Numerical detection of symmetry protected and symmetry enriched topological phases

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A topological phase is a phase of matter which cannot be characterized by a local order parameter. We first introduce non-local order parameters that can detect symmetry protected topological (SPT) phases in 1D systems and then show how to generalize the idea to detect symmetry enriched topological (SET) phases in 2D. SET phases are new structures that occur in topologically ordered systems in the presence of symmetries. We introduce simple methods to detect the SET order directly from a complete set of topologically degenerate ground state wave functions. 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 non-local order parameter can be measured to detect SETs. The usefulness of the proposed approached is demonstrated by examining two concrete model states which exhibit SET: (i) a spin-1 model on the honeycomb lattice and (ii) the resonating valence bond state on a kagome lattice. We conclude that the spin-1 model and the RVB state are in the same SET phases.