  We construct new many-body invariants for 2d Chern and 3d chiral hinge
insulators, which are characterized by quantized pumping of dipole and
quadrupole moments. The invariants that we devise are written entirely in terms
of many-body ground state wavefunctions on a torus geometry with a set of
unitary operators. We provide a number of supporting evidences for our
invariants via topological field theory interpretation, adiabatic pumping
argument, and direct mapping to free-fermion band indices. We finally confirm
our invariants by numerical computations including infinite density matrix
renormalization group on a quasi-one-dimensional system. The many-body
invariants therefore explicitly encircle several different pillars of
theoretical descriptions of topological phases.
