Geometry of Degeneracy in Potential and Density Space

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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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.

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Cited by

[1] Julia Liebert, Adam Yanis Chaou, and Christian Schilling, "Refining and relating fundamentals of functional theory", The Journal of Chemical Physics 158 21, 214108 (2023).

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