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Physical Review Letters

Dynamic Response of Inhomogeneous Fermi Systems

Sudip Chakravarty, Martin B. Fogel, Walter Kohn

Published Sep 10, 1979• 30 citations
Publisher

Abstract

This paper presents a theory of the linear density response to an external time-dependent scalar potential for a Fermi system whose unperturbed density varies slowly. Simple local-density-functional response theory is valid except in a region of small $q$ and $\ensuremath{\omega}$ (wave number and frequency of the perturbation). For this region we have worked out a generalization, to the case of inhomogeneous systems of Landau Fermi-liquid theory.

DOI
10.1103/physrevlett.43.775

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This paper presents a theory of the linear density response to an external time-dependent scalar potential for a Fermi system whose unperturbed density varies slowly. Simple local-density-functional response theory is valid except in a region of small qqq and \ensuremathω\ensuremath{\omega}\ensuremathω (wave number and frequency of the perturbation). For this region we have worked out a generalization, to the case of inhomogeneous systems of Landau Fermi-liquid theory.
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We consider the dynamics of a class of weakly interacting, gapless 1d1d1d fermionic systems, in presence of small external perturbations slowly varying in space and in time. We consider the evolution of the expectation values of the charge density and of the current density, in the thermodynamic limit and for low enough temperatures. We prove the validity and the asymptotic exactness of linear response in the limit of vanishing space-time variation of the perturbation, and we provide the explicit expression of the response of the system. The proof relies on the representation of the real time Duhamel expansion in terms of Euclidean correlation functions, for which we provide sharp estimates using rigorous renormalization group methods. The asymptotic exactness of linear response holds thanks to a cancellation for the scaling limit of the correlations that is reminiscent of bosonization, and which is derived rigorously using emergent chiral Ward identities.
The dynamics of a confined fluid of Bose atoms is treated within the linear response regime, with a view to establishing a current-density functional formalism for an inhomogeneous superfluid state. After evaluating in full detail a simplified case of an external coupling to the density and phase of the condensate, the theory is extended to include the coupling to the total current density. The Kohn-Sham response functions of the condensate and all the exchange-correlation kernels for the superfluid are introduced from the microscopic equations of motion and are expressed in a physically transparent way through functional derivatives of correlation functions. A microscopic formula for the superfluid density is derived and used to introduce a generalized hydrodynamic approach for a weakly inhomogeneous two-fluid model in isothermal conditions. Local-density expressions are thereby derived for the velocities of first and second sound in the weakly inhomogeneous superfluid and for visco-elastic functions describing the transition from the hydrodynamic to the collisionless regime. Landau's hydrodynamic theory and known results in Green's functions language are recovered in the limiting case of a homogeneous superfluid.
The frequency-dependent exchange-correlation potential, which appears in the usual Kohn-Sham formulation of a time-dependent linear response problem, is a strongly nonlocal functional of the density, so that a consistent local density approximation generally does not exist. This problem can be avoided by choosing the current density as the basic variable in a generalized Kohn-Sham theory. This theory admits a local approximation which, for fixed frequency, is exact in the limit of slowly varying densities and perturbing potentials.
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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.
The general expressions for the number density and current are first considered under the action of an external time-dependent perturbing potential of arbitrary strength and cast in terms of the single-particle lesser Green’s function. The expansion of the number density up to linear order in the perturbing potential is then considered, yielding the density–density correlation function of linear-response theory. A connection is also considered with the temperature correlation function of the Matsubara formalism via an analytic continuation in frequency space.