Excitations in the Higher Lattice Gauge Theory Model for Topological Phases II: The (2+1)-Dimensional Case

TL;DR

This paper explores the 2+1-dimensional higher lattice gauge theory model for topological phases, constructing operators for excitations, analyzing their braiding and topological properties, and relating the model to symmetry enriched string-net models.

Contribution

It introduces the construction of ribbon and membrane operators for excitations in the 2+1d higher lattice gauge theory model and connects it to symmetry enriched topological phases.

Findings

Ribbon operators produce point-like excitations with braiding properties.

Loop-like excitations can act as domain walls between different ground states.

The model maps to symmetry enriched string-net models in certain cases.

Abstract

In this work, the second paper of this series, we study the 2+1d version of a Hamiltonian model for topological phases based on higher lattice gauge theory. We construct the ribbon operators that produce the point-like excitations. These ribbon operators are used to find the braiding properties and topological charge carried by the point-like excitations. The model also hosts loop-like excitations, which are produced by membrane operators. By considering a change of basis, we show that, in certain cases, some loop-like excitations represent domain walls between patches corresponding to different symmetry-related ground states, and we find this symmetry. We also map the higher lattice gauge theory Hamiltonian to the symmetry enriched string-net model for symmetry enriched topological phases (SETs) described by Heinrich, Burnell, Fidkowski and Levin [Phys. Rev. B, 94, 235136 (2016)], again in a subset of cases.