String-nets, single- and double-stranded quantum loop gases for non-Abelian anyons

Abstract

String-nets and quantum loop gases are two prominent microscopic lattice models to describe topological phases. String-net condensation can give rise to both Abelian and non-Abelian anyons, whereas loop condensation usually produces Abelian anyons. It has been proposed, however, that generalized quantum loop gases with non-orthogonal inner products could support non-Abelian anyons. We detail an exact mapping between the string-net and these generalized loop models and explain how the non-orthogonal products arise. We also introduce an equivalent loop model of double-stranded nets where quantum loops with an orthogonal inner product and local interactions supports non-Abelian Fibonacci anyons. Finally, we emphasize the origin of the sign problem in systems with non-Abelian excitations and its consequences on the complexity of their ground state wavefunctions.