Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators

The Berry phase provides a modern formulation of electric polarization in crystals. Recently, this formulation was extended by the authors to higher electric multipole moments. In quantum mechanical crystalline insulators, such higher multipole moments manifest themselves by the presence of boundary-localized moments of lower dimension. In the presence of certain symmetries, these moments are quantized, and their boundary signatures are fractionalized. Here, the authors elaborate in detail on the theory of higher multipole moments and discuss associated topological pumping phenomena, as well as higher-order topological insulators derived from them.