Sector Theory of Levin-Wen Models II : Fusion and Braiding

TL;DR

This paper characterizes the superselection sectors of Levin-Wen models, showing they are equivalent to the Drinfeld center of the input fusion category, thus advancing understanding of anyonic excitations in topological phases.

Contribution

It establishes an explicit isomorphism between the superselection sectors and the Drinfeld center, providing a complete categorical description of anyons in Levin-Wen models.

Findings

Superselection sectors are equivalent to the Drinfeld center of the input category.

Fusion and braiding properties are captured by the Drinfeld center.

First complete categorical characterization of sectors with non-integer quantum dimensions.

Abstract

This is the continuation of our study of the Levin-Wen model based on an arbitrary unitary fusion category $\mathcal{C}$ on the infinite plane. The ground state of the Levin-Wen model hosts anyonic excitations whose fusion and braiding properties are captured by the associated braided $\rm C^*$-tensor category of superselection sectors $\mathsf{SSS}$. By constructing explicit isomorphisms between the fusion spaces of $\mathsf{SSS}$ and those of the Drinfeld center $Z(\mathcal{C})$, we show that these two categories have isomorphic $F$- and $R$-symbols. It follows that the full subcategory of finite sectors is unitarily braided monoidally equivalent to the Drinfeld center, $$\,\mathsf{SSS}_f \simeq Z(\mathcal{C}).$$ This provides the first complete characterisation of the category of superselection sectors for a class of two-dimensional lattice models supporting anyons with non-integer quantum dimensions.