The Higgs condensate is a symmetry-protected topological phase

Gauge theories are ubiquitous even outside the standard model, arising as effective low-energy descriptions of many-body quantum systems. The Higgs phase arises upon condensing the gauge charges of such a theory. The physical nature of the Higgs phase is sometimes a source of conceptual confusion---its apparent description as a symmetry-breaking phase of matter is in conflict with the unbreakable nature of a gauge symmetry. In this talk, we point out that the Higgs phase can be naturally interpreted as a symmetry-protected topological (SPT) phase when the condensate carries charge under a global symmetry. This condition arises naturally in emergent gauge theories where the Gauss law is energetically enforced. We illustrate this for the toric code model coupled to matter, where we identify the SPT edge mode of the Higgs phase. We also discuss the physical implications for superconductors---the Higgs phase associated to electromagnetism.