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@ -415,15 +415,15 @@ composition rule extends to arbitrary numbers of vortices which gives 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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<p>Takeaway: Wilson loops can always be decomposed into products of
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plaquettes operators unless they are non-contractable.</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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<div id="fig:state_decomposition_animated" class="fignos">
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<figure>
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<img
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src="/assets/thesis/figure_code/amk_chapter/state_decomposition_animated/state_decomposition_animated.gif"
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style="width:114.0%"
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style="width:100.0%"
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alt="Figure 5: (Bond Sector) A state in the bond sector is specified by assigning \pm 1 to each edge of the lattice. However, this description has a substantial gauge degeneracy. We can simplify things by decomposing each state into the product of three kinds of objects: (Vortex Sector) Only a small number of bonds need to be flipped (compared to some arbitrary reference) to reconstruct the vortex sector. Here, the edges are chosen from a spanning tree of the dual lattice, so there are no loops. (Gauge Field) The ‘loopiness’ of the bond sector can be factored out. This gives a network of loops that can always be written as a product of the gauge operators D_j. (Topological Sector) Finally, there are two loops that have no effect on the vortex sector, nor can they be constructed from gauge symmetries. These can be thought of as two fluxes \Phi_{x/y} that thread through the major and minor axes of the torus. Measuring \Phi_{x/y} amounts to constructing Wilson loops around the axes of the torus. We can flip the value of \Phi_{x} by transporting a vortex pair around the torus in the y direction, as shown here. In each of the three figures on the right, black bonds correspond to those that must be flipped. Composing the three together gives back the original bond sector on the left." />
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<figcaption aria-hidden="true"><span>Figure 5:</span> (Bond Sector) A
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state in the bond sector is specified by assigning <span
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@ -543,9 +543,9 @@ by a highly non-local perturbations<span class="citation"
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data-cites="kitaevFaulttolerantQuantumComputation2003"><sup><a
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href="#ref-kitaevFaulttolerantQuantumComputation2003"
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role="doc-biblioref">1</a></sup></span>.</p>
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<p><strong>The Extended Hilbert Space decomposes into a direct product
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<p>Takeaway: The Extended Hilbert Space decomposes into a direct product
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of Flux Sectors, four Topological Sectors and a set of gauge
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symmetries.</strong></p>
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symmetries.</p>
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<div id="fig:flood_fill" class="fignos">
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<figure>
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<img
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@ -732,7 +732,7 @@ single fermion in the lowest level.</p>
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<figure>
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<img
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src="/assets/thesis/figure_code/amk_chapter/loops_and_dual_loops/loops_and_dual_loops.svg"
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style="width:114.0%"
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style="width:100.0%"
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alt="Figure 10: (Left) The two topological flux operators of the toroidal lattice. These do not 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. Yet, 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 amount to creating a vortex pair, transporting one of them around the major or minor diameters of the torus and, then, annihilating them again." />
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<figcaption aria-hidden="true"><span>Figure 10:</span> (Left) The two
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topological flux operators of the toroidal lattice. These do not
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@ -1043,8 +1043,11 @@ href="#fig:braiding">12</a>).</p>
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<div id="fig:braiding" class="fignos">
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<figure>
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<img src="/assets/thesis/figure_code/amk_chapter/braiding.png"
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style="width:71.0%" alt="Figure 12: " />
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<figcaption aria-hidden="true"><span>Figure 12:</span> </figcaption>
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style="width:71.0%"
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alt="Figure 12: Worldlines of particles in two dimensions can become tangled or braided with one another." />
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<figcaption aria-hidden="true"><span>Figure 12:</span> Worldlines of
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particles in two dimensions can become tangled or <em>braided</em> with
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one another.</figcaption>
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</figure>
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</div>
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<p>From this fact flows a whole of behaviours. The quantum state can
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@ -522,7 +522,7 @@ 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/figure_code/amk_chapter/regular_plaquettes/regular_plaquettes.svg"
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src="/assets/thesis/figure_code/amk_chapter/intro/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 the number of bonds in its enclosing path." />
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<figcaption aria-hidden="true"><span>Figure 2:</span> The eigenvalues of
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@ -335,7 +335,7 @@ the topology of our lattices.</p>
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<div id="fig:bloch" class="fignos">
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<figure>
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<img src="/assets/thesis/figure_code/amk_chapter/methods/bloch.png"
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style="width:57.0%"
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style="width:100.0%"
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alt="Figure 2: Bloch’s theorem can be thought of as transforming from a periodic Hamiltonian on the place to the unit cell defined an torus. In addition we get some phase factors e^{i\vec{k}\cdot\vec{r}} associated with bonds that cross unit cells that depend on the sense in which they do so \vec{r} = (\pm1, \pm1). Representing graphs on the torus turns out to require a similar idea, we unwrap the torus to one unit cell and keep track of which bonds cross the cell boundaries." />
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<figcaption aria-hidden="true"><span>Figure 2:</span> Bloch’s theorem
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can be thought of as transforming from a periodic Hamiltonian on the
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@ -349,17 +349,6 @@ which bonds cross the cell boundaries.</figcaption>
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</figure>
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</div>
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<h2 id="colouring-the-bonds">Colouring the Bonds</h2>
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<div id="fig:multiple_colourings" class="fignos">
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<figure>
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<img
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src="/assets/thesis/figure_code/amk_chapter/multiple_colourings/multiple_colourings.svg"
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style="width:100.0%"
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alt="Figure 3: Three different valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted." />
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<figcaption aria-hidden="true"><span>Figure 3:</span> Three different
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valid 3-edge-colourings of amorphous lattices. Colors that differ from
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the leftmost panel are highlighted.</figcaption>
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</figure>
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</div>
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<p>The Kitaev Model requires that each edge in the lattice be assigned a
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label <span class="math inline">\(x\)</span>, <span
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class="math inline">\(y\)</span> or <span
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@ -416,6 +405,17 @@ role="doc-biblioref">12</a></sup></span>. An <span
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class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm
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robertson1996efficiently exists here. However, it is not clear whether
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this extends to cubic, <strong>toroidal</strong> bridgeless graphs.</p>
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<div id="fig:multiple_colourings" class="fignos">
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<figure>
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<img
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src="/assets/thesis/figure_code/amk_chapter/multiple_colourings/multiple_colourings.svg"
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style="width:100.0%"
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alt="Figure 3: Three different valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted." />
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<figcaption aria-hidden="true"><span>Figure 3:</span> Three different
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valid 3-edge-colourings of amorphous lattices. Colors that differ from
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the leftmost panel are highlighted.</figcaption>
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</figure>
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</div>
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<h3 id="four-colourings-and-three-colourings">Four-colourings and
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three-colourings</h3>
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<p>A four-face-colouring can be converted into a three-edge-colouring
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@ -572,7 +572,7 @@ href="#fig:fermion_gap_vs_L">4</a>.</p>
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<figure>
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<img
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src="/assets/thesis/figure_code/amk_chapter/results/fermion_gap_vs_L/fermion_gap_vs_L.svg"
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style="width:114.0%"
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style="width:100.0%"
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alt="Figure 4: Within a flux sector, the fermion gap \Delta_f measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size L is defined \sqrt{N} where N is the number of plaquettes in the system. Error bars shown are 3 times the standard error of the mean. The lines shown are fits of \tfrac{\Delta_f}{J} = aL ^ b with fit parameters: Ground State: a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004 Random Flux Sector: a = 1.8 \pm 0.2, b = -2.21 \pm 0.03" />
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<figcaption aria-hidden="true"><span>Figure 4:</span> Within a flux
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sector, the fermion gap <span class="math inline">\(\Delta_f\)</span>
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Reference in New Issue
Block a user