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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Updated version: The authors have uploaded version v3 of this work to the arXiv which may contain updates or corrections not contained in the published version v2. The authors left the following comment on the arXiv:
This version includes an appendix on extensions to complex Hamiltonians and arbitrary observables that is not included in the journal-published paper

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.

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► References

[1] U. von Barth, Basic density-functional theory—an overview, Phys. Scr. 2004, 9 (2004).
https:/​/​doi.org/​10.1238/​Physica.Topical.109a00009

[2] K. Burke and friends, The ABC of DFT, (2007).
https:/​/​dft.uci.edu/​doc/​g1.pdf

[3] R. M. Dreizler and E. K. Gross, Density functional theory: An approach to the quantum many-body problem (Springer, 2012).

[4] H. Eschrig, The fundamentals of density functional theory, 2nd ed. (Springer, 2003).

[5] C. A. Ullrich, Time-dependent density-functional theory: Concepts and applications (OUP Oxford, 2011).

[6] C. A. Ullrich and Z. Yang, A brief compendium of time-dependent density functional theory, Braz. J. Phys. 44, 154 (2014).
https:/​/​doi.org/​10.1007/​s13538-013-0141-2

[7] G. Vignale and M. Rasolt, Density-functional theory in strong magnetic fields, Phys. Rev. Lett. 59, 2360 (1987).
https:/​/​doi.org/​10.1103/​PhysRevLett.59.2360

[8] G. Vignale, Mapping from current densities to vector potentials in time-dependent current density functional theory, Phys. Rev. B 70, 201102 (2004).
https:/​/​doi.org/​10.1103/​PhysRevB.70.201102

[9] M. Ruggenthaler, J. Flick, C. Pellegrini, H. Appel, I. V. Tokatly, and A. Rubio, Quantum-electrodynamical density-functional theory: Bridging quantum optics and electronic-structure theory, Phys. Rev. A 90, 012508 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.012508

[10] C. A. Ullrich and W. Kohn, Degeneracy in density functional theory: Topology in the v and n spaces, Phys. Rev. Lett. 89, 156401 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.156401

[11] L. Garrigue, Some properties of the potential-to-ground state map in quantum mechanics, Commun. Math. Phys. 386, 1803 (2021).
https:/​/​doi.org/​10.1007/​s00220-021-04140-9.pdf

[12] D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, The Hubbard model, Annu. Rev. Condens. Matter Phys. 13, 239 (2022).
https:/​/​doi.org/​10.1146/​annurev-conmatphys-031620-102024

[13] M. Qin, T. Schäfer, S. Andergassen, P. Corboz, and E. Gull, The Hubbard model: A computational perspective, Annu. Rev. Condens. Matter Phys. 13, 275 (2022).
https:/​/​doi.org/​10.1146/​annurev-conmatphys-090921-033948

[14] F. Flores, D. Soler-Polo, and J. Ortega, A closed local-orbital unified description of dft and many-body effects, J. Phys. Condens. Matter 34, 304006 (2022).
https:/​/​doi.org/​10.1088/​1361-648X/​ac6eae

[15] M. Penz and R. van Leeuwen, Density-functional theory on graphs, J. Chem. Phys. 155, 244111 (2021).
https:/​/​doi.org/​10.1063/​5.0074249

[16] E. H. Lieb, Density functionals for Coulomb-systems, Int. J. Quantum Chem. 24, 243 (1983).
https:/​/​doi.org/​10.1002/​qua.560240302

[17] E. I. Tellgren, A. Laestadius, T. Helgaker, S. Kvaal, and A. M. Teale, Uniform magnetic fields in density-functional theory, J. Chem. Phys. 148, 024101 (2018).
https:/​/​doi.org/​10.1063/​1.5007300

[18] M. Penz, E. I. Tellgren, M. A. Csirik, M. Ruggenthaler, and A. Laestadius, The structure of the density-potential mapping. Part I: Standard density-functional theory, arXiv preprint (2022), arXiv:2211.16627 [physics.chem-ph].
arXiv:2211.16627

[19] M. Lewin, E. H. Lieb, and R. Seiringer, Universal functionals in density functional theory, arXiv preprint (2019), arXiv:1912.10424 [math-ph].
arXiv:1912.10424

[20] L. Garrigue, Unique continuation for many-body Schrödinger operators and the Hohenberg–Kohn theorem, Math. Phys. Anal. Geom. 21, 27 (2018).
https:/​/​doi.org/​10.1007/​s11040-018-9287-z

[21] I. Bárány and R. Karasev, Notes about the Carathéodory number, Discrete Comput. Geom. 48, 783 (2012).
https:/​/​doi.org/​10.1007/​s00454-012-9439-z

[22] M. C. Beltrametti, E. Carletti, D. Gallarati, and G. Monti Bragadin, Lectures on curves, surfaces and projective varieties (European Mathematical Society, 2009).

[23] J. Harris, Algebraic geometry: A first course (Springer, 1992).

[24] W. L. F. Degen, The types of triangular Bézier surfaces, Proceedings of the 6th IMA Conference on the Mathematics of Surfaces , 153 (1994).
https:/​/​doi.org/​10.5555/​646872.709694

[25] L. Garrigue, Building Kohn-Sham potentials for ground and excited states, Arch. Rational Mech. Anal. 245, 949 (2022).
https:/​/​doi.org/​10.1007/​s00205-022-01804-1

[26] F. Apéry, Models of the Real Projective Plane (Vieweg, 1987).

[27] E. Fortuna, R. Frigerio, and R. Pardini, Projective Geometry: Solved Problems and Theory Review, Vol. 104 (Springer, 2016).

[28] T. Sederberg and D. Anderson, Steiner surface patches, IEEE Comput. Graph. Appl. 5, 23 (1985).
https:/​/​doi.org/​10.1109/​MCG.1985.276391

[29] A. Coffman, A. Schwartz, and C. Stanton, The algebra and geometry of Steiner and other quadratically parametrizable surfaces, Comput. Aided Geom. Des. 13, 257 (1996).
https:/​/​doi.org/​10.1016/​0167-8396(95)00026-7

[30] C. Michel, Compléments de géométrie moderne (Vuibert, 1926).

[31] A. Clebsch, Ueber die Steinersche Fläche. Journal für die reine und angewandte Mathematik 67, 1 (1867).

[32] C. Cayley, On Steiner's surface, Proc. Lond. Math. Soc. 1, 14 (1873).
https:/​/​doi.org/​10.1112/​plms/​s1-5.1.14

[33] E. Lacour, Sur la surface de Steiner, Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale 17, 437 (1898).

[34] D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Vol. 87 (American Mathematical Society, 2021).

[35] G. Liu, M. Pi, L. Zhou, Z. Liu, X. Shen, X. Ye, S. Qin, X. Mi, X. Chen, L. Zhao, et al., Physical realization of topological Roman surface by spin-induced ferroelectric polarization in cubic lattice, Nature Comm. 13, 2373 (2022).
https:/​/​doi.org/​10.1038/​s41467-022-29764-w

[36] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, 4th ed. (Springer, 2012).

[37] S. Kvaal, U. Ekström, A. M. Teale, and T. Helgaker, Differentiable but exact formulation of density-functional theory, J. Chem. Phys. 140, 18A518 (2014).
https:/​/​doi.org/​10.1063/​1.4867005

[38] A. Laestadius, M. Penz, E. I. Tellgren, M. Ruggenthaler, S. Kvaal, and T. Helgaker, Generalized Kohn–Sham iteration on Banach spaces, J. Chem. Phys. 149, 164103 (2018).
https:/​/​doi.org/​10.1063/​1.5037790

[39] M. Levy, Electron densities in search of Hamiltonians, Phys. Rev. A 26, 1200 (1982).
https:/​/​doi.org/​10.1103/​PhysRevA.26.1200

[40] F. Rellich, Störungstheorie der Spektralzerlegung, I. Mitteilung, Mathematische Annalen 113, 600 (1937).
https:/​/​doi.org/​10.1007/​BF01571652

[41] F. Rellich, Perturbation theory of eigenvalue problems (Gordon and Breach Science Publishers, 1969).

[42] T. Kato, Perturbation theory for linear operators (Springer, 1995).

[43] M. Penz, A. Laestadius, E. I. Tellgren, and M. Ruggenthaler, Guaranteed convergence of a regularized Kohn–Sham iteration in finite dimensions, Phys. Rev. Lett. 123, 037401 (2019).
https:/​/​doi.org/​10.1103/​physrevlett.123.037401

[44] M. Penz, A. Laestadius, E. I. Tellgren, M. Ruggenthaler, and P. E. Lammert, Erratum: Guaranteed convergence of a regularized Kohn–Sham iteration in finite dimensions, Phys. Rev. Lett. 125, 249902 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.249902

[45] A. Laestadius, E. I. Tellgren, M. Penz, M. Ruggenthaler, S. Kvaal, and T. Helgaker, Kohn–Sham theory with paramagnetic currents: Compatibility and functional differentiability, J. Chem. Theory Comput. 15, 4003 (2019).
https:/​/​doi.org/​10.1021/​acs.jctc.9b00141

[46] A. Laestadius and E. I. Tellgren, Density–wave-function mapping in degenerate current-density-functional theory, Phys. Rev. A 97, 022514 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.022514

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[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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