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Make bound states an animation

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Tom 2025-02-20 08:54:57 +00:00
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_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html
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<p>with a twofold global chiral degeneracy (picking either <span class="math inline">\(+i\)</span> or <span class="math inline">\(-i\)</span> in eq. <a href="#eq:gs-flux-sector">1</a>).</p> <p>with a twofold global chiral degeneracy (picking either <span class="math inline">\(+i\)</span> or <span class="math inline">\(-i\)</span> in eq. <a href="#eq:gs-flux-sector">1</a>).</p>
<p>To verify numerically that Lieb’s theorem generalises to the AK model, the obvious approach would be via exhaustive checking of flux configurations. However, this is problematic because the number of states to check scales exponentially with system size. We side-step this by gluing together two methods, we first work with lattices small enough that we can fully enumerate their flux sectors but tile them to reduce finite size effects. We then show that the effect of tiling scales away with system size.</p> <p>To verify numerically that Lieb’s theorem generalises to the AK model, the obvious approach would be via exhaustive checking of flux configurations. However, this is problematic because the number of states to check scales exponentially with system size. We side-step this by gluing together two methods, we first work with lattices small enough that we can fully enumerate their flux sectors but tile them to reduce finite size effects. We then show that the effect of tiling scales away with system size.</p>
<figure> <figure>
<img src="/assets/thesis/amk_chapter/intro/majorana_bound_states_animated/majorana_bound_states_animated.svg" id="fig-majorana_bound_states_animated" data-short-caption="Majorana Bound States" style="width:100.0%" alt="Figure 3: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object." /> <img src="/assets/thesis/amk_chapter/intro/majorana_bound_states_animated/majorana_bound_states_animated.gif" id="fig-majorana_bound_states_animated" data-short-caption="Majorana Bound States" style="width:100.0%" alt="Figure 3: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object." />
<figcaption aria-hidden="true">Figure 3: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object.</figcaption> <figcaption aria-hidden="true">Figure 3: (Left) A large amorphous lattice in the ground state save for a single pair of vortices shown in red, separated by the string of bonds that we flipped to create them. (Right) The density of the lowest energy Majorana state in this vortex sector. These Majorana states are bound to the vortices. They ‘dress’ the vortices to create a composite object.</figcaption>
</figure> </figure>
<p>In order to evaluate the Chern marker later, we need a way to evaluate the model on open boundary conditions. Simply removing bonds from the lattice leaves behind unpaired <span class="math inline">\(b^\alpha\)</span> operators that must be paired in some way to arrive at fermionic modes. To fix a pairing, we always start from a lattice defined on the torus and generate a lattice with open boundary conditions by defining the bond coupling <span class="math inline">\(J^{\alpha}_{ij} = 0\)</span> for sites joined by bonds <span class="math inline">\((i,j)\)</span> that we want to remove. This creates fermionic zero modes <span class="math inline">\(u_{ij}\)</span> associated with these cut bonds which we set to 1 when calculating the projector. Alternatively, since all the fermionic zero modes are degenerate anyway, an arbitrary pairing of the unpaired <span class="math inline">\(b^\alpha\)</span> operators can be performed.</p> <p>In order to evaluate the Chern marker later, we need a way to evaluate the model on open boundary conditions. Simply removing bonds from the lattice leaves behind unpaired <span class="math inline">\(b^\alpha\)</span> operators that must be paired in some way to arrive at fermionic modes. To fix a pairing, we always start from a lattice defined on the torus and generate a lattice with open boundary conditions by defining the bond coupling <span class="math inline">\(J^{\alpha}_{ij} = 0\)</span> for sites joined by bonds <span class="math inline">\((i,j)\)</span> that we want to remove. This creates fermionic zero modes <span class="math inline">\(u_{ij}\)</span> associated with these cut bonds which we set to 1 when calculating the projector. Alternatively, since all the fermionic zero modes are degenerate anyway, an arbitrary pairing of the unpaired <span class="math inline">\(b^\alpha\)</span> operators can be performed.</p>