Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids

Abstract

We study 2+1 dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry enriched topological (SET) phases. Here we present a K-matrix Chern-Simons approach to identify all distinct phases with Abelian topological order, in the presence of unitary or anti-unitary global symmetries . A key step is the identification of an edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method for the case of $Z_2$ topological order ($Z_2$ spin liquids), in the presence of an internal Z${}_2$ global symmetry. We find 6 distinct phases. The well known quantum number fractionalization patterns account for half of these states. Phases also differ due to the addition of a symmetry protected topological (SPT) phase. Also, we allow for the unconventional possibility that anyons are exchanged by the symmetry. This leads to 2 additional phases with symmetry protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry enriched Laughlin states and double semion theories are also discussed. Two surprising lessons that emerge are: (i) gauging the global symmetry of distinct SET phases always lead to different topological orders with the same total quantum dimension, (ii) gauge theories with distinct Dijkgraaf-Witten topological terms may have the same topological order.

36 pages, 5 figures, 7 tables, published version

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