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We extend the theory of dipole moments in crystalline insulators to highermultipole moments. In this paper, we expand in great detail the theorypresented in Ref. 1, and extend it to cover associated topological pumpingphenomena, and a novel class of 3D insulator with chiral hinge states. Inquantum-mechanical crystalline insulators, higher multipole bulk momentsmanifest themselves by the presence of boundary-localized moments of lowerdimension, in exact correspondence with the electromagnetic theory of classicalcontinuous dielectrics. In the presence of certain symmetries, these momentsare quantized, and their boundary signatures are fractionalized. Thesemultipole moments then correspond to new SPT phases. The topological structureof these phases is described by "nested" Wilson loops, which reflect thebulk-boundary correspondence in a way that makes evident a hierarchicalclassification of the multipole moments. Just as a varying dipole generatescharge pumping, a varying quadrupole generates dipole pumping, and a varyingoctupole generates quadrupole pumping. For non-trivial adiabatic cycles, thetransport of these moments is quantized. An analysis of these interconnectedphenomena leads to the conclusion that a new kind of Chern-type insulatorexists, which has chiral, hinge-localized modes in 3D. We provide the minimalmodels for the quantized multipole moments, the non-trivial pumping processesand the hinge Chern insulator, and describe the topological invariants thatprotect them.

Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators