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 and chiral or nonchiral. It can systematically treat a general symmetry, which may include both lattice symmetry and internal symmetry, may contain antiunitary 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 quantum anomalies, which are sometimes known as Lieb-Schultz-Mattis anomalies. We apply this framework to classify $\mathrm{U}(1{)}_{2N}$ chiral states and non-Abelian ${\mathrm{Ising}}^{(\nu )}$ states enriched by a $p6\times \mathrm{SO}(3)$ or $p4\times \mathrm{SO}(3)$ symmetry and ${\mathbb{Z}}_N$ topological orders and $\mathrm{U}(1{)}_{2N}\times \mathrm{U}(1{)}_{-2N}$ topological orders enriched by a $p6m\times \mathrm{SO}(3)\times {\mathbb{Z}}_2^T$, $p4m\times \mathrm{SO}(3)\times {\mathbb{Z}}_2^T$, $p6m\times {\mathbb{Z}}_2^T$, 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.

中文翻译：

---

富含对称性的拓扑量子自旋液体的分类


我们提出了一个系统框架来对二维空间维度上富含对称性的拓扑量子自旋液体进行分类。该框架可以处理所有拓扑量子自旋液体，这些液体可以是阿贝尔的或非阿贝尔的，手性的或非手性的。它可以系统地处理一般对称性，该对称性可能包括晶格对称性和内部对称性，可能包含反酉对称性，并且可能会置换任意子。该框架适用于所有类型的晶格，并且可以利用其量子异常（有时称为 Lieb-Schultz-Mattis 异常）系统地区分具有相同对称群的不同晶格系统。我们应用此框架对 $\mathrm{U}(1{)}_{2N}$ 手性态和通过 $p6\times \mathrm{SO}(3)$ 或 $p4\times \mathrm{SO}(3)$ 对称性和 ${\mathbb{Z}}_N$ 态进行分类 拓扑序和 $\mathrm{U}(1{)}_{2N}\times \mathrm{U}(1{)}_{-2N}$ 拓扑序由 $p6m\times \mathrm{SO}(3)\times {\mathbb{Z}}_2^T$ 、 $p4m\times \mathrm{SO}(3)\times {\mathbb{Z}}_2^T$ 、 $p6m\times {\mathbb{Z}}_2^T$ 或 $p4m\times {\mathbb{Z}}_2^T$ 对称性，其中 $p6$ 、 $p4$ 、 $p6m$ 和 $p4m$ 是晶格对称性，而 SO(3) 和 ${\mathbb{Z}}_2^T$ 拓扑顺序的例子。