Exact emergent higher-form symmetries and their physical consequences

Generalized global symmetries provide a unifying perspective of quantum-many body systems. However, since condensed matter systems rarely have these symmetries, it is natural to wonder if the phases of a lattice model without generalized symmetries can be meaningfully characterized by emergent generalized symmetries. Here we explicitly demonstrate this possibility in two bosonic Hamiltonian models. While these UV models have no higher-form symmetries, there is a region of parameter space where a higher-form symmetry emerges in the mid-IR. This emergent symmetry is robust against local UV perturbations and is an exact symmetry of the full mid-IR effective theory in the thermodynamic limit. Therefore, it is an exact emergent symmetry: it is not a UV symmetry but constrains the IR as if it were. We discuss in detail the physical consequences of this. For instance, there is a spontaneous symmetry breaking (SSB) phase with an exactly gapless Goldstone mode or exact ground state degeneracy. Additionally, there can be an SPT phase protected by the exact emergent symmetry. Furthermore, there are topological defects that are gapped in the SSB phase but condensed in the symmetric phase, and at energies below their gap there is a new exact emergent higher-form symmetry and an emergent mixed 't Hooft anomaly. We conclude by discussing these properties of exact emergent generalized symmetries in a more general, model-independent way.