From 348dca9767d725327d156bdd34610c60dfc859dc Mon Sep 17 00:00:00 2001 From: Tom Date: Thu, 20 Feb 2025 08:54:57 +0000 Subject: [PATCH] Make bound states an animation --- _thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html b/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html index 2d56bcb..9be43bb 100644 --- a/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html +++ b/_thesis/4_Amorphous_Kitaev_Model/4.1_AMK_Model.html @@ -117,7 +117,7 @@ image:

with a twofold global chiral degeneracy (picking either \(+i\) or \(-i\) in eq. 1).

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.

-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. +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.

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 \(b^\alpha\) 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 \(J^{\alpha}_{ij} = 0\) for sites joined by bonds \((i,j)\) that we want to remove. This creates fermionic zero modes \(u_{ij}\) 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 \(b^\alpha\) operators can be performed.