Emergent discrete 3-form symmetry and domain walls

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 Zk 3-form symmetry for k>1 in addition to a conventional Zk 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.