In this article we extend the celebrated Berry-phase formulation of electric
polarization in crystals to higher electric multipole moments. We determine the
necessary conditions under which, and minimal models in which, the quadrupole
and octupole moments are topologically quantized electromagnetic observables.
Such systems exhibit gapped boundaries that are themselves lower-dimensional
topological phases. Furthermore, they manifest topologically protected corner
states carrying fractional charge, i.e., fractionalization at the boundary of
the boundary. To characterize these new insulating phases of matter, we
introduce a new paradigm whereby `nested' Wilson loops give rise to a large
number of new topological invariants that have been previously overlooked. We
propose three realistic experimental implementations of this new topological
behavior that can be immediately tested.Comment: Main text: 9 pages, 6 figures. Supplementary Material: 37 pages, 15
figures. Submitted on Jul 25, 201
