Time-dependent current density functional theory on a lattice

TL;DR

This paper establishes a rigorous foundation for time-dependent current density functional theory on lattices, proving local and global ${ m V}$-representability under certain conditions, thus enabling precise studies of lattice models.

Contribution

It provides a rigorous formulation of lattice TDCDFT, reducing key problems to nonlinear Schrödinger equations and proving ${ m V}$-representability for continuous current densities.

Findings

Any continuous in time current density is locally ${ m V}$-representable with nonzero initial kinetic energy.

Results apply to both interacting and noninteracting systems.

Global ${ m V}$-representability is expected in most physical cases.

Abstract

A rigorous formulation of time-dependent current density functional theory (TDCDFT) on a lattice is presented. The density-to-potential mapping and the ${\cal V}$-representability problems are reduced to a solution of a certain nonlinear lattice Schr\"odinger equation, to which the standard existence and uniqueness results for nonliner differential equations are applicable. For two versions of the lattice TDCDFT we prove that any continuous in time current density is locally ${\cal V}$-representable (both interacting and noninteracting), provided in the initial state the local kinetic energy is nonzero everywhere. In most cases of physical interest the ${\cal V}$-representability should also hold globally in time. These results put the application of TDCDFT to any lattice model on a firm ground, and open a way for studying exact properties of exchange correlation potentials.