A Unified Framework of Topological Phases with Symmetry

Abstract

In topological phases in $2+1$ dimensions, anyons fall into representations of quantum group symmetries. As proposed in our work (arXiv:1308.4673), physics of a symmetry enriched phase can be extracted by the Mathematics of (hidden) quantum group symmetry breaking of a "parent phase". This offers a unified framework and classification of the symmetry enriched (topological) phases, including symmetry protected trivial phases as well. In this paper, we extend our investigation to the case where the "parent" phases are non-Abelian topological phases. We show explicitly how one can obtain the topological data and symmetry transformations of the symmetry enriched phases from that of the "parent" non-Abelian phase. Two examples are computed: (1) the $\text{Ising}\times\overline{\text{Ising}}$ phase breaks into the $\mathbb{Z}_2$ toric code with $\mathbb{Z}_2$ global symmetry; (2) the $SU(2)_8$ phase breaks into the chiral Fibonacci $\times$ Fibonacci phase with a $\mathbb{Z}_2$ symmetry, a first non-Abelian example of symmetry enriched topological phase beyond the gauge theory construction.