富对称拓扑相的分类和性质：Chern-Simons方法及其在Z2自旋液体中的应用
Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids
摘要：
在全局对称性存在的情况下，我们研究了具有拓扑序的2 + 1维相，如分数量子霍尔态和带隙自旋液体。共享相同拓扑顺序的相然后可以根据对称性的作用而不同，导致对称性富集拓扑（SET）相。在这里，我们提出了一个K-矩阵Chern-Simons方法来识别所有不同的相位与阿贝尔拓扑顺序，在酉或反酉的全球对称性的存在。一个关键的步骤是识别边缘缝纫条件，用于检查两个假定的阶段是否确实不同。我们说明了这种方法的情况下，$Z_2$拓扑顺序（$Z_2$自旋液体），在存在内部Z $_2$全球对称性。我们发现了6个不同的阶段。众所周知的量子数分馏模式占这些状态的一半。相位也由于添加了对称保护拓扑（SPT）相位而不同。此外，我们允许非常规的可能性，任意子交换的对称性。这导致具有对称保护的马约拉纳边缘模式的2个附加相位。其他路线，以实现保护边缘状态在SET阶段确定。讨论了对称性丰富的Laughlin态和双半子理论。出现了两个令人惊讶的教训：（i）测量不同SET相位的全局对称性总是导致具有相同总量子维数的不同拓扑序，（ii）具有不同Dijkgraaf-Witten拓扑项的规范理论可能具有相同的拓扑序。
英文摘要：
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.
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