Multipole Insulators and Higher-Form symmetries*

We present a suitable refinement of the notion of an insulator by investigating a class of systems that conserve both the total charge and the total dipole moment. In particular, we consider microscopic models for systems that conserve dipole moments exactly and show that one can divide charge insulators into two new classes: (i) a dipole metal, which is a charge-insulating system that supports dipole-moment currents, or (ii) a dipole insulator which is a charge-insulating system that does not allow dipole currents and thus, conserves an overall quadrupole moment. I this work we discuss a more formal description of dipole-conserving systems and show that a conservation of the overall dipole moment can be naturally attributed to a global 1-form electric U(1) symmetry, which is in direct analogy to how the electric charge conservation is guaranteed by the global U(1) phase-rotation symmetry for electrically charged particles. This new approach allows us to construct a topological response action which is especially useful for characterizing Higher-Order Topological phases carrying quantized quadrupole moments.