Many-Body Electric Multipole Operators in Extended Systems

Understand

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.