Emergent discrete 3-form symmetry and domain walls

TL;DR

The paper shows that in (3+1)D axion models with domain-wall number k>1, an emergent ${\mathbb Z}_k$ 3-form symmetry arises in addition to a ${\mathbb Z}_k$ 0-form symmetry. By dualizing the low-energy effective theory to a 3-form gauge theory, it demonstrates a topological coupling between a 3-form field and the axion, yielding a fractional linking phase between domain-wall worldvolumes and point operators. The resulting phase features spontaneous breaking of both 0-form and 3-form symmetries in a type-B pattern, akin to topological order but with distinctive dimensionalities of charged objects and ground-state structure. The work also extends the mechanism to other domain-wall contexts and discusses connections and distinctions from conventional topological phases, with implications for axion cosmology and broader higher-form symmetry classifications.

Abstract

We show that axion models with the domain wall number $k$ in $(3+1)$ dimensions, i.e., periodic scalar field theories admitting $k$ axion domain walls, exhibit an emergent ${\mathbb Z}_k$ 3-form symmetry for $k >1$ in addition to a conventional ${\mathbb Z}_k$ 0-form symmetry. The emergent 3-form symmetry is explicitly shown by establishing a low-energy dual transformation between the scalar field theory and a 3-form gauge theory. We further argue that the emergent 3-form symmetry is spontaneously broken, and the breaking pattern is so-called the type-B spontaneous symmetry breaking. We discuss similar and different points between the phase admitting the domain walls and topologically ordered phases.