Anomalous continuous symmetries and quantum topology of Goldstone modes

TL;DR

This work develops a unified framework to understand topological terms in the Goldstone sector when the underlying continuous symmetry $G$ is potentially anomalous and is spontaneously broken to a subgroup $H$. The central concept is a compatibility relation $p \triangleleft \alpha$ between $H_0$-pump invariants over the order-parameter space $\Lambda=G/H$ and the $G$-anomaly $\alpha$, which constrains which topological terms can appear in the Goldstone action. The authors introduce a compatibility spectral sequence to compute this relation in the invertible case, relate it to generalized cohomology classifications, and discuss extensions to non-invertible, topologically ordered families via SET data. They illustrate the framework with detailed examples such as the Thouless pump, QSH boundary fermion parity pump, superconducting proximity effects on TI surfaces, and quantum Hall ferromagnets, highlighting how anomalies and symmetry-breaking patterns govern pumping and topological responses. The results provide a systematic tool for predicting when topological pumping and anomaly-induced terms can arise, with implications for boundary physics, higher Chern-number pumps, and symmetry-enriched topological phases.

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.

Figures (1)

• Figure 1: Schematic illustration of differentials in the compatibility AHSS.