  We discuss a large class of classical field theories with continuous
translation symmetry. In the quantum theory, a new anomaly explicitly breaks
this translation symmetry to a discrete symmetry. Furthermore, this discrete
translation symmetry is extended by a $d-2$-form global symmetry. All these
theories can be described as $U(1)$ gauge theories where Gauss law states that
the system has nonzero charge density. Special cases of such systems can be
phrased as theories with a compact phase space. Examples are ferromagnets and
lattices in the lowest Landau level. In some cases, the broken continuous
translation symmetry can be resurrected as a noninvertible symmetry. We clarify
the relation between the discrete translation symmetry of the continuum theory
and the discrete translation symmetry of an underlying lattice model. Our
treatment unifies, clarifies, and extends earlier works on the same subject.
