Higgs Condensates are Symmetry-Protected Topological Phases: II. $U(1)$ Gauge Theory and Superconductors

Understand

Classifying Higgs phases within the landscape of gapped and symmetry preserving states of matter presents a conceptual challenge.

• We argue that $U(1)$ Higgs phases are symmetry-protected topological (SPT) phases and we derive their topological response theory and boundary anomaly -- applicable to superconductors treated with dynamical electromagnetic field.
• This generalizes the discussion of discrete gauge theories by Verresen et al., arXiv:2211.01376.
• We show that a Higgs phase in $d$ spatial dimensions is in a non-trivial SPT class protected by a global $U(1)$ symmetry associated with the Higgs field, and a $d-2$ form $U(1)$ magnetic symmetry, associated with the absence of magnetic monopoles.