The Kitaev honeycomb model on surfaces of genus $g\ge 2$

Abstract

We present a construction of the Kitaev honeycomb lattice model on an arbitrary higher genus surface. We first generalize the exact solution of the model based on the Jordan-Wigner fermionization to a surface with genus $g=2$, and then use this as a basic module to extend the solution to lattices of arbitrary genus. We demonstrate our method by calculating the ground states of the model in both the Abelian doubled ${\mathbb{Z}}_2$ phase and the non-Abelian Ising topological phase on lattices with the genus up to $g=6$. We verify the expected ground state degeneracy of the system in both topological phases and further illuminate the role of fermionic parity in the Abelian phase.

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