Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometry of Degeneracy in Potential and Density Space

Markus Penz, Robert van Leeuwen

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.

Geometry of Degeneracy in Potential and Density Space

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.
Paper Structure (9 sections, 10 theorems, 37 equations, 7 figures)

This paper contains 9 sections, 10 theorems, 37 equations, 7 figures.

Table of Contents

  1. Introduction
  2. State subspaces under the density map
  3. Further classification and examples for density regions
  4. Non-pure-state $v$-representable densities
  5. Degeneracy-preserving potential mani-folds
  6. Geometrization of the potential-density mapping
  7. Summary and open questions
  8. Extension to complex hermitian Hamiltonians
  9. Extension to sets of arbitrary observables

Key Result

Lemma 1

If $H_v$ is real symmetric and has a $g$-dimensional eigenspace $\mathcal{U}$ with eigenvalue $E$ then this space is spanned by $g$real orthonormal vectors $\{\Phi_k\}_{k=1}^g$ with complex coefficients, $\mathcal{U} = \mathop{\mathrm{span}}\nolimits_\mathbb{C} \{ \Phi_1,\ldots,\Phi_g \}$.

Figures (7)

  • Figure 1: The three-fold degeneracy region $D_\mathbb{R}$ of the tetrahedron graph inside the octahedronal density domain. The corners of the octahedron correspond to the extreme density $(1,1,0,0)$ and its permutations, generalized barycentric coordinates are used to display densities.
  • Figure 2: Three different parametrizations for the Roman surface that all show how it can be constructed from ellipses. The first corresponds to introducing polar coordinates for $x\in\mathbb{R}^3$ in \ref{['eq:P-nu-map']}, the second to doing the same in \ref{['eq:D_R-Roman-surface']}, while the third is from apery-book.
  • Figure 3: The union of all degeneracy regions (degeneracy structure) of the tetrahedron graph inside the octahedral set of densities $\mathcal{P}_{4,2}$. The four degeneracy bundles in the shape of cylinders reaching out from the flat sides of the convexified Roman surface have to be imagined as filled.
  • Figure 4: The two situations of level crossings in the proof of Theorem \ref{['th:geometry']}.
  • Figure 5: Left, the union of all degeneracy regions of the square graph inside the octahedral set of densities $\mathcal{P}_{4,2}$. Right, four elliptical degeneracy regions from it are displayed that mutually touch at the diagonals drawn in red.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • Corollary 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 7 more