Three Dimensional Z2 Topological Phases enriched by Time-Reversal Symmetry

Abstract

We discuss three dimensional $Z_2$ topological phases enriched by the $Z_2^T$ time-reversal symmetry with bosonic bulk excitations. Some of these phases can be constructed by coupling the three dimensional symmetry protected topological phases with $Z_2 \times Z_2^T$ symmetry to a deconfined dynamical $Z_2$ gauge field. Besides these simple phases, we also construct a special root phase (phase 3) which cannot be interpreted as gauged $Z_2 \times Z_2^T$ SPT phase. The $2d$ boundary of these phases can have an extra $Z_2$ topological order in addition to their bulk topological order. In one of the possible boundary states of this special root phase (phase 3), $Z_2^T$ acts as the "self-dual" transformation of the boundary topological order, in the sense that the boundary $e$ and $m$ anyons are interchanged under $Z_2^T$.