  Insulating systems are characterized by their insensitivity to twisted
boundary conditions as quantified by the charge stiffness and charge
localization length. The latter quantity was shown to be related to the
expectation value of the many-body position operator and serves as a universal
criterion to distinguish between metals and insulators. In this work we extend
these concepts to a new class of quantum systems having conserved charge and
dipole moments. We refine the concept of a charge insulator by introducing
notions of multipolar insulators, e.g., a charge insulator could be a dipole
insulator or dipole metal. We develop a universal criterion to distinguish
between these phases by extending the concept of charge stiffness and
localization to analogous versions for multipole moments, but with our focus on
dipoles. We are able relate the dipole localization scale to the expectation
value of a recently introduced many-body quadrupole operator. This refined
structure allows for the identification of phase transitions where charge
remains localized but, e.g., dipoles delocalize. We illustrate the proposed
criterion using several exactly solvable models that exemplify these concepts,
and discuss a possible realization in cold-atom systems.
