  We extend the theory of dipole moments in crystalline insulators to higher
multipole moments. In this paper, we expand in great detail the theory
presented in Ref. 1, and extend it to cover associated topological pumping
phenomena, and a novel class of 3D insulator with chiral hinge states. In
quantum-mechanical crystalline insulators, higher multipole bulk moments
manifest themselves by the presence of boundary-localized moments of lower
dimension, in exact correspondence with the electromagnetic theory of classical
continuous dielectrics. In the presence of certain symmetries, these moments
are quantized, and their boundary signatures are fractionalized. These
multipole moments then correspond to new SPT phases. The topological structure
of these phases is described by "nested" Wilson loops, which reflect the
bulk-boundary correspondence in a way that makes evident a hierarchical
classification of the multipole moments. Just as a varying dipole generates
charge pumping, a varying quadrupole generates dipole pumping, and a varying
octupole generates quadrupole pumping. For non-trivial adiabatic cycles, the
transport of these moments is quantized. An analysis of these interconnected
phenomena leads to the conclusion that a new kind of Chern-type insulator
exists, which has chiral, hinge-localized modes in 3D. We provide the minimal
models for the quantized multipole moments, the non-trivial pumping processes
and the hinge Chern insulator, and describe the topological invariants that
protect them.
