  We propose new dipole and quadrupole indices for interacting insulators with
point group symmetries. The proposed indices are defined in terms of many-body
quantum multipole operators combined with the generator of the point group
symmetry. Unlike the original multipole operators, these combined operators
commute with Hamiltonian under the symmetry and therefore their eigenvalues are
quantized. This enables a clear identification of nontrivial multipolar states.
We calculate the multipole indices in representative models and show their
effectiveness as order parameters. Furthermore, we demonstrate a bulk-boundary
correspondence: a non-zero index implies the existence of edge/corner states
under the the point group symmetry.
