Geometry of Degeneracy in Potential and Density Space

Geometry of Degeneracy in Potential and Density Space

Time Stamp: February 9, 2023 9:26 AM
Source Node: 1950118

Markus Penz1 and Robert van Leeuwen2

1Basic Research Community for Physics, Innsbruck, Austria
2Department of Physics, Nanoscience Center, University of Jyväskylä, Finland

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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.

Featured image: This algebraic variety, called the Roman surface, is formed by all one-particle densities corresponding to wave functions from the real span of a three-dimensional eigenspace in the Hilbert space of two particles on a tetrahedron graph. The densities for all states from the same eigenspace form the convex hull of this variety. The octahedron around it is the full density domain, where the corners correspond to the extreme
density (1, 1, 0, 0) and its permutations on the tetrahedron graph.

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