Topological Goldstone phases of matter

We consider the possibility for phases of matter in which a continuous symmetry is spontaneously broken in a topologically nontrivial way, which, roughly, means that the action for the Goldstone modes contains a quantized topological term, and could manifest in, for example, nontrivial quantum numbers of topological defects of the order parameter. We show that, in fact, such a scenario can occur only when the system is in a nontrivial symmetry-protected topological or symmetry-enriched topological phase with respect to the residual symmetry or, alternatively, if the original symmetry before spontaneous symmetry breaking acts on the system in an anomalous way. Our arguments are based on a general correspondence between topological defects of the order parameter and topological defects of a background gauge field for the residual symmetry.

Automated summary

In some phases of matter, continuous symmetry can be spontaneously broken in a topologically nontrivial way, but this can only occur when the system is in a nontrivial symmetry-protected topological or symmetry-enriched topological phase, or when the original symmetry acts on the system in an anomalous way. This is based on a general correspondence between topological defects of the order parameter and background gauge field for the residual symmetry.