Many-body electric multipole operators in extended systems

Abstract

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 1 , 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 n-th order multipole operators whose expectation values can be used to calculate the n-th 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 − 1-st multipole currents, and encode the adiabatic evolution of the system in the presence of an n − 1-st gradient of the electric field. Finally, we test our operators on a set of tightbinding 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.

First-Principles Theory of Spatial Dispersion: Dynamical Quadrupoles and Flexoelectricity

“…[36] On a different note, our theory might also prove itself very helpful in establishing higher-order multipolar generalizations [37] of the Berry-phase theory of polarization [38]. Indeed, our expressions for the dynamical quadrupole and flexoelectric tensors can be regarded as the linear variation of the "bulk quadrupolization" [39] with respect to a zone-center lattice distortion or uniform strain, respectively. There are intriguing parallels to the theory of multipolar magnetic orders [40,41] as well, which will certanly stimulate further studies.…”

Plasmon-polaritonic quadrupole topological insulators

“…With such geometric control, the relative strength of the intra-unit-cell coupling and the inter-unitcell coupling can be switched. We thus determine the topological phase diagram of the plasmon-polaritonic lattice using the phase diagram of the BBH model [2,12].…”

Topological states in generalized electric quadrupole insulators

“…In addition, the nested Wilson loop approach [16] originally used to obtain the topological invariant from the momentum space is no longer applicable in the disordered systems. Here, we prove that the quadrupole moment defined in the real space, given by [51][52][53]…”