Quantum loop models and the non-abelian toric code

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.