Many-body electric multipole operators in extended systems

The quantum mechanical position operators, and their products, are not well-defined in systems obeying periodic boundary conditions. Here we extend the work of Resta [Phys. Rev. Lett. 80, 1800 (1998)], who developed a formalism to calculate the electronic polarization as an expectation value of a many-body operator, to include higher multipole moments, e.g., quadrupole and octupole. We define nth-order multipole operators whose expectation values can be used to calculate the nth multipole moment when all of the lower moments are vanishing (modulo a quantum). We show that changes in our operators are tied to flows of n-1st multipole currents, and encode the adiabatic evolution of the system in the presence of an n-1st gradient of the electric field. Finally, we test our operators on a set of tight-binding models to show that they correctly determine the phase diagrams of topological quadrupole and octupole models, capture an adiabatic quadrupole pump, and distinguish a bulk quadrupole moment from other mechanisms that generate corner charges.