Topological Goldstone phases of matter

TL;DR

This paper explores phases of matter where continuous symmetry breaking occurs with non-trivial topological features, linking such phases to SPT/SET states or anomalous symmetries, and establishing a correspondence between defects.

Contribution

It demonstrates that topologically non-trivial Goldstone phases can only arise in SPT/SET phases or with anomalous symmetries, providing a new theoretical framework.

Findings

Topological Goldstone phases require non-trivial SPT or SET order.

Such phases are linked to anomalous residual symmetries.

A correspondence between defects of order parameters and gauge fields is established.

Abstract

We consider the possibility for phases of matter in which a continuous symmetry is spontaneously broken in a \emph{topologically non-trivial} way, which, roughly, means that the action for the Goldstone modes contains a quantized topological term, and could manifest in, for example, non-trivial 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 non-trivial symmetry-protected topological (SPT) or symmetry-enriched topological (SET) 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.