math inline">\(V\)</span> symmetry operators <span class="math inline">\(D_i\)</span> because <span class="math inline">\(\prod_{j} D_j = \mathbb{I}\)</span> is enforced by the projector. Finally, the two topological fluxes <span class="math inline">\(\Phi_x\)</span> and <span class="math inline">\(\Phi_y\)</span> account for the last factor of <span class="math inline">\(2^2\)</span>.</p>
<p>In addition, the fact that we only work with trivalent lattices implies that each vertex shares three bonds with other vertices so effectively comes with <span class="math inline">\(\tfrac{3}{2}\)</span> bonds. This is consistent with the fact that, in the Majorana representation on the torus, each vertex brings three <span class="math inline">\(b^\alpha\)</span> operators which then pair along bonds to give <span class="math inline">\(3/2\)</span> bonds per vertex. Substituting <span class="math inline">\(3V = 2B\)</span> into Euler’s equation tells us that any trivalent lattice on the torus with <span class="math inline">\(N\)</span> plaquettes has <span class="math inline">\(2N\)</span> vertices and <span class="math inline">\(3N\)</span> bonds. Since each bond is part of two plaquettes this implies that the mean number of sides of a plaquette is exactly six and that odd sides plaquettes must come in pairs.</p>
<p>In addition, the fact that we only work with trivalent lattices implies that each vertex shares three bonds with other vertices so effectively comes with <span class="math inline">\(3/2\)</span> bonds. This is consistent with the fact that, in the Majorana representation on the torus, each vertex brings three <span class="math inline">\(b^\alpha\)</span> operators which then pair along bonds to give <span class="math inline">\(3/2\)</span> bonds per vertex. Substituting <span class="math inline">\(3V = 2B\)</span> into Euler’s equation tells us that any trivalent lattice on the torus with <span class="math inline">\(N\)</span> plaquettes has <span class="math inline">\(2N\)</span> vertices and <span class="math inline">\(3N\)</span> bonds. Since each bond is part of two plaquettes this implies that the mean number of sides of a plaquette is exactly six and that odd sides plaquettes must come in pairs.</p>
<p>Next Section: <a href="../4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html">Methods</a></p>
</section>
</section>

17
_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html
View File

@ -76,15 +76,14 @@ image:
</div>
<section id="amk-methods" class="level1">
<h1>Methods</h1>
<p>This section describes the novel methods we developed to simulate the AK model including lattice generation, bond colouring and the inverse mapping between flux sector and gauge sector. Implementations are available online as a Python package called Koala (Kitaev On Amorphous LAttices) <span class="citation" data-cites="hodsonKoalaKitaevAmorphous2022"> [<a href="#ref-hodsonKoalaKitaevAmorphous2022" role="doc-biblioref">1</a>]</span>. All results and figures herein were generated with Koala.</p>
<p>This section describes the novel methods we developed to simulate the AK model including lattice generation, bond colouring and the inverse mapping between flux sector and gauge sector. All results and figures herein were generated with Koala <span class="citation" data-cites="hodsonKoalaKitaevAmorphous2022"> [<a href="#ref-hodsonKoalaKitaevAmorphous2022" role="doc-biblioref">1</a>]</span>.</p>
<section id="voronisation" class="level2">
<h2>Voronisation</h2>
<figure>
<img src="/assets/thesis/amk_chapter/lattice_construction_animated/lattice_construction_animated.gif" id="fig-lattice_construction_animated" data-short-caption="Lattice Construction" style="width:100.0%" alt="Figure 1: (Left) Lattice construction begins with the Voronoi partition of the plane with respect to a set of seed points (black points) sampled uniformly from \mathbb{R}^2. (Center) However, we want the Voronoi partition of the torus, so we tile the seed points into a three by three grid. The boundaries of each tile are shown in light grey. (Right) Finally, we identify edges corresponding to each other across the boundaries to produce a graph on the torus." />
<figcaption aria-hidden="true">Figure 1: (Left) Lattice construction begins with the Voronoi partition of the plane with respect to a set of seed points (black points) sampled uniformly from <span class="math inline">\(\mathbb{R}^2\)</span>. (Center) However, we want the Voronoi partition of the torus, so we tile the seed points into a three by three grid. The boundaries of each tile are shown in light grey. (Right) Finally, we identify edges corresponding to each other across the boundaries to produce a graph on the torus.</figcaption>
</figure>
<p>The lattices we use are Voronoi partitions of the torus <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020 florescu_designer_2009"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">2</a>–<a href="#ref-florescu_designer_2009" role="doc-biblioref">4</a>]</span>. We start by sampling <em>seed points</em> uniformly (or otherwise) on the torus. As most off the shelf routines for computing Voronoi partitions are defined on the plane rather than the torus, we tile our seed points into a <span class="math inline">\(3\times3\)</span> pr <span class="math inline">\(5\times5\)</span> grid before calling a standard Voronoi routine <span class="citation" data-cites="barberQuickhullAlgorithmConvex1996"> [<a href="#ref-barberQuickhullAlgorithmConvex1996" role="doc-biblioref">5</a>]</span> from the python package Scipy <span class="citation" data-cites="virtanenSciPyFundamentalAlgorithms2020"> [<a href="#ref-virtanenSciPyFundamentalAlgorithms2020" role="doc-biblioref">6</a>]</span>. Finally, we undo the tiling to the grid by identifying edges in the tiled lattice which are identical, yielding a trivalent lattice on the torus. We encode our lattices with edge lists <span class="math inline">\([(i,j), (j,k)\ldots]\)</span> and an additional vector <span class="math inline">\((\{-1,0,+1\}, \{-1,0,+1\})\)</span> for each edge that encodes the sense in which it crosses the periodic boundary conditions, equivalent to how the edge leaves the unit cell were the system to tile the plane, see <a href="../6_Appendices/A.3_Lattice_Generation.html#app-lattice-generation">appendix A.3</a> for more detail.</p>
<p>The graph generated by a Voronoi partition of a two dimensional surface is always planar. This means that no edges cross each other when the graph is embedded into the plane. It is also trivalent in that every vertex is connected to exactly three edges <span class="citation" data-cites="voronoiNouvellesApplicationsParamètres1908 watsonComputingNdimensionalDelaunay1981"> [<a href="#ref-voronoiNouvellesApplicationsParamètres1908" role="doc-biblioref">7</a>,<a href="#ref-watsonComputingNdimensionalDelaunay1981" role="doc-biblioref">8</a>]</span>.</p>
<p>The lattices we use are Voronoi partitions of the torus <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020 florescu_designer_2009"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">2</a>–<a href="#ref-florescu_designer_2009" role="doc-biblioref">4</a>]</span>. We start by sampling <em>seed points</em> uniformly on the torus. As most off the shelf routines for computing Voronoi partitions are defined on the plane rather than the torus, we tile our seed points into a <span class="math inline">\(3\times3\)</span> pr <span class="math inline">\(5\times5\)</span> grid before calling a standard Voronoi routine <span class="citation" data-cites="barberQuickhullAlgorithmConvex1996"> [<a href="#ref-barberQuickhullAlgorithmConvex1996" role="doc-biblioref">5</a>]</span> from the python package Scipy <span class="citation" data-cites="virtanenSciPyFundamentalAlgorithms2020"> [<a href="#ref-virtanenSciPyFundamentalAlgorithms2020" role="doc-biblioref">6</a>]</span>. Finally, we undo the tiling to the grid by identifying edges in the tiled lattice which are identical, yielding a trivalent lattice on the torus. We encode our lattices with edge lists <span class="math inline">\([(i,j), (k,l)\ldots]\)</span> and an additional 2D vector <span class="math inline">\(\vec{v} \in \{-1,0,+1\}^2\)</span> for each edge that encodes the sense in which it crosses the periodic boundary conditions. This is equivalent to how the edge would leave the unit cell were the system to tile the plane, see <a href="../6_Appendices/A.3_Lattice_Generation.html#app-lattice-generation">appendix A.3</a> for more detail. The graph generated by a Voronoi partition of a 2D surface is always planar. This means that no edges cross each other when the graph is embedded into the plane. It is also trivalent in that every vertex is connected to exactly three edges <span class="citation" data-cites="voronoiNouvellesApplicationsParamètres1908 watsonComputingNdimensionalDelaunay1981"> [<a href="#ref-voronoiNouvellesApplicationsParamètres1908" role="doc-biblioref">7</a>,<a href="#ref-watsonComputingNdimensionalDelaunay1981" role="doc-biblioref">8</a>]</span>.</p>
</section>
<section id="colouring-the-bonds" class="level2">
<h2>Colouring the Bonds</h2>
@ -92,11 +91,11 @@ image:
<img src="/assets/thesis/amk_chapter/multiple_colourings/multiple_colourings.svg" id="fig-multiple_colourings" data-short-caption="Colourings of an Amorphous Lattice" style="width:100.0%" alt="Figure 2: Different valid three-edge-colourings of an amorphous lattice. Colors that differ from the leftmost panel are highlighted in the other panels." />
<figcaption aria-hidden="true">Figure 2: Different valid three-edge-colourings of an amorphous lattice. Colors that differ from the leftmost panel are highlighted in the other panels.</figcaption>
</figure>
<p>To be solvable the AK model requires that each edge in the lattice be assigned a label <span class="math inline">\(x\)</span>, <span class="math inline">\(y\)</span> or <span class="math inline">\(z\)</span>, such that each vertex has exactly one edge of each type connected to it. This problem must be distinguished from that considered by the famous four-colour theorem <span class="citation" data-cites="appelEveryPlanarMap1989"> [<a href="#ref-appelEveryPlanarMap1989" role="doc-biblioref">9</a>]</span>. The four-colour theorem is concerned with assigning colours to the <strong>vertices</strong> of planar graphs, such that no vertices that share an edge have the same colour. Here we are instead concerned with finding an edge colouring.</p>
<p>For a graph of maximum degree <span class="math inline">\(\Delta\)</span>, <span class="math inline">\(\Delta + 1\)</span> colours are always enough to edge-colour it. An <span class="math inline">\(\mathcal{O}(mn)\)</span> algorithm exists to do this for a graph with <span class="math inline">\(m\)</span> edges and <span class="math inline">\(n\)</span> vertices <span class="citation" data-cites="gEstimateChromaticClass1964"> [<a href="#ref-gEstimateChromaticClass1964" role="doc-biblioref">10</a>]</span>. Restricting ourselves to graphs with <span class="math inline">\(\Delta = 3\)</span>, these graphs are known as cubic graphs. Cubic graphs can be four-edge-coloured in linear time <span class="citation" data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"> [<a href="#ref-skulrattanakulchai4edgecoloringGraphsMaximum2002" role="doc-biblioref">11</a>]</span>. However we need a three-edge-colouring of our cubic graphs, which turns out to be more difficult. Cubic, planar, <em>bridgeless</em> graphs can be three-edge-coloured if and only if they can be four-face-coloured <span class="citation" data-cites="tait1880remarks"> [<a href="#ref-tait1880remarks" role="doc-biblioref">12</a>]</span>. Bridges are edges that connect otherwise disconnected components. An <span class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists for these <span class="citation" data-cites="robertson1996efficiently"> [<a href="#ref-robertson1996efficiently" role="doc-biblioref">13</a>]</span>. However, it is not clear whether this extends to cubic, <strong>toroidal</strong> bridgeless graphs.</p>
<p>To be solvable, the AK model requires that each edge in the lattice be assigned a label <span class="math inline">\(x\)</span>, <span class="math inline">\(y\)</span> or <span class="math inline">\(z\)</span>, such that each vertex has exactly one edge of each type connected to it, a three-edge-colouring. This problem must be distinguished from that considered by the famous four-colour theorem <span class="citation" data-cites="appelEveryPlanarMap1989"> [<a href="#ref-appelEveryPlanarMap1989" role="doc-biblioref">9</a>]</span>. The four-colour theorem is concerned with assigning colours to the <strong>vertices</strong> of planar graphs, such that no vertices that share an edge have the same colour.</p>
<p>For a graph of maximum degree <span class="math inline">\(\Delta\)</span>, <span class="math inline">\(\Delta + 1\)</span> colours are always enough to edge-colour it. An <span class="math inline">\(\mathcal{O}(mn)\)</span> algorithm exists to do this for a graph with <span class="math inline">\(m\)</span> edges and <span class="math inline">\(n\)</span> vertices <span class="citation" data-cites="gEstimateChromaticClass1964"> [<a href="#ref-gEstimateChromaticClass1964" role="doc-biblioref">10</a>]</span>. Graphs with <span class="math inline">\(\Delta = 3\)</span> are known as cubic graphs. Cubic graphs can be four-edge-coloured in linear time <span class="citation" data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"> [<a href="#ref-skulrattanakulchai4edgecoloringGraphsMaximum2002" role="doc-biblioref">11</a>]</span>. However we need a three-edge-colouring of our cubic graphs, which turns out to be more difficult. Cubic, planar, <em>bridgeless</em> graphs can be three-edge-coloured if and only if they can be four-face-coloured <span class="citation" data-cites="tait1880remarks"> [<a href="#ref-tait1880remarks" role="doc-biblioref">12</a>]</span>. Bridges are edges that connect otherwise disconnected components. An <span class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists for these <span class="citation" data-cites="robertson1996efficiently"> [<a href="#ref-robertson1996efficiently" role="doc-biblioref">13</a>]</span>. However, it is not clear whether this extends to cubic, <strong>toroidal</strong> bridgeless graphs.</p>
<p>A four-face-colouring is equivalent to a four-vertex-colouring of the dual graph, see <a href="../6_Appendices/A.3_Lattice_Generation.html#app-lattice-generation">appendix A.3</a>. So if we could find a four-vertex-colouring of the dual graph we would be done. However vertex-colouring a toroidal graph may require up to seven colours <span class="citation" data-cites="heawoodMapColouringTheorems"> [<a href="#ref-heawoodMapColouringTheorems" role="doc-biblioref">14</a>]</span>! The complete graph of seven vertices <span class="math inline">\(K_7\)</span> is a good example of a toroidal graph that requires seven colours.</p>
<p>Luckily, some problems are harder in theory than in practice. Three-edge-colouring cubic toroidal graphs appears to be one of those things. To find colourings, we use a Boolean Satisfiability Solver or SAT solver. A SAT problem is a set of statements about a set of boolean variables, such as “<span class="math inline">\(x_1\)</span> or not <span class="math inline">\(x_3\)</span> is true”. A solution to a SAT problem is a assignment <span class="math inline">\(x_i \in {0,1}\)</span> that satisfies all the statements <span class="citation" data-cites="Karp1972"> [<a href="#ref-Karp1972" role="doc-biblioref">15</a>]</span>. General purpose, high performance programs for solving SAT problems have been an area of active research for decades <span class="citation" data-cites="alounehComprehensiveStudyAnalysis2019"> [<a href="#ref-alounehComprehensiveStudyAnalysis2019" role="doc-biblioref">16</a>]</span>. Such programs are useful because, by the Cook-Levin theorem, any NP problem can be encoded (in polynomial time) as an instance of a SAT problem . This property is what makes SAT one of the subset of NP problems called NP-Complete <span class="citation" data-cites="cookComplexityTheoremprovingProcedures1971 levin1973universal"> [<a href="#ref-cookComplexityTheoremprovingProcedures1971" role="doc-biblioref">17</a>,<a href="#ref-levin1973universal" role="doc-biblioref">18</a>]</span>. Thus, it is a relatively standard technique in the computer science community to solve NP problems by first transforming them to SAT instances and then using an off the shelf SAT solver. The output of this can then be mapped back to the original problem domain.</p>
<p>Whether graph colouring problems are in NP or P seems to depend delicately on the class of graphs considered, the maximum degree and the number of colours used. Since we I didn’t know of any better algorithm for the problem at hand using a SAT solver appeared to be a reasonable first method to try and it turns out to be fast enough in practice that it is by no means to rate limiting step for solving instances of our model. In <a href="../6_Appendices/A.3_Lattice_Generation.html#app-lattice-generation">appendix A.3</a> I detail the specifics of how I mapped edge-colouring problems to SAT instances and show a breakdown of where the computational effort is spent, the majority being on diagonalisation.</p>
<p>Luckily, some problems are easier in practice. Three-edge-colouring cubic toroidal graphs appears to be one of those things. To find colourings, we use a Boolean Satisfiability Solver or SAT solver. A SAT problem is a set of statements about a set of boolean variables <span class="math inline">\([x_1, x_2\ldots]\)</span>, such as “<span class="math inline">\(x_1\)</span> or not <span class="math inline">\(x_3\)</span> is true”. A solution to a SAT problem is a assignment <span class="math inline">\(x_i \in {0,1}\)</span> that satisfies all the statements <span class="citation" data-cites="Karp1972"> [<a href="#ref-Karp1972" role="doc-biblioref">15</a>]</span>. General purpose, high performance programs for solving SAT problems have been an area of active research for decades <span class="citation" data-cites="alounehComprehensiveStudyAnalysis2019"> [<a href="#ref-alounehComprehensiveStudyAnalysis2019" role="doc-biblioref">16</a>]</span>. Such programs are useful because, by the Cook-Levin theorem <span class="citation" data-cites="cookComplexityTheoremprovingProcedures1971 levin1973universal"> [<a href="#ref-cookComplexityTheoremprovingProcedures1971" role="doc-biblioref">17</a>,<a href="#ref-levin1973universal" role="doc-biblioref">18</a>]</span>, any NP problem can be encoded (in polynomial time) as an instance of a SAT problem. This property is what makes SAT one of the subset of NP problems called NP-Complete. It is a relatively standard technique in the computer science community to solve NP problems by first transforming them to SAT instances and then using an off-the-shelf SAT solver. The output of this can then be mapped back to the original problem domain.</p>
<p>Whether graph colouring problems are in NP or P seems to depend delicately on the class of graphs considered, the maximum degree and the number of colours used. It is therefore possible that a polynomial time algorithm may exist for our problem. However using a SAT solver turns out to be fast enough in practice that it is by no means the rate limiting step for generating and solving instances of the AK model. In <a href="../6_Appendices/A.3_Lattice_Generation.html#app-lattice-generation">appendix A.3</a> I detail the specifics of how I mapped edge-colouring problems to SAT instances and show a breakdown of where the computational effort is spent, the majority being on diagonalisation.</p>
</section>
<section id="mapping-between-flux-sectors-and-bond-sectors" class="level2">
<h2>Mapping between flux sectors and bond sectors</h2>
@ -106,11 +105,11 @@ image:
</figure>
<p>In the AK model, going from the bond sector to flux sector is done simply from the definition of the fluxes</p>
<p><span class="math display">\[ \phi_i = \prod_{(j,k) \; \in \; \partial \phi_i} i u_{jk}.\]</span></p>
<p>The reverse, constructing the bond sector <span class="math inline">\(\{u_{jk}\}\)</span> that corresponds to a particular flux sector <span class="math inline">\(\{\{\Phi_i\}\)</span> is not so trivial. The algorithm, shown visually in fig. <a href="#fig:flux_finding">3</a> is this:</p>
<p>The reverse, constructing a bond sector <span class="math inline">\(\{u_{jk}\}\)</span> that corresponds to a particular flux sector <span class="math inline">\(\{\{\Phi_i\}\)</span> is not so trivial. The algorithm I used, shown visually in fig. <a href="#fig:flux_finding">3</a> is this:</p>
<ol type="1">
<li><p>Fix the gauge by choosing some arbitrary <span class="math inline">\(u_{jk}\)</span> configuration. In practice, we use <span class="math inline">\(u_{jk} = +1\)</span>. This chooses an arbitrary one of the four topological sectors.</p></li>
<li><p>Compute the current flux configuration and how it differs from the target one. Consider any plaquette that differs from the target as a defect.</p></li>
<li><p>Find any adjacent pairs of defects and flip the <span class="math inline">\(u_jk\)</span> between them. This leaves a set of isolated defects.</p></li>
<li><p>Find any adjacent pairs of defects and flip the <span class="math inline">\(u_{jk}\)</span> between them. This leaves a set of isolated defects.</p></li>
<li><p>Pair the defects up using a greedy algorithm and compute paths along the dual lattice between each pair of plaquettes using A*. Flipping the corresponding set of bonds transports one flux to the other and annihilates both.</p></li>
</ol>
<p>Next Section: <a href="../4_Amorphous_Kitaev_Model/4.3_AMK_Results.html">Results</a></p>

96
_thesis/4_Amorphous_Kitaev_Model/4.3_AMK_Results.html
View File

@ -44,12 +44,16 @@ image:
<li><a href="#the-ground-state-flux-sector" id="toc-the-ground-state-flux-sector">The Ground State Flux Sector</a></li>
<li><a href="#ground-state-phase-diagram" id="toc-ground-state-phase-diagram">Ground State Phase Diagram</a>
<ul>
<li><a href="#abelian-or-non-abelian-of-the-gapped-phase" id="toc-abelian-or-non-abelian-of-the-gapped-phase">Abelian or non-Abelian of the Gapped Phase</a></li>
<li><a href="#abelian-or-non-abelian-statistics-of-the-gapped-phase" id="toc-abelian-or-non-abelian-statistics-of-the-gapped-phase">Abelian or non-Abelian statistics of the Gapped Phase</a></li>
<li><a href="#edge-modes" id="toc-edge-modes">Edge Modes</a></li>
</ul></li>
<li><a href="#anderson-transition-to-a-thermal-metal" id="toc-anderson-transition-to-a-thermal-metal">Anderson Transition to a Thermal Metal</a></li>
</ul></li>
<li><a href="#sec:AMK-Conclusion" id="toc-sec:AMK-Conclusion">Conclusion</a></li>
<li><a href="#sec:AMK-Conclusion" id="toc-sec:AMK-Conclusion">Discussion and Conclusion</a>
<ul>
<li><a href="#experimental-realisations-and-signatures" id="toc-experimental-realisations-and-signatures">Experimental Realisations and Signatures</a></li>
<li><a href="#thermodynamics" id="toc-thermodynamics">Thermodynamics</a></li>
</ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
</nav>
@ -68,12 +72,16 @@ image:
<li><a href="#the-ground-state-flux-sector" id="toc-the-ground-state-flux-sector">The Ground State Flux Sector</a></li>
<li><a href="#ground-state-phase-diagram" id="toc-ground-state-phase-diagram">Ground State Phase Diagram</a>
<ul>
<li><a href="#abelian-or-non-abelian-of-the-gapped-phase" id="toc-abelian-or-non-abelian-of-the-gapped-phase">Abelian or non-Abelian of the Gapped Phase</a></li>
<li><a href="#abelian-or-non-abelian-statistics-of-the-gapped-phase" id="toc-abelian-or-non-abelian-statistics-of-the-gapped-phase">Abelian or non-Abelian statistics of the Gapped Phase</a></li>
<li><a href="#edge-modes" id="toc-edge-modes">Edge Modes</a></li>
</ul></li>
<li><a href="#anderson-transition-to-a-thermal-metal" id="toc-anderson-transition-to-a-thermal-metal">Anderson Transition to a Thermal Metal</a></li>
</ul></li>
<li><a href="#sec:AMK-Conclusion" id="toc-sec:AMK-Conclusion">Conclusion</a></li>
<li><a href="#sec:AMK-Conclusion" id="toc-sec:AMK-Conclusion">Discussion and Conclusion</a>
<ul>
<li><a href="#experimental-realisations-and-signatures" id="toc-experimental-realisations-and-signatures">Experimental Realisations and Signatures</a></li>
<li><a href="#thermodynamics" id="toc-thermodynamics">Thermodynamics</a></li>
</ul></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
</nav>
@ -86,28 +94,28 @@ image:
</div>
<section id="amk-results" class="level1">
<h1>Results</h1>
<p>This section contains our results on the AK model, we first look at how we checked numerically that Lieb’s theorem generalises to our model. Next we compute the ground state diagram and look at the two phases that arise there. We then use a local Chern marker and the presence of edge modes to characterise these phases as having Abelian or non-Abelian statistics. Finally we look at the finite temperature behaviour of the model.</p>
<section id="the-ground-state-flux-sector" class="level2">
<h2>The Ground State Flux Sector</h2>
<p>Here I will discuss the numerical evidence that our guess for the ground state flux sector is correct. We will do this by enumerating all the flux sectors of many separate system realisations. However we have two seemingly irreconcilable problems. Finite size effects have a large energetic contribution for small systems <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span> so we would like to perform our analysis for very large lattices. However for an amorphous system with <span class="math inline">\(N\)</span> plaquettes, <span class="math inline">\(2N\)</span> edges and <span class="math inline">\(3N\)</span> vertices we have <span class="math inline">\(2^{N-1}\)</span> flux sectors to check and diagonalisation scales with <span class="math inline">\(\mathcal{O}(N^3)\)</span>. That exponential scaling makes it infeasible to work with lattices much larger than <span class="math inline">\(16\)</span> plaquettes.</p>
<p>To get around this we instead look at periodic systems with amorphous unit cells. For a similarly sized periodic system with <span class="math inline">\(A\)</span> unit cells and <span class="math inline">\(B\)</span> plaquettes in each unit cell where <span class="math inline">\(N \sim AB\)</span> things get much better. We can use Bloch’s theorem to diagonalise this system in about <span class="math inline">\(\mathcal{0}(A B^3)\)</span> operations, and more importantly there are only <span class="math inline">\(2^{B-1}\)</span> flux sectors to check. We fully enumerated the flux sectors of ~25,000 periodic systems with disordered unit cells of up to <span class="math inline">\(B = 16\)</span> plaquettes and <span class="math inline">\(A = 100\)</span> unit cells. However, showing that our guess is correct for periodic systems with disordered unit cells is not quite convincing on its own as we have effectively removed longer-range disorder from our lattices.</p>
<p>The second part of the argument is to show that the energetic effect of introducing periodicity scales away as we go to larger system sizes and has already diminished to a small enough value at 16 plaquettes, which is indeed what we find. From this we argue that the results for small periodic systems generalise to large amorphous systems. In the isotropic case (<span class="math inline">\(J^\alpha = 1\)</span>), our conjecture correctly predicted the ground state flux sector for all of the lattices we tested. For the toric code phase (<span class="math inline">\(J^x = J^y = 0.25, J^z = 1\)</span>) all but around (<span class="math inline">\(\sim 0.5 \%\)</span>) lattices had ground states conforming to our conjecture. In these cases, the energy difference between the true ground state and our prediction was on the order of <span class="math inline">\(10^{-6} J\)</span>. It is unclear whether this is a finite size effect or something else.</p>
<p>The spin Kitaev Hamiltonian is real and therefore has time reversal symmetry (TRS). However in the ground state the flux <span class="math inline">\(\phi_p\)</span> through any plaquette with an odd number of sides has imaginary eigenvalues <span class="math inline">\(\pm i\)</span>. Thus, states with a fixed flux sector spontaneously break time reversal symmetry. This was first described by Yao and Kivelson for a translation invariant Kitaev model with odd sided plaquettes <span class="citation" data-cites="Yao2011"> [<a href="#ref-Yao2011" role="doc-biblioref">2</a>]</span>.</p>
<p>We will check that Lieb’s theorem generalises to our model by enumerating all the flux sectors of many separate amorphous lattice realisations. However we have two seemingly irreconcilable problems. Finite size effects have a large energetic contribution for small systems <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span> so we would like to perform our analysis for very large lattices. However for an amorphous system with <span class="math inline">\(N\)</span> plaquettes, <span class="math inline">\(2N\)</span> edges and <span class="math inline">\(3N\)</span> vertices we have <span class="math inline">\(2^{N-1}\)</span> flux sectors to check and diagonalisation scales with <span class="math inline">\(\mathcal{O}(N^3)\)</span>. That exponential scaling makes it difficult to work with lattices much larger than <span class="math inline">\(16\)</span> plaquettes with the resources.</p>
<p>To get around this we instead look at periodic systems with amorphous unit cells. For a similarly sized periodic system with <span class="math inline">\(A\)</span> unit cells and <span class="math inline">\(B\)</span> plaquettes in each unit cell where <span class="math inline">\(N \sim AB\)</span> things get much better. We can use Bloch’s theorem to diagonalise this system in about <span class="math inline">\(\mathcal{O}(A B^3)\)</span> operations, and more importantly there are only <span class="math inline">\(2^{B-1}\)</span> flux sectors to check. We fully enumerated the flux sectors of <span class="math inline">\(\sim\)</span> 25,000 periodic systems with disordered unit cells of up to <span class="math inline">\(B = 16\)</span> plaquettes and <span class="math inline">\(A = 100\)</span> unit cells. However, showing that our guess is correct for periodic systems with disordered unit cells is not quite convincing on its own as we have effectively removed longer-range disorder from our lattices.</p>
<p>The second part of the argument is to show that the energetic effect of introducing periodicity scales away as we go to larger system sizes and has already diminished to a small enough value at 16 plaquettes, which is indeed what we find. From this we argue that the results for small periodic systems generalise to large amorphous systems. In the isotropic case (<span class="math inline">\(J^\alpha = 1\)</span>), Lieb’s theorem correctly predicts the ground state flux sector for all of the lattices we tested. For the toric code phase (<span class="math inline">\(J^x = J^y = 0.25, J^z = 1\)</span>) all but around (<span class="math inline">\(\sim 0.5 \%\)</span>) lattices had ground states conforming to our conjecture. In these cases, the energy difference between the true ground state and our prediction was on the order of <span class="math inline">\(10^{-6} J\)</span>.</p>
<p>The spin Kitaev Hamiltonian is real and therefore has time reversal symmetry. However in the ground state the flux <span class="math inline">\(\phi_p\)</span> through any plaquette with an odd number of sides has imaginary eigenvalues <span class="math inline">\(\pm i\)</span>. Thus, states with a fixed flux sector spontaneously break time reversal symmetry. Kiteav noted this in his original paper but it was first explored in a concrete model by Yao and Kivelson for a translation invariant Kitaev model with odd sided plaquettes <span class="citation" data-cites="Yao2011"> [<a href="#ref-Yao2011" role="doc-biblioref">2</a>]</span>.</p>
<p>So we have flux sectors that come in degenerate pairs, where time reversal is equivalent to inverting the flux through every odd plaquette, a general feature for lattices with odd plaquettes <span class="citation" data-cites="yaoExactChiralSpin2007 Peri2020"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>,<a href="#ref-Peri2020" role="doc-biblioref">4</a>]</span>. This spontaneously broken symmetry serves a role analogous to the external magnetic field in the original honeycomb model, leading the AK model to have a non-Abelian anyonic phase without an external magnetic field.</p>
</section>
<section id="ground-state-phase-diagram" class="level2">
<h2>Ground State Phase Diagram</h2>
<p>The triangular phase <span class="math inline">\(J_x, J_y, J_z\)</span> phase diagram of this family of models arises from setting the energy scale with <span class="math inline">\(J_x + J_y + J_z = 1\)</span>, the intersection of this plane and the unit cube is what yields the equilateral triangles seen in diagrams like fig. <a href="#fig:phase_diagram">1</a>. The KH model has an Abelian, gapped phase in the anisotropic region (the A phase) and is gapless in the isotropic region. The introduction of a magnetic field breaks the chiral symmetry, leading to the isotropic region becoming a gapped, non-Abelian phase, the B phase.</p>
<p>Similar to the Kitaev Honeycomb model with a magnetic field, we find that the amorphous model is only gapless along critical lines, see fig. <a href="#fig:phase_diagram">1</a> (Left). Interestingly, in finite size systems the gap closing exists in only one of the four topological sectors though the sectors must become degenerate in the thermodynamic limit. Nevertheless this could be a useful way to define the (0, 0) topological flux sector for the amorphous model which otherwise has no natural way to choose it.</p>
<p>In the honeycomb model, the phase boundaries are located on the straight lines <span class="math inline">\(|J^x| = |J^y| + |J^x|\)</span> and permutations of <span class="math inline">\(x,y,z\)</span>. These are shown as dotted lines on ~fig. <a href="#fig:phase_diagram">1</a> (Right). We find that on the amorphous lattice these boundaries exhibit an inward curvature, similar to honeycomb Kitaev models with flux or bond disorder <span class="citation" data-cites="knolle_dynamics_2016 Nasu_Thermal_2015 lahtinenPerturbedVortexLattices2014 willansDisorderQuantumSpin2010 zschockePhysicalStatesFinitesize2015 kaoDisorderDisorderLocalization2021"> [<a href="#ref-knolle_dynamics_2016" role="doc-biblioref">5</a>–<a href="#ref-kaoDisorderDisorderLocalization2021" role="doc-biblioref">10</a>]</span>.</p>
<p>The triangular <span class="math inline">\(J_x, J_y, J_z\)</span> phase diagram of this family of models arises from setting the energy scale with <span class="math inline">\(J_x + J_y + J_z = 1\)</span>. The intersection of this plane and the unit cube is what yields the equilateral triangles seen in diagrams like fig. <a href="#fig:phase_diagram">1</a>. The KH model has an Abelian, gapped phase in the anisotropic region (the A phase) and is gapless in the isotropic region. The introduction of a magnetic field breaks the chiral symmetry, leading to the isotropic region becoming a gapped, non-Abelian phase, the B phase.</p>
<p>Similar to the KH model with a magnetic field, we find that the amorphous model is only gapless along critical lines, see fig. <a href="#fig:phase_diagram">1</a> (Left). Interestingly, in finite size systems the gap closing exists in only one of the four topological sectors though the sectors become degenerate in the thermodynamic limit. Nevertheless this could be a useful way to define the (0, 0) topological flux sector for the amorphous model which otherwise has no natural way to choose it.</p>
<p>In the honeycomb model, the phase boundaries are located on the straight lines <span class="math inline">\(|J^x| = |J^y| \;+ \;|J^x|\)</span> and permutations of <span class="math inline">\(x,y,z\)</span>. These are shown as dotted lines in fig. <a href="#fig:phase_diagram">1</a> (Right). We find that on the amorphous lattice these boundaries exhibit an inward curvature, similar to honeycomb Kitaev models with flux or bond disorder <span class="citation" data-cites="knolle_dynamics_2016 Nasu_Thermal_2015 lahtinenPerturbedVortexLattices2014 willansDisorderQuantumSpin2010 zschockePhysicalStatesFinitesize2015 kaoDisorderDisorderLocalization2021"> [<a href="#ref-knolle_dynamics_2016" role="doc-biblioref">5</a>–<a href="#ref-kaoDisorderDisorderLocalization2021" role="doc-biblioref">10</a>]</span>.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/phase_diagram/phase_diagram.svg" id="fig-phase_diagram" data-short-caption="The Ground State Phase Diagram" style="width:100.0%" alt="Figure 1: The phase diagram of the model can be characterised by an equilateral triangle whose corners indicate points where J_\alpha = 1, J_\beta = J_\gamma = 0 while the centre denotes J_x = J_y = J_z. (Center) To compute critical lines efficiently in this space we evaluate the order parameter of interest along rays shooting from the corners of the phase diagram. The ray highlighted in red defines the values of J used for the left figure. (Left) The fermion gap as a function of J for an amorphous system with 20 plaquettes, where the x axis is the position on the red line in the central figure from 0 to 1. For finite size systems the four topological sectors are not degenerate and only one of them (in green) has a true gap closing. (Right) The Abelian A_\alpha phases of the model and the non-Abelian B phase separated by critical lines where the fermion gap closes. Later we will show that the Chern number \nu changes from 0 to \pm 1 from the A phases to the B phase. Indeed the gap must close in order for the Chern number to change  [11]." />
<figcaption aria-hidden="true">Figure 1: The phase diagram of the model can be characterised by an equilateral triangle whose corners indicate points where <span class="math inline">\(J_\alpha = 1, J_\beta = J_\gamma = 0\)</span> while the centre denotes <span class="math inline">\(J_x = J_y = J_z\)</span>. (Center) To compute critical lines efficiently in this space we evaluate the order parameter of interest along rays shooting from the corners of the phase diagram. The ray highlighted in red defines the values of J used for the left figure. (Left) The fermion gap as a function of J for an amorphous system with 20 plaquettes, where the x axis is the position on the red line in the central figure from 0 to 1. For finite size systems the four topological sectors are not degenerate and only one of them (in green) has a true gap closing. (Right) The Abelian <span class="math inline">\(A_\alpha\)</span> phases of the model and the non-Abelian B phase separated by critical lines where the fermion gap closes. Later we will show that the Chern number <span class="math inline">\(\nu\)</span> changes from <span class="math inline">\(0\)</span> to <span class="math inline">\(\pm 1\)</span> from the A phases to the B phase. Indeed the gap <em>must</em> close in order for the Chern number to change <span class="citation" data-cites="ezawaTopologicalPhaseTransition2013"> [<a href="#ref-ezawaTopologicalPhaseTransition2013" role="doc-biblioref">11</a>]</span>.</figcaption>
</figure>
<section id="abelian-or-non-abelian-of-the-gapped-phase" class="level3">
<h3>Abelian or non-Abelian of the Gapped Phase</h3>
<p>The two phases of the amorphous model are clearly gapped, though later I’ll double check this with finite size scaling.</p>
<p>The next question is: do these phases support excitations with trivial, Abelian or non-Abelian statistics? To answer that we turn to Chern numbers <span class="citation" data-cites="berryQuantalPhaseFactors1984 simonHolonomyQuantumAdiabatic1983 thoulessQuantizedHallConductance1982"> [<a href="#ref-berryQuantalPhaseFactors1984" role="doc-biblioref">12</a>–<a href="#ref-thoulessQuantizedHallConductance1982" role="doc-biblioref">14</a>]</span>. As discussed earlier the Chern number is a quantity intimately linked to both the topological properties and the anyonic statistics of a model. Here we will make use of the fact that the Abelian/non-Abelian character of a model is linked to its Chern number <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. The Chern number is only defined for the translation invariant case because it relies on integrals defined in k-space. We instead use a family of real space generalisations of the Chern number that work for amorphous systems exist called local topological markers <span class="citation" data-cites="bianco_mapping_2011 Hastings_Almost_2010 mitchellAmorphousTopologicalInsulators2018"> [<a href="#ref-bianco_mapping_2011" role="doc-biblioref">15</a>–<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">17</a>]</span>, indeed Kitaev defines one in his original paper on the KH model <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>.</p>
<p>Here we use the crosshair marker of <span class="citation" data-cites="peru_preprint"> [<a href="#ref-peru_preprint" role="doc-biblioref">18</a>]</span> because it works well on smaller systems. We calculate the projector <span class="math inline">\(P = \sum_i |\psi_i\rangle \langle \psi_i|\)</span> onto the occupied fermion eigenstates of the system in open boundary conditions. The projector encodes local information about the occupied eigenstates of the system and in gapped systems it is exponentially localised <span class="citation" data-cites="hastingsLiebSchultzMattisHigherDimensions2004"> [<a href="#ref-hastingsLiebSchultzMattisHigherDimensions2004" role="doc-biblioref">19</a>]</span>. The name <em>crosshair</em> comes from the fact that the marker is defined with respect to a particular point <span class="math inline">\((x_0, y_0)\)</span> by step functions in x and y</p>
<section id="abelian-or-non-abelian-statistics-of-the-gapped-phase" class="level3">
<h3>Abelian or non-Abelian statistics of the Gapped Phase</h3>
<p>The two phases of the amorphous model are gapped as we can see from the finite size scaling of fig. <a href="#fig:fermion_gap_vs_L">4</a>. The next question is: do these phases support excitations with trivial, Abelian or non-Abelian statistics? To answer that we turn to Chern numbers <span class="citation" data-cites="berryQuantalPhaseFactors1984 simonHolonomyQuantumAdiabatic1983 thoulessQuantizedHallConductance1982"> [<a href="#ref-berryQuantalPhaseFactors1984" role="doc-biblioref">12</a>–<a href="#ref-thoulessQuantizedHallConductance1982" role="doc-biblioref">14</a>]</span>. As discussed earlier the Chern number is a quantity intimately linked to both the topological properties and the anyonic statistics of a model. Here we will make use of the fact that the Abelian/non-Abelian character of a model is linked to its Chern number <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. The Chern number is only defined for the translation invariant case so we instead use a family of real space generalisations of the Chern number that work for amorphous systems called local topological markers <span class="citation" data-cites="bianco_mapping_2011 Hastings_Almost_2010 mitchellAmorphousTopologicalInsulators2018"> [<a href="#ref-bianco_mapping_2011" role="doc-biblioref">15</a>–<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">17</a>]</span>.</p>
<p>There are many possible choices here, indeed Kitaev defines one in his original paper on the KH model <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. Here we use the crosshair marker of <span class="citation" data-cites="peru_preprint"> [<a href="#ref-peru_preprint" role="doc-biblioref">18</a>]</span> because it works well on smaller systems. We calculate the projector <span class="math inline">\(P = \sum_i |\psi_i\rangle \langle \psi_i|\)</span> onto the occupied fermion eigenstates of the system in open boundary conditions. The projector encodes local information about the occupied eigenstates of the system and in gapped systems it is exponentially localised <span class="citation" data-cites="hastingsLiebSchultzMattisHigherDimensions2004"> [<a href="#ref-hastingsLiebSchultzMattisHigherDimensions2004" role="doc-biblioref">19</a>]</span>. The name <em>crosshair</em> comes from the fact that the marker is defined with respect to a particular point <span class="math inline">\((x_0, y_0)\)</span> by step functions in x and y</p>
<p><span class="math display">\[\begin{aligned}
\nu (x, y) = 4\pi \; \Im\; \mathrm{Tr}_{\mathrm{B}}
\left (
@ -115,16 +123,16 @@ image:
\right ),
\end{aligned}\]</span></p>
<p>when the trace is taken over a region <span class="math inline">\(B\)</span> around <span class="math inline">\((x_0, y_0)\)</span> that is large enough to include local information about the system but does not come too close to the edges. If these conditions are met then then this quantity will be very close to quantised to the Chern number, see fig. <a href="#fig:phase_diagram_chern">2</a>. We’ll use the crosshair marker to assess the Abelian/non-Abelian character of the phases.</p>
<p>In the A phase of the amorphous model we find that <span class="math inline">\(\nu=0\)</span> and hence the excitations have Abelian character, similar to the honeycomb model. This phase is thus the amorphous analogue of the Abelian toric-code quantum spin liquid <span class="citation" data-cites="kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">20</a>]</span>. The B phase has <span class="math inline">\(\nu=\pm1\)</span> so is a non-Abelian <em>chiral spin liquid</em> (CSL) similar to that of the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>]</span>. The CSL state is the the magnetic analogue of the fractional quantum Hall state <span class="citation" data-cites="laughlinPropertiesChiralspinliquidState1990"> [<a href="#ref-laughlinPropertiesChiralspinliquidState1990" role="doc-biblioref">21</a>]</span>. Hereafter we focus our attention on this phase.</p>
<p>In the A phase of the amorphous model we find that <span class="math inline">\(\nu=0\)</span> and hence the excitations have Abelian character, similar to the honeycomb model. This phase is thus the amorphous analogue of the Abelian toric-code QSL <span class="citation" data-cites="kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">20</a>]</span>. The B phase has <span class="math inline">\(\nu=\pm1\)</span> so is a non-Abelian Chiral Spin Liquid (CSL) similar to that of the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>]</span>. The CSL state is the magnetic analogue of the fractional quantum Hall state <span class="citation" data-cites="laughlinPropertiesChiralspinliquidState1990"> [<a href="#ref-laughlinPropertiesChiralspinliquidState1990" role="doc-biblioref">21</a>]</span>. Hereafter we focus our attention on this phase.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/phase_diagram_chern/phase_diagram_chern.svg" id="fig-phase_diagram_chern" data-short-caption="Local Chern Markers" style="width:100.0%" alt="Figure 2: (Center) The crosshair marker  [18], a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian." />
<figcaption aria-hidden="true">Figure 2: (Center) The crosshair marker <span class="citation" data-cites="peru_preprint"> [<a href="#ref-peru_preprint" role="doc-biblioref">18</a>]</span>, a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime <span class="math inline">\(J_\alpha = 1\)</span> in red has <span class="math inline">\(\nu = \pm 1\)</span> implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has <span class="math inline">\(\nu = 0\)</span> implying it has Abelian statistics. (Right) Extending this analysis to the whole <span class="math inline">\(J_\alpha\)</span> phase diagram with fixed <span class="math inline">\(r = 0.3\)</span> nicely confirms that the isotropic phase is non-Abelian.</figcaption>
<img src="/assets/thesis/amk_chapter/results/phase_diagram_chern/phase_diagram_chern.svg" id="fig-phase_diagram_chern" data-short-caption="Local Chern Markers" style="width:100.0%" alt="Figure 2: (Center) The crosshair marker  [18], a local topological marker, evaluated on the Amorphous Kitaev model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian." />
<figcaption aria-hidden="true">Figure 2: (Center) The crosshair marker <span class="citation" data-cites="peru_preprint"> [<a href="#ref-peru_preprint" role="doc-biblioref">18</a>]</span>, a local topological marker, evaluated on the Amorphous Kitaev model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the centre) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime <span class="math inline">\(J_\alpha = 1\)</span> in red has <span class="math inline">\(\nu = \pm 1\)</span> implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has <span class="math inline">\(\nu = 0\)</span> implying it has Abelian statistics. (Right) Extending this analysis to the whole <span class="math inline">\(J_\alpha\)</span> phase diagram with fixed <span class="math inline">\(r = 0.3\)</span> nicely confirms that the isotropic phase is non-Abelian.</figcaption>
</figure>
</section>
<section id="edge-modes" class="level3">
<h3>Edge Modes</h3>
<p>Chiral Spin Liquids support topological protected edge modes on open boundary conditions <span class="citation" data-cites="qi_general_2006"> [<a href="#ref-qi_general_2006" role="doc-biblioref">22</a>]</span>. Fig. <a href="#fig:edge_modes">3</a> shows the probability density of one such edge mode. It is near zero energy and exponentially localised to the boundary of the system. While the model is gapped in periodic boundary conditions (i.e on the torus) these edge modes appear in the gap when the boundary is cut.</p>
<p>The localization of the edge modes can be quantified by their inverse participation ratio (IPR), <span class="math display">\[\mathrm{IPR} = \int d^2r|\psi(\mathbf{r})|^4 \propto L^{-\tau},\]</span> where <span class="math inline">\(L\sim\sqrt{N}\)</span> is the linear dimension of the amorphous lattices and <span class="math inline">\(\tau\)</span> the dimensional scaling exponent of IPR. This is relevant because localised in-gap states do not participate in transport and hence do not turn band insulators into metals. It is only when the gap fills with extended states that we get a metallic state.</p>
<p>The localisation of the edge modes can be quantified by their inverse participation ratio (IPR) and its scaling with system size <span class="math inline">\(\tau\)</span>, <span class="math display">\[\mathrm{IPR} = \int d^2r|\psi(\mathbf{r})|^4 \propto L^{-\tau},\]</span> where <span class="math inline">\(L\sim\sqrt{N}\)</span> is the linear dimension of the amorphous lattices. This is relevant because localised in-gap states do not participate in transport and hence do not turn band insulators into conductive metals. It is only when the gap fills with extended states that we get a conductive state.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/edge_modes/edge_modes.svg" id="fig-edge_modes" data-short-caption="Edges States and Density of States" style="width:100.0%" alt="Figure 3: (a) The density of one of the topologically protected edge states in the B phase. (Below) the log density plotted along the black path showing that the state is exponentially localised. (a)/(b) The density of states of the corresponding lattice in (a) periodic boundary conditions, (b) open boundary conditions. The colour of the bars shows the mean log IPR for each energy window. Cutting the boundary fills the gap with localised states." />
<figcaption aria-hidden="true">Figure 3: (a) The density of one of the topologically protected edge states in the B phase. (Below) the log density plotted along the black path showing that the state is exponentially localised. (a)/(b) The density of states of the corresponding lattice in (a) periodic boundary conditions, (b) open boundary conditions. The colour of the bars shows the mean log IPR for each energy window. Cutting the boundary fills the gap with localised states.</figcaption>
@ -133,45 +141,45 @@ image:
</section>
<section id="anderson-transition-to-a-thermal-metal" class="level2">
<h2>Anderson Transition to a Thermal Metal</h2>
<p>Previous work on the honeycomb model at finite temperature has shown that the B phase undergoes a thermal transition from a quantum spin liquid phase a to a <strong>thermal metal</strong> phase <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>]</span>.</p>
<p>This happens because at finite temperature, thermal fluctuations lead to spontaneous vortex-pair formation. As discussed previously these fluxes are dressed by Majorana bounds states and the composite object is an Ising-type non-Abelian anyon <span class="citation" data-cites="Beenakker2013"> [<a href="#ref-Beenakker2013" role="doc-biblioref">24</a>]</span>. The interactions between these anyons are oscillatory similar to the RKKY exchange and decay exponentially with separation <span class="citation" data-cites="Laumann2012 Lahtinen_2011 lahtinenTopologicalLiquidNucleation2012"> [<a href="#ref-Laumann2012" role="doc-biblioref">25</a>–<a href="#ref-lahtinenTopologicalLiquidNucleation2012" role="doc-biblioref">27</a>]</span>. At sufficient density, the anyons hybridise to a macroscopically degenerate state known as <em>thermal metal</em> <span class="citation" data-cites="Laumann2012"> [<a href="#ref-Laumann2012" role="doc-biblioref">25</a>]</span>. At close range the oscillatory behaviour of the interactions can be modelled by a random sign which forms the basis for a random matrix theory description of the thermal metal state.</p>
<p>The amorphous chiral spin liquid undergoes the same form of Anderson transition to a thermal metal state. Markov Chain Monte Carlo would be necessary to simulate this in full detail <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>]</span> but in order to avoid that complexity in the current work we instead opted to use vortex density <span class="math inline">\(\rho\)</span> as a proxy for temperature. We give each plaquette probability <span class="math inline">\(\rho\)</span> of being a vortex, possibly with one additional adjustment to preserve overall vortex parity. This approximation is exact in the limits <span class="math inline">\(T = 0\)</span> (corresponding to <span class="math inline">\(\rho = 0\)</span>) and <span class="math inline">\(T \to \infty\)</span> (corresponding to <span class="math inline">\(\rho = 0.5\)</span>) while at intermediate temperatures there may be vortex-vortex correlations that are not captured by positioning vortices using uncorrelated random variables.</p>
<p>First we performed a finite size scaling to that the presence of a gap in the CSL ground state and absence of a gap in the thermal phase are both robust as we go to larger systems, see fig. <a href="#fig:fermion_gap_vs_L">4</a>.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/fermion_gap_vs_L/fermion_gap_vs_L.svg" id="fig-fermion_gap_vs_L" data-short-caption="Finite Size Scaing of the Fermion Gap" style="width:100.0%" alt="Figure 4: Within a flux sector, the fermion gap \Delta_f measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size L is defined \sqrt{N} where N is the number of plaquettes in the system. Error bars shown are 3 times the standard error of the mean. The lines shown are fits of \tfrac{\Delta_f}{J} = aL ^ b with fit parameters: Ground State: a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004 Random Flux Sector: a = 1.8 \pm 0.2, b = -2.21 \pm 0.03" />
<figcaption aria-hidden="true">Figure 4: Within a flux sector, the fermion gap <span class="math inline">\(\Delta_f\)</span> measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size <span class="math inline">\(L\)</span> is defined <span class="math inline">\(\sqrt{N}\)</span> where N is the number of plaquettes in the system. Error bars shown are <span class="math inline">\(3\)</span> times the standard error of the mean. The lines shown are fits of <span class="math inline">\(\tfrac{\Delta_f}{J} = aL ^ b\)</span> with fit parameters: Ground State: <span class="math inline">\(a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004\)</span> Random Flux Sector: <span class="math inline">\(a = 1.8 \pm 0.2, b = -2.21 \pm 0.03\)</span></figcaption>
</figure>
<p>Next we evaluated the fermionic density of states (DOS), Inverse Participation Ratio (IPR) and IPR scaling exponent <span class="math inline">\(\tau\)</span> as functions of the vortex density <span class="math inline">\(\rho\)</span>, see fig. <a href="#fig:DOS_vs_rho">5</a>. This leads to a nice picture of what happens as we raise the temperature of the system away from the gapped, insulating CSL phase. At small <span class="math inline">\(\rho\)</span>, states begin to populate the gap but they have <span class="math inline">\(\tau\approx0\)</span>, indicating that they are localised states pinned to the vortices, and the system remains insulating. At large <span class="math inline">\(\rho\)</span>, the in-gap states merge with the bulk band and become extensive, closing the gap, and the system transitions to the thermal metal phase.</p>
<p>Previous work on the honeycomb model, at finite temperature has shown that the B phase undergoes a thermal transition from a QSL phase to a <em>thermal metal</em> phase <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>]</span>. This happens because at finite temperature, thermal fluctuations lead to spontaneous vortex-pair formation. As discussed previously, these fluxes are dressed by Majorana bounds states and the composite object is an Ising-type non-Abelian anyon <span class="citation" data-cites="Beenakker2013"> [<a href="#ref-Beenakker2013" role="doc-biblioref">24</a>]</span>. The interactions between these anyons are oscillatory, similar to the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange and decay exponentially with separation <span class="citation" data-cites="Laumann2012 Lahtinen_2011 lahtinenTopologicalLiquidNucleation2012"> [<a href="#ref-Laumann2012" role="doc-biblioref">25</a>–<a href="#ref-lahtinenTopologicalLiquidNucleation2012" role="doc-biblioref">27</a>]</span>. At sufficient density, the anyons hybridise to a macroscopically degenerate state known as <em>thermal metal</em> <span class="citation" data-cites="Laumann2012"> [<a href="#ref-Laumann2012" role="doc-biblioref">25</a>]</span>. At close range the oscillatory behaviour of the interactions can be modelled by a random sign which forms the basis for a random matrix theory description of the thermal metal state.</p>
<p>The amorphous chiral spin liquid undergoes the same form of Anderson transition to a thermal metal state. Markov Chain Monte Carlo would be necessary to simulate this in full detail <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>]</span> but in order to avoid that complexity in the current work we instead opted to use vortex density <span class="math inline">\(\rho\)</span> as a proxy for temperature. We give each plaquette the probability <span class="math inline">\(\rho\)</span> of being a vortex, possibly with one additional adjustment to preserve overall vortex parity. This approximation is exact in the limits <span class="math inline">\(T = 0\)</span> (corresponding to <span class="math inline">\(\rho = 0\)</span>) and <span class="math inline">\(T \to \infty\)</span> (corresponding to <span class="math inline">\(\rho = 0.5\)</span>) while at intermediate temperatures there may be vortex-vortex correlations that are not captured by our uncorrelated vortex placement.</p>
<p>First we performed a finite size scaling to show that the presence of a gap in the CSL ground state and absence of a gap in the thermal metal phase are both robust as we go to larger systems, see fig. <a href="#fig:fermion_gap_vs_L">4</a>.</p>
<p>Next we evaluated the fermionic density of states (DOS), Inverse Participation Ratio and IPR scaling exponent <span class="math inline">\(\tau\)</span> as functions of the vortex density <span class="math inline">\(\rho\)</span>, see fig. <a href="#fig:DOS_vs_rho">5</a>. This leads to a nice picture of what happens as we raise the temperature of the system away from the gapped, insulating CSL phase. At small <span class="math inline">\(\rho\)</span>, states begin to populate the gap but they have <span class="math inline">\(\tau\approx0\)</span>, indicating that they are localised states pinned to the vortices, and the system remains insulating. At large <span class="math inline">\(\rho\)</span>, the in-gap states merge with the bulk band and become extensive, closing the gap, and the system transitions to the thermal metal phase.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/DOS_vs_rho/DOS_vs_rho.svg" id="fig-DOS_vs_rho" data-short-caption="Transition to a Thermal Metal" style="width:100.0%" alt="Figure 5: (Top) Density of states and (Bottom) scaling exponent \tau of the amorphous Kitaev model as a vortex density \rho is increased. The scaling exponent \tau is the exponent with which the inverse participation ratio scales with system size. It gives a measure of the degree of localisation of the states in each (E/J, \rho) bin. At zero \rho we have the gapped ground state. At small \rho, states begin to populate the gap. These states have \tau\approx0, indicating that they are localised states pinned to fluxes, and the system remains insulating. As \rho increases further, the in-gap states merge with the bulk band and become extensive, fully closing the gap, and the system transitions to a thermal metal phase." />
<figcaption aria-hidden="true">Figure 5: (Top) Density of states and (Bottom) scaling exponent <span class="math inline">\(\tau\)</span> of the amorphous Kitaev model as a vortex density <span class="math inline">\(\rho\)</span> is increased. The scaling exponent <span class="math inline">\(\tau\)</span> is the exponent with which the inverse participation ratio scales with system size. It gives a measure of the degree of localisation of the states in each <span class="math inline">\((E/J, \rho)\)</span> bin. At zero <span class="math inline">\(\rho\)</span> we have the gapped ground state. At small <span class="math inline">\(\rho\)</span>, states begin to populate the gap. These states have <span class="math inline">\(\tau\approx0\)</span>, indicating that they are localised states pinned to fluxes, and the system remains insulating. As <span class="math inline">\(\rho\)</span> increases further, the in-gap states merge with the bulk band and become extensive, fully closing the gap, and the system transitions to a thermal metal phase.</figcaption>
</figure>
<p>The thermal metal phase has a signature logarithmic divergence at zero energy and oscillations in the DOS. These signatures can be shown to occur by a recursive argument that involves mapping the original model onto a Majorana model with interactions that take random signs which can itself be mapped onto a coarser lattice with lower energy excitations and so on. This can be repeating indefinitely, showing the model must have excitations at arbitrarily low energies in the thermodynamic limit <span class="citation" data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>,<a href="#ref-bocquet_disordered_2000" role="doc-biblioref">28</a>]</span>.</p>
<p>These signatures for our model and for the honeycomb model are shown in fig. <a href="#fig:DOS_oscillations">6</a>. They do not occur in the honeycomb model unless the chiral symmetry is broken by a magnetic field.</p>
<p>The thermal metal phase has a signature logarithmic divergence at zero energy and oscillations in the DOS. These signatures can be shown to occur by a recursive argument that involves mapping the original model onto a Majorana model with interactions that take random signs which can itself be mapped onto a coarser lattice with lower energy excitations and so on. This can be repeating indefinitely, showing the model must have excitations at arbitrarily low energies in the thermodynamic limit <span class="citation" data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>,<a href="#ref-bocquet_disordered_2000" role="doc-biblioref">28</a>]</span>. These signatures are shown in fig. <a href="#fig:DOS_oscillations">6</a> for our model and for the KH model. They do not occur in the KH model unless the chiral symmetry is broken by a magnetic field.</p>
<figure>
<img src="/assets/thesis/amk_chapter/results/DOS_oscillations/DOS_oscillations.svg" id="fig-DOS_oscillations" data-short-caption="Distinctive Oscillations in the Density of States" style="width:100.0%" alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state  [23,28]. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is \rho = 0.5 corresponding to the T = \infty limit." />
<figcaption aria-hidden="true">Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state <span class="citation" data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>,<a href="#ref-bocquet_disordered_2000" role="doc-biblioref">28</a>]</span>. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is <span class="math inline">\(\rho = 0.5\)</span> corresponding to the <span class="math inline">\(T = \infty\)</span> limit.</figcaption>
</figure>
<p>We found a small number of lattices for which the ground state conjecture did not correctly predict the true ground state flux sector. I see two possibilities for what could cause this. Firstly it could be a a finite size effect that is amplified by certain rare lattice configurations. It would be interesting to try to elucidate what lattice features are present when the ground state conjecture fails. Alternatively, it might be telling that the ground state conjecture failed in the toric code A phase where the couplings are anisotropic. We showed that the colouring does not matter in the B phase. However an avenue that I did not explore was whether the particular choice of colouring for a lattice affects the physical properties in the toric code A phase. It is possible that some property of the particular colouring chosen is what leads to failure of the ground state conjecture here.</p>
</section>
</section>
<section id="sec:AMK-Conclusion" class="level1">
<h1>Conclusion</h1>
<p>In this chapter we have looked at an extension of the Kitaev honeycomb model to amorphous lattices with coordination number three. We discussed a method to construct arbitrary trivalent lattices using Voronoi partitions, how to embed them onto the torus and how to edge-colour them using a SAT solver. We showed numerically that the ground state flux sector of the model is given by a simple extension of Lieb’s theorem. The model has two gapped quantum spin liquid phases. The two phases support excitations with different anyonic statistics, Abelian and non-Abelian, distinguished using a real-space generalisation of the Chern number <span class="citation" data-cites="peru_preprint"> [<a href="#ref-peru_preprint" role="doc-biblioref">18</a>]</span>. The presence of odd-sided plaquettes in the model results in spontaneous breaking of time reversal symmetry, leading to the emergence of a chiral spin liquid phase. Finally we showed evidence that the amorphous system undergoes an Anderson transition to a thermal metal phase, driven by the proliferation of vortices with increasing temperature.</p>
<p>The AK model is an exactly solvable model of the chiral QSL state, one of the first models to exhibit a topologically non-trivial quantum many-body phase on an amorphous lattice. As such this study provides a number of future lines of research.</p>
<p><strong>Experimental Realisations and Signatures</strong></p>
<p>We should also consider whether a physical amorphous system that supports a QSL ground state could exist. The search for translation invariant Kitaev systems is already motivated by the prospect of a physically realised QSL state, Majorana fermions and direct access to a system with emergent <span class="math inline">\(\mathbb{Z}_2\)</span> gauge physics <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">29</a>]</span>. An amorphous Kitaev model would provide all this but in addition the possibility of exploring the CSL state as well as potentially very different routes to a physical realisation. One route would be to ask if any crystalline Kitaev material candidates can be heated and rapidly quenched <span class="citation" data-cites="Weaire1976 Petrakovski1981 Kaneyoshi2018"> [<a href="#ref-Weaire1976" role="doc-biblioref">30</a>–<a href="#ref-Kaneyoshi2018" role="doc-biblioref">32</a>]</span> to produce amorphous analogues that might preserve enough of their local structure to support a QSL state.</p>
<p>Instead looking to more designer materials, metal organic frameworks (MOFs) could present a platform for a synthetic Kitaev material. These materials are composed of repeating units of large organic molecular frameworks coordinated with metal ions. Amorphous MOFs can be generated with mechanical processes that introduce disorder into crystalline MOFs <span class="citation" data-cites="bennett2014amorphous"> [<a href="#ref-bennett2014amorphous" role="doc-biblioref">33</a>]</span>. There have been recent proposals for realizing strong Kitaev interactions <span class="citation" data-cites="yamadaDesigningKitaevSpin2017"> [<a href="#ref-yamadaDesigningKitaevSpin2017" role="doc-biblioref">34</a>]</span> in them as potential signatures of a resonating valence bond QSL state in MOFs with Kagome geometry <span class="citation" data-cites="misumiQuantumSpinLiquid2020"> [<a href="#ref-misumiQuantumSpinLiquid2020" role="doc-biblioref">35</a>]</span>. Finally MOFs are composed of large synthetic molecules so may provide more opportunity for fine tuning to target particular physics than than elemental compounds. There have also been proposals to realise Kitaev physics in optical lattice experiments <span class="citation" data-cites="duanControllingSpinExchange2003 micheliToolboxLatticespinModels2006"> [<a href="#ref-duanControllingSpinExchange2003" role="doc-biblioref">36</a>,<a href="#ref-micheliToolboxLatticespinModels2006" role="doc-biblioref">37</a>]</span> which can also support amorphous lattices <span class="citation" data-cites="sadeghiAmorphousTwodimensionalOptical2005"> [<a href="#ref-sadeghiAmorphousTwodimensionalOptical2005" role="doc-biblioref">38</a>]</span>.</p>
<p>A physical realisation in either an amorphous compound or a MOF would likely entail a high degree of defects. Amorphous silicon for instance tends to contain a high degree of dangling bonds which must be passivated by hydrogenation to improve its physical properties <span class="citation" data-cites="streetHydrogenatedAmorphousSilicon1991"> [<a href="#ref-streetHydrogenatedAmorphousSilicon1991" role="doc-biblioref">39</a>]</span>. In both cases, if we assume that Kitaev physics can be realised by crystalline systems, it is not clear if the necessary superexchange couplings would survive the addition of disorder to the MOF lattice. It would therefore make sense to examine how robust the CSL ground state of the AK model is to additional disorder in the Hamiltonian, for example mis-colourings of the bonds, vertex degree disorder and disorder in coupling strengths. Relatedly, one could look at perturbations to the Hamiltonian that break integrability <span class="citation" data-cites="Rau2014 Chaloupka2010 Chaloupka2013 Chaloupka2015 Winter2016"> [<a href="#ref-Rau2014" role="doc-biblioref">40</a>–<a href="#ref-Winter2016" role="doc-biblioref">44</a>]</span>.</p>
<p>Considering experimental signatures, we expect that the chiral amorphous QSL will display a half-quantized thermal Hall effect similar to the magnetic field induced behaviour of KH materials <span class="citation" data-cites="Kasahara2018 Yokoi2021 Yamashita2020 Bruin2022"> [<a href="#ref-Kasahara2018" role="doc-biblioref">45</a>–<a href="#ref-Bruin2022" role="doc-biblioref">48</a>]</span>. Alternatively, the CSL state could be characterized by local probes such as spin-polarized scanning tunnelling microscopy <span class="citation" data-cites="Feldmeier2020 Konig2020 Udagawa2021"> [<a href="#ref-Feldmeier2020" role="doc-biblioref">49</a>–<a href="#ref-Udagawa2021" role="doc-biblioref">51</a>]</span> and the thermal metal phase displays characteristic longitudinal heat transport signatures <span class="citation" data-cites="Beenakker2013"> [<a href="#ref-Beenakker2013" role="doc-biblioref">24</a>]</span>.</p>
<p>Local perturbations such as those that might come from an atomic force microscope could potentially be used to create and control vortices <span class="citation" data-cites="jangVortexCreationControl2021"> [<a href="#ref-jangVortexCreationControl2021" role="doc-biblioref">52</a>]</span> To this end it may make sense to look at how the move to amorphous lattices affects vortex time dynamics in perturbed KH models <span class="citation" data-cites="joyDynamicsVisonsThermal2022"> [<a href="#ref-joyDynamicsVisonsThermal2022" role="doc-biblioref">53</a>]</span>.</p>
<p>Given the lack of unambiguous signatures of the QSL state it can be hard to distinguish the effects of the QSL state from the effect of disorder. So introducing topological disorder from amorphous lattices may pose considerable experimental challenges. Three dimensional realisations could get around this as they would be expected to have a true FTPT to the thermal metal state that could be a useful experimental signature <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 OBrienPRB2016"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">54</a>,<a href="#ref-OBrienPRB2016" role="doc-biblioref">55</a>]</span>. Three dimensional Kitaev systems can also support CSL ground states <span class="citation" data-cites="mishchenkoChiralSpinLiquids2020"> [<a href="#ref-mishchenkoChiralSpinLiquids2020" role="doc-biblioref">56</a>]</span>.</p>
<p><strong>Thermodynamics</strong></p>
<p>The KH model can be extended to three dimensional tri-coordinate lattices <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 OBrienPRB2016"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">54</a>,<a href="#ref-OBrienPRB2016" role="doc-biblioref">55</a>]</span> or it can be generalised to an exactly solvable spin-<span class="math inline">\(\tfrac{3}{2}\)</span> model on four-coordinate lattices <span class="citation" data-cites="yaoAlgebraicSpinLiquid2009 wenQuantumOrderStringnet2003 ryuThreedimensionalTopologicalPhase2009 Baskaran2008 Nussinov2009 Yao2011 Chua2011 Natori2020 Chulliparambil2020 Chulliparambil2021 Seifert2020 WangHaoranPRB2021 Wu2009"> [<a href="#ref-Yao2011" role="doc-biblioref">2</a>,<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">57</a>–<a href="#ref-Wu2009" role="doc-biblioref">68</a>]</span>. In <span class="citation" data-cites="yaoAlgebraicSpinLiquid2009"> [<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">57</a>]</span> the two dimensional square lattice with 4 bond types (<span class="math inline">\(J_w, J_x, J_y, J_z\)</span>) is considered. Since Voronoi partitions in three dimensions produce lattices of degree four, one interesting generalisation of this work would be to look at the spin-<span class="math inline">\(\tfrac{3}{2}\)</span> Kitaev model on amorphous lattices.</p>
<p>We did not perform a full MCMC simulation of the AK model at finite temperature but the possible extension of the model to three dimensions with an FTPT would motive this full analysis in different dimensions. This would be a numerically challenging task but poses no conceptual barriers <span class="citation" data-cites="Laumann2012 lahtinenTopologicalLiquidNucleation2012 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>,<a href="#ref-Laumann2012" role="doc-biblioref">25</a>,<a href="#ref-lahtinenTopologicalLiquidNucleation2012" role="doc-biblioref">27</a>]</span>. Doing this would, firstly, allow one to look for possible violations of the Harris criterion <span class="citation" data-cites="harrisEffectRandomDefects1974"> [<a href="#ref-harrisEffectRandomDefects1974" role="doc-biblioref">69</a>]</span> for the Ising transition of the flux sector. Recall that topological disorder in two dimensions has radically different properties to that of other kinds of disorder due to the constraints imposed by the Euler equation and maintaining coordination number which allows it to violate otherwise quite general rules like the Harris criterion <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">70</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">71</a>]</span>. Second, incorporating the projector in addition to MCMC would allow for a full investigation of whether the effect of topological degeneracy is apparent at finite temperatures as is done in <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>]</span>.</p>
<p>Next, one could investigate whether a QSL phase may exist for for other models defined on amorphous lattices with a view to more realistic prospects of observation. For instance, it would be interesting to see if the properties of the Kitaev-Heisenberg model generalise from the honeycomb to the amorphous case [<span class="citation" data-cites="Chaloupka2010"> [<a href="#ref-Chaloupka2010" role="doc-biblioref">41</a>]</span>; <span class="citation" data-cites="Chaloupka2015"> [<a href="#ref-Chaloupka2015" role="doc-biblioref">43</a>]</span>; <span class="citation" data-cites="Jackeli2009"> [<a href="#ref-Jackeli2009" role="doc-biblioref">72</a>]</span>; <span class="citation" data-cites="Kalmeyer1989"> [<a href="#ref-Kalmeyer1989" role="doc-biblioref">73</a>]</span>; <span class="citation" data-cites="manousakisSpinTextonehalfHeisenberg1991"> [<a href="#ref-manousakisSpinTextonehalfHeisenberg1991" role="doc-biblioref">74</a>]</span>;]. Alternatively we might look at other lattice construction techniques. For instance we could construct lattices by linking close points <span class="citation" data-cites="agarwala2019topological"> [<a href="#ref-agarwala2019topological" role="doc-biblioref">75</a>]</span> or create simplices from random sites <span class="citation" data-cites="christRandomLatticeField1982"> [<a href="#ref-christRandomLatticeField1982" role="doc-biblioref">76</a>]</span>. Lattices constructed these ways would like have a large number of lattice defects <span class="math inline">\(z \neq 3\)</span> in the bulk, leading to persistent zero modes.</p>
<p>Overall, there has been surprisingly little research on amorphous quantum many body phases despite there being plenty of material candidates. I expect the exact chiral amorphous spin liquid to find many generalisation to realistic amorphous quantum magnets.</p>
<p>Next Chapter: <a href="../5_Conclusion/5_Conclusion.html">5 Conclusion</a></p>
<h1>Discussion and Conclusion</h1>
<p>In this chapter we have looked at an extension of the KH model to amorphous lattices with coordination number three. We discussed a method to construct arbitrary trivalent lattices using Voronoi partitions, how to embed them onto the torus and how to edge-colour them using a SAT solver. We showed numerically that the ground state flux sector of the model is given by a simple extension of Lieb’s theorem. The model has two gapped QSL phases. The two phases support excitations with different anyonic statistics, Abelian and non-Abelian, distinguished using a real-space generalisation of the Chern number <span class="citation" data-cites="peru_preprint"> [<a href="#ref-peru_preprint" role="doc-biblioref">18</a>]</span>. The presence of odd-sided plaquettes in the model resulted in spontaneous breaking of time reversal symmetry, leading to the emergence of a chiral spin liquid phase. Finally we showed evidence that the amorphous system undergoes an Anderson transition to a thermal metal phase, driven by the proliferation of vortices with increasing temperature. The AK model is an exactly solvable model of the chiral QSL state, one of the first models to exhibit a topologically non-trivial quantum many-body phase on an amorphous lattice. As such this study provides a number of future lines of research.</p>
<section id="experimental-realisations-and-signatures" class="level3">
<h3>Experimental Realisations and Signatures</h3>
<p>We should consider whether a physical amorphous system that supports a QSL ground state could exist. The search for translation invariant Kitaev systems is already motivated by the prospect of a physically realised QSL state, Majorana fermions and direct access to a system with emergent <span class="math inline">\(\mathbb{Z}_2\)</span> gauge physics <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">29</a>]</span>. An amorphous Kitaev model would provide all this and in addition the possibility of exploring the CSL state and potentially very different routes to a physical realisation. One route would be to ask if any crystalline Kitaev material candidates can be heated and rapidly quenched <span class="citation" data-cites="Weaire1976 Petrakovski1981 Kaneyoshi2018"> [<a href="#ref-Weaire1976" role="doc-biblioref">30</a>–<a href="#ref-Kaneyoshi2018" role="doc-biblioref">32</a>]</span> to produce amorphous analogues that might preserve enough of their local structure to support a QSL state.</p>
<p>Considering more designer materials, metal organic frameworks (MOFs) could present a platform for a synthetic Kitaev material. These materials are composed of repeating units of large organic molecules coordinated with metal ions. Amorphous MOFs can be generated with mechanical processes that introduce disorder into crystalline MOFs <span class="citation" data-cites="bennett2014amorphous"> [<a href="#ref-bennett2014amorphous" role="doc-biblioref">33</a>]</span> and there have been recent proposals for realising strong Kitaev interactions <span class="citation" data-cites="yamadaDesigningKitaevSpin2017"> [<a href="#ref-yamadaDesigningKitaevSpin2017" role="doc-biblioref">34</a>]</span> in them as potential signatures of a resonating valence bond QSL state in MOFs with Kagome geometry <span class="citation" data-cites="misumiQuantumSpinLiquid2020"> [<a href="#ref-misumiQuantumSpinLiquid2020" role="doc-biblioref">35</a>]</span>. Finally, MOFs are composed of large synthetic molecules so may provide more opportunity for fine tuning to target particular physics than with ionic compounds. There have also been proposals to realise Kitaev physics in optical lattice experiments <span class="citation" data-cites="duanControllingSpinExchange2003 micheliToolboxLatticespinModels2006"> [<a href="#ref-duanControllingSpinExchange2003" role="doc-biblioref">36</a>,<a href="#ref-micheliToolboxLatticespinModels2006" role="doc-biblioref">37</a>]</span> which can also support amorphous lattices <span class="citation" data-cites="sadeghiAmorphousTwodimensionalOptical2005"> [<a href="#ref-sadeghiAmorphousTwodimensionalOptical2005" role="doc-biblioref">38</a>]</span>.</p>
<p>A physical realisation in either an amorphous compound or a MOF would likely entail a high degree of defects. Amorphous silicon, for instance, tends to contain dangling bonds which must be passivated by hydrogenation to improve its physical properties <span class="citation" data-cites="streetHydrogenatedAmorphousSilicon1991"> [<a href="#ref-streetHydrogenatedAmorphousSilicon1991" role="doc-biblioref">39</a>]</span>. In both cases, if we assume that Kitaev physics can be realised by crystalline systems, it is not clear if the necessary superexchange couplings would survive the addition of disorder to the lattice. It would therefore make sense theoretically to examine how robust the CSL ground state of the AK model is to additional disorder in the Hamiltonian, for example mis-colourings of the bonds, vertex degree disorder and disorder in coupling strengths. Relatedly, one could look at perturbations to the Hamiltonian that break integrability <span class="citation" data-cites="Rau2014 Chaloupka2010 Chaloupka2013 Chaloupka2015 Winter2016"> [<a href="#ref-Rau2014" role="doc-biblioref">40</a>–<a href="#ref-Winter2016" role="doc-biblioref">44</a>]</span>.</p>
<p>Considering experimental signatures, we expect that the chiral amorphous QSL will display a half-quantised thermal Hall effect similar to the magnetic field induced behaviour of KH materials <span class="citation" data-cites="Kasahara2018 Yokoi2021 Yamashita2020 Bruin2022"> [<a href="#ref-Kasahara2018" role="doc-biblioref">45</a>–<a href="#ref-Bruin2022" role="doc-biblioref">48</a>]</span>. Alternatively, the CSL state could be characterised by local probes such as spin-polarised scanning tunnelling microscopy <span class="citation" data-cites="Feldmeier2020 Konig2020 Udagawa2021"> [<a href="#ref-Feldmeier2020" role="doc-biblioref">49</a>–<a href="#ref-Udagawa2021" role="doc-biblioref">51</a>]</span> while the thermal metal phase displays characteristic longitudinal heat transport signatures <span class="citation" data-cites="Beenakker2013"> [<a href="#ref-Beenakker2013" role="doc-biblioref">24</a>]</span>. Local perturbations, such as those that might come from an atomic force microscope, could potentially be used to create and control vortices <span class="citation" data-cites="jangVortexCreationControl2021"> [<a href="#ref-jangVortexCreationControl2021" role="doc-biblioref">52</a>]</span>. To this end it one could look at how the move to amorphous lattices affects vortex time dynamics in perturbed KH models <span class="citation" data-cites="joyDynamicsVisonsThermal2022"> [<a href="#ref-joyDynamicsVisonsThermal2022" role="doc-biblioref">53</a>]</span>.</p>
<p>Given the lack of unambiguous signatures of the QSL state, it can be hard to distinguish the effects of the QSL state from the effect of disorder. So introducing topological disorder may only increase the experimental challenges. Three dimensional realisations could get around this as they would be expected to have a true Finite-Temperature Phase Transition (FTPT) to the thermal metal state that could be a useful experimental signature <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 OBrienPRB2016"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">54</a>,<a href="#ref-OBrienPRB2016" role="doc-biblioref">55</a>]</span>. Three dimensional Kitaev systems can also support CSL ground states <span class="citation" data-cites="mishchenkoChiralSpinLiquids2020"> [<a href="#ref-mishchenkoChiralSpinLiquids2020" role="doc-biblioref">56</a>]</span>.</p>
</section>
<section id="thermodynamics" class="level3">
<h3>Thermodynamics</h3>
<p>The KH model can be extended to 3D either on trivalent lattices <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 OBrienPRB2016"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">54</a>,<a href="#ref-OBrienPRB2016" role="doc-biblioref">55</a>]</span> or it can be generalised to an exactly solvable spin-<span class="math inline">\(\tfrac{3}{2}\)</span> model on 3D four-coordinate lattices <span class="citation" data-cites="yaoAlgebraicSpinLiquid2009 wenQuantumOrderStringnet2003 ryuThreedimensionalTopologicalPhase2009 Baskaran2008 Nussinov2009 Yao2011 Chua2011 Natori2020 Chulliparambil2020 Chulliparambil2021 Seifert2020 WangHaoranPRB2021 Wu2009"> [<a href="#ref-Yao2011" role="doc-biblioref">2</a>,<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">57</a>–<a href="#ref-Wu2009" role="doc-biblioref">68</a>]</span>. In <span class="citation" data-cites="yaoAlgebraicSpinLiquid2009"> [<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">57</a>]</span>, the 2D square lattice with 4 bond types (<span class="math inline">\(J_w, J_x, J_y, J_z\)</span>) is considered. Since Voronoi partitions in 3D produce lattices of degree four, one interesting generalisation of this work would be to look at the spin-<span class="math inline">\(\tfrac{3}{2}\)</span> Kitaev model on amorphous lattices.</p>
<p>We did not perform a full Markov Chain Monte Carlo (MCMC) simulation of the AK model at finite temperature but the possible extension to a 3D model with an FTPT would motivate this full analysis. This MCMC simulation would be a numerically challenging task but poses no conceptual barriers <span class="citation" data-cites="Laumann2012 lahtinenTopologicalLiquidNucleation2012 selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>,<a href="#ref-Laumann2012" role="doc-biblioref">25</a>,<a href="#ref-lahtinenTopologicalLiquidNucleation2012" role="doc-biblioref">27</a>]</span>. Doing this would, first, allow one to look for possible violations of the Harris criterion <span class="citation" data-cites="harrisEffectRandomDefects1974"> [<a href="#ref-harrisEffectRandomDefects1974" role="doc-biblioref">69</a>]</span> for the Ising transition of the flux sector. Recall that topological disorder in 2D has radically different properties to that of other kinds of disorder due to the constraints imposed by the Euler equation and maintaining coordination number which allows it to violate otherwise quite general rules like the Harris criterion <span class="citation" data-cites="barghathiPhaseTransitionsRandom2014 schrauthViolationHarrisBarghathiVojtaCriterion2018"> [<a href="#ref-barghathiPhaseTransitionsRandom2014" role="doc-biblioref">70</a>,<a href="#ref-schrauthViolationHarrisBarghathiVojtaCriterion2018" role="doc-biblioref">71</a>]</span>. Second, incorporating the projector in addition to MCMC would allow for a full investigation of whether the effect of topological degeneracy is apparent at finite temperatures, this is done for the KH model in <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">23</a>]</span>.</p>
<p>Next, one could investigate whether a QSL phase may exist for other models defined on amorphous lattices with a view to more realistic prospects of observation. Do the properties of the Kitaev-Heisenberg model generalise from the honeycomb to the amorphous case? <span class="citation" data-cites="Chaloupka2010 Chaloupka2015 Jackeli2009 Kalmeyer1989 manousakisSpinTextonehalfHeisenberg1991"> [<a href="#ref-Chaloupka2010" role="doc-biblioref">41</a>,<a href="#ref-Chaloupka2015" role="doc-biblioref">43</a>,<a href="#ref-Jackeli2009" role="doc-biblioref">72</a>–<a href="#ref-manousakisSpinTextonehalfHeisenberg1991" role="doc-biblioref">74</a>]</span> Alternatively we might look at other lattice construction techniques. For instance we could construct lattices by linking close points <span class="citation" data-cites="agarwala2019topological"> [<a href="#ref-agarwala2019topological" role="doc-biblioref">75</a>]</span> or create simplices from random sites <span class="citation" data-cites="christRandomLatticeField1982"> [<a href="#ref-christRandomLatticeField1982" role="doc-biblioref">76</a>]</span>. Lattices constructed using these methods would likely have a large number of lattice defects where <span class="math inline">\(z \neq 3\)</span> in the bulk, leading to many localised Majorana zero modes.</p>
<p>We found a small number of lattices for which Lieb’s theorem did not correctly predict the true ground state flux sector. I see two possibilities for what could cause this. Firstly it could be a a finite size effect that is amplified by certain rare lattice configurations. It would be interesting to try to elucidate what lattice features are present when Lieb’s theorem fails. Alternatively, it might be telling that the ground state conjecture failed in the toric code A phase where the couplings are anisotropic. We showed that the colouring does not matter in the B phase. However an avenue that I did not explore was whether the particular choice of colouring for a lattice affects the physical properties in the toric code A phase. It is possible that some property of the particular colouring chosen is what leads to these rare failures of Lieb’s theorem.</p>
<p>Overall, there has been surprisingly little research on amorphous quantum many-body phases despite there being plenty of material candidates. I expect the exact chiral amorphous spin liquid to find many generalisations to realistic amorphous quantum magnets.</p>
<p>Next Chapter: <a href="../6_Appendices/A.1.2_Fermion_Free_Energy.html">5 Conclusion</a></p>
</section>
</section>
<section id="bibliography" class="level1 unnumbered">
<h1 class="unnumbered">Bibliography</h1>

119
_thesis/5_Conclusion/5_Conclusion.html
View File

@ -38,14 +38,6 @@ image:
<br>
<nav aria-label="Table of Contents" class="page-table-of-contents">
<ul>
<li><a href="#chap:5-conclusion" id="toc-chap:5-conclusion">5 Conclusion</a></li>
<li><a href="#material-realisations" id="toc-material-realisations">Material Realisations</a>
<ul>
<li><a href="#amorphous-materials" id="toc-amorphous-materials">Amorphous Materials</a></li>
<li><a href="#metal-organic-frameworks" id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
</ul></li>
<li><a href="#discussion" id="toc-discussion">Discussion</a></li>
<li><a href="#outlook" id="toc-outlook">Outlook</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
</nav>
@ -59,65 +51,84 @@ image:
<!-- Table of Contents -->
<!-- <nav id="TOC" role="doc-toc">
<ul>
<li><a href="#chap:5-conclusion" id="toc-chap:5-conclusion">5 Conclusion</a></li>
<li><a href="#material-realisations" id="toc-material-realisations">Material Realisations</a>
<ul>
<li><a href="#amorphous-materials" id="toc-amorphous-materials">Amorphous Materials</a></li>
<li><a href="#metal-organic-frameworks" id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
</ul></li>
<li><a href="#discussion" id="toc-discussion">Discussion</a></li>
<li><a href="#outlook" id="toc-outlook">Outlook</a></li>
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
</ul>
</nav>
-->
<!-- Main Page Body -->
<div id="page-header">
<p>5 Conclusion</p>
<hr />
</div>
<section id="chap:5-conclusion" class="level1">
<h1>5 Conclusion</h1>
<p>Using exactly solvable systems as a way to look at the physics of many-body interacting systems. Another theme from the two models is that longer range correlations from criticality in the LRFK model and anti-correlations in the topological disorder in the AK model, lead to a wider range of effects that short range correlations.</p>
<p>paragraph about topological order as new addition to the pantheon of spontaneously broken symmetries</p>
<p>FK model as a way to probe the Mott insulator state. Also the Mott insulator gives rise to the QSl and the doped Mott Insulator may be the source of the sought after High-<span class="math inline">\(T_c\)</span> superconductor. The concept of quantum orders is relevant because for instance, if we can classify the kinds of order in the MI state, we can classify the kinds of high-<span class="math inline">\(T_c\)</span> theories that can emerge from them.</p>
<p>Xiao-Gang Wen <span class="citation" data-cites="wenQuantumOrdersSymmetric2002"> [<a href="#ref-wenQuantumOrdersSymmetric2002" role="doc-biblioref">1</a>]</span> when talks about quantum orders as a those that arise within quantum states at zero temperature, included QSLs, FQH states and superconductors<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a>. He also argues that the High-<span class="math inline">\(T_c\)</span> superconductors is in terms of them being doped Mott insulators so that we should try to understand the QSL which emerges from the undoped Mott insulator (at half filling).</p>
<p>The existence of distinct, spatially limited quasiparticle excitations is not obvious.</p>
<p>emergent gauge physics, could condensed matter systems be useful in understanding the standard model too?</p>
<p>Electron-electron interactions play a dominant role in determining electronic and thermodynamic properties in these strongly correlated materials.</p>
<p>Specific examples where strongly correlated materials may figure prominently are high temperature superconductors and hard magnets without rare earth elements.</p>
</section>
<section id="material-realisations" class="level1">
<h1>Material Realisations</h1>
<section id="amorphous-materials" class="level2">
<h2>Amorphous Materials</h2>
</section>
<section id="metal-organic-frameworks" class="level2">
<h2>Metal Organic Frameworks</h2>
</section>
</section>
<section id="discussion" class="level1">
<h1>Discussion</h1>
</section>
<section id="outlook" class="level1">
<h1>Outlook</h1>
<p>Next Chapter: <a href="../6_Appendices/A.1.2_Fermion_Free_Energy.html">Appendices</a></p>
</section>
<p>This thesis has focussed on two strongly correlated systems. As is the case with many strongly correlated systems, their many-body ground states can be complex and often cannot be reduced to or even adiabatically connected to a product state. In this work, we looked at the Falicov-Kimball (FK) model and the Kitaev Honeycomb (KH) model and defined extensions to them: the Long-Range Falicov-Kimball (LRFK) and amorphous Kitaev (AK) models.</p>
<p>These models are all exactly solvable. They contain extensively many conserved charges which allow their Hamiltonians, and crucially, the interaction terms within them, to be written in quadratic form. This allows them to be solved using the theoretical machinery of non-interacting systems. In the case of the FK and LRFK models, this solvability arises from what is essentially a separation of timescales. The heavy particles move so slowly that they can be treated as stationary. In the KH and AK models, on the other hand, the origin of the conserved degrees of freedom is more complex. Here, the algebra of the Pauli matrices interacts with the trivalent lattices on which the models are defined, to give rise to an emergent <span class="math inline">\(\mathbb{Z}_2\)</span> gauge field whose fluxes are conserved. This latter case is a beautiful example of emergence at play in condensed matter. The gauge and Majorana physics of the Kitaev models seem to arise spontaneously from nothing. Though, of course, this physics was hidden within the structure and local symmetries of the spin Hamiltonian all along.</p>
<p>At first glance, exactly solvable models can seem a little too fine tuned to be particularly relevant to the real world. Surely these models don’t spontaneously arise in nature? The models studied here provide two different ways to answer this. As we saw, the FK model arises quite naturally as a limit of the Hubbard model which is not exactly solvable. In fact, the FK model has been used as a way to understand more about the behaviour of the Hubbard model itself and of the Mott insulating state. We’ve also seen that it can provide insight into other phenomena such as disorder-free localisation. The KH model was not originally proposed as a model of any particular physical system. It was nevertheless a plausible microscopic Hamiltonian and, given its remarkable properties, it is little wonder that material candidates for Kitaev physics were quickly found. In neither case is the model expected to be a perfect description of any material, indeed more realistic corrections to each model are likely to break their integrability. Despite this, exactly solvable models, by virtue of being solvable, can provide important insights into the diverse physics of strongly correlated materials.</p>
<p>In <a href="../3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">chapter 3</a>, we looked at a generalised FK model in 1D, the LRFK model. Metal-insulator transitions are a key theme of work on the FK model and our 1D extension to it was no exception. With the addition of long-range interactions, the model showed a similarly rich phase diagram as its higher dimensional cousins, allowing us to look at transitions between metallic states and, band, Anderson and Mott insulators. We also looked at thermodynamics in 1D and how thermal fluctuations of the conserved charges can lead to disorder-free localisation in the FK and LRFK models. Though the initial surprising results suggested the presence of a mobility edge in 1D it turned out to be a weak localisation effect present in finite sized systems. We propose that a topological variant of the LRFK model akin to the SSH model could be interesting as a point for further study. It could also be an interesting target for cold atom experiments that can naturally generate long-range interactions <span class="citation" data-cites="lepersLongrangeInteractionsUltracold2017"> [<a href="#ref-lepersLongrangeInteractionsUltracold2017" role="doc-biblioref">1</a>]</span>.</p>
<p>That the Mott insulating state is a key part of the work on the LRFK model is fitting because Mott insulators are the main route to the formation of Quantum Spin Liquid (QSL) states which are themselves a primary driver of interest in the KH model and our extension, the AK model. In <a href="../4_Amorphous_Kitaev_Model/4.1_AMK_Model.html">chapter 4</a>, we addressed the question of whether frustrated magnetic interactions on amorphous lattices can give rise to quantum phases such as the QSL state and found that indeed they can. The AK model, a generalisation of the KH model to random lattices with fixed coordination number three, supports a kind of symmetry broken QSL state called a chiral spin liquid. We showed numerically that the ground state of the model follows a simple generalisation of Lieb’s theorem <span class="citation" data-cites="lieb_flux_1994 OBrienPRB2016 eschmannThermodynamicClassificationThreedimensional2020"> [<a href="#ref-lieb_flux_1994" role="doc-biblioref">2</a>–<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">4</a>]</span>. As with other extensions of the KH model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">5</a>]</span>, we found that removing the chiral symmetry of the lattice allows the model to support a gapped phase with non-Abelian anyon excitations. The broken lattice symmetry plays the role of the external magnetic field in the original KH model. Finally, like the KH model, finite temperature causes vortex defects to proliferate, causing a transition to a thermal metal state. We have discussed the prospect of whether AK model physics might be realisable in amorphous versions of known KH candidate materials <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">6</a>]</span>. Alternatively, we might be able to engineer them in synthetic materials such as Metal Organic Frameworks where both Kitaev interactions and amorphous lattices have already been proposed <span class="citation" data-cites="yamadaDesigningKitaevSpin2017 bennett2014amorphous"> [<a href="#ref-yamadaDesigningKitaevSpin2017" role="doc-biblioref">7</a>,<a href="#ref-bennett2014amorphous" role="doc-biblioref">8</a>]</span>.</p>
<p>Unlike the 2D FK model and 1D LRFK models, the KH and AK models don’t have a Finite-Temperature Phase Transition (FTPT). They immediately disorder at any finite temperature <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">4</a>]</span>. However, generalisations of the KH model to 3D do in general have an FTPT. Indeed, the role of dimensionality has been a key theme in this work. Both localisation and thermodynamic phenomena depend crucially on dimensionality, with thermodynamic order generally suppressed and localisation effects strengthened in low dimensions. The graph theory that underpins the KH and AK models itself also changes strongly with dimension. Voronisation in 2D produces trivalent lattices, on which the spin-<span class="math inline">\(1/2\)</span> AK model is exactly solvable. Meanwhile in 3D, Voronisation gives us <span class="math inline">\(z=4\)</span> lattices upon which a spin-<span class="math inline">\(3/2\)</span> generalisation to the KH model is exactly solvable <span class="citation" data-cites="yaoAlgebraicSpinLiquid2009 wenQuantumOrderStringnet2003 ryuThreedimensionalTopologicalPhase2009"> [<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">9</a>–<a href="#ref-ryuThreedimensionalTopologicalPhase2009" role="doc-biblioref">11</a>]</span>. Similarly, planar graphs are a uniquely 2D construct. Satisfying planarity imposes constraints on the connectivity of planar graphs leading amorphous planar graphs to have strong anti-correlations which can violate otherwise robust bounds like the Harris criterion <span class="citation" data-cites="harrisEffectRandomDefects1974"> [<a href="#ref-harrisEffectRandomDefects1974" role="doc-biblioref">12</a>]</span>. Contrast this with Anderson localisation in 1D where only longer range correlations in the disorder can produce surprising effects <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990 izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">13</a>–<a href="#ref-izrailevAnomalousLocalizationLowDimensional2012" role="doc-biblioref">18</a>]</span>.</p>
<p>Xiao-Gang Wen makes the case that one of the primary reasons to study QSLs is as a stepping stone to understanding the High-<span class="math inline">\(T_c\)</span> superconductors <span class="citation" data-cites="wenQuantumOrdersSymmetric2002"> [<a href="#ref-wenQuantumOrdersSymmetric2002" role="doc-biblioref">19</a>]</span>. His logic is that since the High-<span class="math inline">\(T_c\)</span> superconductors are believed to arise from doped Mott insulators, the QSLs, which arise from undoped Mott insulators, make a good jumping off point. This is where exactly solvable models like the FK and KH models shine. The FK model provides a tractable means to study superconductor like states in doped Mott insulators <span class="citation" data-cites="caiVisualizingEvolutionMott2016"> [<a href="#ref-caiVisualizingEvolutionMott2016" role="doc-biblioref">20</a>]</span>, while the KH model gives us a tangible QSL state to play with. The extensions introduced in this work serve to explore how far we can push these models. Overall, I believe that the results presented here show that exactly solvable models can be a useful theoretical tool for understanding the behaviour of generic strongly correlated materials.</p>
<section id="bibliography" class="level1 unnumbered">
<h1 class="unnumbered">Bibliography</h1>
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-lepersLongrangeInteractionsUltracold2017" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">M. Lepers and O. Dulieu, <em><a href="https://doi.org/10.48550/arXiv.1703.02833">Long-Range Interactions Between Ultracold Atoms and Molecules</a></em>, arXiv:1703.02833.</div>
</div>
<div id="ref-lieb_flux_1994" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">E. H. Lieb, <em><a href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux Phase of the Half-Filled Band</a></em>, Physical Review Letters <strong>73</strong>, 2158 (1994).</div>
</div>
<div id="ref-OBrienPRB2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">K. O’Brien, M. Hermanns, and S. Trebst, <em><a href="https://doi.org/10.1103/PhysRevB.93.085101">Classification of Gapless Z₂ Spin Liquids in Three-Dimensional Kitaev Models</a></em>, Phys. Rev. B <strong>93</strong>, 085101 (2016).</div>
</div>
<div id="ref-eschmannThermodynamicClassificationThreedimensional2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">T. Eschmann, P. A. Mishchenko, K. O’Brien, T. A. Bojesen, Y. Kato, M. Hermanns, Y. Motome, and S. Trebst, <em><a href="https://doi.org/10.1103/PhysRevB.102.075125">Thermodynamic Classification of Three-Dimensional Kitaev Spin Liquids</a></em>, Phys. Rev. B <strong>102</strong>, 075125 (2020).</div>
</div>
<div id="ref-yaoExactChiralSpin2007" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">H. Yao and S. A. Kivelson, <em><a href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett. <strong>99</strong>, 247203 (2007).</div>
</div>
<div id="ref-TrebstPhysRep2022" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">S. Trebst and C. Hickey, <em><a href="https://doi.org/10.1016/j.physrep.2021.11.003">Kitaev Materials</a></em>, Physics Reports <strong>950</strong>, 1 (2022).</div>
</div>
<div id="ref-yamadaDesigningKitaevSpin2017" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">M. G. Yamada, H. Fujita, and M. Oshikawa, <em><a href="https://doi.org/10.1103/PhysRevLett.119.057202">Designing Kitaev Spin Liquids in Metal-Organic Frameworks</a></em>, Phys. Rev. Lett. <strong>119</strong>, 057202 (2017).</div>
</div>
<div id="ref-bennett2014amorphous" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">T. D. Bennett and A. K. Cheetham, <em><a href="https://doi.org/10.1021/ar5000314">Amorphous Metal–Organic Frameworks</a></em>, Acc. Chem. Res. <strong>47</strong>, 1555 (2014).</div>
</div>
<div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">H. Yao, S.-C. Zhang, and S. A. Kivelson, <em><a href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett. <strong>102</strong>, 217202 (2009).</div>
</div>
<div id="ref-wenQuantumOrderStringnet2003" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">X.-G. Wen, <em><a href="https://doi.org/10.1103/PhysRevD.68.065003">Quantum Order from String-Net Condensations and the Origin of Light and Massless Fermions</a></em>, Phys. Rev. D <strong>68</strong>, 065003 (2003).</div>
</div>
<div id="ref-ryuThreedimensionalTopologicalPhase2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">S. Ryu, <em><a href="https://doi.org/10.1103/PhysRevB.79.075124">Three-Dimensional Topological Phase on the Diamond Lattice</a></em>, Phys. Rev. B <strong>79</strong>, 075124 (2009).</div>
</div>
<div id="ref-harrisEffectRandomDefects1974" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">A. B. Harris, <em><a href="https://doi.org/10.1088/0022-3719/7/9/009">Effect of Random Defects on the Critical Behaviour of Ising Models</a></em>, J. Phys. C: Solid State Phys. <strong>7</strong>, 1671 (1974).</div>
</div>
<div id="ref-aubryAnalyticityBreakingAnderson1980" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">S. Aubry and G. André, <em>Analyticity Breaking and Anderson Localization in Incommensurate Lattices</em>, Proceedings, VIII International Colloquium on Group-Theoretical Methods in Physics <strong>3</strong>, 18 (1980).</div>
</div>
<div id="ref-dassarmaLocalizationMobilityEdges1990" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">S. Das Sarma, S. He, and X. C. Xie, <em><a href="https://doi.org/10.1103/PhysRevB.41.5544">Localization, Mobility Edges, and Metal-Insulator Transition in a Class of One-Dimensional Slowly Varying Deterministic Potentials</a></em>, Phys. Rev. B <strong>41</strong>, 5544 (1990).</div>
</div>
<div id="ref-dunlapAbsenceLocalizationRandomdimer1990" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">D. H. Dunlap, H.-L. Wu, and P. W. Phillips, <em><a href="https://doi.org/10.1103/PhysRevLett.65.88">Absence of Localization in a Random-Dimer Model</a></em>, Phys. Rev. Lett. <strong>65</strong>, 88 (1990).</div>
</div>
<div id="ref-izrailevLocalizationMobilityEdge1999" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">F. M. Izrailev and A. A. Krokhin, <em><a href="https://doi.org/10.1103/PhysRevLett.82.4062">Localization and the Mobility Edge in One-Dimensional Potentials with Correlated Disorder</a></em>, Phys. Rev. Lett. <strong>82</strong>, 4062 (1999).</div>
</div>
<div id="ref-croyAndersonLocalization1D2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">A. Croy, P. Cain, and M. Schreiber, <em><a href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B <strong>82</strong>, 107 (2011).</div>
</div>
<div id="ref-izrailevAnomalousLocalizationLowDimensional2012" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">F. M. Izrailev, A. A. Krokhin, and N. M. Makarov, <em><a href="https://doi.org/10.1016/j.physrep.2011.11.002">Anomalous Localization in Low-Dimensional Systems with Correlated Disorder</a></em>, Physics Reports <strong>512</strong>, 125 (2012).</div>
</div>
<div id="ref-wenQuantumOrdersSymmetric2002" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">X.-G. Wen, <em><a href="https://doi.org/10.1103/PhysRevB.65.165113">Quantum Orders and Symmetric Spin Liquids</a></em>, Phys. Rev. B <strong>65</strong>, 165113 (2002).</div>
<div class="csl-left-margin">[19] </div><div class="csl-right-inline">X.-G. Wen, <em><a href="https://doi.org/10.1103/PhysRevB.65.165113">Quantum Orders and Symmetric Spin Liquids</a></em>, Phys. Rev. B <strong>65</strong>, 165113 (2002).</div>
</div>
<div id="ref-caiVisualizingEvolutionMott2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[20] </div><div class="csl-right-inline">P. Cai, W. Ruan, Y. Peng, C. Ye, X. Li, Z. Hao, X. Zhou, D.-H. Lee, and Y. Wang, <em><a href="https://doi.org/10.1038/nphys3840">Visualizing the Evolution from the Mott Insulator to a Charge-Ordered Insulator in Lightly Doped Cuprates</a></em>, Nature Phys <strong>12</strong>, 11 (2016).</div>
</div>
</div>
</section>
<section class="footnotes footnotes-end-of-document" role="doc-endnotes">
<hr />
<ol>
<li id="fn1" role="doc-endnote"><p>Wen argues that superconductors cannot be characterised be a local order parameter in the way that superfluids can.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
</main>

2
_thesis/6_Appendices/A.3_Lattice_Generation.html
View File

@ -103,7 +103,7 @@ image:
<p>To require that exactly one of the variables be true, we can enforce that no pair of variables be true: <code>-(r and b) -(r and g) -(b and g)</code></p>
<p>However, these clauses are not in CNF form. Therefore, we also have to use the fact that <code>-(a and b) = (-a OR -b)</code>. To enforce that at least one of these is true we simply OR them all together <code>(r or b or g)</code></p>
<p>To encode the fact that no adjacent edges can have the same colour, we emit a clause that, for each pair of adjacent edges, they cannot be both red, both green or both blue.</p>
<p>We get a solution or set of solutions from the solver, which we can map back to a labelling of the edges. fig. <strong>¿fig:multiple_colourings?</strong> shows some examples.</p>
<p>We get a solution or set of solutions from the solver, which we can map back to a labelling of the edges.</p>
<p>The solution presented here works well enough for our purposes. It does not take up a substantial fraction of the overall computation time, see +fig:times but other approaches could likely work.</p>
<p>When translating problems to CNF form, there is often some flexibility. For instance, we used three boolean variables to encode the colour of each edge and, then, additional constraints to require that only one of these variables be true. An alternative method which we did not try would be to encode the label of each edge using two variables, yielding four states per edge, and then add a constraint that one of the states, say (true, true) is disallowed. This would, however, have added some complexity to the encoding of the constraint that no adjacent edges can have the same colour.</p>
<p>The popular <em>Networkx</em> Python library uses a greedy graph colouring algorithm. It simply iterates over the vertices/edges/faces of a graph and assigns them a colour that is not already disallowed. This does not work for our purposes because it is not designed to look for a particular n-colouring. However, it does include the option of using a heuristic function that determine the order in which vertices will be coloured <span class="citation" data-cites="kosowski2004classical matulaSmallestlastOrderingClustering1983"> [<a href="#ref-kosowski2004classical" role="doc-biblioref">2</a>,<a href="#ref-matulaSmallestlastOrderingClustering1983" role="doc-biblioref">3</a>]</span>. Perhaps</p>

21
_thesis/6_Appendices/A.5_The_Projector.html
View File

@ -66,10 +66,10 @@ image:
<section id="app-the-projector" class="level1">
<h1>The Projector</h1>
<p>The projection from the extended space to the physical space will not be particularly important for the results presented here. However, the theory remains useful to explain why this is.</p>
<figure>
<img src="/assets/thesis/amk_chapter/hilbert_spaces.svg" id="fig-hilbert_spaces" data-short-caption="How the different Hilbert Spaces relate to one another" style="width:100.0%" alt="Figure 1: The relationship between the different Hilbert spaces used in the solution. needs updating" />
<figcaption aria-hidden="true">Figure 1: The relationship between the different Hilbert spaces used in the solution. <strong>needs updating</strong></figcaption>
</figure>
<!-- <figure>
<img src="../../figure_code/amk_chapter/hilbert_spaces.svg" style="max-width:700px;" title="How the different Hilbert Spaces relate to one another">
<figcaption>The relationship between the different Hilbert spaces used in the solution. __needs updating__ </figcaption>
</figure> -->
<p>The physical states are defined as those for which <span class="math inline">\(D_i |\phi\rangle = |\phi\rangle\)</span> for all <span class="math inline">\(D_i\)</span>. Since <span class="math inline">\(D_i\)</span> has eigenvalues <span class="math inline">\(\pm1\)</span>, the quantity <span class="math inline">\(\tfrac{(1+D_i)}{2}\)</span> has eigenvalue <span class="math inline">\(1\)</span> for physical states and <span class="math inline">\(0\)</span> for extended states so is the local projector onto the physical subspace.</p>
<p>Therefore, the global projector is <span class="math display">\[ \mathcal{P} = \prod_{i=1}^{2N} \left( \frac{1 + D_i}{2}\right)\]</span></p>
<p>for a toroidal trivalent lattice with <span class="math inline">\(N\)</span> plaquettes <span class="math inline">\(2N\)</span> vertices and <span class="math inline">\(3N\)</span> edges. As discussed earlier, the product over <span class="math inline">\((1 + D_j)\)</span> can also be thought of as the sum of all possible subsets <span class="math inline">\(\{i\}\)</span> of the <span class="math inline">\(D_j\)</span> operators, which is the set of all possible gauge symmetry operations.</p>
@ -82,12 +82,12 @@ image:
<p><span class="math display">\[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i \prod_i^{2N} b^y_i \prod_i^{2N} b^z_i \prod_i^{2N} c_i\]</span></p>
<p>The product over <span class="math inline">\(c_i\)</span> operators reduces to a determinant of the Q matrix and the fermion parity, see <span class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a href="#ref-pedrocchiPhysicalSolutionsKitaev2011" role="doc-biblioref">1</a>]</span>. The only difference from the honeycomb case is that we cannot explicitly compute the factors <span class="math inline">\(p_x,p_y,p_z = \pm\;1\)</span> that arise from reordering the b operators such that pairs of vertices linked by the corresponding bonds are adjacent.</p>
<p><span class="math display">\[\prod_i^{2N} b^\alpha_i = p_\alpha \prod_{(i,j)}b^\alpha_i b^\alpha_j\]</span></p>
<p>However, they are simply the parity of the permutation from one ordering to the other and can be computed in linear time with a cycle decomposition <strong>cite</strong>.</p>
<p>However, they are simply the parity of the permutation from one ordering to the other and can be computed in linear time with a cycle decomposition <span class="citation" data-cites="sedgewickPermutationGenerationMethods1977"> [<a href="#ref-sedgewickPermutationGenerationMethods1977" role="doc-biblioref">2</a>]</span>.</p>
<p>We find that <span class="math display">\[\mathcal{P}_0 = 1 + p_x\;p_y\;p_z\; \hat{\pi} \; \mathrm{det}(Q^u) \; \prod_{\{i,j\}} -iu_{ij}\]</span></p>
<p>where <span class="math inline">\(p_x\;p_y\;p_z = \pm 1\)</span> are lattice structure factors and <span class="math inline">\(\mathrm{det}(Q^u)\)</span> is the determinant of the matrix mentioned earlier that maps <span class="math inline">\(c_i\)</span> operators to normal mode operators <span class="math inline">\(b&#39;_i, b&#39;&#39;_i\)</span>. These depend only on the lattice structure.</p>
<p><span class="math inline">\(\hat{\pi} = \prod{i}^{N} (1 - 2\hat{n}_i)\)</span> is the parity of the particular many body state determined by fermionic occupation numbers <span class="math inline">\(n_i\)</span>. As discussed in <span class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a href="#ref-pedrocchiPhysicalSolutionsKitaev2011" role="doc-biblioref">1</a>]</span>, <span class="math inline">\(\hat{\pi}\)</span> is gauge invariant in the sense that <span class="math inline">\([\hat{\pi}, D_i] = 0\)</span>.</p>
<p>This implies that <span class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> is also a gauge invariant quantity. In translation invariant models this quantity which can be related to the parity of the number of vortex pairs in the system <span class="citation" data-cites="yaoAlgebraicSpinLiquid2009"> [<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">2</a>]</span>.</p>
<p>All these factors take values <span class="math inline">\(\pm 1\)</span> so <span class="math inline">\(\mathcal{P}_0\)</span> is 0 or 1 for a particular state. Since <span class="math inline">\(\mathcal{S}\)</span> corresponds to symmetrising over all the gauge configurations and cannot be 0, once we have determined the single particle eigenstates of a bond sector, the true many body ground state has the same energy as either the empty state with <span class="math inline">\(n_i = 0\)</span> or a state with a single fermion in the lowest level.</p>
<p><span class="math inline">\(\hat{\pi} = \prod{i}^{N} (1 - 2\hat{n}_i)\)</span> is the parity of the particular many-body state determined by fermionic occupation numbers <span class="math inline">\(n_i\)</span>. As discussed in <span class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a href="#ref-pedrocchiPhysicalSolutionsKitaev2011" role="doc-biblioref">1</a>]</span>, <span class="math inline">\(\hat{\pi}\)</span> is gauge invariant in the sense that <span class="math inline">\([\hat{\pi}, D_i] = 0\)</span>.</p>
<p>This implies that <span class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> is also a gauge invariant quantity. In translation invariant models this quantity which can be related to the parity of the number of vortex pairs in the system <span class="citation" data-cites="yaoAlgebraicSpinLiquid2009"> [<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">3</a>]</span>.</p>
<p>All these factors take values <span class="math inline">\(\pm 1\)</span> so <span class="math inline">\(\mathcal{P}_0\)</span> is 0 or 1 for a particular state. Since <span class="math inline">\(\mathcal{S}\)</span> corresponds to symmetrising over all the gauge configurations and cannot be 0, once we have determined the single particle eigenstates of a bond sector, the true many-body ground state has the same energy as either the empty state with <span class="math inline">\(n_i = 0\)</span> or a state with a single fermion in the lowest level.</p>
</section>
<section id="bibliography" class="level1 unnumbered">
<h1 class="unnumbered">Bibliography</h1>
@ -95,8 +95,11 @@ image:
<div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">F. L. Pedrocchi, S. Chesi, and D. Loss, <em><a href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of the Kitaev honeycomb model</a></em>, Phys. Rev. B <strong>84</strong>, 165414 (2011).</div>
</div>
<div id="ref-sedgewickPermutationGenerationMethods1977" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">R. Sedgewick, <em><a href="https://doi.org/10.1145/356689.356692">Permutation Generation Methods</a></em>, ACM Comput. Surv. <strong>9</strong>, 137 (1977).</div>
</div>
<div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">H. Yao, S.-C. Zhang, and S. A. Kivelson, <em><a href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett. <strong>102</strong>, 217202 (2009).</div>
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">H. Yao, S.-C. Zhang, and S. A. Kivelson, <em><a href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett. <strong>102</strong>, 217202 (2009).</div>
</div>
</div>
</section>

12
_thesis/toc.html
View File

@ -13,23 +13,17 @@
</ul>
<li><a href="./3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falicov-Kimball Model</a></li>
<ul>
<li><a href="./3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html">The Model</a></li>
<li><a href="./3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html#the-model">The Model</a></li>
<li><a href="./3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html#methods">Methods</a></li>
<li><a href="./3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html#results">Results</a></li>
<li><a href="./3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html#discussion-and-conclusion">Discussion and Conclusion</a></li>
</ul>
<li><a href="./4_Amorphous_Kitaev_Model/4.1_AMK_Model.html">4 The Amorphous Kitaev Model</a></li>
<ul>
<li><a href="./4_Amorphous_Kitaev_Model/4.1_AMK_Model.html">The Model</a></li>
<li><a href="./4_Amorphous_Kitaev_Model/4.1_AMK_Model.html#the-model">The Model</a></li>
<li><a href="./4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html#methods">Methods</a></li>
<li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#results">Results</a></li>
<li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#conclusion">Conclusion</a></li>
</ul>
<li><a href="./5_Conclusion/5_Conclusion.html">5 Conclusion</a></li>
<ul>
<li><a href="./5_Conclusion/5_Conclusion.html">Material Realisations</a></li>
<li><a href="./5_Conclusion/5_Conclusion.html#discussion">Discussion</a></li>
<li><a href="./5_Conclusion/5_Conclusion.html#outlook">Outlook</a></li>
<li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#discussion-and-conclusion">Discussion and Conclusion</a></li>
</ul>
<li><a href="./6_Appendices/A.1.2_Fermion_Free_Energy.html">Appendices</a></li>
<ul>

56
assets/thesis/amk_chapter/visual_kitaev_1.svg
View File

@ -6,7 +6,7 @@
version="1.1"
id="svg5942"
sodipodi:docname="visual_kitaev_1.svg"
inkscape:version="1.2.1 (9c6d41e4, 2022-07-14)"
inkscape:version="1.3-dev (77bc73e, 2022-05-18)"
inkscape:export-filename="visual_kitaev_1.pdf"
inkscape:export-xdpi="96"
inkscape:export-ydpi="96"
@ -30,12 +30,12 @@
inkscape:document-units="pt"
showgrid="false"
inkscape:zoom="1.2049729"
inkscape:cx="241.49921"
inkscape:cx="230.29564"
inkscape:cy="143.15675"
inkscape:window-width="1422"
inkscape:window-height="819"
inkscape:window-x="18"
inkscape:window-y="48"
inkscape:window-y="25"
inkscape:window-maximized="0"
inkscape:current-layer="layer2" />
<metadata
@ -3686,46 +3686,49 @@
inkscape:label="a_panel">
<text
xml:space="preserve"
style="font-size:9px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-linecap:butt;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:1"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-linecap:butt;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:1"
x="267.21384"
y="30.871222"
id="text9210-3"><tspan
id="text9210-3"
transform="matrix(1.2,0,0,1.2,-59.567251,-5.5685443)"><tspan
sodipodi:role="line"
style="fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
x="267.21384"
y="30.871222"
id="tspan675-6">The Hamiltonian</tspan></text>
<text
xml:space="preserve"
style="font-size:9px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
style="font-size:9px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1;-inkscape-font-specification:'STIXGeneral, Normal';font-family:STIXGeneral;font-weight:normal;font-style:normal;font-stretch:normal;font-variant:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal"
x="32.469551"
y="69.977478"
id="text10460"><tspan
sodipodi:role="line"
id="tspan10458"
style="stroke-width:6.685"
style="stroke-width:6.685;-inkscape-font-specification:'STIXGeneral, Normal';font-family:STIXGeneral;font-weight:normal;font-style:normal;font-stretch:normal;font-variant:normal;font-size:9px;font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal"
x="32.469551"
y="69.977478" /></text>
<text
xml:space="preserve"
style="font-size:9px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
x="122.17456"
y="107.52657"
id="text10464"><tspan
id="text10464"
transform="matrix(1.2,0,0,1.2,-30.078791,-21.092215)"><tspan
sodipodi:role="line"
id="tspan10462"
style="stroke-width:6.685"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;stroke-width:6.685"
x="122.17456"
y="107.52657">Loop Operators</tspan></text>
<text
xml:space="preserve"
style="font-size:9px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-linecap:butt;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:1"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-linecap:butt;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:1"
x="122.17456"
y="30.161995"
id="text10464-7"><tspan
id="text10464-7"
transform="matrix(1.2,0,0,1.2,-31.513388,-5.612999)"><tspan
sodipodi:role="line"
id="tspan10462-5"
style="stroke-width:6.685"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;stroke-width:6.685"
x="122.17456"
y="30.161995">Plaquette Operators</tspan></text>
<g
@ -3810,18 +3813,19 @@
xml:space="preserve"
transform="scale(0.75)"
id="text1537"
style="white-space:pre;shape-inside:url(#rect1539);display:inline;fill:#000000;stroke-width:8.91333" />
style="white-space:pre;shape-inside:url(#rect1539);display:inline;fill:#000000;stroke-width:8.91333;-inkscape-font-specification:'STIXGeneral, Normal';font-family:STIXGeneral;font-weight:normal;font-style:normal;font-stretch:normal;font-variant:normal;font-size:12px;font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal" />
<g
id="g32152"
transform="translate(-61.160339)">
<text
xml:space="preserve"
style="font-size:9px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
x="80.457794"
y="30.871222"
id="text9210"><tspan
id="text9210"
transform="matrix(1.2,0,0,1.2,-21.790339,-5.7548443)"><tspan
sodipodi:role="line"
style="fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
x="80.457794"
y="30.871222"
id="tspan675">Bond Operators</tspan></text>
@ -4235,26 +4239,28 @@
</g>
<text
xml:space="preserve"
style="font-size:9px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-opacity:1"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-opacity:1"
x="123.05347"
y="9.8608837"
id="text5607"><tspan
id="text5607"
transform="matrix(1.2,0,0,1.2,-35.056957,-1.5536767)"><tspan
sodipodi:role="line"
id="tspan5605"
style="stroke:none;stroke-width:6.685"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;stroke:none;stroke-width:6.685"
x="123.05347"
y="9.8608837">The Kitavev Model, Visually</tspan></text>
y="9.8608837">The Kitaev Model, Visually</tspan></text>
<g
id="g32052"
transform="translate(-60)">
<text
xml:space="preserve"
style="font-size:9px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-linecap:butt;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:1"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-linecap:butt;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:1"
x="79.297455"
y="90.782036"
id="text9210-7"><tspan
id="text9210-7"
transform="matrix(1.2,0,0,1.2,-20.168676,-17.737907)"><tspan
sodipodi:role="line"
style="fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9px;font-family:STIXGeneral;-inkscape-font-specification:'STIXGeneral, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:6.685;stroke-dasharray:none;stroke-opacity:1"
x="79.297455"
y="90.782036"
id="tspan675-3">Bond Types</tspan></text>

Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 219 KiB

After

Width:  |  Height:  |  Size: 223 KiB

242
assets/thesis/background_chapter/localisation_radius_vs_length.svg
View File

@ -1,12 +1,12 @@
<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<svg
width="350pt"
height="162pt"
viewBox="0 0 350 162"
height="100pt"
viewBox="0 0 350 100"
version="1.1"
id="svg5942"
sodipodi:docname="localisation_radius_vs_length.svg"
inkscape:version="1.2.1 (9c6d41e4, 2022-07-14)"
inkscape:version="1.3-dev (77bc73e, 2022-05-18)"
inkscape:export-filename="visual_kitaev_1.pdf"
inkscape:export-xdpi="96"
inkscape:export-ydpi="96"
@ -31,14 +31,14 @@
inkscape:document-units="pt"
showgrid="false"
inkscape:zoom="2.4099458"
inkscape:cx="269.50814"
inkscape:cy="206.64365"
inkscape:cx="224.69385"
inkscape:cy="96.267725"
inkscape:window-width="2304"
inkscape:window-height="1246"
inkscape:window-x="185"
inkscape:window-y="26"
inkscape:window-y="25"
inkscape:window-maximized="0"
inkscape:current-layer="layer1" /><metadata
inkscape:current-layer="g1" /><metadata
id="metadata5878"><rdf:RDF><cc:Work><dc:type
rdf:resource="http://purl.org/dc/dcmitype/StillImage" /><dc:date /><dc:format /><dc:creator><cc:Agent><dc:title /></cc:Agent></dc:creator></cc:Work></rdf:RDF></metadata><defs
id="defs5882"><marker
@ -48,9 +48,9 @@
refY="0"
orient="auto-start-reverse"
inkscape:stockid="TriangleStart"
markerWidth="4.99163344"
markerHeight="5.66661785"
viewBox="0 0 4.99163342 5.66661783"
markerWidth="4.9916334"
markerHeight="5.6666179"
viewBox="0 0 4.9916334 5.6666178"
inkscape:isstock="true"
inkscape:collect="always"
preserveAspectRatio="xMidYMid"><path
@ -1628,161 +1628,161 @@
width="120.13624"
height="90.261147"
id="rect34606"
transform="translate(-411.95, -106.442)" /></clipPath><clipPath
transform="translate(-411.95,-106.442)" /></clipPath><clipPath
id="clipPath21187"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect28141"
transform="translate(-413.147, -98.8495)" /></clipPath><clipPath
transform="translate(-413.147,-98.8495)" /></clipPath><clipPath
id="clipPath49992"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect47147"
transform="translate(-415.844, -92.0826)" /></clipPath><clipPath
transform="translate(-415.844,-92.0826)" /></clipPath><clipPath
id="clipPath19067"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect26819"
transform="translate(-419.265, -86.4245)" /></clipPath><clipPath
transform="translate(-419.265,-86.4245)" /></clipPath><clipPath
id="clipPath27735"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect44215"
transform="translate(-421.877, -82.6042)" /></clipPath><clipPath
transform="translate(-421.877,-82.6042)" /></clipPath><clipPath
id="clipPath45926"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect27439"
transform="translate(-426.096, -77.1178)" /></clipPath><clipPath
transform="translate(-426.096,-77.1178)" /></clipPath><clipPath
id="clipPath7289"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect66322"
transform="translate(-431.537, -71.4239)" /></clipPath><clipPath
transform="translate(-431.537,-71.4239)" /></clipPath><clipPath
id="clipPath58174"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect64821"
transform="translate(-439.197, -67.3684)" /></clipPath><clipPath
transform="translate(-439.197,-67.3684)" /></clipPath><clipPath
id="clipPath64024"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect80472"
transform="translate(-433.229, -70.06)" /></clipPath><clipPath
transform="translate(-433.229,-70.06)" /></clipPath><clipPath
id="clipPath2833"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect43588"
transform="translate(-424.813, -71.6405)" /></clipPath><clipPath
transform="translate(-424.813,-71.6405)" /></clipPath><clipPath
id="clipPath66495"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect13314"
transform="translate(-417.065, -72.1117)" /></clipPath><clipPath
transform="translate(-417.065,-72.1117)" /></clipPath><clipPath
id="clipPath83583"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect2005"
transform="translate(-411.892, -72.1966)" /></clipPath><clipPath
transform="translate(-411.892,-72.1966)" /></clipPath><clipPath
id="clipPath90333"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect5478"
transform="translate(-404.87, -72.0601)" /></clipPath><clipPath
transform="translate(-404.87,-72.0601)" /></clipPath><clipPath
id="clipPath57852"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect19527"
transform="translate(-396.905, -71.3773)" /></clipPath><clipPath
transform="translate(-396.905,-71.3773)" /></clipPath><clipPath
id="clipPath40275"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect86105"
transform="translate(-388.612, -69.0926)" /></clipPath><clipPath
transform="translate(-388.612,-69.0926)" /></clipPath><clipPath
id="clipPath96059"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect90729"
transform="translate(-386.42, -68.5044)" /></clipPath><clipPath
transform="translate(-386.42,-68.5044)" /></clipPath><clipPath
id="clipPath35636"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect94464"
transform="translate(-393.041, -73.5162)" /></clipPath><clipPath
transform="translate(-393.041,-73.5162)" /></clipPath><clipPath
id="clipPath63499"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect40464"
transform="translate(-398.214, -78.979)" /></clipPath><clipPath
transform="translate(-398.214,-78.979)" /></clipPath><clipPath
id="clipPath36392"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect54703"
transform="translate(-402.337, -84.1558)" /></clipPath><clipPath
transform="translate(-402.337,-84.1558)" /></clipPath><clipPath
id="clipPath27861"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect93728"
transform="translate(-405.025, -87.897)" /></clipPath><clipPath
transform="translate(-405.025,-87.897)" /></clipPath><clipPath
id="clipPath23537"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect37560"
transform="translate(-408.539, -93.7235)" /></clipPath><clipPath
transform="translate(-408.539,-93.7235)" /></clipPath><clipPath
id="clipPath89676"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect88240"
transform="translate(-410.994, -100.707)" /></clipPath><clipPath
transform="translate(-410.994,-100.707)" /></clipPath><clipPath
id="clipPath49973"><rect
x="351.82376"
y="22.235085"
width="120.13624"
height="90.261147"
id="rect61786"
transform="translate(-411.888, -108.393)" /></clipPath><g
transform="translate(-411.888,-108.393)" /></clipPath><g
id="id-53922e9c-4038-4a9a-978c-4aad615c5605-5"><g
id="id-a857f4ec-4441-493c-bbdb-4ae422b9ecaa-9" /><g
id="id-ff1c1e07-7dbc-4d44-828e-8fc52c8d88d5-7"><path
@ -1814,38 +1814,40 @@
transform="matrix(0.47253439,0,0,0.47253439,5.9054299,16.212057)" /></g><g
inkscape:groupmode="layer"
id="layer1"
inkscape:label="2"><path
style="fill:#000000;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947;marker-start:url(#TriangleStart);marker-end:url(#TriangleStart)"
d="M 80.394862,26.407071 H 374.98041"
id="path3410"
transform="matrix(0.74255011,0,0,0.75,6.3354814,29.487896)" /><path
style="fill:#000000;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linecap:butt;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947;marker-start:url(#TriangleStart);marker-end:url(#TriangleStart)"
d="m 33.570668,113.25699 -41.4817721,0.58683"
id="path3410-3"
transform="matrix(0.74255011,0,0,0.75,43.96556,29.778015)"
sodipodi:nodetypes="cc" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="m 22.299304,129.10124 c 30.280009,0.90059 45.705671,-7.94214 56.931514,-22.30891 3.964145,-2.03876 7.069342,-10.900917 11.519023,-8.028349 4.495794,2.902339 7.841325,8.093369 13.117969,9.211599 5.53096,1.17212 10.65875,-6.77124 16.04354,-6.14935 5.73237,0.66204 9.91616,4.36797 14.04968,9.02457 2.68493,3.0247 10.17171,-2.46048 13.90511,-4.68669 3.51965,-2.09875 7.26865,-10.987342 10.54255,-8.681463 4.46002,3.141293 6.95769,7.826533 6.44124,13.305633 -0.65106,6.9072 6.99846,6.04275 11.47824,5.85577 4.41859,-0.18442 7.50551,-6.8763 9.90239,-11.02107 3.89774,-6.740121 7.80901,-1.18935 11.7353,1.51205 4.8791,3.35696 10.57071,1.56066 14.41377,-2.06137 4.82541,-4.54789 8.06681,-4.10026 11.11697,1.89023 2.22284,4.36564 7.5216,8.53325 12.01607,8.32714 5.34004,-0.24489 10.35456,-6.16128 15.48783,-5.02152 5.02627,1.116 9.27955,4.31853 13.46119,-0.83644 3.27424,-4.03636 8.18886,-9.16707 10.23134,-1.06294 2.11595,8.39564 6.49594,1.26556 10.03175,-2.27025 L 295.7592,95.065457 c 5.20315,-5.203151 2.27936,17.988923 9.38918,19.952013 6.18499,1.70773 9.00086,-10.31315 15.25742,-11.73648 5.76629,-1.31179 11.5057,5.3919 17.21719,4.9757 4.85491,-0.35378 10.02485,-6.36222 13.35527,-0.22349 3.07736,5.67229 9.72655,1.31286 14.19861,-0.4782 6.23822,-2.4984 8.85627,2.95858 12.27568,6.41366 14.64403,13.37791 40.57276,17.12287 63.25213,16.30623"
id="path3414"
transform="matrix(0.75,0,0,0.75,6.1796467,0.88023539)"
sodipodi:nodetypes="ccssssssssssssssssssssssscc" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="m 32.275309,88.023572 c 29.928015,-1.173648 51.932848,43.958518 74.526621,41.664488 40.2629,-4.08805 39.26823,18.76362 59.85603,19.95201 20.69814,1.19475 32.353,-17.7817 60.44285,-14.67059 29.46589,3.26351 63.50462,1.98657 88.6104,-7.62871 25.37459,-9.71823 23.49676,11.24702 63.37697,-35.209432 C 389.07595,80.496588 421.3395,73.9398 421.3395,73.9398"
id="path4095"
transform="matrix(0.75,0,0,0.75,7.5,0)"
sodipodi:nodetypes="csssssc" /><g
ns1:version="1.8.1"
ns1:texconverter="pdflatex"
ns1:pdfconverter="inkscape"
ns1:text="$$V(x)$$"
ns1:preamble="/Users/tom/Library/Application Support/org.inkscape.Inkscape/config/inkscape/extensions/textext/default_packages.tex"
ns1:scale="1.0"
ns1:alignment="middle center"
ns1:stroke-to-path="0"
ns1:inkscapeversion="1.2.1"
transform="translate(214.30438,104.59686)"
ns1:jacobian_sqrt="1.0"
id="g6901"><defs
inkscape:label="2"><g
id="g1"
transform="matrix(0.80330028,0,0,0.80330028,30.560421,-17.719266)"><path
style="fill:#000000;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947;marker-start:url(#TriangleStart);marker-end:url(#TriangleStart)"
d="M 80.394862,26.407071 H 374.98041"
id="path3410"
transform="matrix(0.74255011,0,0,0.75,6.3354814,29.487896)" /><path
style="fill:#000000;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linecap:butt;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947;marker-start:url(#TriangleStart);marker-end:url(#TriangleStart)"
d="m 33.570668,113.25699 -41.4817721,0.58683"
id="path3410-3"
transform="matrix(0.74255011,0,0,0.75,43.96556,29.778015)"
sodipodi:nodetypes="cc" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="m 22.299304,129.10124 c 30.280009,0.90059 45.705671,-7.94214 56.931514,-22.30891 3.964145,-2.03876 7.069342,-10.900917 11.519023,-8.028349 4.495794,2.902339 7.841325,8.093369 13.117969,9.211599 5.53096,1.17212 10.65875,-6.77124 16.04354,-6.14935 5.73237,0.66204 9.91616,4.36797 14.04968,9.02457 2.68493,3.0247 10.17171,-2.46048 13.90511,-4.68669 3.51965,-2.09875 7.26865,-10.987342 10.54255,-8.681463 4.46002,3.141293 6.95769,7.826533 6.44124,13.305633 -0.65106,6.9072 6.99846,6.04275 11.47824,5.85577 4.41859,-0.18442 7.50551,-6.8763 9.90239,-11.02107 3.89774,-6.740121 7.80901,-1.18935 11.7353,1.51205 4.8791,3.35696 10.57071,1.56066 14.41377,-2.06137 4.82541,-4.54789 8.06681,-4.10026 11.11697,1.89023 2.22284,4.36564 7.5216,8.53325 12.01607,8.32714 5.34004,-0.24489 10.35456,-6.16128 15.48783,-5.02152 5.02627,1.116 9.27955,4.31853 13.46119,-0.83644 3.27424,-4.03636 8.18886,-9.16707 10.23134,-1.06294 2.11595,8.39564 6.49594,1.26556 10.03175,-2.27025 L 295.7592,95.065457 c 5.20315,-5.203151 2.27936,17.988923 9.38918,19.952013 6.18499,1.70773 9.00086,-10.31315 15.25742,-11.73648 5.76629,-1.31179 11.5057,5.3919 17.21719,4.9757 4.85491,-0.35378 10.02485,-6.36222 13.35527,-0.22349 3.07736,5.67229 9.72655,1.31286 14.19861,-0.4782 6.23822,-2.4984 8.85627,2.95858 12.27568,6.41366 14.64403,13.37791 40.57276,17.12287 63.25213,16.30623"
id="path3414"
transform="matrix(0.75,0,0,0.75,6.1796467,0.88023539)"
sodipodi:nodetypes="ccssssssssssssssssssssssscc" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="m 32.275309,88.023572 c 29.928015,-1.173648 51.932848,43.958518 74.526621,41.664488 40.2629,-4.08805 39.26823,18.76362 59.85603,19.95201 20.69814,1.19475 32.353,-17.7817 60.44285,-14.67059 29.46589,3.26351 63.50462,1.98657 88.6104,-7.62871 25.37459,-9.71823 23.49676,11.24702 63.37697,-35.209432 C 389.07595,80.496588 421.3395,73.9398 421.3395,73.9398"
id="path4095"
transform="matrix(0.75,0,0,0.75,7.5,0)"
sodipodi:nodetypes="csssssc" /><g
ns1:version="1.8.1"
ns1:texconverter="pdflatex"
ns1:pdfconverter="inkscape"
ns1:text="$$V(x)$$"
ns1:preamble="/Users/tom/Library/Application Support/org.inkscape.Inkscape/config/inkscape/extensions/textext/default_packages.tex"
ns1:scale="1.0"
ns1:alignment="middle center"
ns1:stroke-to-path="0"
ns1:inkscapeversion="1.2.1"
transform="matrix(1.21,0,0,1.21,212.60172,105.87307)"
ns1:jacobian_sqrt="1.0"
id="g6901"><defs
id="id-f93bc5a9-0f53-44ee-a968-9f8ba62920ff">
<g
id="id-e346f289-c4fc-4265-abbc-c7b1704db368">
@ -1924,18 +1926,18 @@
id="id-8e8d5363-2b8c-4bd6-901c-979a07881cb8" />
</g></g>
</g><g
ns1:version="1.8.1"
ns1:texconverter="pdflatex"
ns1:pdfconverter="inkscape"
ns1:text="$$\\psi(x)$$"
ns1:preamble="/Users/tom/Library/Application Support/org.inkscape.Inkscape/config/inkscape/extensions/textext/default_packages.tex"
ns1:scale="1.0"
ns1:alignment="middle center"
ns1:stroke-to-path="0"
ns1:inkscapeversion="1.2.1"
transform="translate(180.43219,70.267668)"
ns1:jacobian_sqrt="1.0"
id="g9167"><defs
ns1:version="1.8.1"
ns1:texconverter="pdflatex"
ns1:pdfconverter="inkscape"
ns1:text="$$\\psi(x)$$"
ns1:preamble="/Users/tom/Library/Application Support/org.inkscape.Inkscape/config/inkscape/extensions/textext/default_packages.tex"
ns1:scale="1.0"
ns1:alignment="middle center"
ns1:stroke-to-path="0"
ns1:inkscapeversion="1.2.1"
transform="matrix(1.21,0,0,1.21,179.21277,67.282315)"
ns1:jacobian_sqrt="1.0"
id="g9167"><defs
id="id-b1e11367-492a-4111-bf16-ab33939cfc38">
<g
id="id-6190f0e5-80e1-4171-83d4-727c598c7bb0">
@ -2014,18 +2016,18 @@
id="id-8c34accc-a04a-436e-b07a-12f33964d290" />
</g></g>
</g><g
ns1:version="1.8.1"
ns1:texconverter="pdflatex"
ns1:pdfconverter="inkscape"
ns1:text="$$\\lambda$$"
ns1:preamble="/Users/tom/Library/Application Support/org.inkscape.Inkscape/config/inkscape/extensions/textext/default_packages.tex"
ns1:scale="1.0"
ns1:alignment="middle center"
ns1:stroke-to-path="0"
ns1:inkscapeversion="1.2.1"
transform="translate(51.023657,122.79034)"
ns1:jacobian_sqrt="1.0"
id="g11453"><defs
ns1:version="1.8.1"
ns1:texconverter="pdflatex"
ns1:pdfconverter="inkscape"
ns1:text="$$\\lambda$$"
ns1:preamble="/Users/tom/Library/Application Support/org.inkscape.Inkscape/config/inkscape/extensions/textext/default_packages.tex"
ns1:scale="1.0"
ns1:alignment="middle center"
ns1:stroke-to-path="0"
ns1:inkscapeversion="1.2.1"
transform="matrix(1.21,0,0,1.21,50.505167,122.05039)"
ns1:jacobian_sqrt="1.0"
id="g11453"><defs
id="id-66104870-d8ff-4326-9f0b-bba513d2a55c">
<g
id="id-8229c4a0-e594-4bc6-86c5-c48769b94087">
@ -2051,18 +2053,18 @@
id="id-74bb4d22-59a6-477d-9695-5bf37f4e093f" />
</g></g>
</g><g
ns1:version="1.8.1"
ns1:texconverter="pdflatex"
ns1:pdfconverter="inkscape"
ns1:text="$$R$$"
ns1:preamble="/Users/tom/Library/Application Support/org.inkscape.Inkscape/config/inkscape/extensions/textext/default_packages.tex"
ns1:scale="1.0"
ns1:alignment="middle center"
ns1:stroke-to-path="0"
ns1:inkscapeversion="1.2.1"
transform="translate(177.56383,33.894532)"
ns1:jacobian_sqrt="1.0"
id="g13741"><defs
ns1:version="1.8.1"
ns1:texconverter="pdflatex"
ns1:pdfconverter="inkscape"
ns1:text="$$R$$"
ns1:preamble="/Users/tom/Library/Application Support/org.inkscape.Inkscape/config/inkscape/extensions/textext/default_packages.tex"
ns1:scale="1.0"
ns1:alignment="middle center"
ns1:stroke-to-path="0"
ns1:inkscapeversion="1.2.1"
transform="matrix(1.21,0,0,1.21,176.81566,33.156146)"
ns1:jacobian_sqrt="1.0"
id="g13741"><defs
id="id-9f003370-ad74-4e77-80e6-50315d824045">
<g
id="id-f6a843fa-3ecd-4db1-9cdc-39fd525d2a8a">
@ -2088,22 +2090,22 @@
id="id-c34cad69-d07b-4f50-be4d-59fa167232f0" />
</g></g>
</g><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="m 78.425001,50.208598 0,50.623542"
id="path15082"
transform="matrix(0.75,0,0,0.75,2.4896826,-4.6681548)"
sodipodi:nodetypes="cc" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="M 397.51932,26.971561 V 73.030689"
id="path15086"
transform="matrix(0.75,0,0,0.75,-8.0914684,11.514782)" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="m 97.09762,117.01508 0,61.82712"
id="path15088"
transform="matrix(0.75,0,0,0.75,0,-8.4026787)"
sodipodi:nodetypes="cc" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="m 44.399339,135.27275 0,30.70609"
id="path15090"
transform="scale(0.75)"
sodipodi:nodetypes="cc" /></g></svg>
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="M 78.425001,50.208598 V 100.83214"
id="path15082"
transform="matrix(0.75,0,0,0.75,2.4896826,-4.6681548)"
sodipodi:nodetypes="cc" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="M 397.51932,26.971561 V 73.030689"
id="path15086"
transform="matrix(0.75,0,0,0.75,-8.0914684,11.514782)" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="M 97.09762,117.01508 V 178.8422"
id="path15088"
transform="matrix(0.75,0,0,0.75,0,-8.4026787)"
sodipodi:nodetypes="cc" /><path
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1.33333;stroke-linejoin:round;stroke-dasharray:none;stroke-opacity:0.997947"
d="m 44.399339,135.27275 v 30.70609"
id="path15090"
transform="scale(0.75)"
sodipodi:nodetypes="cc" /></g></g></svg>

Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 158 KiB

After

Width:  |  Height:  |  Size: 158 KiB

233
assets/thesis/intro_chapter/venn_diagram.svg
View File

@ -4,10 +4,10 @@
<svg
version="1.1"
width="486pt"
height="320pt"
height="220pt"
id="svg2"
sodipodi:docname="venn_diagram.svg"
inkscape:version="1.2.1 (9c6d41e4, 2022-07-14)"
inkscape:version="1.3-dev (77bc73e, 2022-05-18)"
xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
xmlns="http://www.w3.org/2000/svg"
@ -22,13 +22,13 @@
inkscape:pagecheckerboard="0"
inkscape:deskcolor="#d1d1d1"
showgrid="false"
inkscape:zoom="1.2805486"
inkscape:cx="300.26193"
inkscape:cy="250.28336"
inkscape:zoom="1"
inkscape:cx="289"
inkscape:cy="251.5"
inkscape:window-width="1749"
inkscape:window-height="1205"
inkscape:window-x="609"
inkscape:window-y="125"
inkscape:window-y="97"
inkscape:window-maximized="0"
inkscape:current-layer="layer3"
inkscape:document-units="pt" />
@ -82,153 +82,182 @@
id="layer2"
inkscape:label="Venn"
transform="matrix(0.90965985,0,0,0.90965985,-19.730949,-27.726575)">
<path
d="m 480,292.36218 c 0,77.31987 -62.68014,140 -140,140 -77.31986,0 -140,-62.68013 -140,-140 0,-77.31986 62.68014,-140 140,-140 77.31986,0 140,62.68014 140,140 z"
id="path2818"
style="fill:#000000;fill-opacity:0.0472279;stroke:#000000;stroke-width:1;stroke-linecap:round;stroke-miterlimit:4;stroke-dashoffset:0;stroke-opacity:0.997947"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-43.358497)" />
<path
d="m 406,412.36218 c 0,77.31987 -62.68014,140 -140,140 -77.31986,0 -140,-62.68013 -140,-140 0,-77.31986 62.68014,-140 140,-140 77.31986,0 140,62.68014 140,140 z"
id="path2818-1"
style="fill:#000000;fill-opacity:0.0472279;stroke:#000000;stroke-width:1;stroke-linecap:round;stroke-miterlimit:4;stroke-dashoffset:0;stroke-opacity:0.997947"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-43.358497)" />
<path
d="m 554,412.36218 c 0,77.31987 -62.68014,140 -140,140 -77.31986,0 -140,-62.68013 -140,-140 0,-77.31986 62.68014,-140 140,-140 77.31986,0 140,62.68014 140,140 z"
id="path2818-1-0"
style="fill:#000000;fill-opacity:0.0472279;stroke:#000000;stroke-width:1;stroke-linecap:round;stroke-miterlimit:4;stroke-dashoffset:0;stroke-opacity:0.997947"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-43.358497)" />
<path
d="m 340,292.36218 c -39.62859,24.72433 -66,68.7254 -66,118.875 0,1.1766 0.002,2.3303 0.0312,3.5 19.65735,10.5221 42.11343,16.5 65.96875,16.5 23.85532,0 46.3114,-5.9779 65.96875,-16.5 0.0289,-1.1697 0.0312,-2.3234 0.0312,-3.5 0,-50.1496 -26.37141,-94.15067 -66,-118.875 z"
id="path389"
style="fill:#ffffff;fill-opacity:1;stroke:#000000;stroke-width:1;stroke-linecap:round;stroke-miterlimit:4;stroke-dashoffset:0;stroke-opacity:1"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-42.603803)"
inkscape:label="center_white" />
<path
d="m 340,292.36218 c -39.62859,24.72433 -66,68.7254 -66,118.875 0,1.1766 0.002,2.3303 0.0312,3.5 19.65735,10.5221 42.11343,16.5 65.96875,16.5 23.85532,0 46.3114,-5.9779 65.96875,-16.5 0.0289,-1.1697 0.0312,-2.3234 0.0312,-3.5 0,-50.1496 -26.37141,-94.15067 -66,-118.875 z"
id="path2818-9"
style="fill:#ffbc1d;fill-opacity:0.755647;stroke:#000000;stroke-width:1;stroke-linecap:round;stroke-miterlimit:4;stroke-dashoffset:0;stroke-opacity:1"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-42.603803)"
inkscape:label="center_translucent" />
<g
id="g1"
transform="matrix(0.75791305,0,0,0.75791305,86.881656,-7.994375)">
<path
d="m 480,292.36218 c 0,77.31987 -62.68014,140 -140,140 -77.31986,0 -140,-62.68013 -140,-140 0,-77.31986 62.68014,-140 140,-140 77.31986,0 140,62.68014 140,140 z"
id="path2818"
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1;stroke-linecap:round;stroke-miterlimit:4;stroke-dashoffset:0;stroke-opacity:0.997947"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-43.358497)" />
<path
d="m 406,412.36218 c 0,77.31987 -62.68014,140 -140,140 -77.31986,0 -140,-62.68013 -140,-140 0,-77.31986 62.68014,-140 140,-140 77.31986,0 140,62.68014 140,140 z"
id="path2818-1"
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1;stroke-linecap:round;stroke-miterlimit:4;stroke-dashoffset:0;stroke-opacity:0.997947"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-43.358497)" />
<path
d="m 554,412.36218 c 0,77.31987 -62.68014,140 -140,140 -77.31986,0 -140,-62.68013 -140,-140 0,-77.31986 62.68014,-140 140,-140 77.31986,0 140,62.68014 140,140 z"
id="path2818-1-0"
style="fill:none;fill-opacity:0.0472279;stroke:#000000;stroke-width:1;stroke-linecap:round;stroke-miterlimit:4;stroke-dashoffset:0;stroke-opacity:0.997947"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-43.358497)" />
<path
d="m 340,292.36218 c -39.62859,24.72433 -66,68.7254 -66,118.875 0,1.1766 0.002,2.3303 0.0312,3.5 19.65735,10.5221 42.11343,16.5 65.96875,16.5 23.85532,0 46.3114,-5.9779 65.96875,-16.5 0.0289,-1.1697 0.0312,-2.3234 0.0312,-3.5 0,-50.1496 -26.37141,-94.15067 -66,-118.875 z"
id="path389"
style="fill:#ffffff;fill-opacity:1;stroke:#000000;stroke-width:1;stroke-linecap:round;stroke-miterlimit:4;stroke-dashoffset:0;stroke-opacity:1"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-42.603803)"
inkscape:label="center_white" />
<path
d="m 340,292.36218 c -39.62859,24.72433 -66,68.7254 -66,118.875 0,1.1766 0.002,2.3303 0.0312,3.5 19.65735,10.5221 42.11343,16.5 65.96875,16.5 23.85532,0 46.3114,-5.9779 65.96875,-16.5 0.0289,-1.1697 0.0312,-2.3234 0.0312,-3.5 0,-50.1496 -26.37141,-94.15067 -66,-118.875 z"
id="path2818-9"
style="fill:#f49900;fill-opacity:0.646217;stroke:#000000;stroke-width:2.24795;stroke-linecap:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
transform="matrix(0.89844426,0,0,0.89844426,61.476642,-42.603803)"
inkscape:label="center_translucent" />
</g>
</g>
<g
inkscape:groupmode="layer"
id="layer3"
inkscape:label="Text"
transform="translate(-39.713457,-46.524858)">
transform="translate(-39.713457,-46.524858)"
style="display:inline">
<text
xml:space="preserve"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:8.68596px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;inline-size:65.7594;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="125.72736"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.62325px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;text-anchor:middle;white-space:pre;inline-size:65.7594;display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="139.9279"
y="80.043816"
id="text1123"
inkscape:label="text1123"
transform="matrix(2.415731,0,0,2.415731,10.08029,-111.5283)"><tspan
x="125.72736"
transform="matrix(2.415731,0,0,2.415731,112.34125,-112.91)"><tspan
x="139.9279"
y="80.043816"
id="tspan495">Interacting</tspan></text>
id="tspan1">Interacting
</tspan><tspan
x="139.9279"
y="88.322874"
id="tspan2">Theories
</tspan></text>
<text
xml:space="preserve"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.1245px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;inline-size:65.7594;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="125.72736"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:9.33101px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;text-anchor:middle;white-space:pre;inline-size:65.7594;display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="143.19339"
y="80.043816"
id="text1123-03"
inkscape:label="text1123"
transform="matrix(1.5718199,0,0,1.5718199,232.84681,231.02987)"><tspan
x="125.72736"
transform="matrix(1.5718199,0,0,1.5718199,212.04837,141.24214)"><tspan
x="143.19339"
y="80.043816"
id="tspan497">Ideal Gas</tspan></text>
id="tspan3">Ideal Gas</tspan></text>
<text
xml:space="preserve"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.1245px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;inline-size:65.7594;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="125.72736"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:8.80124px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;text-anchor:middle;white-space:pre;inline-size:65.7594;display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="142.85497"
y="80.043816"
id="text1123-03-5"
inkscape:label="text1123"
transform="matrix(1.6664321,0,0,1.6664321,110.01032,217.64802)"><tspan
x="125.72736"
transform="matrix(1.6664321,0,0,1.6664321,114.02269,133.66902)"><tspan
x="142.85497"
y="80.043816"
id="tspan499">Band
id="tspan4">Band
</tspan><tspan
x="125.72736"
y="93.949442"
id="tspan501">Insulators</tspan></text>
x="142.85497"
y="91.045364"
id="tspan5">Theory
</tspan></text>
<text
xml:space="preserve"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:8.68596px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;inline-size:65.7594;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="125.72736"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.62325px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;text-anchor:middle;white-space:pre;inline-size:65.7594;display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="141.38319"
y="80.043816"
id="text1123-0"
inkscape:label="text1123"
transform="matrix(2.415731,0,0,2.415731,236.19784,235.72061)"><tspan
x="125.72736"
transform="matrix(2.415731,0,0,2.415731,181.80934,0.223041)"><tspan
x="141.38319"
y="80.043816"
id="tspan503">Many Body</tspan></text>
id="tspan6">Many Body
</tspan><tspan
x="141.38319"
y="88.322874"
id="tspan7">Theories</tspan></text>
<text
xml:space="preserve"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:8.68596px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;inline-size:65.7594;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="125.72736"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:6.62325px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;text-anchor:middle;white-space:pre;inline-size:65.7594;display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
x="137.93736"
y="80.043816"
id="text1123-0-3"
inkscape:label="text1123"
transform="matrix(2.415731,0,0,2.415731,-182.71073,236.14068)"><tspan
x="125.72736"
transform="matrix(2.415731,0,0,2.415731,-146.32222,95.647)"><tspan
x="137.93736"
y="80.043816"
id="tspan505">Quantum</tspan></text>
id="tspan8">Quantum
</tspan><tspan
x="137.93736"
y="88.322874"
id="tspan9">Theories</tspan></text>
<text
xml:space="preserve"
id="text1814"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:13.3333px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;shape-inside:url(#rect1816);fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,159.01555,204.35697)"><tspan
x="31.236328"
y="101.6396"
id="tspan507">Hydrogen </tspan><tspan
x="31.236328"
y="118.30623"
id="tspan509">Atom</tspan></text>
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.1837px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;white-space:pre;shape-inside:url(#rect1816);shape-padding:8.33653;display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,173.84223,125.33066)"
x="22.318253"
y="0"><tspan
x="51.824585"
y="108.06946"
id="tspan10">Hydrogen </tspan><tspan
x="61.449229"
y="122.04908"
id="tspan11">Atom</tspan></text>
<text
xml:space="preserve"
id="text1814-4"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:13.3333px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;shape-inside:url(#rect1816-6);fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,211.60906,101.49702)"><tspan
x="31.236328"
y="101.6396"
id="tspan511">Molecules</tspan></text>
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.1837px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;white-space:pre;shape-inside:url(#rect1816-6);display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,204.06475,43.587823)"
x="23.003582"
y="0"><tspan
x="50.89352"
y="99.731572"
id="tspan12">Molecules</tspan></text>
<text
xml:space="preserve"
id="text1814-9"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:13.3333px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;shape-inside:url(#rect1816-4);fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,281.65359,136.91623)"><tspan
x="31.236328"
y="101.6396"
id="tspan513">Strongly </tspan><tspan
x="31.236328"
y="118.30623"
id="tspan515">Correlated </tspan><tspan
x="31.236328"
y="134.97285"
id="tspan517">Materials</tspan></text>
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.1837px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;white-space:pre;shape-inside:url(#rect1816-4);display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,265.69622,68.538429)"
x="23.519627"
y="0"><tspan
x="46.33374"
y="99.731572"
id="tspan13">Strongly </tspan><tspan
x="41.997872"
y="113.71119"
id="tspan14">Correlated </tspan><tspan
x="44.479802"
y="127.69082"
id="tspan15">Materials</tspan></text>
<text
xml:space="preserve"
id="text1814-9-9"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:13.3333px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;shape-inside:url(#rect1816-4-1);fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,355.28094,100.49922)"><tspan
x="31.236328"
y="101.6396"
id="tspan519">Classical </tspan><tspan
x="31.236328"
y="118.30623"
id="tspan521">Phases</tspan></text>
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.1837px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;white-space:pre;shape-inside:url(#rect1816-4-1);display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,330.59545,43.587823)"
x="20.095711"
y="0"><tspan
x="36.427667"
y="99.731572"
id="tspan16">Classical </tspan><tspan
x="41.394257"
y="113.71119"
id="tspan17">Phases</tspan></text>
<text
xml:space="preserve"
id="text1814-9-9-2"
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:13.3333px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;shape-inside:url(#rect1816-4-1-5);fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,276.57599,21.896637)"><tspan
x="31.236328"
y="101.6396"
id="tspan523">Classical </tspan><tspan
x="31.236328"
y="118.30623"
id="tspan525">Mechanics</tspan></text>
style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.1837px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;text-align:center;white-space:pre;shape-inside:url(#rect1816-4-1-5);display:inline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
transform="matrix(1.3114339,0,0,1.3114339,256.89798,-20.177366)"
x="23.931917"
y="0"><tspan
x="52.436456"
y="99.731572"
id="tspan18">Classical </tspan><tspan
x="48.403662"
y="113.71119"
id="tspan19">Mechanics</tspan></text>
</g>
<g
transform="matrix(0.84345288,0,0,0.84345288,-145.5668,-111.83806)"

Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 12 KiB

After

Width:  |  Height:  |  Size: 13 KiB

BIN
assets/thesis/sandpile.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 301 KiB