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14
_sass/figures.scss
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@ -1,14 +0,0 @@
figure {
display: flex;
flex-direction: column;
align-items: center;
}
figure img {
max-width: 900px;
width: 80%;
margin-bottom: 2em;
}
figcaption {
aria-hidden: true;
max-width: 700px;
}

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_sass/main.scss
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@ -4,7 +4,7 @@
@import "header";
@import "article";
@import "cv";
@import "figures";
@import "thesis";
* {
box-sizing: border-box;
@ -66,29 +66,6 @@ img {
margin-bottom: 1em;
}
// For the thesis table of contents, should probably put this in a container
li {
margin-bottom: 0.2em;
}
main > ul > li {
margin-top: 1em;
}
main > ul > ul > li {
margin-top: 0.5em;
}
div.csl-entry {
margin-bottom: 0.5em;
}
// div.csl-entry a {
// text-decoration: none;
// }
div.csl-entry div {
display: inline;
}
@media
only screen and (max-width: $horizontal_breakpoint),
only screen and (max-height: $vertical_breakpoint)

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_sass/thesis.scss Normal file
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// Make figures looks nice
figure {
display: flex;
flex-direction: column;
align-items: center;
}
figure img {
max-width: 900px;
width: 100%;
margin-bottom: 2em;
}
figcaption {
aria-hidden: true;
max-width: 700px;
}
// For the table of contents, should probably put this in a container
// remove underline from toc links
nav a {
text-decoration: none;
}
// modify the spacing of the various levels
li {
margin-bottom: 0.2em;
}
main > ul > li {
margin-top: 1em;
}
main > ul > ul > li {
margin-top: 0.5em;
}
// Mess with the formatting of the citations
div.csl-entry {
margin-bottom: 0.5em;
}
// div.csl-entry a {
// text-decoration: none;
// }
div.csl-entry div {
display: inline;
}

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_thesis/2.1.2_AMK_Intro.html
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@ -205,8 +205,7 @@ image:
<main>
<nav id="TOC" role="doc-toc">
<ul>
<li><a href="#properties-of-the-gauge-field"
id="toc-properties-of-the-gauge-field">Properties of the Gauge Field</a>
<li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a>
<ul>
<li><a href="#vortices-and-their-movements"
id="toc-vortices-and-their-movements">Vortices and their
@ -223,26 +222,29 @@ plaquettes and vertices</a></li>
</ul></li>
<li><a href="#the-projector" id="toc-the-projector">The
Projector</a></li>
<li><a href="#open-boundary-conditions"
id="toc-open-boundary-conditions">Open boundary conditions</a></li>
<li><a href="#the-ground-state-vortex-sector"
id="toc-the-ground-state-vortex-sector">The Ground State Vortex
Sector</a>
<li><a href="#the-ground-state" id="toc-the-ground-state">The Ground
State</a>
<ul>
<li><a href="#finite-size-effects" id="toc-finite-size-effects">Finite
size effects</a></li>
</ul></li>
<li><a href="#chiral-symmetry" id="toc-chiral-symmetry">Chiral
Symmetry</a></li>
</ul></li>
<li><a href="#phases-of-the-kitaev-model"
id="toc-phases-of-the-kitaev-model">Phases of the Kitaev Model</a></li>
<li><a href="#whats-so-great-about-two-dimensions"
id="toc-whats-so-great-about-two-dimensions">What’s so great about two
dimensions?</a>
<ul>
<li><a href="#topology-chirality-and-edge-modes"
id="toc-topology-chirality-and-edge-modes">Topology, chirality and edge
modes</a></li>
<li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic
Statistics</a></li>
</ul></li>
</ul>
</nav>
<h2 id="properties-of-the-gauge-field">Properties of the Gauge
Field</h2>
<h2 id="gauge-fields">Gauge Fields</h2>
<p>The bond operators <span class="math inline">\(u_{ij}\)</span> are
useful because they label a bond sector <span
class="math inline">\(\mathcal{\tilde{L}}_u\)</span> in which we can
@ -513,9 +515,9 @@ take each set of <span class="math inline">\(\prod_{i \in \{i\}}
D_j\)</span> operators and gives us the complement of that set. I said
earlier that <span class="math inline">\(C\)</span> is the identity in
the physical subspace and we will shortly see why.</p>
<p>W use the complement operator to rewrite the projector as a sum over
half the subsets <span class="math inline">\(\{\}\)</span> let’s call
that <span class="math inline">\(\Lambda\)</span>. The complement
<p>We use the complement operator to rewrite the projector as a sum over
half the subsets of <span class="math inline">\(\{i\}\)</span> let’s
call that <span class="math inline">\(\Lambda\)</span>. The complement
operator deals with the other half</p>
<p><span class="math display">\[ \mathcal{P} = \left(
\frac{1}{2^{2N-1}} \sum_{\Lambda} \prod_{i\in\{i\}} D_i\right)
@ -538,7 +540,7 @@ reduces to a determinant of the Q matrix and the fermion parity,
see<span class="citation"
data-cites="pedrocchiPhysicalSolutionsKitaev2011b"><sup><a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011b"
role="doc-biblioref">1</a></sup></span> . The only difference from the
role="doc-biblioref">1</a></sup></span>. The only difference from the
honeycomb case is that we cannot explicitely 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
@ -551,19 +553,53 @@ decomposition<span class="citation"
data-cites="app:cycle_decomp"><sup><a href="#ref-app:cycle_decomp"
role="doc-biblioref"><strong>app:cycle_decomp?</strong></a></sup></span>.</p>
<p>We find that <span class="math display">\[\mathcal{P}_0 = 1 +
p_x\;p_y\;p_z\; \mathrm{det}(Q^u) \; \hat{\pi} \; \prod_{\{i,j\}}
p_x\;p_y\;p_z\; \hat{\pi} \; \mathrm{det}(Q^u) \; \prod_{\{i,j\}}
-iu_{ij}\]</span></p>
<p>where <span class="math inline">\(p_x\;p_y\;p_z = \pm 1\)</span> are
lattice structure factors. <span class="math inline">\(Q^u\)</span> is
the determinant of the matrix mentioned earlier that maps <span
lattice structure factors. <span class="math inline">\(det(Q^u)\)</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. <span class="math inline">\(\prod
-i \; u_{ij}\)</span> depend on the lattice and the particular vortex
sector. <span class="math inline">\(\hat{\pi} = \prod{i}^{N} (1 -
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>.</p>
class="math inline">\(n_i\)</span>. As discussed in +<span
class="citation"
data-cites="pedrocchiPhysicalSolutionsKitaev2011b"><sup><a
href="#ref-pedrocchiPhysicalSolutionsKitaev2011b"
role="doc-biblioref">1</a></sup></span> is <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 guage 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"><sup><a
href="#ref-yaoAlgebraicSpinLiquid2009"
role="doc-biblioref">2</a></sup></span>. However it is not so simple to
evaluate in the amorphous case.</p>
<p>More general arguments<span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"><sup><a
href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">3</a>,<a
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
role="doc-biblioref">4</a></sup></span> imply that <span
class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> has an
interesting relationship to the topological fluxes. In the non-Abelian
phase we expect that it will change sign in exactly on of the four
topological sectors. This forces that sector that contain a fermion and
hence gives the model a three-fold degerenate ground state. In the
Abelian phase this doesn’t happen and we get a fourfold degerate ground
state. Whether this analysis generalises to the amorphous case in
unclear.</p>
<p>An alternate way to view this is to consider the adiabatic insertion
of the fluxes <span class="math inline">\(\Phi_{x,y}\)</span> as the
operations that undo vortex transport around the lattice. In this
picture the three fold degeneracy occurs because transporting a vortex
around <strong>both</strong> the major and minor axes of the torus
changes its fusion channel such that the two vortices fuse into a
fermion excition rather than the vacuum.</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
@ -602,32 +638,26 @@ fermions in the system grows.</p>
basis we would need to include the full symmetrisation over the gauge
fields. However this was not necessary for any of the results that will
be presented here.</p>
<h2 id="open-boundary-conditions">Open boundary conditions</h2>
<p>Care must be taken in the definition of open boundary conditions.
Simply removing bonds from the lattice leaves behind unpaired <span
class="math inline">\(b^\alpha\)</span> operators that need to be paired
in some way to arrive at fermionic modes. In order 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.</p>
<p>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.
<strong>Is is possible that a lattice constructed and coloured like this
would have unequal numbers of <span class="math inline">\(b^x\)</span>
<span class="math inline">\(b^y\)</span> and <span
class="math inline">\(b^z\)</span> operators?</strong></p>
<h2 id="the-ground-state-vortex-sector">The Ground State Vortex
Sector</h2>
<h2 id="the-ground-state">The Ground State</h2>
<p>As we have shown that the Hamiltonian is gauge invariant, only the
flux sector and the two topological fluxes affect the spectrum of the
Hamiltonian. Thus we can label many body ground state by a combination
of flux sector and fermionic occupation numbers.</p>
<p>By studying the projector we saw that the fermionic occupation
numbers of the ground state will always be either <span
class="math inline">\(n_m = 0\)</span> or <span
class="math inline">\(n_0 = 1, n_{m&gt;1} = 0\)</span> because the
projector really just enforces vortex and fermion parity.</p>
<p>I refer to the flux sector that contains the ground state as the
ground state flux sector. Recall that we call the excitations of the
fluxes away from the ground ground state configuration
<strong>vortices</strong>, so that the ground state flux sector is the
vortex free sector by definition.</p>
<p>On the Honeycomb, Lieb’s theorem implies that the the ground state
corresponds to the state where all <span class="math inline">\(u_jk =
corresponds to the state where all <span class="math inline">\(u_{jk} =
1\)</span> implying that the flux free sector is the ground state
sector<span class="citation" data-cites="lieb_flux_1994"><sup><a
href="#ref-lieb_flux_1994" role="doc-biblioref">2</a></sup></span>.</p>
href="#ref-lieb_flux_1994" role="doc-biblioref">5</a></sup></span>.</p>
<p>Lieb’s theorem does not generalise easily to the amorphous case.
However we can get some intuition by examining the problem that will
lead to a guess for the ground state. We will then provide numerical
@ -713,7 +743,7 @@ that form each plaquette and the choice of sign gives a twofold chiral
ground state degeneracy.</p>
<p>This conjecture is consistent with Lieb’s theorem on regular
lattices<span class="citation" data-cites="lieb_flux_1994"><sup><a
href="#ref-lieb_flux_1994" role="doc-biblioref">2</a></sup></span> and
href="#ref-lieb_flux_1994" role="doc-biblioref">5</a></sup></span> and
is supported by numerical evidence. As noted before, any flux that
differs from the ground state is an excitation which I call a
vortex.</p>
@ -723,8 +753,36 @@ effects. In order to rigorously test it we would like to directly
enumerate the <span class="math inline">\(2^N\)</span> vortex sectors
for a smaller lattice and check that the lowest state found is the
vortex sector predicted by ???.</p>
<p>To do this we tile an amorphous lattice onto a repeating <span
class="math inline">\(NxN\)</span> grid. The use of a fourier series
<p>To do this we tile use an amorphous lattice as the unit cell of a
periodic <span class="math inline">\(N\times N\)</span> system. Bonds
that originally crossed the periodic boundaries now connect adjacent
unit cells. Using Bloch’s theorem the problem then essnetially reduces
back to the single amorphous unit cell but now the edges that cross the
periodic boundaries pick up a phase dependent on the crystal momentum
<span class="math inline">\(\vex{q} = (q_x, q_y)\)</span> and the
lattice vector of the bond <span class="math inline">\(\vec{x} = (+1, 0,
-1, +1, 0, -1)\)</span>. Assigning these lattice vectors to each bond is
also a very conveninent way to store and plot toroidal graphs.</p>
<p>This can then be solved using Bloch’s theorem. For a given crystal
momentum <span class="math inline">\(\textbf{q} \in [0,2\pi)^2\)</span>,
we are left with a Bloch Hamiltonian, which is identical to the original
Hamiltonian aside from an extra phase on edges that cross the periodic
boundaries in the <span class="math inline">\(x\)</span> and <span
class="math inline">\(y\)</span> directions, <span
class="math display">\[\begin{aligned}
M_{jk}(\textbf{q}) = \frac{i}{2} J^{\alpha} u_{jk} e^{i
q_{jk}},\end{aligned}\]</span> where <span class="math inline">\(q_{jk}
= q_x\)</span> for a bond that crosses the <span
class="math inline">\(x\)</span>-periodic boundary in the positive
direction, with the analogous definition for <span
class="math inline">\(y\)</span>-crossing bonds. We also have <span
class="math inline">\(q_{jk} = -q_{kj}\)</span>. Finally <span
class="math inline">\(q_{jk} = 0\)</span> if the edge does not cross any
boundaries at all – in essence we are imposing twisted boundary
conditions on our system. The total energy of the tiled system can be
calculated by summing the energy of <span class="math inline">\(M(
\textbf{q})\)</span> for every value of <span
class="math inline">\(\textbf{q}\)</span>. The use of a fourier series
then allows us to compute the diagonalisation with a penalty only linear
in the number of tiles used compared to diagonalising a single lattice.
With this technique the finite size effects related to the
@ -736,16 +794,16 @@ class="math inline">\(\phi_0\)</span> correctly predicts the ground
state for hundreds of thousands of lattices with upto 20 plaquettes. For
larger lattices we verified that random perturbations around the
predicted ground state never yield a lower energy state.</p>
<h2 id="chiral-symmetry">Chiral Symmetry</h2>
<h3 id="chiral-symmetry">Chiral Symmetry</h3>
<p>In the discussion above we see that the ground state has a twofold
<strong>chiral</strong> degeneracy that comes about because the global
sign of the odd plaquettes does not matter.</p>
<p>This happens because by adding odd plaquettes we have broken the time
reversal symmetry of the original model<span class="citation"
data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"><sup><a
href="#ref-Chua2011" role="doc-biblioref">3</a>–<a
href="#ref-Chua2011" role="doc-biblioref">6</a>–<a
href="#ref-WangHaoranPRB2021"
role="doc-biblioref">10</a></sup></span>.</p>
role="doc-biblioref">13</a></sup></span>.</p>
<p>Similar to the behaviour of the original Kitaev model in response to
a magnetic field, we get two degenerate ground states of different
handedness. Practicaly speaking, one ground state is related to the
@ -753,9 +811,16 @@ other by inverting the imaginary <span
class="math inline">\(\phi\)</span> fluxes<span class="citation"
data-cites="yaoExactChiralSpin2007"><sup><a
href="#ref-yaoExactChiralSpin2007"
role="doc-biblioref">4</a></sup></span>.</p>
<h2 id="topology-chirality-and-edge-modes">Topology, chirality and edge
modes</h2>
role="doc-biblioref">7</a></sup></span>.</p>
<h2 id="phases-of-the-kitaev-model">Phases of the Kitaev Model</h2>
<p>discuss the abelian A phase / toric code phase / anisotropic
phase</p>
<p>the isotropic gapless phase of the standard model</p>
<p>The isotropic gapped phase with the addition of a magnetic field</p>
<h2 id="whats-so-great-about-two-dimensions">What’s so great about two
dimensions?</h2>
<h3 id="topology-chirality-and-edge-modes">Topology, chirality and edge
modes</h3>
<p>Most thermodynamic and quantum phases studied can be characterised by
a local order parameter. That is, a function or operator that only
requires knowledge about some fixed sized patch of the system that does
@ -772,7 +837,7 @@ breaking.</p>
looked at it defined on a graph that is embedded either into the plane
or onto the torus. The extension to surfaces like the torus but with
more than one handle is relatively easy.</p>
<h2 id="anyonic-statistics">Anyonic Statistics</h2>
<h3 id="anyonic-statistics">Anyonic Statistics</h3>
<p>In dimensions greater than two, the quantum state of a system must
pick up a factor of <span class="math inline">\(-1\)</span> or <span
class="math inline">\(+1\)</span> if two identical particles are
@ -884,23 +949,23 @@ class="math inline">\((+1, +1), (+1, -1), (-1, +1), (-1,
<figure>
<img src="/assets/thesis/figure_code/amk_chapter/topological_fluxes.png"
style="width:57.0%"
alt="Figure 8: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the donut/torus or through the filling. If they made donuts that had both a jam filling and a hole this analogy would be a lot easier to make11." />
alt="Figure 8: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the donut/torus or through the filling. If they made donuts that had both a jam filling and a hole this analogy would be a lot easier to make14." />
<figcaption aria-hidden="true"><span>Figure 8:</span> Wilson loops that
wind the major or minor diameters of the torus measure flux winding
through the hole of the donut/torus or through the filling. If they made
donuts that had both a jam filling and a hole this analogy would be a
lot easier to make<span class="citation"
data-cites="parkerWhyDoesThis"><sup><a href="#ref-parkerWhyDoesThis"
role="doc-biblioref">11</a></sup></span>.</figcaption>
role="doc-biblioref">14</a></sup></span>.</figcaption>
</figure>
</div>
<p>However in the non-Abelian phase we have to wrangle with
monodromy<span class="citation"
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"><sup><a
href="#ref-chungExplicitMonodromyMoore2007"
role="doc-biblioref">12</a>,<a
role="doc-biblioref">3</a>,<a
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
role="doc-biblioref">13</a></sup></span>. Monodromy is behaviour of
role="doc-biblioref">4</a></sup></span>. Monodromy is behaviour of
objects as they move around a singularity. This manifests here in that
the identity of a vortex and cloud of Majoranas can change as we wind
them around the torus in such a way that rather than anhilating to the
@ -910,7 +975,7 @@ the non-Abelian phase <span class="math inline">\((+1, +1), (+1, -1),
(-1, +1)\)</span><span class="citation"
data-cites="chungTopologicalQuantumPhase2010"><sup><a
href="#ref-chungTopologicalQuantumPhase2010"
role="doc-biblioref">14</a></sup></span>. The way that this shows up
role="doc-biblioref">15</a></sup></span>. The way that this shows up
concretly is that the projector enforces both flux and fermion parity.
When we wind a vortex around both non-contractible loops of the torus,
it flips the flux parity which forces means we have to introduce a
@ -928,10 +993,10 @@ passively fault tolerant and actively stabilised quantum computations
[<span class="citation"
data-cites="kitaevFaulttolerantQuantumComputation2003"><sup><a
href="#ref-kitaevFaulttolerantQuantumComputation2003"
role="doc-biblioref">15</a></sup></span>;<span class="citation"
role="doc-biblioref">16</a></sup></span>;<span class="citation"
data-cites="poulinStabilizerFormalismOperator2005"><sup><a
href="#ref-poulinStabilizerFormalismOperator2005"
role="doc-biblioref">16</a></sup></span>;
role="doc-biblioref">17</a></sup></span>;
hastingsDynamicallyGeneratedLogical2021].</p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
@ -943,8 +1008,34 @@ href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of
the <span>Kitaev</span> honeycomb model</a>. <em>Phys. Rev. B</em>
<strong>84</strong>, 165414 (2011).</div>
</div>
<div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div class="csl-right-inline">Yao,
H., Zhang, S.-C. &amp; Kivelson, S. A. <a
href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic
<span>Spin Liquid</span> in an <span>Exactly Solvable Spin
Model</span></a>. <em>Phys. Rev. Lett.</em> <strong>102</strong>, 217202
(2009).</div>
</div>
<div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div
class="csl-right-inline">Chung, S. B. &amp; Stone, M. <a
href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit monodromy of
<span>Moore</span> wavefunctions on a torus</a>. <em>J. Phys. A: Math.
Theor.</em> <strong>40</strong>, 4923–4947 (2007).</div>
</div>
<div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div
class="csl-right-inline">Oshikawa, M., Kim, Y. B., Shtengel, K., Nayak,
C. &amp; Tewari, S. <a
href="https://doi.org/10.1016/j.aop.2006.08.001">Topological degeneracy
of non-<span>Abelian</span> states for dummies</a>. <em>Annals of
Physics</em> <strong>322</strong>, 1477–1498 (2007).</div>
</div>
<div id="ref-lieb_flux_1994" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">2. </div><div
<div class="csl-left-margin">5. </div><div
class="csl-right-inline">Lieb, E. H. <a
href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux
<span>Phase</span> of the <span>Half-Filled Band</span></a>.
@ -952,7 +1043,7 @@ href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux
(1994).</div>
</div>
<div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">3. </div><div
<div class="csl-left-margin">6. </div><div
class="csl-right-inline">Chua, V., Yao, H. &amp; Fiete, G. A. <a
href="https://doi.org/10.1103/PhysRevB.83.180412">Exact chiral spin
liquid with stable spin <span>Fermi</span> surface on the kagome
@ -961,21 +1052,21 @@ lattice</a>. <em>Phys. Rev. B</em> <strong>83</strong>, 180412
</div>
<div id="ref-yaoExactChiralSpin2007" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">4. </div><div class="csl-right-inline">Yao,
<div class="csl-left-margin">7. </div><div class="csl-right-inline">Yao,
H. &amp; Kivelson, S. A. <a
href="https://doi.org/10.1103/PhysRevLett.99.247203">An exact chiral
spin liquid with non-<span>Abelian</span> anyons</a>. <em>Phys. Rev.
Lett.</em> <strong>99</strong>, 247203 (2007).</div>
</div>
<div id="ref-ChuaPRB2011" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">5. </div><div
<div class="csl-left-margin">8. </div><div
class="csl-right-inline">Chua, V. &amp; Fiete, G. A. <a
href="https://doi.org/10.1103/PhysRevB.84.195129">Exactly solvable
topological chiral spin liquid with random exchange</a>. <em>Phys. Rev.
B</em> <strong>84</strong>, 195129 (2011).</div>
</div>
<div id="ref-Fiete2012" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">6. </div><div
<div class="csl-left-margin">9. </div><div
class="csl-right-inline">Fiete, G. A. <em>et al.</em> <a
href="https://doi.org/10.1016/j.physe.2011.11.011">Topological
insulators and quantum spin liquids</a>. <em>Physica E: Low-dimensional
@ -983,20 +1074,20 @@ Systems and Nanostructures</em> <strong>44</strong>, 845–859
(2012).</div>
</div>
<div id="ref-Natori2016" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">7. </div><div
<div class="csl-left-margin">10. </div><div
class="csl-right-inline">Natori, W. M. H., Andrade, E. C., Miranda, E.
&amp; Pereira, R. G. Chiral spin-orbital liquids with nodal lines.
<em>Phys. Rev. Lett.</em> <strong>117</strong>, 017204 (2016).</div>
</div>
<div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">8. </div><div class="csl-right-inline">Wu,
<div class="csl-left-margin">11. </div><div class="csl-right-inline">Wu,
C., Arovas, D. &amp; Hung, H.-H. <span><span
class="math inline">\(\Gamma\)</span></span>-matrix generalization of
the <span>Kitaev</span> model. <em>Physical Review B</em>
<strong>79</strong>, 134427 (2009).</div>
</div>
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">9. </div><div
<div class="csl-left-margin">12. </div><div
class="csl-right-inline">Peri, V. <em>et al.</em> <a
href="https://doi.org/10.1103/PhysRevB.101.041114">Non-<span>Abelian</span>
chiral spin liquid on a simple non-<span>Archimedean</span> lattice</a>.
@ -1004,7 +1095,7 @@ chiral spin liquid on a simple non-<span>Archimedean</span> lattice</a>.
</div>
<div id="ref-WangHaoranPRB2021" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">10. </div><div
<div class="csl-left-margin">13. </div><div
class="csl-right-inline">Wang, H. &amp; Principi, A. <a
href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana edge and
corner states in square and kagome quantum spin-<span><span
@ -1013,30 +1104,13 @@ class="math inline">\(^{3}\fracslash_2\)</span></span> liquids</a>.
</div>
<div id="ref-parkerWhyDoesThis" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">11. </div><div
<div class="csl-left-margin">14. </div><div
class="csl-right-inline">Parker, M. Why does this balloon have -1
holes?</div>
</div>
<div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">12. </div><div
class="csl-right-inline">Chung, S. B. &amp; Stone, M. <a
href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit monodromy of
<span>Moore</span> wavefunctions on a torus</a>. <em>J. Phys. A: Math.
Theor.</em> <strong>40</strong>, 4923–4947 (2007).</div>
</div>
<div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">13. </div><div
class="csl-right-inline">Oshikawa, M., Kim, Y. B., Shtengel, K., Nayak,
C. &amp; Tewari, S. <a
href="https://doi.org/10.1016/j.aop.2006.08.001">Topological degeneracy
of non-<span>Abelian</span> states for dummies</a>. <em>Annals of
Physics</em> <strong>322</strong>, 1477–1498 (2007).</div>
</div>
<div id="ref-chungTopologicalQuantumPhase2010" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">14. </div><div
<div class="csl-left-margin">15. </div><div
class="csl-right-inline">Chung, S. B., Yao, H., Hughes, T. L. &amp; Kim,
E.-A. <a href="https://doi.org/10.1103/PhysRevB.81.060403">Topological
quantum phase transition in an exactly solvable model of a chiral spin
@ -1045,7 +1119,7 @@ liquid at finite temperature</a>. <em>Phys. Rev. B</em>
</div>
<div id="ref-kitaevFaulttolerantQuantumComputation2003"
class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">15. </div><div
<div class="csl-left-margin">16. </div><div
class="csl-right-inline">Kitaev, A. Yu. <a
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-tolerant
quantum computation by anyons</a>. <em>Annals of Physics</em>
@ -1053,7 +1127,7 @@ quantum computation by anyons</a>. <em>Annals of Physics</em>
</div>
<div id="ref-poulinStabilizerFormalismOperator2005" class="csl-entry"
role="doc-biblioentry">
<div class="csl-left-margin">16. </div><div
<div class="csl-left-margin">17. </div><div
class="csl-right-inline">Poulin, D. <a
href="https://doi.org/10.1103/PhysRevLett.95.230504">Stabilizer
<span>Formalism</span> for <span>Operator Quantum Error

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@ -209,39 +209,29 @@ image:
<ul>
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
Systems</a></li>
<li><a href="#chapter-outline" id="toc-chapter-outline">Chapter
outline</a></li>
</ul></li>
<li><a href="#an-in-depth-look-at-the-kitaev-model"
id="toc-an-in-depth-look-at-the-kitaev-model">An in-depth look at the
Kitaev Model</a>
<li><a href="#the-kitaev-model" id="toc-the-kitaev-model">The Kitaev
Model</a>
<ul>
<li><a href="#commutation-relations"
id="toc-commutation-relations">Commutation relations</a>
<ul>
<li><a href="#spins" id="toc-spins">Spins</a></li>
<li><a href="#fermions-and-majoranas"
id="toc-fermions-and-majoranas">Fermions and Majoranas</a></li>
</ul></li>
id="toc-commutation-relations">Commutation relations</a></li>
<li><a href="#the-hamiltonian" id="toc-the-hamiltonian">The
Hamiltonian</a></li>
<li><a href="#from-spins-to-majorana-operators"
id="toc-from-spins-to-majorana-operators">From Spins to Majorana
operators</a>
<ul>
<li><a href="#for-a-single-spin" id="toc-for-a-single-spin">For a single
spin</a></li>
<li><a href="#for-multiple-spins" id="toc-for-multiple-spins">For
multiple spins</a></li>
</ul></li>
operators</a></li>
<li><a href="#partitioning-the-hilbert-space-into-bond-sectors"
id="toc-partitioning-the-hilbert-space-into-bond-sectors">Partitioning
the Hilbert Space into Bond sectors</a></li>
</ul></li>
<li><a href="#the-majorana-hamiltonian"
id="toc-the-majorana-hamiltonian">The Majorana Hamiltonian</a></li>
id="toc-the-majorana-hamiltonian">The Majorana Hamiltonian</a>
<ul>
<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace"
id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
back from Bond Sectors to the Physical Subspace</a></li>
<li><a href="#open-boundary-conditions"
id="toc-open-boundary-conditions">Open boundary conditions</a></li>
</ul></li>
</ul></li>
</ul>
</nav>
@ -278,10 +268,6 @@ href="#ref-kitaevAnyonsExactlySolved2006"
role="doc-biblioref">4</a></sup></span>. Its solubility comes about
because the model has extensively many conserved degrees of freedom that
mediate the interactions between quantum degrees of freedom.</p>
<h2 id="amorphous-systems">Amorphous Systems</h2>
<p><strong>Insert discussion of why a generalisation to the amorphous
case is intersting</strong></p>
<h2 id="chapter-outline">Chapter outline</h2>
<p>In this chapter I will discuss the physics of the Kitaev Model on
amorphous lattices.</p>
<p>I’ll start by discussing the physics of the Kitaev model in much more
@ -320,13 +306,15 @@ addtional small Heisenberg term <span class="math display">\[H_{KH} = -
\sum_{\langle j,k\rangle_\alpha}
J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} +
\sigma_j\sigma_k\]</span></p>
<h1 id="an-in-depth-look-at-the-kitaev-model">An in-depth look at the
Kitaev Model</h1>
<h2 id="commutation-relations">Commutation relations</h2>
<h2 id="amorphous-systems">Amorphous Systems</h2>
<p><strong>Insert discussion of why a generalisation to the amorphous
case is intersting</strong></p>
<h2 id="the-kitaev-model">The Kitaev Model</h2>
<h3 id="commutation-relations">Commutation relations</h3>
<p>Before diving into the Hamiltonian of the Kitaev Model, here is a
quick refresher of the key commutation relations of spins, fermions and
Majoranas.</p>
<h3 id="spins">Spins</h3>
<h4 id="spins">Spins</h4>
<p>Skip this is you’re super familiar with the algebra of the Pauli
martrices. Scalars like <span class="math inline">\(\delta_{ij}\)</span>
should be understood to be multiplied by an implicit identity <span
@ -359,7 +347,7 @@ class="math display">\[\sigma^\alpha \sigma^\beta \sigma^\gamma = i
\epsilon^{\alpha\beta\gamma}\]</span> and <span
class="math display">\[[\sigma^\alpha \sigma^\beta, \sigma^\gamma] =
0\]</span></p>
<h3 id="fermions-and-majoranas">Fermions and Majoranas</h3>
<h4 id="fermions-and-majoranas">Fermions and Majoranas</h4>
<p>The fermionic creation and anhilation operators are defined by the
canonical anticommutation relations <span
class="math display">\[\begin{aligned}
@ -387,7 +375,7 @@ class="math inline">\(c_i\)</span>. The property that must be preserved
however is that the Majoranas still anticommute:</p>
<p><span class="math display">\[ \{c_i, c_j\} =
2\delta_{ij}\]</span></p>
<h2 id="the-hamiltonian">The Hamiltonian</h2>
<h3 id="the-hamiltonian">The Hamiltonian</h3>
<p>To get down to brass tacks, the Kitaev Honeycomb model is a model of
interacting spin<span class="math inline">\(-1/2\)</span>s on the
vertices of a honeycomb lattice. Each bond in the lattice is assigned a
@ -497,9 +485,9 @@ of a plaqutte operator away from the ground state as
Hilbert space into a set of ‘vortex sectors’ labelled by that particular
flux configuration <span class="math inline">\(\phi_i = \pm 1,\pm
i\)</span>.</p>
<h2 id="from-spins-to-majorana-operators">From Spins to Majorana
operators</h2>
<h3 id="for-a-single-spin">For a single spin</h3>
<h3 id="from-spins-to-majorana-operators">From Spins to Majorana
operators</h3>
<h4 id="for-a-single-spin">For a single spin</h4>
<p>Let’s start by considering just one site and its <span
class="math inline">\(\sigma^x, \sigma^y\)</span> and <span
class="math inline">\(\sigma^z\)</span> operators which live in a two
@ -560,7 +548,7 @@ alt="Figure 4: " />
<figcaption aria-hidden="true"><span>Figure 4:</span> </figcaption>
</figure>
</div>
<h3 id="for-multiple-spins">For multiple spins</h3>
<h4 id="for-multiple-spins">For multiple spins</h4>
<p>This construction generalises easily to the case of multiple spins:
we get a set of 4 Majoranas <span class="math inline">\(b^x_j,\;
b^y_j,\;b^z_j,\; c_j\)</span> and a <span class="math inline">\(D_j =
@ -612,8 +600,8 @@ degree of degeneracy.</p>
<p>In summary Majorana bond operators <span
class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span
class="math inline">\(\mathbb{Z_2}\)</span> gauge field!</p>
<h2 id="partitioning-the-hilbert-space-into-bond-sectors">Partitioning
the Hilbert Space into Bond sectors</h2>
<h3 id="partitioning-the-hilbert-space-into-bond-sectors">Partitioning
the Hilbert Space into Bond sectors</h3>
<p>Similar to the story with the plaquette operators from the spin
language, we can break the Hilbert space <span
class="math inline">\(\mathcal{L}\)</span> up into sectors labelled by
@ -713,8 +701,8 @@ can take half the absolute value of the whole set to recover <span
class="math inline">\(\sum_m \epsilon_m\)</span> easily.</p>
<p><strong>The Majorana Hamiltonian is quadratic within a Bond
Sector.</strong></p>
<h2 id="mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
back from Bond Sectors to the Physical Subspace</h2>
<h3 id="mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
back from Bond Sectors to the Physical Subspace</h3>
<p>At this point, given a particular bond configuration <span
class="math inline">\(u_{ij} = \pm 1\)</span> we are able to construct a
quadratic Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span>
@ -771,6 +759,25 @@ class="math inline">\(u_{ij}\)</span>, these will turn out to just be
the plaquette operators.</p>
<p><strong>The Bond Sectors overlap with the physical subspace but are
not contained within it.</strong></p>
<h3 id="open-boundary-conditions">Open boundary conditions</h3>
<p>Care must be taken in the definition of open boundary conditions.
Simply removing bonds from the lattice leaves behind unpaired <span
class="math inline">\(b^\alpha\)</span> operators that need to be paired
in some way to arrive at fermionic modes. In order 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.</p>
<p>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.
<strong>Is is possible that a lattice constructed and coloured like this
would have unequal numbers of <span class="math inline">\(b^x\)</span>
<span class="math inline">\(b^y\)</span> and <span
class="math inline">\(b^z\)</span> operators?</strong></p>
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
role="doc-bibliography">
<div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry"

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<li><a href="./1.1_FK_Intro.html#introduction">Introduction</a></li>
<ul>
<li><a href="./1.1_FK_Intro.html#localisation">Localisation</a></li>
<li><a href="./1.1_FK_Intro.html#falikov-kimball-and-hubbard-models">Falikov Kimball and Hubbard models</a></li>
<li><a href="./1.1_FK_Intro.html#localisation">Localisation</a></li>
<li><a href="./1.1_FK_Intro.html#numerical-methods">Numerical Methods</a></li>
<li><a href="./1.1_FK_Intro.html#markov-chain-monte-carlo-in-practice}">Markov Chain Monte-Carlo in Practice}</a></li>
</ul></ul>
<li>Chapter 2: The Amorphous Kitaev Model</li>
<ul>
<li><a href="./2.1_AMK_Intro.html#introduction">Introduction</a></li>
<ul>
<li><a href="./2.1_AMK_Intro.html#amorphous-systems">Amorphous Systems</a></li>
<li><a href="./2.1_AMK_Intro.html#chapter-outline">Chapter outline</a></li>
</ul>
<li><a href="./2.1_AMK_Intro.html#an-in-depth-look-at-the-kitaev-model">An in-depth look at the Kitaev Model</a></li>
<ul>
<li><a href="./2.1_AMK_Intro.html#commutation-relations">Commutation relations</a></li>
<li><a href="./2.1.2_AMK_Intro.html#properties-of-the-gauge-field">Properties of the Gauge Field</a></li>
<li><a href="./2.1_AMK_Intro.html#the-kitaev-model">The Kitaev Model</a></li>
<li><a href="./2.1_AMK_Intro.html#the-majorana-hamiltonian">The Majorana Hamiltonian</a></li>
<li><a href="./2.1.2_AMK_Intro.html#gauge-fields">Gauge Fields</a></li>
<li><a href="./2.1.2_AMK_Intro.html#the-projector">The Projector</a></li>
<li><a href="./2.1.2_AMK_Intro.html#the-ground-state">The Ground State</a></li>
<li><a href="./2.1.2_AMK_Intro.html#phases-of-the-kitaev-model">Phases of the Kitaev Model</a></li>
<li><a href="./2.1.2_AMK_Intro.html#what's-so-great-about-two-dimensions?">What's so great about two dimensions?</a></li>
</ul>
<li><a href="./2.2_AMK_Methods.html#methods">Methods</a></li>
<ul>
<li><a href="./2.2_AMK_Methods.html#voronisation">Voronisation</a></li>
<li><a href="./2.2_AMK_Methods.html#graph-representation">Graph Representation</a></li>
<li><a href="./2.2_AMK_Methods.html#coloring-the-bonds">Coloring the Bonds</a></li>
<li><a href="./2.2_AMK_Methods.html#mapping-between-flux-sectors-and-bond-sectors">Mapping between flux sectors and bond sectors</a></li>
<li><a href="./2.2_AMK_Methods.html#chern-markers">Chern Markers</a></li>
</ul>
<li><a href="./2.3_AMK_Results.html#results">Results</a></li>
<ul>
<li><a href="./2.3_AMK_Results.html#the-ground-state">The Ground State</a></li>
<li><a href="./2.3_AMK_Results.html#the-ground-state-flux-sector">The Ground State Flux Sector</a></li>
<li><a href="./2.3_AMK_Results.html#spontaneous-chiral-symmetry-breaking">Spontaneous Chiral Symmetry Breaking</a></li>
<li><a href="./2.3_AMK_Results.html#ground-state-phase-diagram">Ground State Phase Diagram</a></li>
<li><a href="./2.3_AMK_Results.html#the-flux-gap">The Flux Gap</a></li>
<li><a href="./2.3_AMK_Results.html#anderson-transition-to-a-thermal-metal">Anderson Transition to a Thermal Metal</a></li>
</ul>
<li><a href="./2.3_AMK_Results.html#conclusion">Conclusion</a></li>
<li><a href="./2.3_AMK_Results.html#discussion">Discussion</a></li>
<ul>
<li><a href="./2.3_AMK_Results.html#discussion">Discussion</a></li>
<li><a href="./2.3_AMK_Results.html#future-work">Future Work</a></li>
<li><a href="./2.3_AMK_Results.html#fluxes-and-the-ground-state">Fluxes and the Ground State</a></li>
<li><a href="./2.3_AMK_Results.html#zero-temperature-phase-diagram">Zero Temperature Phase Diagram</a></li>
<li><a href="./2.3_AMK_Results.html#failure-of-the-ground-state-conjecture">Failure of the ground state conjecture</a></li>
<li><a href="./2.3_AMK_Results.html#full-monte-carlo">Full Monte Carlo</a></li>
</ul>
<li><a href="./2.3_AMK_Results.html#outlook">Outlook</a></li>
<ul>
<li><a href="./2.3_AMK_Results.html#experimental-realisations-and-signatures">Experimental Realisations and Signatures</a></li>
<li><a href="./2.3_AMK_Results.html#generalisations">Generalisations</a></li>
</ul></ul>
<li>Conclusion</li>
<ul><ul>

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This is my work-in-progress thesis. It will be available as a traditional PDF too but I wanted to make it available as nicely rendered website too!
<h2>Contents</h2>
{% include_relative _thesis/toc.html %}
<nav>
{% include_relative _thesis/toc.html %}
</nav>