  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.
