  Systems with conserved dipole moment have drawn considerable interest in
light of their realization in recent experiments on tilted optical lattices. An
important question for such systems is delineating the conditions under which
they admit a unique gapped ground state that is consistent with all symmetries.
Here, we study one-dimensional translation-invariant lattices that conserve
U(1) charge and $\mathbb{Z}_L$ dipole moment, where discreteness of the dipole
symmetry is enforced by periodic boundary conditions, with $L$ the system size.
We show that in these systems, a symmetric, gapped, and non-degenerate ground
state requires not only integer charge filling, but also a fixed value of the
dipole filling, while other fractional dipole fillings enforce either a gapless
or symmetry-breaking ground state. In contrast with prior results in the
literature, we find that the dipole filling constraint depends both on the
charge filling as well as the system size, emphasizing the subtle interplay of
dipole symmetry with boundary conditions. We support our results with numerical
simulations and exact results.

Strongly Correlated Electrons;Quantum Physics