Geometry of Degeneracy in Potential and Density Space

TL;DR

This paper explores the geometric structure of degeneracy regions in density and potential space within density-functional theory, revealing their complex shapes and rare intersections, which challenge the fundamental Hohenberg-Kohn theorem.

Contribution

It characterizes the geometry of degeneracy regions as convex hulls of algebraic varieties and analyzes their interactions with potential spaces, extending previous findings to continuum systems.

Findings

Degeneracy regions are convex hulls of algebraic varieties.

Rare intersections occur where degeneracy regions touch or boundary of density domain.

Examples include shapes like the Roman surface illustrating complex geometries.

Abstract

In a previous work [J. Chem. Phys. 155, 244111 (2021)], we found counterexamples to the fundamental Hohenberg-Kohn theorem from density-functional theory in finite-lattice systems represented by graphs. Here, we demonstrate that this only occurs at very peculiar and rare densities, those where density sets arising from degenerate ground states, called degeneracy regions, touch each other or the boundary of the whole density domain. Degeneracy regions are shown to generally be in the shape of the convex hull of an algebraic variety, even in the continuum setting. The geometry arising between density regions and the potentials that create them is analyzed and explained with examples that, among other shapes, feature the Roman surface.