Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1903.01516 · Physical review. B./Physical review. B

Reduced density matrix functional theory for superconductors

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

Published Jun 5, 2019• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.99.224502

Related papers

Time-Dependent Density-Functional Theory for Superconductors

Oliver Wacker, R. Kümmel, Eberhard K. U. Gross

A density-functional theory is established for strongly correlated inhomogeneous superconductors subject to time-dependent external scalar, vector, and pairing potentials. Hohenberg-Kohn and Kohn-Sham type theorems are formulated for gauge-invariant densities. The central result is a set of time-dependent Bogoliubov-de Gennes equations which include exchange-correlation effects.

Physical Review Letters•Nov 21, 1994•74 citations

Reduced-density-matrix-functional theory at finite temperature: Theoretical foundations

Tim Baldsiefen, Attila Cangi, Eberhard K. U. Gross

We present an ab initio approach for grand-canonical ensembles in thermal equilibrium (eq) with local or nonlocal external potentials based on the one-reduced density matrix (1RDM). We show that equilibrium properties of a grand-canonical ensemble are determined uniquely by the eq-1RDM and establish a variational principle for the grand potential with respect to its 1RDM. We further prove the existence of a Kohn-Sham system capable of reproducing the 1RDM of an interacting system at finite temperature. Utilizing this Kohn-Sham system as an unperturbed system, we deduce a many-body approach to iteratively construct approximations to the correlation contribution of the grand potential.

arXiv:1208.4703•Physical Review A•Nov 20, 2015•36 citations

Representability problem of density functional theory for superconductors

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

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.

arXiv:1810.08391•Physical review. B./Physical review. B•Jan 4, 2019•6 citations

Expressions for the Exchange Correlation Potential and Exchange Correlation Functional of Kohn--Sham Density Functional Theory

James P. Finley

The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of $v_s$ and $ϕ_1$, where $v_s$ is the local, one-body potential from the Kohn--Sham equations, and $ϕ_1$ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential $v_s$. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.

arXiv:physics/0506109•Jun 12, 2005•0 citations

Microscopic derivation of density functional theory for superfluid systems based on effective action formalism

Takeru Yokota, Haruki Kasuya, Kenichi Yoshida +1 more

Abstract A density-functional theory for superfluid systems is developed in the framework of the functional renormalization group based on the effective action formalism. We introduce the effective action for the particle-number and non-local pairing densities and demonstrate that the Hohenberg–Kohn theorem for superfluid systems is established in terms of the effective action. The flow equation for the effective action is then derived, where the flow parameter runs from $0$ to $1$, corresponding to the non-interacting and interacting systems. From the flow equation and the variational equation that the equilibrium density satisfies, we obtain the exact expression for the Kohn–Sham potential generalized to include the pairing potentials. The resultant Kohn–Sham potential has a nice feature in that it expresses the microscopic formulae of the external, Hartree, pairing and exchange–correlation terms separately. It is shown that our Kohn–Sham potential gives the ground-state energy of the Hartree–Fock–Bogoliubov theory by neglecting the correlations. An advantage of our exact formalism lies in the fact that it provides ways to improve the correlation part systematically.

arXiv:2008.05919•Progress of Theoretical and Experimental Physics•Dec 1, 2020•10 citations

Approximations based on density-matrix embedding theory for density-functional theories

Iris Theophilou, Teresa E. Reinhard, Ángel Rubio +1 more

Abstract Recently a novel approach to find approximate exchange–correlation functionals in density-functional theory was presented (Mordovina et al 2019 J. Chem. Theory Comput. 15 5209), which relies on approximations to the interacting wave function using density-matrix embedding theory (DMET). This approximate interacting wave function is constructed by using a projection determined by an iterative procedure that makes parts of the reduced density matrix of an auxiliary system the same as the approximate interacting density matrix. If only the diagonal of both systems are connected this leads to an approximation of the interacting-to-non-interacting mapping of the Kohn–Sham approach to density-functional theory. Yet other choices are possible and allow to connect DMET with other density-functional theories such as kinetic-energy density functional theory or reduced density-matrix functional theory. In this work we give a detailed review of the basics of the DMET procedure from a density-functional perspective and show how both approaches can be used to supplement each other. We do not present a specific realization of combining density-functional methods with DMET but rather provide common grounds to facilitate future developments that encompass both approaches. We do so explicitly for the case of a one-dimensional lattice system, as this is the simplest setting where we can apply DMET and the one that was originally presented. Among others we highlight how the mappings of density-functional theories can be used to identify uniquely defined auxiliary systems and projections in DMET and how to construct approximations for different density-functional theories using DMET inspired projections. Such alternative approximation strategies become especially important for density-functional theories that are based on non-linearly coupled observables such as kinetic-energy density-functional theory, where the Kohn–Sham fields are no longer obtainable by functional differentiation of an energy expression, or for reduced density-matrix functional theories, where a straightforward Kohn–Sham construction is not feasible.

arXiv:2103.02027•Electronic Structure•Jul 20, 2021•3 citations
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 \ensuremathγ\ensuremath{\gamma}\ensuremathγ and the anomalous density \ensuremathχ\ensuremath{\chi}\ensuremathχ, on the other. This formalism for superconductivity yields the existence of a universal functional F\ensuremathβ[\ensuremathγ,\ensuremathχ]{\mathfrak{F}}_{\ensuremath{\beta}}[\ensuremath{\gamma},\ensuremath{\chi}]F\ensuremathβ​[\ensuremathγ,\ensuremathχ] 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.
We present an ab initio approach for grand-canonical ensembles in thermal equilibrium (eq) with local or nonlocal external potentials based on the one-reduced density matrix (1RDM). We show that equilibrium properties of a grand-canonical ensemble are determined uniquely by the eq-1RDM and establish a variational principle for the grand potential with respect to its 1RDM. We further prove the existence of a Kohn-Sham system capable of reproducing the 1RDM of an interacting system at finite temperature. Utilizing this Kohn-Sham system as an unperturbed system, we deduce a many-body approach to iteratively construct approximations to the correlation contribution of the grand potential.
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.
The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of vsv_svs​ and ϕ1ϕ_1ϕ1​, where vsv_svs​ is the local, one-body potential from the Kohn--Sham equations, and ϕ1ϕ_1ϕ1​ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential vsv_svs​. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.
Abstract A density-functional theory for superfluid systems is developed in the framework of the functional renormalization group based on the effective action formalism. We introduce the effective action for the particle-number and non-local pairing densities and demonstrate that the Hohenberg–Kohn theorem for superfluid systems is established in terms of the effective action. The flow equation for the effective action is then derived, where the flow parameter runs from 000 to 111, corresponding to the non-interacting and interacting systems. From the flow equation and the variational equation that the equilibrium density satisfies, we obtain the exact expression for the Kohn–Sham potential generalized to include the pairing potentials. The resultant Kohn–Sham potential has a nice feature in that it expresses the microscopic formulae of the external, Hartree, pairing and exchange–correlation terms separately. It is shown that our Kohn–Sham potential gives the ground-state energy of the Hartree–Fock–Bogoliubov theory by neglecting the correlations. An advantage of our exact formalism lies in the fact that it provides ways to improve the correlation part systematically.
Abstract Recently a novel approach to find approximate exchange–correlation functionals in density-functional theory was presented (Mordovina et al 2019 J. Chem. Theory Comput. 15 5209), which relies on approximations to the interacting wave function using density-matrix embedding theory (DMET). This approximate interacting wave function is constructed by using a projection determined by an iterative procedure that makes parts of the reduced density matrix of an auxiliary system the same as the approximate interacting density matrix. If only the diagonal of both systems are connected this leads to an approximation of the interacting-to-non-interacting mapping of the Kohn–Sham approach to density-functional theory. Yet other choices are possible and allow to connect DMET with other density-functional theories such as kinetic-energy density functional theory or reduced density-matrix functional theory. In this work we give a detailed review of the basics of the DMET procedure from a density-functional perspective and show how both approaches can be used to supplement each other. We do not present a specific realization of combining density-functional methods with DMET but rather provide common grounds to facilitate future developments that encompass both approaches. We do so explicitly for the case of a one-dimensional lattice system, as this is the simplest setting where we can apply DMET and the one that was originally presented. Among others we highlight how the mappings of density-functional theories can be used to identify uniquely defined auxiliary systems and projections in DMET and how to construct approximations for different density-functional theories using DMET inspired projections. Such alternative approximation strategies become especially important for density-functional theories that are based on non-linearly coupled observables such as kinetic-energy density-functional theory, where the Kohn–Sham fields are no longer obtainable by functional differentiation of an energy expression, or for reduced density-matrix functional theories, where a straightforward Kohn–Sham construction is not feasible.