# Kohn-Sham theory with paramagnetic currents: compatibility and functional differentiability

## TL;DR

This paper extends Moreau-Yosida regularization to paramagnetic current-density-functional theory, providing a rigorous mathematical framework and a Kohn-Sham iteration scheme for magnetic field effects, with numerical illustrations.

## Contribution

It introduces a compatible convex formulation for paramagnetic current-density-functional theory using Moreau-Yosida regularization, including a well-defined Kohn-Sham scheme and numerical results.

## Key findings

- Regularized functionals computed for a quantum ring.
- Compatibility conditions for function spaces analyzed.
- Numerical performance demonstrated for different regularizations.

## Abstract

Recent work has established Moreau-Yosida regularization as a mathematical tool to achieve rigorous functional differentiability in density-functional theory. In this article, we extend this tool to paramagnetic current-density-functional theory, the most common density-functional framework for magnetic field effects. The extension includes a well-defined Kohn-Sham iteration scheme with a partial convergence result. To this end, we rely on a formulation of Moreau-Yosida regularization for reflexive and strictly convex function spaces. The optimal $L^p$-characterization of the paramagnetic current density $L^1\cap L^{3/2}$ is derived from the $N$-representability conditions. A crucial prerequisite for the convex formulation of paramagnetic current-density-functional theory, termed compatibility between function spaces for the particle density and the current density, is pointed out and analyzed. Several results about compatible function spaces are given, including their recursive construction. The regularized, exact functionals are calculated numerically for a Kohn-Sham iteration on a quantum ring, illustrating their performance for different regularization parameters.