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arXiv:1810.08391 · Physical review. B./Physical review. B

Representability problem of density functional theory for superconductors

Jonathan Schmidt, Carlos L. Benavides-Riveros, Miguel A. L. Marques

Published Jan 4, 2019• 6 citations
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Abstract

Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross, and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn- Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system [Phys. Rev. Lett. 60, 2430 (1988)]. However, the question of noninteracting $v$-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.

DOI
10.1103/physrevb.99.024502

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Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross, and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn- Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system [Phys. Rev. Lett. 60, 2430 (1988)]. However, the question of noninteracting vvv-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.
Inspired by earlier work on the band-gap problem in insulators, we reexamine the treatment of strongly correlated Hubbard-type models within density-functional theory. In contrast to previous studies, the density is fully parametrized by occupation numbers and overlap of orbitals centered at neighboring atomic sites, as is the local potential by the hopping matrix. This corresponds to a good formal agreement between density-functional theory in real space and second quantization. It is shown that density-functional theory is formally applicable to such systems and the theoretical framework is provided. The question of noninteracting v representability is studied numerically for finite one-dimnsional clusters, for which exact results are available, and qualitatively for infinite systems. This leads to the conclusion that the electron density corresponding to interacting systems of the type studied here is in fact not noninteracting v representable because the Kohn-Sham electrons are unable to reproduce the correlation-induced localization correctly.
We present an ab initio theory for superconductors, based on a unique mapping between the statistical density operator at equilibrium, on the one hand, and the corresponding one-body reduced density matrix $\ensuremath{\gamma}$ and the anomalous density $\ensuremath{\chi}$, on the other. This formalism for superconductivity yields the existence of a universal functional ${\mathfrak{F}}_{\ensuremath{\beta}}[\ensuremath{\gamma},\ensuremath{\chi}]$ for the superconductor ground state, whose unique properties we derive. We then prove the existence of a Kohn-Sham system at finite temperature and derive the corresponding Bogoliubov--de Gennes--type single-particle equations. By adapting the decoupling approximation from density functional theory for superconductors we bring these equations into a computationally feasible form. Finally, we use the existence of the Kohn-Sham system to extend the Sham-Schl\"uter connection and derive a first exchange-correlation functional for our theory. This reduced density matrix functional theory for superconductors has the potential of overcoming some of the shortcomings and fundamental limitations of density functional theory of superconductivity.
On the non-interacting vvv-representability conjecture within density functional theory
Density functional theory together with the Kohn-Sham (KS) scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary system (one-body with a local potential that is a density functional). However, to fully achieve establishing the KS scheme, a general proof of the validity of the so-called vvv-representability conjecture is still required. In this article, we propose such a proof. We reduce the demonstration to proving (1) that the KS potential is differentiable for all non-interacting vvv-representable densities and (2) that its derivative is bounded. Then, we demonstrate points (1) and (2) applying static linear response to the non-interacting system.
A particle-density distribution is said to be v-representable if it is the ground-state density for some potential v(r). By developing the idea of proper order, proposed earlier [J. Chen and M. J. Stott, Phys. Rev. A 44, 2816 (1991)], to any number of energy levels which are involved in possible ground-state configurations, we show that any reasonable density with an arbitrary number of spinless fermions is either pure-state or ensemble v-representable. This result supports the general applicability of the Kohn-Sham method [Phys. Rev. 140, A1133 (1965)] in density-functional theory.
Notes on open questions within density functional theory (existence of a derivative of the Lieb functional in a restricted sense and non-interacting vvv-representability)
Density functional theory together with the Kohn-Sham scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary ``non-interacting'' system (one-body with a local potential). However, theoretical questions remain open. An important one is related to the existence of a derivative of the Lieb functional in some restricted (weak) sense (as pointed out by Lammert). Then, a further one would be related to the validity of the ``non-interacting vvv-representability'' conjecture. We gather here elements on these questions, providing reminders and thoughts that are non-conclusive but hopefully contribute to the reflection. This document must be considered as notes (it does not represent an article).
The treatment of degenerate states within Kohn-Sham density functional theory is a problem of long-standing and current interest. We propose a solution to this mapping from the interacting degenerate system to that of the noninteracting fermion model whereby the equivalent density and energy are obtained via the unifying physical framework of quantal density functional theory. We describe the quantal theory of both ground and excited degenerate states, and for the cases of both pure state and ensemble v-representable densities. The quantal description further provides a rigorous physical interpretation of the corresponding Kohn-Sham energy functionals of the density, ensemble density, bidensity and ensemble bidensity, and of their respective functional derivatives. We conclude with examples of the mappings within the quantal theory.