In crystals, quantum electrons can be spatially distributed in a way that the bulk solid supports macroscopic electric multipole moments, which are deeply related with emergence of topology in condensed matter systems, such as the topological insulators. However, unlike the classical multipoles in open space, defining multipoles in crystals is a non-trivial task, and only the dipolar moment, namely polarization, has been successfully defined so far. This polarization, materialized as Su-Schrieffer-Heeger chain, served as a classic example of modern discussions of topological insulators.
In this talk, we propose the many-body invariants, i.e., the general definition, for electric multipoles in crystals, which is related with recently-discovererd higher-order topological insulators. We generalize Resta’s pioneering work on polarizations to the multipoles, which are designed to measure the distribution of electron charge in unit cells and thus can detect multipole moments purely from the bulk ground state wavefunctions. We provide analytic as well as numerical supports for our invariants. Application of our invariants to spin systems as well as various other aspects of the many-body invariants will be discussed.