Classification of symmetry-enriched topological quantum spin liquids

Abstract

We present a systematic framework to classify symmetry-enriched topological quantum spin liquids in two spatial dimensions. This framework can deal with all topological quantum spin liquids, which may be either Abelian or non-Abelian, chiral or non-chiral. It can systematically treat a general symmetry, which may include both lattice symmetry and internal symmetry, may contain anti-unitary symmetry, and may permute anyons. The framework applies to all types of lattices, and can systematically distinguish different lattice systems with the same symmetry group using their Lieb-Schultz-Mattis anomalies. We apply this framework to classify $U(1{)}_{2N}$ chiral states and non-Abelian Ising${}^{(\nu )}$ states enriched by a $p6\times SO(3)$ or $p4\times SO(3)$ symmetry, and ${\mathbb{Z}}_N$ topological orders and $U(1{)}_{2N}\times U(1{)}_{-2N}$ topological orders enriched by a $p6m\times SO(3)\times {\mathbb{Z}}_2^T$, $p4m\times SO(3)\times {\mathbb{Z}}_2^T$, $p6m\times {\mathbb{Z}}_2^T$ or $p4m\times {\mathbb{Z}}_2^T$ symmetry, where $p6$, $p4$, $p6m$ and $p4m$ are lattice symmetries, while $SO(3)$ and ${\mathbb{Z}}_2^T$ are spin rotation and time reversal symmetries, respectively. In particular, we identify symmetry-enriched topological quantum spin liquids that are not easily captured by the usual parton-mean-field approach, including examples with the familiar ${\mathbb{Z}}_2$ topological order.

arXiv commentMinor modification, nearly identical to the published version in PRX, Reference Fixed

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