Coarse-grained spin density-functional theory: infinite-volume limit via the hyperfinite

TL;DR

This paper develops a rigorous mathematical framework for coarse-grained spin density functional theory (SDFT) using nonstandard analysis, establishing existence, uniqueness, and regularity results for representing potentials in the infinite-volume limit.

Contribution

It introduces a nonstandard analysis approach to coarse-grained SDFT, proving existence and regularity of potentials and analyzing non-uniqueness issues in the infinite-volume limit.

Findings

Every nowhere spin-saturated density is V-representable.

Representing potentials form the generalized functional derivative of the Lieb energy.

Non-uniqueness occurs for collinear eigenstates of S_z.

Abstract

Coarse-grained spin density functional theory (SDFT) is a version of SDFT which works with number/spin densities specified to a limited resolution --- averages over cells of a regular spatial partition --- and external potentials constant on the cells. This coarse-grained setting facilitates a rigorous investigation of the mathematical foundations which goes well beyond what is currently possible in the conventional formualation. Problems of existence, uniqueness and regularity of representing potentials in the coarse-grained SDFT setting are here studied using techniques of (Robinsonian) nonstandard analysis. Every density which is nowhere spin-saturated is V-representable, and the set of representing potentials is the functional derivative, in an appropriate generalized sense, of the Lieb interal energy functional. Quasi-continuity and closure properties of the set-valued representing potentials map are also established. The extent of possible non-uniqueness is similar to that found in non-rigorous studies of the conventional theory, namely non-uniqueness can occur for states of collinear magnetization which are eigenstates of $S_z$.