Majorana representations of spin and an alternative solution of the Kitaev honeycomb model

TL;DR

This paper explores different Majorana fermion representations of spin, compares their properties, and introduces a new unphysical-state-free solution to the Kitaev honeycomb model, extending it to a more general gauge theory.

Contribution

It presents a new solution to the Kitaev honeycomb model using the SO(3) Majorana representation that avoids unphysical states, unlike previous methods.

Findings

The SO(3) representation requires pairing of sites.

The new solution involves no unphysical states.

Extension to a generalized Z2 gauge theory.

Abstract

Based on the Dirac spinor representation of the SO(4) group, we discuss the relationship between three types of representation of spin in terms of Majorana fermions, namely the Kitaev representation, the SO(3) representation and the SO(4) chiral representation. Comparing the three types, we show that the Hilbert space of the SO(3) representation is different from the other two by requiring pairing of sites, but it has the advantage over the other two in that no unphysical states are involved. As an example of its application, we present a new alternative solution of the Kitaev honeycomb model. Our new solution involves no unphysical states which enables a systematic calculation of physical observables. Finally, we discuss an extension of the model to a more general exactly soluble $Z_{2}$ gauge theory interacting with complex fermions.