Quantized Electric Multipole Insulators

Understand

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.