A mathematical formulation of the random phase approximation for crystals

TL;DR

This paper rigorously extends the mathematical understanding of the nonlinear Hartree dynamics, known as the random phase approximation, for crystals, including existence, uniqueness, and derivation of macroscopic equations.

Contribution

It provides a rigorous mathematical formulation of the time-dependent RPA for crystals, including existence, uniqueness, and derivation of macroscopic Maxwell equations.

Findings

Proved existence and uniqueness of nonlinear Hartree dynamics in crystals.

Defined the microscopic frequency-dependent polarization matrix rigorously.

Derived macroscopic Maxwell-Gauss equations from the Hartree model using homogenization.

Abstract

This works extends the recent study on the dielectric permittivity of crystals within the Hartree model [E. Cances and M. Lewin, Arch. Rational Mech. Anal., 197 (2010) 139--177] to the time-dependent setting. In particular, we prove the existence and uniqueness of the nonlinear Hartree dynamics (also called the random phase approximation in the physics literature), in a suitable functional space allowing to describe a local defect embedded in a perfect crystal. We also give a rigorous mathematical definition of the microscopic frequency-dependent polarization matrix, and derive the macroscopic Maxwell-Gauss equation for insulating and semiconducting crystals, from a first order approximation of the nonlinear Hartree model, by means of homogenization arguments.