Time-dependent density functional theory on a lattice

TL;DR

This paper rigorously formulates a time-dependent density functional theory for lattice systems, proving key theorems on the uniqueness and existence of solutions, applicable to any lattice configuration.

Contribution

It establishes the first rigorous proof of the density-to-potential mapping and $v$-representability conditions for lattice TDDFT, extending the theory's applicability.

Findings

Proves the uniqueness of the density-to-potential mapping.

Shows $v$-representability for densities with continuous second derivatives.

Demonstrates the theory's validity for any connected lattice regardless of size or geometry.

Abstract

A time-dependent density functional theory (TDDFT) for a quantum many-body system on a lattice is formulated rigorously. We prove the uniqueness of the density-to-potential mapping and demonstrate that a given density is $v$-representable if the initial many-body state and the density satisfy certain well defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is $v$-representable and therefore the lattice TDDFT is guaranteed to exist. The TDDFT existence and uniqueness theorem is valid for any connected lattice, independently of its size, geometry and/or spatial dimensionality. The general statements of the existence theorem are illustrated on a pedagogical exactly solvable example which displays all details and subtleties of the proof in a transparent form. In conclusion we briefly discuss remaining open problems and directions for a future research.