  Systems with dipole moment conservation have been of recent interest, as they
realize both novel quantum dynamics and exotic ground state phases. In this
work, we study some generic properties of 1-D and 2-D dipole-conserving
fermionic models at integer fillings. We find that a dipolar symmetry-breaking
phase can result in a mean-field band insulator whose topological indices can
strongly affect the low-energy physics of the dipolar Goldstone modes. We study
the 2-D topological phase transition of the mean-field ground states in the
presence of the Goldstone modes. The critical theory resembles the 2+1d quantum
electrodynamics coupled to massless Dirac fermions with some crucial
differences and shows a novel quantum critical point featuring a nontrivial
dynamical exponent. We also discuss the analogous case of 1-D dipole-conserving
models and the role of topological invariants.
