Spectrum of the non-abelian phase in Kitaev's honeycomb lattice model

TL;DR

This paper analyzes the spectral properties of Kitaev's honeycomb model in the non-abelian phase, combining analytical and numerical methods to explore vortex effects, degeneracies, and phase transitions.

Contribution

It provides a detailed analytical and numerical study of the non-abelian phase, including vortex interactions and ground state degeneracies, in Kitaev's honeycomb lattice model.

Findings

Explicit demonstration of $2^n$-fold ground state degeneracy with well-separated vortices

Derivation of vortex-vortex interaction as a function of separation

Numerical results confirm analytical predictions for finite systems

Abstract

The spectral properties of Kitaev's honeycomb lattice model are investigated both analytically and numerically with the focus on the non-abelian phase of the model. After summarizing the fermionization technique which maps spins into free Majorana fermions, we evaluate the spectrum of sparse vortex configurations and derive the interaction between two vortices as a function of their separation. We consider the effect vortices can have on the fermionic spectrum as well as on the phase transition between the abelian and non-abelian phases. We explicitly demonstrate the $2^n$-fold ground state degeneracy in the presence of $2n$ well separated vortices and the lifting of the degeneracy due to their short-range interactions. The calculations are performed on an infinite lattice. In addition to the analytic treatment, a numerical study of finite size systems is performed which is in exact agreement with the theoretical considerations. The general spectral properties of the non-abelian phase are considered for various finite toroidal systems.