Density-functional theory for internal magnetic fields

TL;DR

This paper develops a density-functional theory incorporating internal magnetic fields based on the Maxwell–Schrödinger equation, providing a convex, variational framework that generalizes existing current-density functional theories.

Contribution

It introduces a new physical current-density functional theory with strong convexity properties, extending Lieb's formulation and enabling differentiable energy functionals for magnetic systems.

Findings

Formulates a convex energy functional for magnetic fields

Establishes a Hohenberg–Kohn-like mapping for ground states

Uses Moreau–Yosida regularization to ensure differentiability

Abstract

A density-functional theory is developed based on the Maxwell--Schr\"odinger equation with an internal magnetic field in addition to the external electromagnetic potentials. The basic variables of this theory are the electron density and the total magnetic field, which can equivalently be represented as a physical current density. Hence, the theory can be regarded as a physical current-density functional theory and an alternative to the paramagnetic current density-functional theory due to Vignale and Rasolt. The energy functional has strong enough convexity properties to allow a formulation that generalizes Lieb's convex analysis-formulation of standard density-functional theory. Several variational principles as well as a Hohenberg--Kohn-like mapping between potentials and ground-state densities follow from the underlying convex structure. Moreover, the energy functional can be regarded as the result of a standard approximation technique (Moreau--Yosida regularization) applied to the conventional Schr\"odinger ground state energy, which imposes limits on the maximum curvature of the energy (w.r.t.\ the magnetic field) and enables construction of a (Fr\'echet) differentiable universal density functional.