Partition function of the Kitaev quantum double model

Abstract

We compute the degeneracy of energy levels in the Kitaev quantum double model for any discrete group $G$ on any planar graph forming the skeleton of a closed orientable surface of arbitrary genus. The derivation is based on the fusion rules of the properly identified vertex and plaquette excitations, which are selected among the anyons, i.e., the simple objects of the Drinfeld center $\mathcal{Z}(\mathrm{Vec}_G)$. These degeneracies are given in terms of the corresponding $S$-matrix elements and allow one to obtain the exact finite-temperature partition function of the model, valid for any finite-size system.

Figures (15)

• Figure 1: Fusion tree for degeneracies for a eigenenergy level of the KQD model defined on a surface of genus $\mathfrak{g}$, with $n$ vertex excitations $A_1, A_2, \ldots, A_n\in$ Ch and $m$ plaquette excitations $B_1, B_2, \ldots, B_m \in$ Fl. Each vertex excitation $A$ can exist in $n_{A}^\text{Ve} = d_A$ subtypes.
• Figure 2: The octagon shown here is the fundamental domain for a genus 2 surface. Pairs of edges carrying inverse Aharonov-Bohm fluxes, such as $\Phi_{a_1}$ and $\Phi_{a_1}^{-1}$, are identified, with reversed orientation. The red circles inside correspond to $f=3$ plaquette excitations, carrying fluxes $\Phi_i$, for $i=1, 2, 3$. These fluxes have to be evaluated along closed paths starting and ending at the same origin $o$. One way to do this is to connect the inner red circles with finite segments ending at $o$. The blue contour can be continuously shrunk from the outer boundary of the fundamental domain until it coincides with the union of the red paths, without changing its flux. At the beginning of this process, the flux along this contour is equal to the left-hand side of Eq. (\ref{['genus_g_flux_constraint']}), and to the right-hand side at the end.
• Figure 3: This example geometry has 9 vertices, 12 edges, 5 plaquettes and genus $\mathfrak{g}= 0$. The 5th plaquette is the region outside of the figure and includes the point at infinity. If there is no ambiguity we may leave off the dot at the vertex in some diagrams.
• Figure 4: Geometry on a torus (top) specified using periodic boundary conditions. This picture has 2 vertices, 3 edges, and 1 plaquette. Bottom: same picture but without using periodic boundary conditions and instead allowing the lines to go out of the plane. In the bottom figure it is harder to see the single plaquette. Note that the bottom figure has the vertices correctly oriented when comparing with the top figure.
• Figure 5: Same torus geometry as Figure \ref{['fig:periodic-boundary']} except here we use 1 vertex, 2 edges, and 1 plaquette.