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Physical review. B, Condensed matter

Time-dependent functional theory for superconductors

A. K. Rajagopal, F. A. Buot

Published Sep 1, 1995• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.52.6769

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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.
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Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).
We develop a nonequilibrium time-dependent functional theory of coupled interacting fields in terms of the Liouvillian quantum field dynamics. In this theory the expectation values of the appropriate field operators are treated as the functional variables. Thus, the formalism proposed here goes beyond the existing time-dependent density-functional method, in that we now have a mixed-state multivariable dynamical theory. A large class of condensed matter problems, with the given initial condition that the system be in thermodynamic equilibrium and subjected to time-dependent external forces, can now be examined. In particular, we formulate the self-consistent coupled equations for the electron, ion, and the electromagnetic fields. The above formalism provides a firm physical and mathematical basis for time-dependent dynamical simulation procedures such as those envisaged in the Car-Parrinello approach by replacing the fictitious Lagrangian by a precise action principle. The theory involves a certain functional representing the mutual interactions and correlations among these fields. In parallel with this, we also generalize the Baym \ensuremath{\Phi}-derivable method in many-body theory to investigate the class of problems mentioned above. This involves a functional of the exact one-particle Green functions. We compare the two methods and suggest the use of the latter in the former scheme as a way of going beyond the simple mean-field schemes. We illustrate the utility of the framework by citing a variety of illustrative examples.
Approximations for the ground-state exchange-correlation potential of density-functional theory have reached a high level of sophistication. By contrast, time- or frequency-dependent exchange-correlation potentials are still being treated in a local approximation. Here we propose a novel approximation scheme, which effectively brings the power of the generalized gradient approximation (GGA) and meta-GGA to time-dependent density-functional theory. The theory should allow a more accurate treatment of strongly inhomogeneous electronic systems (e.g. molecular junctions) while remaining essentially exact for slowly varying densities and slowly varying external potentials.