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@ -79,6 +79,16 @@ main > ul > ul > li {
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margin-top: 0.5em;
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}
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div.csl-entry {
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margin-bottom: 0.5em;
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}
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// div.csl-entry a {
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// text-decoration: none;
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// }
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div.csl-entry div {
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display: inline;
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}
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@media
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only screen and (max-width: $horizontal_breakpoint),
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only screen and (max-height: $vertical_breakpoint)
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@ -780,8 +780,8 @@ the states in the unperturbed bands remain extended, evidence for a
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mobility edge.</p>
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<div id="fig:binder" class="fignos">
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<figure>
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<img src="/assets/thesis/fk_chapter/binder.png" style="width:100.0%"
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alt="Figure 1: Hello I am the figure caption!" />
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<img src="/assets/thesis/figure_code/fk_chapter/binder.png"
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style="width:100.0%" alt="Figure 1: Hello I am the figure caption!" />
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<figcaption aria-hidden="true"><span>Figure 1:</span> Hello I am the
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figure caption!</figcaption>
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</figure>
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_thesis/2.1.2_AMK_Intro.html
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_thesis/2.1.2_AMK_Intro.html
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@ -207,10 +207,10 @@ image:
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<ul>
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<li><a href="#introduction" id="toc-introduction">Introduction</a>
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<ul>
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<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
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Systems</a></li>
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<li><a href="#chapter-outline" id="toc-chapter-outline">Chapter
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outline</a></li>
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<li><a href="#kitaev-heisenberg-model"
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id="toc-kitaev-heisenberg-model">Kitaev-Heisenberg Model</a></li>
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</ul></li>
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<li><a href="#an-in-depth-look-at-the-kitaev-model"
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id="toc-an-in-depth-look-at-the-kitaev-model">An in-depth look at the
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@ -242,40 +242,6 @@ id="toc-the-majorana-hamiltonian">The Majorana Hamiltonian</a></li>
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<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace"
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id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
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back from Bond Sectors to the Physical Subspace</a></li>
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<li><a href="#properties-of-the-gauge-field"
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id="toc-properties-of-the-gauge-field">Properties of the Gauge Field</a>
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<ul>
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<li><a href="#vortices-and-their-movements"
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id="toc-vortices-and-their-movements">Vortices and their
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movements</a></li>
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<li><a href="#composition-of-u_jk-loops"
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id="toc-composition-of-u_jk-loops">Composition of <span
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class="math inline">\(u_{jk}\)</span> loops</a></li>
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<li><a href="#gauge-degeneracy-and-the-euler-equation"
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id="toc-gauge-degeneracy-and-the-euler-equation">Gauge Degeneracy and
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the Euler Equation</a></li>
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<li><a href="#counting-edges-plaquettes-and-vertices"
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id="toc-counting-edges-plaquettes-and-vertices">Counting edges,
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plaquettes and vertices</a></li>
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</ul></li>
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<li><a href="#the-projector" id="toc-the-projector">The
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Projector</a></li>
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<li><a href="#open-boundary-conditions"
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id="toc-open-boundary-conditions">Open boundary conditions</a></li>
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<li><a href="#the-ground-state-vortex-sector"
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id="toc-the-ground-state-vortex-sector">The Ground State Vortex
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Sector</a>
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<ul>
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<li><a href="#finite-size-effects" id="toc-finite-size-effects">Finite
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size effects</a></li>
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</ul></li>
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<li><a href="#chiral-symmetry" id="toc-chiral-symmetry">Chiral
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Symmetry</a></li>
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<li><a href="#topology-chirality-and-edge-modes"
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id="toc-topology-chirality-and-edge-modes">Topology, chirality and edge
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modes</a></li>
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<li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic
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Statistics</a></li>
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</ul></li>
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</ul>
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</nav>
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@ -312,6 +278,7 @@ href="#ref-kitaevAnyonsExactlySolved2006"
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role="doc-biblioref">4</a></sup></span>. Its solubility comes about
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because the model has extensively many conserved degrees of freedom that
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mediate the interactions between quantum degrees of freedom.</p>
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<h2 id="amorphous-systems">Amorphous Systems</h2>
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<p><strong>Insert discussion of why a generalisation to the amorphous
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case is intersting</strong></p>
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<h2 id="chapter-outline">Chapter outline</h2>
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@ -348,10 +315,10 @@ amorphous case, the key property that will remain is that each vertex
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interacts with exactly three others via an x, y and z edge. However the
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lattice will no longer be bipartite, breaking chiral symmetry among
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other things.</p>
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<h2 id="kitaev-heisenberg-model">Kitaev-Heisenberg Model</h2>
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<p>In real materials there will generally be an addtional small
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Heisenberg term <span class="math display">\[H_{KH} = - \sum_{\langle
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j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} +
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<p>Kitaev-Heisenberg Model In real materials there will generally be an
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addtional small Heisenberg term <span class="math display">\[H_{KH} = -
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\sum_{\langle j,k\rangle_\alpha}
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J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} +
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\sigma_j\sigma_k\]</span></p>
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<h1 id="an-in-depth-look-at-the-kitaev-model">An in-depth look at the
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Kitaev Model</h1>
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@ -447,7 +414,7 @@ class="math inline">\(\alpha\)</span> is a function of <span
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class="math inline">\(i,j\)</span> that picks the correct bond type.</p>
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<div id="fig:visual_kitaev_1" class="fignos">
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<figure>
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<img src="/assets/thesis/amk_chapter/visual_kitaev_1.svg"
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<img src="/assets/thesis/figure_code/amk_chapter/visual_kitaev_1.svg"
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style="width:100.0%" alt="Figure 1: " />
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<figcaption aria-hidden="true"><span>Figure 1:</span> </figcaption>
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</figure>
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@ -486,7 +453,7 @@ i\)</span>.</p>
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<div id="fig:regular_plaquettes" class="fignos">
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<figure>
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<img
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src="/assets/thesis/amk_chapter/regular_plaquettes/regular_plaquettes.svg"
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src="/assets/thesis/figure_code/amk_chapter/regular_plaquettes/regular_plaquettes.svg"
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style="width:86.0%"
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alt="Figure 2: The eigenvalues of a loop or plaquette operators depend on how many bonds in its enclosing path." />
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<figcaption aria-hidden="true"><span>Figure 2:</span> The eigenvalues of
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@ -508,7 +475,7 @@ of the loop</p>
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algebra, outlined in fig. <a href="#fig:visual_kitaev_2">3</a>.</p>
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<div id="fig:visual_kitaev_2" class="fignos">
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<figure>
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<img src="/assets/thesis/amk_chapter/visual_kitaev_2.svg"
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<img src="/assets/thesis/figure_code/amk_chapter/visual_kitaev_2.svg"
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style="width:143.0%" alt="Figure 3: " />
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<figcaption aria-hidden="true"><span>Figure 3:</span> </figcaption>
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</figure>
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@ -588,7 +555,7 @@ to the physical subspace <span class="math inline">\(D = 1\)</span> and
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this all makes sense.</p>
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<div id="fig:majorana" class="fignos">
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<figure>
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<img src="/assets/thesis/majorana.png" style="width:71.0%"
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<img src="/assets/thesis/figure_code/majorana.png" style="width:71.0%"
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alt="Figure 4: " />
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<figcaption aria-hidden="true"><span>Figure 4:</span> </figcaption>
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</figure>
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@ -674,7 +641,7 @@ the Majorana Hamiltonian a little more.</p>
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<div id="fig:intro_figure_template" class="fignos">
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<figure>
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<img
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src="/assets/thesis/amk_chapter/honeycomb_zoom/intro_figure_template.svg"
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src="/assets/thesis/figure_code/amk_chapter/honeycomb_zoom/intro_figure_template.svg"
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style="width:100.0%"
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alt="Figure 5: (a) The standard Kitaev Model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solveable it is that each vertex is joined by exactly three bonds i.e the lattice is trivalent. One of three labels is assigned to each (b) We represent the antisymmetric gauge degree of freedom u_{jk} = \pm 1 with arrows that point in the direction u_{jk} = +1 (c) The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \mathbb{Z}_2 gauge field u_{ij} leaving just a single Majorana c_i per site." />
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<figcaption aria-hidden="true"><span>Figure 5:</span>
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@ -804,694 +771,6 @@ class="math inline">\(u_{ij}\)</span>, these will turn out to just be
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the plaquette operators.</p>
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<p><strong>The Bond Sectors overlap with the physical subspace but are
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not contained within it.</strong></p>
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<h2 id="properties-of-the-gauge-field">Properties of the Gauge
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Field</h2>
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<p>The bond operators <span class="math inline">\(u_{ij}\)</span> are
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useful because they label a bond sector <span
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class="math inline">\(\mathcal{\tilde{L}}_u\)</span> in which we can
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easiy solve the Hamiltonian. However the gauge operators move us between
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bond sectors. <strong>Bond sectors are not gauge invariant!</strong></p>
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<p>Let’s consider instead the properties of the plaquette operators
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<span class="math inline">\(\hat{\phi}_i\)</span> that live on the faces
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of the lattice.</p>
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<p>We already showed that they are conserved. And as one might hope and
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expect, the plaquette operators map cleanly on to the bond operators of
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the Majorana representation:</p>
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<p><span class="math display">\[\begin{aligned}
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\tilde{W}_p &= \prod_{\mathrm{i,j}\; \in\; p} \tilde{K}_{ij}\\
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&= \prod_{\mathrm{i,j}\; \in\; p}
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\tilde{\sigma}_i^\alpha \tilde{\sigma}_j^\alpha\\
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&= \prod_{\mathrm{i,j}\; \in\; p} (ib^\alpha_i
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c_i)(ib^\alpha_j c_j)\\
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&= \prod_{\mathrm{i,j}\; \in\; p} i u_{ij} c_i c_j\\
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&= \prod_{\mathrm{i,j}\; \in\; p} i u_{ij}
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\end{aligned}\]</span></p>
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<p>Where the last steps holds because each <span
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class="math inline">\(c_i\)</span> appears exactly twice and adjacent to
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its neighbour in each plaquette operator. Note that this is consistent
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with the observation from earlier that each <span
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class="math inline">\(W_p\)</span> takes values <span
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class="math inline">\(\pm 1\)</span> for even paths and <span
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class="math inline">\(\pm i\)</span> for odd paths.</p>
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<h3 id="vortices-and-their-movements">Vortices and their movements</h3>
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<p>Let’s imagine we started from the ground state of the model and
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flipped the sign of a single bond. In doing so we will flip the sign of
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the two plaquettes adjacent to that bond. I’ll call these disturbed
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plaquettes <em>vortices</em>. I’ll refer to a particular choice values
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for the plaquette operators as a vortex sector.</p>
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<p>If we chain multiple bond flips we can create a pair of vortices at
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arbitrary locations. The chain of bonds that we must flip corresponds to
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a path on the dual of the lattice.</p>
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<p>Something else we can do is create a pair of vortices, move one
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around a loop and then anhilate it with its partner. This corresponds to
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a closed loop on the dual lattice and applying such a bond flip leaves
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the vortex sector unchanged.</p>
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<p>Notice that the <span class="math inline">\(D_j\)</span> operators
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flip three bonds around a vertex. This is the smallest closed loop
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around which one can move a vortex pair and anhilate it with itself.</p>
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<p>Such operations compose in the sense that we can build any larger
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loop by applying a series of <span class="math inline">\(D_j\)</span>
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operations. Indeed the symetrisation procedure <span
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class="math inline">\(\prod_i \left( \frac{1 + D_i}{2}\right)\)</span>
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that maps from the bond sector to a physical state is applying
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constructing a superposition over every such loop that leaves the vortex
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sector unchanged.</p>
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<p>The only loops that we cannot build out of <span
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class="math inline">\(D_j\)</span>s are non-contractible loops, such as
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those that span the major or minor circumference of the torus.</p>
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<p><strong>The plaquette operators are the gauge invariant quantity that
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determines the physics of the model</strong></p>
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<h3 id="composition-of-u_jk-loops">Composition of <span
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class="math inline">\(u_{jk}\)</span> loops</h3>
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<div id="fig:plaquette_addition_by_hand" class="fignos">
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<figure>
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<img
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src="/assets/thesis/amk_chapter/plaquette_addition/plaquette_addition_by_hand.svg"
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style="width:57.0%"
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alt="Figure 6: In the product of individual plaquette operators shared bonds cancel out. The product is equal to the enclosing path." />
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<figcaption aria-hidden="true"><span>Figure 6:</span> In the product of
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individual plaquette operators shared bonds cancel out. The product is
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equal to the enclosing path.</figcaption>
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</figure>
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</div>
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<p>Second it is now easy to show that the loops and plaquettes satisfy
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nice composition rules, so long as we stick to loops that wind in a
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particular direction.</p>
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<p>Consider the product of two non-overlapping loops <span
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class="math inline">\(W_a\)</span> and <span
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class="math inline">\(W_b\)</span> that share an edge <span
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class="math inline">\(u_{12}\)</span>. Since the two loops both wind
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clockwise and do not overlap, one will contain a term <span
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class="math inline">\(i u_{12}\)</span> and the other <span
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class="math inline">\(i u_{21}\)</span>. Since the <span
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class="math inline">\(u_{ij}\)</span> commute with one another, they
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square to <span class="math inline">\(1\)</span> and <span
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class="math inline">\(u_{ij} = -u_{ji}\)</span> we see have <span
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class="math inline">\(i u_{12} i u_{21} = 1\)</span> and we can repeat
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this for any number of shared edges. Hence, we get a version of Stokes’
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theorem: the product of <span class="math inline">\(i u_{jk}\)</span>
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around any closed loop <span class="math inline">\(\partial A\)</span>
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is equal to the product of plaquette operators <span
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class="math inline">\(\Phi\)</span> that span the area <span
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class="math inline">\(A\)</span> enclosed by that loop: <span
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class="math display">\[\prod_{u_{jk} \in \partial A} i \; u_{jk} =
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\prod_{\phi_i \in A} \phi_i\]</span></p>
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<div id="fig:stokes_theorem" class="fignos">
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<figure>
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<img src="/assets/thesis/amk_chapter/stokes_theorem/stokes_theorem.svg"
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style="width:71.0%"
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alt="Figure 7: The loop composition rule extends to arbitrary numbers of vortices giving a discrete version of Stoke’s theorem." />
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<figcaption aria-hidden="true"><span>Figure 7:</span> The loop
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composition rule extends to arbitrary numbers of vortices giving a
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discrete version of Stoke’s theorem.</figcaption>
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</figure>
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</div>
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<p><strong>Wilson loops can always be decomposed into products of
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plaquettes operators unless they are non-contractable</strong></p>
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<h3 id="gauge-degeneracy-and-the-euler-equation">Gauge Degeneracy and
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the Euler Equation</h3>
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<p>We can check this analysis with a counting argument. For a lattice
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with <span class="math inline">\(B\)</span> bonds, <span
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class="math inline">\(P\)</span> plaquettes and <span
|
||||
class="math inline">\(V\)</span> vertices we can count how many bond
|
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sectors, vortices sectors and gauge symmetries there are and check them
|
||||
against Euler’s polyhedra equation.</p>
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<p>Euler’s equation states for a closed surface of genus <span
|
||||
class="math inline">\(g\)</span>, i.e that has <span
|
||||
class="math inline">\(g\)</span> holes so <span
|
||||
class="math inline">\(0\)</span> for the sphere, <span
|
||||
class="math inline">\(1\)</span> for the torus and <span
|
||||
class="math inline">\(g\)</span> for <span
|
||||
class="math inline">\(g\)</span> tori stuck together <span
|
||||
class="math display">\[B = P + V + 2 - 2g\]</span></p>
|
||||
<div id="fig:torus" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/torus.jpeg" style="width:86.0%"
|
||||
alt="Figure 8: In periodic boundary conditions the Kitaev model is defined on the surface of a torus. Topologically the torus is distinct from the sphere in that it has a hole that cannot be smoothly deformed away. Associated with each such hole are two non-contractible loops on the surface, here labeled A and B, that cannot be smoothly deformed to a point. These two non-contracible loops can. be used to construct two symmetry operators \hat{A} and \hat{A} that flip u_{jk}s along their paths." />
|
||||
<figcaption aria-hidden="true"><span>Figure 8:</span> In periodic
|
||||
boundary conditions the Kitaev model is defined on the surface of a
|
||||
torus. Topologically the torus is distinct from the sphere in that it
|
||||
has a hole that cannot be smoothly deformed away. Associated with each
|
||||
such hole are two non-contractible loops on the surface, here labeled A
|
||||
and B, that cannot be smoothly deformed to a point. These two
|
||||
non-contracible loops can. be used to construct two symmetry operators
|
||||
<span class="math inline">\(\hat{A}\)</span> and <span
|
||||
class="math inline">\(\hat{A}\)</span> that flip <span
|
||||
class="math inline">\(u_{jk}\)</span>s along their paths.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>For the case of the torus where <span class="math inline">\(g =
|
||||
1\)</span> we can rearrange this to read: <span class="math display">\[B
|
||||
= (P-1) + (V-1) + 2\]</span></p>
|
||||
<p>Each <span class="math inline">\(u_{ij}\)</span> takes two values and
|
||||
there is one associated with each bond so there are exactly <span
|
||||
class="math inline">\(2^B\)</span> distinct configurations of the bond
|
||||
sector. Let’s see if we can factor those configurations out into the
|
||||
cartesian product of vortex sectors, gauge symmetries and
|
||||
non-contractible loop operators.</p>
|
||||
<p>Vortex sectors: each plaquette operator <span
|
||||
class="math inline">\(\phi_i\)</span> takes two values (<span
|
||||
class="math inline">\(\pm 1\)</span> or <span class="math inline">\(\pm
|
||||
i\)</span>) and there are <span class="math inline">\(P\)</span> of them
|
||||
so naively one would think there are <span
|
||||
class="math inline">\(2^P\)</span>. However vortices can only be created
|
||||
on pairs so there are really <span class="math inline">\(\tfrac{2^P}{2}
|
||||
= 2^{P-1}\)</span> vortex sectors.</p>
|
||||
<p>Gauge symmetries: As discussed earlier these correspond to the all
|
||||
possible compositions of the <span class="math inline">\(D_j\)</span>
|
||||
operators. Again there are only <span
|
||||
class="math inline">\(2^{V-1}\)</span> of these because, as we will see
|
||||
in the next section, <span class="math inline">\(\prod_{j} D_j =
|
||||
\mathbb{1}\)</span> in the physical space, and we enforce this by
|
||||
chooising the correct product of single particle fermion states. You can
|
||||
get an intuitive picture for why <span class="math inline">\(\prod_{j}
|
||||
D_j = \mathbb{1}\)</span> by imagining larger and larger patches of
|
||||
<span class="math inline">\(D_j\)</span> operators on the torus. These
|
||||
patches correspond to transporting a vortex pair around the edge of the
|
||||
patch. At some point the patch wraps around and starts to cover the
|
||||
entire torus, as this happens the bounday of the patch disappears and
|
||||
hence it which corresponds to the identity operation. See Fig ??
|
||||
(animated in the HTML version).</p>
|
||||
<p>Finally the torus has two non-contractible loop operators asscociated
|
||||
with its major and minor diameters.</p>
|
||||
<p>Putting this all together we see that there are <strong><span
|
||||
class="math inline">\(2^B\)</span> bond sectors</strong> a space which
|
||||
can be decomposed into the cartesian product of <strong><span
|
||||
class="math inline">\(2^{P-1}\)</span> vortex sectors</strong>,
|
||||
<strong><span class="math inline">\(2^{V-1}\)</span> gauge
|
||||
symmetries</strong> and <strong><span class="math inline">\(2^2 =
|
||||
4\)</span> topological sectors</strong> associated with the
|
||||
non-contractible loop operators. This last factor forms the basis of
|
||||
proposals to construct topologically protected qubits since the 4
|
||||
sectors cold only be mixed by a highly non-local perturbation, ref
|
||||
?????.</p>
|
||||
<p><img
|
||||
src="/assets/thesis/amk_chapter/intro/types_of_dual_loops/types_of_dual_loops.svg"
|
||||
id="fig:types_of_dual_loops" style="width:100.0%"
|
||||
alt="The different kinds of strings and loops that we can make by flipping bond variables. (a) Flipping a single bond makes a pair of vortices on either side. (b) Flipping a string of bonds separates the vortex pair spatially. The flipped bonds form a path in blue on the dual lattice. (c) If we create a vortex-vortex pair, transported one of them around a loop and then anhilate them we can change the bond sector without changing the vortex sector. This is a manifestation of the gauge symmetry of the bond sector. (d) If we transport a vortex around the major or minor axes of the torus we create a non-contractable loop of bonds. These are relevant because they cannot be constructed from the contractable loops created by D_j operators." />
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/intro/gauge_symmetries/gauge_symmetries.svg"
|
||||
id="fig:gauge_symmetries" style="width:100.0%"
|
||||
alt="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) surounding site j. The corresponding edges of the dual lattice (blue lines) form a closed triangle. (middle) Composing multiple adjacent D_j operators produces a large closed loop or multiple disconnected loops. These loops are not directed as they are in the case of the 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 anhilate again. Note that every plaquette has an even number of u_{ij}s flipped on it’s edge and hence all retain the same value. This all works the same way for the amorphous lattice but is much harder to read visually." />
|
||||
<img src="/assets/thesis/amk_chapter/flood_fill/flood_fill.gif"
|
||||
id="fig:flood_fill" style="width:100.0%"
|
||||
alt="In both figures a honeycomb lattice is shown in grey along with its dual in light blue. (Left) Taking a larger and larger set of D_j operators leads to an outward expanding boundary line shown in blue on the dual lattice. Eventually every lattice on the torus is included and the boundary dissapears. This is a visual proof that \prod_i D_i = \mathbb{1}. (Right) In black and blue the edges and dual edges that must be flipped to add vortices at the sites highlighted in orange. Flipping all the plaquettes in the system is not equivalent to the identity." />
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/flood_fill_amorphous/flood_fill_amorphous.gif"
|
||||
id="fig:flood_fill_amorphous" style="width:100.0%" /></p>
|
||||
<h3 id="counting-edges-plaquettes-and-vertices">Counting edges,
|
||||
plaquettes and vertices</h3>
|
||||
<p>It will be useful to know how the trivalent structre of the lattice
|
||||
constraints the number of bonds <span class="math inline">\(B\)</span>,
|
||||
plaquettes <span class="math inline">\(P\)</span> and vertices <span
|
||||
class="math inline">\(V\)</span> it has.</p>
|
||||
<p>We can immediately see that the lattice is built from vertices that
|
||||
each share 3 edges with their neighbours. This means each vertex comes
|
||||
with <span class="math inline">\(\tfrac{3}{2}\)</span> bonds i.e <span
|
||||
class="math inline">\(3V = 2B\)</span>. 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.</p>
|
||||
<p>If we define an integer <span class="math inline">\(N\)</span> such
|
||||
that <span class="math inline">\(V = 2N\)</span> and <span
|
||||
class="math inline">\(B = 3N\)</span> and substitite this into the
|
||||
polyhedra equation for the torus we see that <span
|
||||
class="math inline">\(P = N\)</span>. So if is a trivalent lattice on
|
||||
the torus has <span class="math inline">\(N\)</span> plaquettes, it has
|
||||
<span class="math inline">\(2N\)</span> vertices and <span
|
||||
class="math inline">\(3N\)</span> bonds.</p>
|
||||
<p>We can also consider the sum of the number of bonds in each plaquette
|
||||
<span class="math inline">\(S_p\)</span>, since each bond is a member of
|
||||
exactly two plaquettes <span class="math display">\[S_p = 2B =
|
||||
6N\]</span></p>
|
||||
<p>The mean size of a plaquette in a trivalent lattice on the torus is
|
||||
exactly 6. Since the sum is even, this also tells us that all odd
|
||||
plaquettes must come in pairs.</p>
|
||||
<div id="fig:hilbert_spaces" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/hilbert_spaces.svg"
|
||||
style="width:100.0%"
|
||||
alt="Figure 9: The relationship between the different Hilbert spaces used in the solution is slightly complex." />
|
||||
<figcaption aria-hidden="true"><span>Figure 9:</span> The relationship
|
||||
between the different Hilbert spaces used in the solution is slightly
|
||||
complex.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<h2 id="the-projector">The Projector</h2>
|
||||
<p>It will turn out that the projection from the extended space to the
|
||||
physical space is not actually that important for the results that I
|
||||
will present. However it it useful to go through the theory of it to
|
||||
explain why this is.</p>
|
||||
<p>The physicil 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>The global projector is therefore <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 I pointed out before 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>
|
||||
<p><span class="math display">\[ \mathcal{P} = \frac{1}{2^{2N}}
|
||||
\sum_{\{i\}} \prod_{i\in\{i\}} D_i\]</span></p>
|
||||
<p>Since the gauge operators <span class="math inline">\(D_j\)</span>
|
||||
commute and square to one, we can define the complement operator <span
|
||||
class="math inline">\(C = \prod_{i=1}^{2N} D_i\)</span> and see that it
|
||||
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
|
||||
operator deals with the other half</p>
|
||||
<p><span class="math display">\[ \mathcal{P} = \left(
|
||||
\frac{1}{2^{2N-1}} \sum_{\Lambda} \prod_{i\in\{i\}} D_i\right)
|
||||
\left(\frac{1 + \prod_i^{2N} D_i}{2}\right) = \mathcal{S} \cdot
|
||||
\mathcal{P}_0\]</span></p>
|
||||
<p>To compute <span class="math inline">\(\mathcal{P}_0\)</span> the
|
||||
main quantity needed is the product of the local projectors <span
|
||||
class="math inline">\(D_i\)</span> <span
|
||||
class="math display">\[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i b^y_i b^z_i
|
||||
c_i \]</span> 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.</p>
|
||||
<p>First we reorder the operators by bond type, this doesn’t require any
|
||||
information about the underlying lattice.</p>
|
||||
<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="pedrocchiPhysicalSolutionsKitaev2011b"><sup><a
|
||||
href="#ref-pedrocchiPhysicalSolutionsKitaev2011b"
|
||||
role="doc-biblioref">7</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
|
||||
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<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">\[\mathcal{P}_0 = 1 +
|
||||
p_x\;p_y\;p_z\; \mathrm{det}(Q^u) \; \hat{\pi} \; \prod_{\{i,j\}}
|
||||
-iu_{ij}\]</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
|
||||
class="math inline">\(c_i\)</span> operators to normal mode operators
|
||||
<span class="math inline">\(b'_i, b''_i\)</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">\(\hat{\pi} = \prod{i}^{N} (1 -
|
||||
2\hat{n}_i)\)</span> is the parity of the particular many body state
|
||||
determined by fermionic occupation numbers <span
|
||||
class="math inline">\(n_i\)</span>.</p>
|
||||
<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, this tells use that
|
||||
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>Let’s think about where are with the model now. We can map the spin
|
||||
Hamiltonian to a Majorana Hamiltonian in an extended Hilbert space.
|
||||
Along with that mapping comes a gauge field <span
|
||||
class="math inline">\(u_{jk}\)</span> defining <strong>bond
|
||||
sectors</strong>. The gauge symmetries of <span
|
||||
class="math inline">\(u_{jk}\)</span> are generated by the set of <span
|
||||
class="math inline">\(D_j\)</span> operators. The gauge invariant and
|
||||
therefore physically relevant variables are the plaquette operators
|
||||
<span class="math inline">\(\phi_i\)</span> which define as a
|
||||
<strong>vortex sector</strong>. In order to practically solve the
|
||||
Majorana Hamiltonian we must remove hats from the gauge field by
|
||||
restricting ourselves to a particular bond sector. From there the
|
||||
Majorana Hamiltonian becomes non-interacting and we can solve it like
|
||||
any quadratic theory. This lets us construct the single particle
|
||||
eigenstates from which we can also construct many body states. However
|
||||
the many body states constructed this way are not in the physical
|
||||
subspace!</p>
|
||||
<p>However for the many body states within a particular bond sector,
|
||||
<span class="math inline">\(\mathcal{P}_0 = 0,1\)</span> tells us which
|
||||
of those have some overlap with the physical sector.</p>
|
||||
<p>We see that finding a state that has overlap with a physical state
|
||||
only ever requires the addition or removal of one fermion. There are
|
||||
cases where this can make a difference but for most observables such as
|
||||
ground state energy this correction scales away as the number of
|
||||
fermions in the system grows.</p>
|
||||
<p>If we wanted to construct a full many body wavefunction in the spin
|
||||
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>
|
||||
<p>On the Honeycomb, Lieb’s theorem implies that the the ground state
|
||||
corresponds to the state where all <span class="math inline">\(u_jk =
|
||||
1\)</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">8</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
|
||||
evidence that this guess is in fact correct.</p>
|
||||
<p>Let’s consider the partition function of the Majorana hamiltonian:
|
||||
<span class="math display">\[ \mathcal{Z} = \mathrm{Tr}\left( e^{-\beta
|
||||
H}\right) = \sum_i \exp{-\beta \epsilon_i}\]</span> At low temperatures
|
||||
<span class="math inline">\(\mathcal{Z} \approx \beta
|
||||
\epsilon_0\)</span> where <span
|
||||
class="math inline">\(\epsilon_0\)</span> is the lowest energy fermionic
|
||||
state.</p>
|
||||
<p>How does the <span class="math inline">\(\mathcal{Z}\)</span> depend
|
||||
on the Majorana hamiltonian? Expanding the exponential out gives: <span
|
||||
class="math display">\[ \mathcal{Z} = \sum_n \frac{(-\beta)^n}{n!}
|
||||
\mathrm{Tr(H^k)} \]</span></p>
|
||||
<p>Now there’s an interesting observation to make here. The Hamiltonian
|
||||
is essentially a scaled adjacency matrix. An adjacency matrix being a
|
||||
matrix <span class="math inline">\(g_{ij}\)</span> such that <span
|
||||
class="math inline">\(g_{ij} = 1\)</span> if vertices <span
|
||||
class="math inline">\(i\)</span> and <span
|
||||
class="math inline">\(j\)</span> and joined by an edge and 0
|
||||
otherwise.</p>
|
||||
<p>Powers of adjacency matrices have the property that the entry <span
|
||||
class="math inline">\((g^n)_{ij}\)</span> corresponds to the number of
|
||||
paths of length n on the graph that begin at site <span
|
||||
class="math inline">\(i\)</span> and end at site <span
|
||||
class="math inline">\(j\)</span>. These include somewhat degenerate
|
||||
paths that go back on themselves etc.</p>
|
||||
<p>The trace of an adjacency matrix <span
|
||||
class="math display">\[\mathrm{Tr}(g^n) = \sum_i (g^n)_{ii}\]</span>
|
||||
therefore counts the number number of loops of size <span
|
||||
class="math inline">\(n\)</span> that can be drawn on the graph.</p>
|
||||
<p>Applying the same treatment to our Majorana Hamiltonian, we can
|
||||
interpret <span class="math inline">\(u_ij\)</span> to equal 0 if the
|
||||
two sites are not joined by a bond and we put ourselves in the isotropic
|
||||
phase where <span class="math inline">\(J^\alpha = 1\)</span> <span
|
||||
class="math display">\[ \tilde{H}_{ij} = \tfrac{1}{2} i
|
||||
u_{ij}\]</span></p>
|
||||
<p>We then see that the trace of the nth power of H is a sums over
|
||||
Wilson loops of size <span class="math inline">\(n\)</span> with an
|
||||
additional factor of <span class="math inline">\(2^{-n}\)</span>. We
|
||||
showed earlier that the Wilson loop operators can always be written as
|
||||
products of the plaquette operators that they enclose.</p>
|
||||
<p>Lumping all the prefactors together, we can write: <span
|
||||
class="math display">\[ \mathcal{Z} = c_A \hat{A} + c_B \hat{B} + \sum_i
|
||||
c_i \hat{\phi}_i + \sum_{ij} c_{ij} \hat{\phi}_i \hat{\phi}_j +
|
||||
\sum_{ijk} c_{ijk} \hat{\phi}_i \hat{\phi}_j \hat{\phi}_k +
|
||||
...\]</span></p>
|
||||
<p>Where the <span class="math inline">\(c\)</span> factors would be
|
||||
something like <span class="math display">\[c_{ijk...} = \sum_n
|
||||
\tfrac{(-\beta)^n}{n!} \tfrac{1}{2^n} K_{ijk...}\]</span> which is a sum
|
||||
over all loop lengths <span class="math inline">\(n\)</span> and for
|
||||
each we have a combinatoral factor <span
|
||||
class="math inline">\(K_{ijk...}\)</span> that counts how many ways
|
||||
there are to draw a loop of length <span
|
||||
class="math inline">\(n\)</span> that only encloses plaquettes <span
|
||||
class="math inline">\(ijk...\)</span>.</p>
|
||||
<p>We also have the pesky non-contractible loop operators <span
|
||||
class="math inline">\(\hat{A}\)</span> and <span
|
||||
class="math inline">\(\hat{B}\)</span>. Again the prefactors for these
|
||||
are very complicated but we can intuitively see that for larger and
|
||||
larger loops lengths there will be a combinatorial explosion of possible
|
||||
ways that they appear in these sums. These are suppressed exponentially
|
||||
with system size but at practical lattice sizes they cause significant
|
||||
finite size effects. The main evidence of this is that the 4 loop
|
||||
sectors spanned by the <span class="math inline">\(\hat{A}\)</span> and
|
||||
<span class="math inline">\(\hat{B}\)</span> operators are degenerate in
|
||||
the infinite system size limit, while that degeneracy is lifted in
|
||||
finite sized systems.</p>
|
||||
<p>We don’t have much hope of actually evaluating this for an amorphous
|
||||
lattice. However it lead us to guess that the ground state vortex sector
|
||||
might be a simple function of the side length of each plaquette.</p>
|
||||
<p>The ground state of the Amorphous Kitaev Model is found by setting
|
||||
the flux through each plaquette <span
|
||||
class="math inline">\(\phi\)</span> to be equal to <span
|
||||
class="math inline">\(\phi^{\mathrm{g.s.}}(n_{\mathrm{sides}})\)</span></p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\phi^{\mathrm{g.s.}}(n_{\mathrm{sides}}) = -(\pm
|
||||
i)^{n_{\mathrm{sides}}},
|
||||
\end{aligned}\]</span> where <span
|
||||
class="math inline">\(n_{\mathrm{sides}}\)</span> is the number of edges
|
||||
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">8</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>
|
||||
<h3 id="finite-size-effects">Finite size effects</h3>
|
||||
<p>This guess only works for larger lattices because of the finite size
|
||||
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
|
||||
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
|
||||
non-contractible loop operators are removed with only a linear penalty
|
||||
in computation time compared to the exponential penalty paid by simply
|
||||
simply diagonalising larger lattices.</p>
|
||||
<p>Using this technique we verified that <span
|
||||
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>
|
||||
<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">9</a>–<a
|
||||
href="#ref-WangHaoranPRB2021"
|
||||
role="doc-biblioref">16</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
|
||||
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">10</a></sup></span>.</p>
|
||||
<h2 id="topology-chirality-and-edge-modes">Topology, chirality and edge
|
||||
modes</h2>
|
||||
<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
|
||||
not scale with system size.</p>
|
||||
<p>However there are quantum phases that cannot be characterised by such
|
||||
a local order parameter. These phases are intead said to posess
|
||||
‘topological order’.</p>
|
||||
<p>One property of topological order that is particularly easy to
|
||||
observe that the ground state degeneracy depends on the topology of the
|
||||
manifold that we put the system on to. This is referred to as
|
||||
topological degeneracy to distinguish it from standard symmetry
|
||||
breaking.</p>
|
||||
<p>The Kitaev model will be a good example of this, we have already
|
||||
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>
|
||||
<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
|
||||
swapped. We call these Fermions and Bosons.</p>
|
||||
<p>This argument is predicated on the idea that performing two swaps is
|
||||
equivalent to doing nothing. Doing nothing should not change the quantum
|
||||
state at all, so doing one swap can at most multiply it by <span
|
||||
class="math inline">\(\pm 1\)</span>.</p>
|
||||
<p>However there are many hidden parts to this argument. Firstly, this
|
||||
argument just isn’t the whole story, if you want to know why Fermions
|
||||
have half integer spin, for instance, you have to go to field
|
||||
theory.</p>
|
||||
<p>There is also a second niggle, why does this argument only work in
|
||||
dimensions greater than two? What we’re really saying when we say that
|
||||
two swaps do nothing is that the world lines of two particles that have
|
||||
been swapped twice can be untangled without crossing. Why can’t they
|
||||
cross? Well because if they cross then the particles can interact and
|
||||
the quantum state could change in an arbitrary way. We’re implcitly
|
||||
using the locality of physics here to argue that if the worldlines stay
|
||||
well separated then the overall quantum state cannot too much.</p>
|
||||
<p>In two dimensions we cannot untangle the worldlines of two particles
|
||||
that have swapped places, they are braided together. See fig. <a
|
||||
href="#fig:braiding">10</a> for a diagram.</p>
|
||||
<div id="fig:braiding" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/braiding.png" style="width:71.0%"
|
||||
alt="Figure 10: " />
|
||||
<figcaption aria-hidden="true"><span>Figure 10:</span> </figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>From this fact flows a whole new world of behaviours, now the quantum
|
||||
state can aquire a phase factor <span
|
||||
class="math inline">\(e^{i\phi}\)</span> upon exchange of two identical
|
||||
particles, which we now call Anyons.</p>
|
||||
<p>The Kitaev Model is a good demonstration of the connection beween
|
||||
Anyons and topological degeneracy. In the Kitaev model we can create a
|
||||
pair of vortices, move one around a non-contractable loop <span
|
||||
class="math inline">\(\mathcal{T}_{x/y}\)</span> and then anhilate them
|
||||
together. Without topology this should leave the quantum state
|
||||
unchanged. Instead it moves us to another ground state in a
|
||||
topologically degenerate ground state subspace. Practically speaking it
|
||||
flips a dual line of bonds <span class="math inline">\(u_{jk}\)</span>
|
||||
going around the loop which we cannot undo with any gauge transformation
|
||||
made from <span class="math inline">\(D_j\)</span> operators.</p>
|
||||
<p>If the ground state subspace is multidimensional, quasiparticle
|
||||
exchange can move us around in the space with an action corresponding to
|
||||
a matrix. These matrices do not in general commmute and so these are
|
||||
known as non-Abelian anyons.</p>
|
||||
<p>From here things get even more complex, the Kitaev model has a
|
||||
non-Abelian phase when exposed to a magnetic field, and the amorphous
|
||||
Kitaev Model has a non-Abelian phase because of its broken chiral
|
||||
symmetry.</p>
|
||||
<p>The way that we have subdivided the Kitaev model into vortex sectors,
|
||||
we have a neat separation beween vortices and fermionic excitations.
|
||||
However if we looked at the full many body picture we would see that a
|
||||
vortex caries with it a cloud of bound majorana states.</p>
|
||||
<div id="fig:majorana_bound_states" class="fignos">
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/majorana_bound_states/majorana_bound_states.svg"
|
||||
style="width:100.0%"
|
||||
alt="Figure 11: (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. The state is clearly bound to the vortices." />
|
||||
<figcaption aria-hidden="true"><span>Figure 11:</span> (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. The state is clearly bound to the vortices.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>Consider two processes</p>
|
||||
<ol type="1">
|
||||
<li><p>We transport one half of a vortex pair around either the x or y
|
||||
loops of the torus before anhilating back to the ground state vortex
|
||||
sector <span class="math inline">\(\mathcal{T}_{x,y}\)</span>.</p></li>
|
||||
<li><p>We flip a line of bond operators coresponding to measuring the
|
||||
flux through either the major or minor axes of the torus <span
|
||||
class="math inline">\(\mathcal{\Phi}_{x,y}\)</span></p></li>
|
||||
</ol>
|
||||
<div id="fig:loops_and_dual_loops" class="fignos">
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/loops_and_dual_loops/loops_and_dual_loops.svg"
|
||||
style="width:114.0%"
|
||||
alt="Figure 12: (Left) The two topological flux operators of the toroidal lattice, these don’t correspond to any face of the lattice, but rather measure flux that threads through the major and minor axes of the torus. This shows a particular choice but any loop that crosses the boundary is gauge equivalent to one of or the sum of these two loop. (Right) The two ways to transport vortices around the diameters. These correspond to creating a vortex pair, transporting one of them around the major or minor diameters of the torus and then anhilating them again." />
|
||||
<figcaption aria-hidden="true"><span>Figure 12:</span> (Left) The two
|
||||
topological flux operators of the toroidal lattice, these don’t
|
||||
correspond to any face of the lattice, but rather measure flux that
|
||||
threads through the major and minor axes of the torus. This shows a
|
||||
particular choice but any loop that crosses the boundary is gauge
|
||||
equivalent to one of or the sum of these two loop. (Right) The two ways
|
||||
to transport vortices around the diameters. These correspond to creating
|
||||
a vortex pair, transporting one of them around the major or minor
|
||||
diameters of the torus and then anhilating them again.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>The plaquette operators <span class="math inline">\(\phi_i\)</span>
|
||||
are associated with fluxes. Wilson loops that wind the torus are
|
||||
associated with the fluxes through its two diameters <span
|
||||
class="math inline">\(\mathcal{\Phi}_{x,y}\)</span>.</p>
|
||||
<p>In the Abelian phase we can move a vortex along any path we like and
|
||||
then when we bring them back together they will anhilate back to the
|
||||
vacuum, where we understand ‘the vacuum’ to refer to one of the ground
|
||||
states, though not necesarily the same one we started in. We can use
|
||||
this to get from the <span class="math inline">\((\Phi_x, \Phi_y) = (+1,
|
||||
+1)\)</span> ground state and construct the set <span
|
||||
class="math inline">\((+1, +1), (+1, -1), (-1, +1), (-1,
|
||||
-1)\)</span>.</p>
|
||||
<div id="fig:topological_fluxes" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/topological_fluxes.png"
|
||||
style="width:57.0%"
|
||||
alt="Figure 13: 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 make17." />
|
||||
<figcaption aria-hidden="true"><span>Figure 13:</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">17</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">18</a>,<a
|
||||
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
|
||||
role="doc-biblioref">19</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
|
||||
vacuum the anhilate to create an excited state instead of a ground
|
||||
state. This means we end up with only three degenerate ground states in
|
||||
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">20</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
|
||||
fermionic excitation to make the state physical. Hence the process does
|
||||
not give a fourth ground state.</p>
|
||||
<p><img src="/assets/thesis/amk_chapter/threefold_degeneracy.png"
|
||||
id="fig:threefold_degeneracy" style="width:86.0%" /> <img
|
||||
src="/assets/thesis/amk_chapter/state_decomposition_animated/state_decomposition_animated.gif"
|
||||
id="fig:state_decomposition_animated" style="width:114.0%"
|
||||
alt="(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 simplfy 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. The edges here 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 giving a network of loops that can always be written as a product the of the gauge operators D_j. (Topolical 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} corresponds 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 and that is what is 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." /></p>
|
||||
<p>One reason the topology has gained interest recently is there have
|
||||
proposals to use this ground state degeneracy to implement both
|
||||
passively fault tolerant and actively stabilised quantum computations
|
||||
[<span class="citation"
|
||||
data-cites="kitaevFaulttolerantQuantumComputation2003"><sup><a
|
||||
href="#ref-kitaevFaulttolerantQuantumComputation2003"
|
||||
role="doc-biblioref">21</a></sup></span>;<span class="citation"
|
||||
data-cites="poulinStabilizerFormalismOperator2005"><sup><a
|
||||
href="#ref-poulinStabilizerFormalismOperator2005"
|
||||
role="doc-biblioref">22</a></sup></span>;
|
||||
hastingsDynamicallyGeneratedLogical2021].</p>
|
||||
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
|
||||
role="doc-bibliography">
|
||||
<div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry"
|
||||
@ -1542,131 +821,6 @@ systems</a>. <em>Physical Review B</em> <strong>79</strong>, 214440
|
||||
class="csl-right-inline">Blaizot, J.-P. & Ripka, G. <em>Quantum
|
||||
theory of finite systems</em>. (<span>The MIT Press</span>, 1986).</div>
|
||||
</div>
|
||||
<div id="ref-pedrocchiPhysicalSolutionsKitaev2011b" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">7. </div><div
|
||||
class="csl-right-inline">Pedrocchi, F. L., Chesi, S. & Loss, D. <a
|
||||
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-lieb_flux_1994" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">8. </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>.
|
||||
<em>Physical Review Letters</em> <strong>73</strong>, 2158–2161
|
||||
(1994).</div>
|
||||
</div>
|
||||
<div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">9. </div><div
|
||||
class="csl-right-inline">Chua, V., Yao, H. & 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
|
||||
lattice</a>. <em>Phys. Rev. B</em> <strong>83</strong>, 180412
|
||||
(2011).</div>
|
||||
</div>
|
||||
<div id="ref-yaoExactChiralSpin2007" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">10. </div><div
|
||||
class="csl-right-inline">Yao, H. & 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">11. </div><div
|
||||
class="csl-right-inline">Chua, V. & 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">12. </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
|
||||
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">13. </div><div
|
||||
class="csl-right-inline">Natori, W. M. H., Andrade, E. C., Miranda, E.
|
||||
& 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">14. </div><div class="csl-right-inline">Wu,
|
||||
C., Arovas, D. & 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">15. </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>.
|
||||
<em>Phys. Rev. B</em> <strong>101</strong>, 041114 (2020).</div>
|
||||
</div>
|
||||
<div id="ref-WangHaoranPRB2021" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">16. </div><div
|
||||
class="csl-right-inline">Wang, H. & 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
|
||||
class="math inline">\(^{3}\fracslash_2\)</span></span> liquids</a>.
|
||||
<em>Phys. Rev. B</em> <strong>104</strong>, 214422 (2021).</div>
|
||||
</div>
|
||||
<div id="ref-parkerWhyDoesThis" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">17. </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">18. </div><div
|
||||
class="csl-right-inline">Chung, S. B. & 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">19. </div><div
|
||||
class="csl-right-inline">Oshikawa, M., Kim, Y. B., Shtengel, K., Nayak,
|
||||
C. & 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">20. </div><div
|
||||
class="csl-right-inline">Chung, S. B., Yao, H., Hughes, T. L. & 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
|
||||
liquid at finite temperature</a>. <em>Phys. Rev. B</em>
|
||||
<strong>81</strong>, 060403 (2010).</div>
|
||||
</div>
|
||||
<div id="ref-kitaevFaulttolerantQuantumComputation2003"
|
||||
class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">21. </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>
|
||||
<strong>303</strong>, 2–30 (2003).</div>
|
||||
</div>
|
||||
<div id="ref-poulinStabilizerFormalismOperator2005" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">22. </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
|
||||
Correction</span></a>. <em>Phys. Rev. Lett.</em> <strong>95</strong>,
|
||||
230504 (2005).</div>
|
||||
</div>
|
||||
</div>
|
||||
<section class="footnotes footnotes-end-of-document"
|
||||
role="doc-endnotes">
|
||||
|
||||
@ -273,7 +273,7 @@ torus.</p>
|
||||
<div id="fig:lattice_construction_animated" class="fignos">
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/lattice_construction_animated/lattice_construction_animated.gif"
|
||||
src="/assets/thesis/figure_code/amk_chapter/lattice_construction_animated/lattice_construction_animated.gif"
|
||||
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 actually 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 indentify edges correspond to each other across the boundaries to produce a graph on the torus. An edge colouring is shown here to help the reader identify corresponding edges." />
|
||||
<figcaption aria-hidden="true"><span>Figure 1:</span> (Left) Lattice
|
||||
@ -435,7 +435,7 @@ examples.</p>
|
||||
<div id="fig:multiple_colourings" class="fignos">
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/multiple_colourings/multiple_colourings.svg"
|
||||
src="/assets/thesis/figure_code/amk_chapter/multiple_colourings/multiple_colourings.svg"
|
||||
style="width:100.0%"
|
||||
alt="Figure 2: Three different valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted." />
|
||||
<figcaption aria-hidden="true"><span>Figure 2:</span> Three different
|
||||
@ -476,7 +476,8 @@ and anhilates them.</p></li>
|
||||
</ol>
|
||||
<div id="fig:flux_finding" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/flux_finding/flux_finding.svg"
|
||||
<img
|
||||
src="/assets/thesis/figure_code/amk_chapter/flux_finding/flux_finding.svg"
|
||||
style="width:100.0%"
|
||||
alt="Figure 3: (Left) The ground state flux sector and bond sector for an amorphous lattice. Bond arrows indicate the direction in which u_{jk} = +1. Plaquettes are coloured blue when \hat{\phi}_i = -1 (-i) for even (odd) plaquettes and orange when \hat{\phi}_i = +1 (+i) for even/odd plaquettes. (Centre) In order to transform this to the target flux sector (all +1/+i) we first flip any u_{jk} that are between two fluxes. This leaves a set of isolated fluxes that need to be anhilated. These are then paired up as indicated by the black lines. (Right) A* search is used to find paths (coloured plaquettes) on the dual lattice between each pair of fluxes and the coresponding u_{jk} (shown in black) are flipped. One flux has will remain because the starting and target flux sectors differed by an odd number of fluxes." />
|
||||
<figcaption aria-hidden="true"><span>Figure 3:</span> (Left) The ground
|
||||
|
||||
@ -243,7 +243,7 @@ Evidence for the Ground State Flux Sector</a></li>
|
||||
<div id="fig:fermion_gap_vs_L" class="fignos">
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/results/fermion_gap_vs_L/fermion_gap_vs_L.svg"
|
||||
src="/assets/thesis/figure_code/amk_chapter/results/fermion_gap_vs_L/fermion_gap_vs_L.svg"
|
||||
style="width:100.0%"
|
||||
alt="Figure 1: 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"><span>Figure 1:</span> Within a flux
|
||||
|
||||
258
_thesis/3.1_Conclusion.html
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|
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|
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|
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|
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|
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|
||||
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|
||||
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|
||||
<ul>
|
||||
<li><a href="#discussion" id="toc-discussion">Discussion</a></li>
|
||||
<li><a href="#future-work" id="toc-future-work">Future Work</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
<h2 id="discussion">Discussion</h2>
|
||||
<h2 id="future-work">Future Work</h2>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
</main>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,30 +1,25 @@
|
||||
---
|
||||
title: Interacting quantum many body systems I have known and loved
|
||||
layout: default
|
||||
permalink: /thesis/
|
||||
---
|
||||
|
||||
<ul>
|
||||
<li>Introduction</li>
|
||||
<ul><ul>
|
||||
<li><a href="./0.1_Intro.html#themes">Themes</a></li>
|
||||
</ul></ul>
|
||||
<li>The Long Range Falikov-Kimball Model</li>
|
||||
<li>Chapter 1: The Long Range Falikov-Kimball Model</li>
|
||||
<ul>
|
||||
<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>
|
||||
</ul></ul>
|
||||
<li>The Amorphous Kitaev Model</li>
|
||||
<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>
|
||||
<li><a href="./2.1_AMK_Intro.html#kitaev-heisenberg-model">Kitaev-Heisenberg Model</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>
|
||||
</ul>
|
||||
<li><a href="./2.2_AMK_Methods.html#methods">Methods</a></li>
|
||||
<ul>
|
||||
@ -44,3 +39,8 @@ permalink: /thesis/
|
||||
<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>
|
||||
</ul></ul>
|
||||
<li>Conclusion</li>
|
||||
<ul><ul>
|
||||
<li><a href="./3.1_Conclusion.html#discussion">Discussion</a></li>
|
||||
<li><a href="./3.1_Conclusion.html#future-work">Future Work</a></li>
|
||||
|
||||
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thesis.html
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thesis.html
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|
||||
---
|
||||
title: Interacting quantum many body systems I have known and loved
|
||||
layout: default
|
||||
permalink: /thesis/
|
||||
---
|
||||
<h1>Interacting quantum many body systems I have known and loved</h1>
|
||||
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 %}
|
||||
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Reference in New Issue
Block a user