Defects in the 3-dimensional toric code model form a braided fusion 2-category

TL;DR

This paper constructs and characterizes all topological defects in the 3D toric code model, revealing they form a braided fusion 2-category, thus deepening the understanding of topological order and defect structures.

Contribution

It explicitly constructs all higher codimension topological defects in the 3D $bZ_2$ topological order and shows they form a braided fusion 2-category, a novel mathematical framework.

Findings

Topological defects form a braided fusion 2-category

Construction of all codimension 2 and higher defects

Verification of braiding non-degeneracy condition

Abstract

It was well known that there are $e$-particles and $m$-strings in the 3-dimensional (spatial dimension) toric code model, which realizes the 3-dimensional $\mathbb{Z}_2$ topological order. Recent mathematical result, however, shows that there are additional string-like topological defects in the 3-dimensional $\mathbb{Z}_2$ topological order. In this work, we construct all topological defects of codimension 2 and higher, and show that they form a braided fusion 2-category satisfying a braiding non-degeneracy condition.