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We construct new many-body invariants for 2d Chern and 3d chiral hingeinsulators, which are characterized by quantized pumping of dipole andquadrupole moments. The invariants that we devise are written entirely in termsof many-body ground state wavefunctions on a torus geometry with a set ofunitary operators. We provide a number of supporting evidences for ourinvariants via topological field theory interpretation, adiabatic pumpingargument, and direct mapping to free-fermion band indices. We finally confirmour invariants by numerical computations including infinite density matrixrenormalization group on a quasi-one-dimensional system. The many-bodyinvariants therefore explicitly encircle several different pillars oftheoretical descriptions of topological phases.

Many-Body Invariants for Chern and Chiral Hinge Insulators