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Physical Review Letters

Time-Dependent Density-Functional Theory for Superconductors

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

Published Nov 21, 1994• 74 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.73.2915

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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.
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We develop a Keldysh-Eliashberg-type formalism based on a functional theory for strongly correlated inhomogeneous superconductors subject to time-dependent external forces on the ions, four-vector, and pairing potentials. This theory incorporates exchange-correlation effects including a self-consistent treatment of the electromagnetic vector field as well as the ion motion on equal footing. Thus our theory not only contains but also goes beyond the scope of a recently published time-dependent density-functional theory for superconductors.
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.
A time-dependent density-functional formalism is developed for many-electron systems subjected to external electric and magnetic fields with arbitrary time dependence. The single-particle current density is shown to determine uniquely the time-dependent scalar and vector potentials characterizing the system, and hence also the many-particle wave function. A Levy-type universal functional is defined, and practical schemes for the calculation of electron density and the current density through hydrodynamical as well as a set of single-particle Kohn-Sham-like equations are proposed. The gauge invariance of the present self-consistent formalism is also proved.
A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ leading to a given vvv-representable density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ is uniquely determined so that the corresponding map $v\ensuremath{\rightarrow}n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schr\"odinger equation.