Three-dimensionalZ2topological phases enriched by time-reversal symmetry

Abstract

We study three-dimensional ${Z}_{2}$ topological phases enriched by ${Z}_{2}^{T}$ time-reversal symmetry with bosonic bulk excitations. Some of these phases can be constructed by simply coupling the three-dimensional symmetry protected topological phases with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T}$ symmetry to a deconfined ${Z}_{2}$ gauge field. Besides these simple phases, we also construct two special root phases, whose boundary can have an extra ${Z}_{2}$ topological order in addition to their bulk topological order, and the boundary anyon excitations can have fractional time-reversal transformation $\mathcal{T}$ with ${\mathcal{T}}^{4}=\ensuremath{-}1$. In particular, the boundary $e$ and $m$ anyons of one of the two root phases are interchanged under time-reversal transformation, and they must be degenerate and orthogonal with each other. Eventually we obtain ${({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$ classification (8 different phases) for ${Z}_{2}^{T}$-enriched ${Z}_{2}$ topological phases with bosonic bulk excitations in three dimensions.