RVB gauge theory and the Topological degeneracy in the Honeycomb Kitaev model

TL;DR

This paper connects the Z₂ gauge theory of the Kitaev honeycomb model with RVB physics, reformulates a Jordan-Wigner transformation as a gauge fixing, and demonstrates four-fold topological degeneracy of the ground state.

Contribution

It provides a unified gauge theory framework for the Kitaev model and explicitly constructs operators proving topological degeneracy in the thermodynamic limit.

Findings

Ground state is four-fold degenerate on a torus.

Exact eigenstates constructed using gauge-invariant Jordan-Wigner fermions.

Fermionic spectrum analyzed for flux-free sector.

Abstract

We relate the Z$_2$ gauge theory formalism of the Kitaev model to the SU(2) gauge theory of the resonating valence bond (RVB) physics. Further, we reformulate a known Jordan-Wigner transformation of Kitaev model on a torus in a general way that shows that it can be thought of as a Z$_2$ gauge fixing procedure. The conserved quantities simplify in terms of the gauge invariant Jordan-Wigner fermions, enabling us to construct exact eigen states and calculate physical quantities. We calculate the fermionic spectrum for flux free sector for different gauge field configurations and show that the ground state is four-fold degenerate on a torus in thermodynamic limit. Further on a torus we construct four mutually anti-commuting operators which enable us to prove that all eigenstates of this model are four fold degenerate in thermodynamic limit.