The choice of basic variables in current-density functional theory

TL;DR

This paper critically examines the foundational choices of basic variables in current-density functional theory, highlighting issues with the gauge-dependent current density and proposing a redefinition to restore mathematical properties like concavity.

Contribution

It demonstrates that redefining the scalar potential restores convexity in current-density functional theory, and critically assesses the mathematical foundations of theories using physical current density.

Findings

Redefinition of scalar potential restores concavity in energy functional.

Theories with physical current density lack solid mathematical foundations.

Standard density-functional theory concepts do not fully transfer to current-density functional theory.

Abstract

The selection of basic variables in current-density functional theory and formal properties of the resulting formulations are critically examined. Focus is placed on the extent to which the Hohenberg--Kohn theorem, constrained-search approach and Lieb's formulation (in terms of convex and concave conjugation) of standard density-functional theory can be generalized to provide foundations for current-density functional theory. For the well-known case with the gauge-dependent paramagnetic current density as a basic variable, we find that the resulting total energy functional is not concave. It is shown that a simple redefinition of the scalar potential restores concavity and enables the application of convex analysis and convex/concave conjugation. As a result, the solution sets arising in potential-optimization problems can be given a simple characterization. We also review attempts to establish theories with the physical current density as a basic variable. Despite the appealing physical motivation behind this choice of basic variables, we find that the mathematical foundations of the theories proposed to date are unsatisfactory. Moreover, the analogy to standard density-functional theory is substantially weaker as neither the constrained-search approach nor the convex analysis framework carry over to a theory making use of the physical current density.