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Physical review. B, Condensed matter

Comments on "Time-dependent Hartree-Fock formalism for the dielectric function"

S. Das Sarma, Werner R. Hanke

Published Jul 15, 1983• 0 citations
Publisher

Abstract

The correct formalism for obtaining a dielectric function consistent with the Hartree-Fock approximation is outlined. It is pointed out that a recent calculation by Brener and Fry does violate Ward identities and is not a conserving approximation. To be consistent with the Hartree-Fock single-particle equation the infinite series of "ladder-bubble" diagrams must be included.

DOI
10.1103/physrevb.28.1134

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The correct formalism for obtaining a dielectric function consistent with the Hartree-Fock approximation is outlined. It is pointed out that a recent calculation by Brener and Fry does violate Ward identities and is not a conserving approximation. To be consistent with the Hartree-Fock single-particle equation the infinite series of "ladder-bubble" diagrams must be included.
In this paper we derive a formula for the time-dependent Hartree-Fock dielectric function by using Green's-function theory. We develop a set of self-consistent equations for the self energy, polarization, and Green's function which can be iterated to obtain successively more accurate approximations for the polarization. In this procedure the Hartree and Hartree-Fock approximations occur naturally as the first and second steps in the iterations. As the iterations are carried out, we obtain not only the expressions for the polarization, but also the one-electron Hartree-Fock equations, so that there is no question as to which polarization formula corresponds to which equation. In this way we obtain an expression for the dielectric function which clearly and directly corresponds to the Hartree-Fock equation and which is therefore called the time-dependent Hartree-Fock (TDHF) dielectric function. In this TDHF formula for the dielectric function, the first term is the time-dependent Hartree or random-phase approximation dielectric function. We show that in order to be consistent with the theory, one should use Hartree wave functions and energies to compute random-phase approximation dielectric functions and Hartree-Fock wave functions and energies to compute TDHF dielectric functions. Numerical results are presented for the free-electron gas.
We present predictions for temperature-dependent shifts and damping rates. They are obtained by applying the dielectric formalism to set up a self-consistent model of a trapped Bose gas which can be shown to satisfy generalized Ward identities. Within the framework of the model we use lowest-order perturbation theory to determine the first-order correction to the results of Hartree-Fock-Bogoliubov-Popov theory for the complex collective excitation frequencies, and present numerical results for the temperature dependence of the damping rates and the frequency shifts. Good agreement with the experimental values measured by Jin et al. [Phys. Rev. Lett. 77, 420 (1996)] are found for the m=2m=2m=2 mode, while we find disagreements in the shifts for m=0.m=0.m=0. The latter point to the necessity of a nonperturbative treatment for an explanation of the temperature dependence of the m=0m=0m=0 shifts.