Localized endomorphisms in Kitaev's toric code on the plane

TL;DR

This paper analyzes the structure and statistics of excitations in Kitaev's toric code on a plane using a C*-algebraic approach, connecting localized endomorphisms with quantum double representations.

Contribution

It introduces a C*-algebraic framework to describe excitations in the toric code and links their statistics to the representation theory of D(Z_2).

Findings

Excitations are described by localized endomorphisms of the observable algebra.

The statistics of excitations are computed via this algebraic approach.

Excitations correspond to representations of the quantum double D(Z_2).

Abstract

We consider various aspects of Kitaev's toric code model on a plane in the C^*-algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher-Haag-Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of D(Z_2), i.e., Drinfel'd's quantum double of the group algebra of Z_2.