Higgs Condensates are Symmetry-Protected Topological Phases: I. Discrete Symmetries

Where in the landscape of many-body phases of matter do we place the Higgs condensate of a gauge theory? On the one hand, the Higgs phase is gapped, has no local order parameter, and for fundamental Higgs fields is adiabatically connected to the confined phase. On the other hand, Higgs phases such as superconductors display rich phenomenology. In this work, we propose a minimal description of the Higgs phase as a symmetry-protected topological (SPT) phase, utilizing conventional and higher-form symmetries. In this first part, we focus on 2+1D $\mathbb Z_2$ gauge theory and find that the Higgs phase is protected by a higher-form magnetic symmetry and a matter symmetry, whose meaning depends on the physical context. While this proposal captures known properties of Higgs phases, it also predicts that the Higgs phase of the Fradkin-Shenker model has SPT edge modes in the symmetric part of the phase diagram, which we confirm analytically. In addition, we argue that this SPT property is remarkably robust upon explicitly breaking the magnetic symmetry. Although the Higgs and confined phases are then connected without a bulk transition, they are separated by a boundary phase transition, which we confirm with tensor network simulations. More generally, the boundary anomaly of the Higgs SPT phase coincides with the emergent anomaly of symmetry-breaking phases, making precise the relation between Higgs phases and symmetry breaking. The SPT nature of the Higgs phase can also manifest in the bulk, e.g., at transitions between distinct Higgs condensates. Finally, we extract insights which are applicable to general SPT phases, such as a 'bulk-defect correspondence' generalizing discrete gauge group analogs of Superconductor-Insulator-Superconductor (SIS) junctions. The sequel to this work will generalize 'Higgs=SPT' to continuous symmetries, interpreting superconductivity as an SPT property.