Geometry of Degeneracy in Potential and Density Space

• URL: http://arxiv.org/abs/2206.12366v3
• Date: Tue, 6 Feb 2024 11:14:43 GMT
• Title: Geometry of Degeneracy in Potential and Density Space
• Authors: Markus Penz, Robert van Leeuwen
• Abstract summary: We show counterexamples to the fundamental Hohenberg-Kohn theorem from density-functional theory in finite-lattice systems represented by graphs. The geometry arising between density regions and the potentials that create them is analyzed and explained.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Geometrical Perspective on Spin-Lattice Density-Functional Theory [0.0]
  A recently developed viewpoint on the fundamentals of density-functional theory for finite interacting spin-lattice systems is presented. It allows for an entirely geometrical description of the Hohenberg-Kohn theorem and v-representability.
  arXiv  Detail & Related papers  (2024-07-24T09:39:44Z)
• Quantum Scattering of Spinless Particles in Riemannian Manifolds [0.0]
  Quantum mechanics is sensitive to the geometry of the underlying space. We present a framework for quantum scattering of a non-relativistic particle confined to a two-dimensional space.
  arXiv  Detail & Related papers  (2024-02-16T10:50:50Z)
• From locality to irregularity: Introducing local quenches in massive scalar field theory [68.8204255655161]
  We consider the dynamics of excited local states in massive scalar field theory in an arbitrary spacetime dimension. We identify different regimes of their evolution depending on the values of the field mass and the quench regularization parameter. We also investigate the local quenches in massive scalar field theory on a cylinder and show that they cause an erratic and chaotic-like evolution of observables.
  arXiv  Detail & Related papers  (2022-05-24T18:00:07Z)
• A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations [77.86290991564829]
  Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. We study a particular sequence of maps between manifold, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between implementing neural networks of practical interest.
  arXiv  Detail & Related papers  (2021-12-17T11:43:30Z)
• Multipartitioning topological phases by vertex states and quantum entanglement [9.519248546806903]
  We discuss multipartitions of the gapped ground states of (2+1)-dimensional topological liquids into three spatial regions. We compute various correlation measures, such as entanglement negativity, reflected entropy, and associated spectra. As specific examples, we consider topological chiral $p$-wave superconductors and Chern insulators.
  arXiv  Detail & Related papers  (2021-10-22T18:01:24Z)
• Density-Functional Theory on Graphs [0.0]
  The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn is found void in general.
  arXiv  Detail & Related papers  (2021-06-29T12:55:54Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)
• Long-distance entanglement of purification and reflected entropy in conformal field theory [58.84597116744021]
  We study entanglement properties of mixed states in quantum field theory via entanglement of purification and reflected entropy. We find an elementary proof that the decay of both, the entanglement of purification and reflected entropy, is enhanced with respect to the mutual information behaviour.
  arXiv  Detail & Related papers  (2021-01-29T19:00:03Z)
• Density profile of a semi-infinite one-dimensional Bose gas and bound states of the impurity [62.997667081978825]
  We study the effect of the boundary on a system of weakly interacting bosons in one dimension. The quantum contribution to the boson density gives rise to small corrections of the bound state energy levels.
  arXiv  Detail & Related papers  (2020-07-21T13:12:33Z)
• Geometry of Similarity Comparisons [51.552779977889045]
  We show that the ordinal capacity of a space form is related to its dimension and the sign of its curvature. More importantly, we show that the statistical behavior of the ordinal spread random variables defined on a similarity graph can be used to identify its underlying space form.
  arXiv  Detail & Related papers  (2020-06-17T13:37:42Z)
• Topological hyperbolic lattices [0.0]
  We show how the quantized curvature and edge dominance of hyperbolic geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory.
  arXiv  Detail & Related papers  (2020-03-16T03:41:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.