# Many-Body Electric Multipole Operators in Extended Systems

## TL;DR

This paper develops a formalism for defining and calculating higher-order multipole moments in extended quantum systems with periodic boundary conditions, enabling better understanding of topological phases and electric multipole responses.

## Contribution

It introduces many-body operators for higher multipole moments and demonstrates their effectiveness in characterizing topological quadrupole and octupole phases.

## Key findings

- Operators correctly identify phase diagrams of topological multipole models
- Capture adiabatic multipole pumping phenomena
- Distinguish bulk multipole moments from corner charge effects

## 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 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.