Symmetry enriched string-nets: Exactly solvable models for SET phases

TL;DR

This paper introduces exactly solvable models for 2D symmetry enriched topological phases, connecting them to string-net models and providing explicit examples including a $Z_2$ symmetry exchanging anyons.

Contribution

The authors develop a construction for exactly solvable models of 2D bosonic SET phases with finite onsite symmetry, conjecturing they realize all such phases with commuting projector Hamiltonians.

Findings

Models realize phases with specific anyon structures

Coupling to gauge fields yields modified string-net models

Explicit example with $Z_2$ symmetry exchanging anyons

Abstract

We construct exactly solvable models for a wide class of symmetry enriched topological (SET) phases. Our construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group $G$ and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: if we couple them to a dynamical lattice gauge field with gauge group $G$, the resulting gauge theories are equivalent to modified string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a $\mathbb{Z}_2$ symmetry which exchanges the $e$ and $m$ type anyons. We further illustrate our construction with a number of additional examples.