Quantum loop models and the non-abelian toric code

TL;DR

This paper introduces quantum loop models based on Ising spins on a square lattice that exhibit non-abelian anyonic excitations, with a topological inner product enabling the realization of non-abelian topological phases.

Contribution

It defines a new class of quantum loop models with a topological inner product that supports non-abelian anyons and demonstrates the possibility of gapped non-abelian phases with simple interactions.

Findings

Deconfined non-abelian anyons are possible for various loop weights d.

A gapped non-abelian topological phase occurs at d=√2 with four-spin interactions.

The model generalizes the toric code to non-abelian anyonic excitations.

Abstract

I define quantum loop models whose degrees of freedom are Ising spins on the square lattice as in the toric code, but where the excitations should have non-abelian statistics. The inner product is topological, allowing a direct implementation of the anyonic fusion matrix on the lattice. It also makes deconfined anyons possible for a variety of values of the weight per loop $d$ in the ground state. For d=\sqrt{2}, a gapped non-abelian topological phase can occur with only four-spin interactions.