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

TL;DR

This paper extends the exact solution of the Kitaev honeycomb model to higher genus surfaces, enabling analysis of topological phases and ground state degeneracies on complex lattice geometries.

Contribution

It generalizes the solution of the Kitaev model to arbitrary higher genus surfaces, allowing exploration of topological phases on complex geometries.

Findings

Verified ground state degeneracy on higher genus surfaces

Extended solution to genus up to 6

Analyzed fermionic parity in Abelian phase

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.