"Since $H$ is a sum of commuting terms, we can calculate the ground state as the simultaneous ground state for all the terms. \n",

"Let us first look at the vertex terms proportional to $A$. If we color all the bonds with $\\sigma_z=-1$ as blue on our lattice, then we find that each vertex in the ground state configuration has an even number of blue lines coming in. Thus, we can think of the blue lines forming loops that can never be open ended. This allows us to view the ground state of the toric code as a loop gas. \n",

"Let us first look at the vertex terms proportional to $A$. If we draw a red line through bond connecting neighboring spins with $\\sigma_z=-1$ on our lattice (as shown below), then we find that each vertex in the ground state configuration has an even number of red lines coming in. Thus, we can think of the red lines forming loops that can never be open ended. This allows us to view the ground state of the toric code as a loop gas. \n",

"\n",

"![](figures/loops.png)"

]

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@@ -644,7 +644,7 @@

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"What if we focused on the large plaquette term limit i.e. $A_v\\ll B_p$ instead? The toric code is fairly symmetric between the vertex and plaquette terms. Clearly focusing on the $\\sigma_z$ diagonal basis was a choice. If we draw loops (red lines) through the dual lattice whenever $\\sigma_x=-1$ on some link. This results in a loop gas picture on the dual lattice, which focusses on the $\\sigma_x$ terms. \n",

"What if we focused on the large plaquette term limit i.e. $A_v\\ll B_p$ instead? The toric code is fairly symmetric between the vertex and plaquette terms. Clearly focusing on the $\\sigma_z$ diagonal basis was a choice. If we draw loops (blue lines) through the dual lattice (whose vertices are in the middle of the original lattice) whenever $\\sigma_x=-1$ on some link. This results in a loop gas picture (blue lines) on the dual lattice, which focusses on the $\\sigma_x$ terms. \n",

"\n",

"Returning to the $\\sigma_z$ representation, it looks like every loop configuration is a ground state wave-function and so is a massively degenerate loop space $L$. But this conclusion doesn't include the plaquette terms (i.e. the $B_p$ coefficient) yet. Since the plaquette terms commute with the vertex terms in the Hamiltonian, the plaquette terms take us between different loop configurations. Considering the plaquette Hamiltonian in the low energy space of closed loops we can show that the ground state wave-function must be the sum of all possible (i.e. ones that can be reached by applying the plaquette terms) loop configurations with equal weight. "