Fermion $N$-representability for prescribed density and paramagnetic current density

Abstract

The $N$-representability problem is the problem of determining whether or not there exists $N$-particle states with some prescribed property. Here we report an affirmative solution to the fermion $N$-representability problem when both the density and paramagnetic current density are prescribed. This problem arises in current-density functional theory and is a generalization of the well-studied corresponding problem (only the density prescribed) in density functional theory. Given any density and paramagnetic current density satisfying a minimal regularity condition (essentially that a von Weizäcker-like the canonical kinetic energy density is locally integrable), we prove that there exist a corresponding $N$-particle state. We prove this by constructing an explicit one-particle reduced density matrix in the form of a position-space kernel, i.e.\ a function of two continuous position variables. In order to make minimal assumptions, we also address mathematical subtleties regarding the diagonal of, and how to rigorously extract paramagnetic current densities from, one-particle reduced density matrices in kernel form.

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• Fractional occupation in Kohn-Sham density-functional theory and the treatment of non-pure-state v-representable densities

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• The structure of the density-potential mapping. Part II: Including magnetic fields
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• Metric space approach to potentials and its relevance to density functional theory
• Metric space analysis of systems immersed in a magnetic field
• Physical spin torques from exactly constrained exchange-correlation torques
• Pure-state $N$-representability in current-spin-density-functional theory