A Description of Kitaev's Honeycomb Model with Toric-Code Stabilizers

TL;DR

This paper provides an exact solution to Kitaev's honeycomb model using fermionization, revealing its topological degeneracy structure and connection to the toric code, with implications for understanding non-Abelian phases.

Contribution

It introduces a non-auxiliary fermionization method to solve Kitaev's model explicitly, linking its ground states and degeneracies to the toric code and analyzing non-Abelian phase behavior.

Findings

Explicit eigenstates and eigenvalues on the torus

Topological degeneracy derived from the toric code

Reduction of degeneracy in non-Abelian phase

Abstract

We present a solution of Kitaev's spin model on the honeycomb lattice and of related topologically ordered spin models. We employ a Jordan-Wigner type fermionization and find that the Hamiltonian takes a BCS type form, allowing the system to be solved by Bogoliubov transformation. Our fermionization does not employ non-physical auxiliary degrees of freedom and the eigenstates we obtain are completely explicit in terms of the spin variables. The ground-state is obtained as a BCS condensate of fermion pairs over a vacuum state which corresponds to the toric code state with the same vorticity. We show in detail how to calculate all eigenstates and eigenvalues of the model on the torus. In particular, we find that the topological degeneracy on the torus descends directly from that of the toric code, which now supplies four vacua for the fermions, one for each choice of periodic vs. anti-periodic boundary conditions. The reduction of the degeneracy in the non-Abelian phase of the model is seen to be due to the vanishing of one of the corresponding candidate BCS ground-states in that phase. This occurs in particular in the fully periodic vortex-free sector. The true ground-state in this sector is exhibited and shown to be gapped away from the three partially anti-periodic ground-states whenever the non-Abelian phase is gapped.