Emergent discrete 3-form symmetry and domain walls

TL;DR

This paper reveals that axion models with multiple domain walls in four dimensions exhibit an emergent discrete 3-form symmetry, which is dual to a 3-form gauge theory and undergoes spontaneous breaking, enriching the understanding of their phase structure.

Contribution

It demonstrates the emergence of a ${f Z}_k$ 3-form symmetry in axion models with multiple domain walls and establishes its duality with a 3-form gauge theory.

Findings

Emergent ${f Z}_k$ 3-form symmetry for $k>1$ in axion models.

Explicit duality between scalar theory and 3-form gauge theory.

Spontaneous breaking of the emergent 3-form symmetry.

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.