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 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.
Highlights
The Kitaev honeycomb model is an example of an exactly solvable two-dimensional model that exhibits both Abelian and non-Abelian topological phases [1]
We demonstrate our method by calculating the ground states of the model in both the Abelian doubled 2 phase and the non-Abelian Ising topological phase on lattices with the genus up to g = 6
The Abelian phase, which is known as the toric code [2], provides a realization of a topological quantum field theory known as doubled- 2 theory
Summary
The Kitaev honeycomb model is an example of an exactly solvable two-dimensional model that exhibits both Abelian and non-Abelian topological phases [1]. We will demonstrate the generalized solution on a number of different surfaces of genus greater than 1 by calculating the ground state degeneracy of the model in both the Abelian and non-Abelian phases. In this context, we investigate additional features of these topological states that are intrinsic to their lattice realizations. We point out that realizing a topological phase of a physical system on a closed oriented surface of some genus represents a realization of an important part of this functor It assigns to the surface (2-manifold) a vector space spanned by the ground states of the relevant physical system.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
