Title: Anomalous continuous symmetries and quantum topology of Goldstone modes Show Chinese title

Title: 异常连续对称性和Goldstone模的量子拓扑

Abstract: We consider systems in which a continuous symmetry $G$, which may be anomalous, is spontaneously broken to an anomaly-free subgroup $H$ such that the effective action for the Goldstone modes contains topologically non-trivial terms. If the original system has trivial $G$ anomaly, it is known that the possible topological terms are fully determined by SPT or SET invariants of the residual $H$ symmetry. Here we address the more general setting in which the $G$ symmetry has an anomaly. We argue that in general, the appropriate concept to consider is the "compatibility relation" between the Goldstone invariants and the $G$ anomaly. In the case where the Goldstone modes can be gapped out to obtain invertible families (i.e. without any topological order), we give an explicit mathematical scheme to construct the desired compatibility relation. We also address the case where gapping out the Goldstone modes leads to a family of topologically ordered states. We discuss several examples including the canonical Thouless pump, the quantum Hall ferromagnet, pumps arising from breaking $\text{U}(1)$ symmetry at the boundary of topological insulators in two and three dimensions, and pumps classified by the higher Chern number. Show Chinese abstract

Abstract: 我们考虑系统中连续对称性$G$被自发地破缺为一个无异常的子群$H$，使得黄金子模式的有效作用包含拓扑非平凡项。如果原始系统具有平凡的$G$异常，已知可能的拓扑项完全由剩余$H$对称性的 SPT 或 SET 不变量决定。在这里，我们处理更一般的情况，其中$G$对称性具有异常。我们认为，在一般情况下，需要考虑的概念是黄金子不变量与$G$异常之间的“兼容性关系”。在黄金子模式可以被禁带从而获得可逆族（即没有任何拓扑序）的情况下，我们给出一个显式的数学方案来构建所需的兼容性关系。我们还讨论了禁带黄金子模式导致拓扑有序态族的情况。我们讨论了几个例子，包括经典的索尔斯泵，量子霍尔铁磁体，在二维和三维拓扑绝缘体边界上破缺$\text{U}(1)$对称性产生的泵，以及由高陈数分类的泵。