Detection of symmetry-enriched topological phases

Topologically ordered systems in the presence of symmetries can exhibit new structures which are referred to as symmetry-enriched topological (SET) phases. We introduce simple methods to detect certain SET orders directly from a complete set of topologically degenerate ground-state wave functions. In particular, we first show how to directly determine the characteristic symmetry fractionalization of the quasiparticles from the reduced density matrix of the minimally entangled states. Second, we show how a simple generalization of a nonlocal order parameter can be measured to detect SET phases. The usefulness of the proposed approach is demonstrated by examining two concrete model states which exhibit SET phases: (i) a spin-1 model on the honeycomb lattice and (ii) the resonating valence bond (RVB) state on a kagome lattice. We conclude that the spin-1 model and the RVB state are in the same SET phases.