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_thesis/1_Introduction/1_Intro.html
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@ -574,61 +574,72 @@ spin couples. In certain transition metal based compounds, such as those
based on Iridium and Rutheniun, crystal field effects, 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 <span class="citation"
data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022"
role="doc-biblioref">42</a>]</span>.</p>
<p>The celebrated Kitaev model <span class="citation"
strongly anisotropic spin-spin couplings known as Kitaev Materials <span
class="citation"
data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a
href="#ref-TrebstPhysRep2022" role="doc-biblioref">42</a>,<a
href="#ref-Jackeli2009" role="doc-biblioref">46</a>–<a
href="#ref-Takagi2019" role="doc-biblioref">49</a>]</span>. Kitaev
materials draw their name from the celebrated Kitaev Honeycomb Model as
it is believed they will realise the QSL state via the mechanisms of the
Kitaev Model.</p>
<p>The Kitaev Honeycomb model <span class="citation"
data-cites="kitaevAnyonsExactlySolved2006"> [<a
href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">46</a>]</span></p>
<p>QSLs are a long range entangled ground state of a highly
frustated</p>
<ul>
<li><p>QSLs introduced by anderson 1973</p></li>
<li><p>Frustration can be geometric, such as AFM couplings on a
triangular lattice. It can also come from anisotropic couplings induced
via spin-orbit coupling.</p></li>
</ul>
<p>Geometric frustration or spin-orbit coupling can prevent magnetic
ordering is an important part of getting a QSL, suggests exploring the
lattice and avenue of interest.</p>
<ul>
<li><p>Spin orbit effect is a relativistic effect that couples electron
spin to orbital angular moment. Very roughly, an electron sees the
electric field of the nucleus as a magnetic field due to its movement
and the electron spin couples to this. Can be strong in heavy
elements</p></li>
<li><p>The Kitaev Model as a canonical QSL</p></li>
<li><p>Kitaev model has extensively many conserved charges too</p></li>
<li><p>anyons</p></li>
<li><p>fractionalisation</p></li>
<li><p>Topology -&gt; GS degeneracy depends on the genus of the
surface</p></li>
<li><p>the chern number</p></li>
</ul>
<p>kinds of mott insulators: Mott-Heisenberg (AFM order below Néel
temperature) Mott-Hubbard (no long-range order of local magnetic
moments) Mott-Anderson (disorder + correlations) Wigner Crystal</p>
role="doc-biblioref">50</a>]</span> was the first concrete model with a
QSL ground state. It is defined on the honeycomb lattice and provides an
exactly solvable model whose ground state is a QSL characterized by a
static <span class="math inline">\(\mathbb Z_2\)</span> gauge field and
Majorana fermion excitations. It can be reduced to a free fermion
problem via a mapping to Majorana fermions which 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, it supports a rich phase diagram hosting gapless,
Abelian and non-Abelian phases and a finite temperature phase transition
to a thermal metal state <span class="citation"
data-cites="selfThermallyInducedMetallic2019"> [<a
href="#ref-selfThermallyInducedMetallic2019"
role="doc-biblioref">51</a>]</span>. It has also been proposed that it
could be used to support topological quantum computing <span
class="citation"
data-cites="freedmanTopologicalQuantumComputation2003"> [<a
href="#ref-freedmanTopologicalQuantumComputation2003"
role="doc-biblioref">52</a>]</span>.</p>
<p>It is by now understood that the Kitaev model on any tri-coordinated
<span class="math inline">\(z=3\)</span> graph has conserved plaquette
operators and local symmetries <span class="citation"
data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007"
role="doc-biblioref">53</a>,<a href="#ref-Baskaran2008"
role="doc-biblioref">54</a>]</span> which allow a mapping onto effective
free Majorana fermion problems in a background of static <span
class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation"
data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a
href="#ref-Nussinov2009" role="doc-biblioref">55</a>–<a
href="#ref-hermanns2015weyl" role="doc-biblioref">58</a>]</span>.
However, depending on lattice symmetries, finding the ground state flux
sector and understanding the QSL properties can still be
challenging <span class="citation"
data-cites="eschmann2019thermodynamics Peri2020"> [<a
href="#ref-eschmann2019thermodynamics" role="doc-biblioref">59</a>,<a
href="#ref-Peri2020" role="doc-biblioref">60</a>]</span>.</p>
<p><strong>paragraph about amorphous lattices</strong></p>
<p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin
liquid by extending the Kitaev honeycomb model to random lattices with
fixed coordination number three. The 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 unearth a rich phase
diagram displaying Abelian as well as a non-Abelian quantum spin liquid
phases with a remarkably simple ground state flux pattern. Furthermore,
I show that the system undergoes a finite-temperature phase transition
to a conducting thermal metal state and discuss possible experimental
realisations.</p>
<h1 id="outline">Outline</h1>
<p>This thesis is composed of two main studies of separate but related
physical models, The Falikov-Kimball Model and the Kitaev-Honeycomb
Model. In this chapter I will discuss the overarching motivations for
looking at these two physical models. I will then review the literature
and methods that are common to both models.</p>
<p>In Chapter 2 I will look at the Falikov-Kimball model. I will review
what it is and why we would want to study it. I’ll survey what is
already known about it and identify the gap in the research that we aim
to fill, namely the model’s behaviour in one dimension. I’ll then
introduce the modified model that we came up with to close this gap. I
will present our results on the thermodynamic phase diagram and
localisation properties of the model</p>
<p>In Chapter 3 I’ll study the Kitaev Honeycomb Model, following the
same structure as Chapter 2 I will motivate the study, survey the
literature and identify a gap. I’ll introduce our Amorphous Kitaev Model
designed to fill this gap and present the results.</p>
<p>Finally in chapter 4 I will summarise the results and discuss what
implications they have for our understanding interacting many-body
quantum systems.</p>
<p>The next chapter, Chapter 2, will introduce some necessary background
to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and
localisation.</p>
<p>In Chapter 3 I introduce the Long Range Falikov-Kimball Model in
greater detail. I will present results that. Chapter 4 focusses on the
Amorphous Kitaev Model.</p>
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-king2012murmurations" class="csl-entry"
role="doc-biblioentry">
@ -984,13 +995,118 @@ class="csl-right-inline">H.-H. Lin, L. Balents, and M. P. A. Fisher,
Symmetry in the Weakly-Interacting Two-Leg Ladder</a></em>, Phys. Rev. B
<strong>58</strong>, 1794 (1998).</div>
</div>
<div id="ref-Jackeli2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[46] </div><div class="csl-right-inline">G.
Jackeli and G. Khaliullin, <em><a
href="https://doi.org/10.1103/PhysRevLett.102.017205">Mott Insulators in
the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum
Compass and Kitaev Models</a></em>, Physical Review Letters
<strong>102</strong>, 017205 (2009).</div>
</div>
<div id="ref-HerrmannsAnRev2018" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[47] </div><div class="csl-right-inline">M.
Hermanns, I. Kimchi, and J. Knolle, <em><a
href="https://doi.org/10.1146/annurev-conmatphys-033117-053934">Physics
of the Kitaev Model: Fractionalization, Dynamic Correlations, and
Material Connections</a></em>, Annual Review of Condensed Matter Physics
<strong>9</strong>, 17 (2018).</div>
</div>
<div id="ref-Winter2017" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[48] </div><div class="csl-right-inline">S.
M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P.
Gegenwart, and R. Valentí, <em>Models and Materials for Generalized
Kitaev Magnetism</em>, Journal of Physics: Condensed Matter
<strong>29</strong>, 493002 (2017).</div>
</div>
<div id="ref-Takagi2019" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[49] </div><div class="csl-right-inline">H.
Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler,
<em>Concept and Realization of Kitaev Quantum Spin Liquids</em>, Nature
Reviews Physics <strong>1</strong>, 264 (2019).</div>
</div>
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[46] </div><div class="csl-right-inline">A.
<div class="csl-left-margin">[50] </div><div class="csl-right-inline">A.
Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons
in an Exactly Solved Model and Beyond</a></em>, Annals of Physics
<strong>321</strong>, 2 (2006).</div>
</div>
<div id="ref-selfThermallyInducedMetallic2019" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[51] </div><div class="csl-right-inline">C.
N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, <em><a
href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally Induced
Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of
the Kitaev Model with Parity Projection</a></em>, Phys. Rev. B
<strong>99</strong>, 045142 (2019).</div>
</div>
<div id="ref-freedmanTopologicalQuantumComputation2003"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[52] </div><div class="csl-right-inline">M.
Freedman, A. Kitaev, M. Larsen, and Z. Wang, <em><a
href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological Quantum
Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31
(2003).</div>
</div>
<div id="ref-Baskaran2007" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[53] </div><div class="csl-right-inline">G.
Baskaran, S. Mandal, and R. Shankar, <em><a
href="https://doi.org/10.1103/PhysRevLett.98.247201">Exact Results for
Spin Dynamics and Fractionalization in the Kitaev Model</a></em>, Phys.
Rev. Lett. <strong>98</strong>, 247201 (2007).</div>
</div>
<div id="ref-Baskaran2008" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[54] </div><div class="csl-right-inline">G.
Baskaran, D. Sen, and R. Shankar, <em><a
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev Model:
Classical Ground States, Order from Disorder, and Exact Correlation
Functions</a></em>, Phys. Rev. B <strong>78</strong>, 115116
(2008).</div>
</div>
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[55] </div><div class="csl-right-inline">Z.
Nussinov and G. Ortiz, <em><a
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and
Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>,
Physical Review B <strong>79</strong>, 214440 (2009).</div>
</div>
<div id="ref-OBrienPRB2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[56] </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-yaoExactChiralSpin2007" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[57] </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-hermanns2015weyl" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[58] </div><div class="csl-right-inline">M.
Hermanns, K. O’Brien, and S. Trebst, <em>Weyl Spin Liquids</em>,
Physical Review Letters <strong>114</strong>, 157202 (2015).</div>
</div>
<div id="ref-eschmann2019thermodynamics" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">[59] </div><div class="csl-right-inline">T.
Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y.
Motome, and S. Trebst, <em>Thermodynamics of a Gauge-Frustrated Kitaev
Spin Liquid</em>, Physical Review Research <strong>1</strong>, 032011(R)
(2019).</div>
</div>
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[60] </div><div class="csl-right-inline">V.
Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R.
Moessner, and R. Thomale, <em><a
href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian Chiral
Spin Liquid on a Simple Non-Archimedean Lattice</a></em>, Phys. Rev. B
<strong>101</strong>, 041114 (2020).</div>
</div>
</div>
</main>
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_thesis/4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html
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@ -266,6 +329,8 @@ Markers</a></li>
</ul>
</nav>
-->
<div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
<h1 id="methods">Methods</h1>
<p>The practical implementation of what is described in this section is
available as a Python package called Koala (Kitaev On Amorphous