Field theories for gauged symmetry-protected topological phases: Non-Abelian anyons with Abelian gauge group ${\mathbb{Z}}_2^{\otimes 3}$

Dijkgraaf-Witten (DW) theories are of recent interest to the condensed matter community, in part because they represent topological phases of matter, but also because they characterize the response theory of certain symmetry protected topological (SPT) phases. However, as yet there has not been a comprehensive treatment of the spectra of these models in the field theoretic setting even for Abelian gauge groups, the goal of this work is to fill the gap in the literature, especially for a selection of DW models with Abelian gauge groups but non-Abelian topological order. Particularly, we focus on the appearance of non-Abelian statistics in type-III twisted DW theories with Abelian gauge groups ${\mathbb{Z}}_2^{\otimes 3}$. There are only 22 distinguishable line operators, and their fusion rules and correlation functions are calculated. The flux insertion operators have quantum dimension 2, which clearly demonstrates the non-Abelian topological order of type-III twisted DW theories.

Figure 1

An illustration of a linking correlation of $V_{100}\left({\gamma }_1\right)$ and $V_{010}\left({\gamma }_2\right)$.

Figure 2

An illustration of self-twist. The left panel is a ribbon without self-twists. The dotted line is the illustration of framing for the solid line. The right panel is a ribbon with one self-twist. The dotted line, the framing, winds around the solid line once [47]. We can simply view the right panel as a link of the solid and dotted line.