Local realizations of anyon exchange symmetries without lattice dislocations

TL;DR

This paper presents a method to realize anyon exchange symmetries locally in the toric code without lattice dislocations, enabling transitions to non-Abelian phases and generalizations for other Abelian groups.

Contribution

The authors introduce an exactly solvable Hamiltonian with defect sites that locally realize anyon exchange symmetries without lattice dislocations, supporting Ising anyon fusion rules.

Findings

Supports states obeying Ising anyon fusion rules

Achieves local realization of anyon symmetries without lattice defects

Generalizes to other Abelian groups for local symmetry realization

Abstract

The global $e$-$m$ exchange symmetry of the toric code is realized locally through an exactly solvable Hamiltonian on a two dimensional lattice which has no lattice dislocations and their associated defect line. The Hamiltonian is still changed locally in selected sites where we wish to realize this anyon symmetry. We refer to these selected sites as defect sites in analogy with the usual lattice defects. The operators on the defect sites condense dyons of the toric code and are shown to support states which obey the fusion rules of Ising anyons just as the lattice dislocations thus achieving the transition to the Ising phase from the toric code phase. They can also be introduced in an entire region leading to an idea of non-localized defects. The method leads to a natural generalization for other Abelian groups where they help realize all the anyon exchange symmetries locally.