Momentum conservation and local field corrections for the response of interacting Fermi gases

Abstract

We reanalyze the recently derived response function for interacting systems in relaxation time approximation respecting density, momentum and energy conservation. We find that momentum conservation leads exactly to the local field corrections for both cases respecting only density conservation and respecting density and energy conservation. This rewriting simplifies the former formulae dramatically. We discuss the small wave vector expansion and find that the response function shows a high frequency dependence of ω −5 which allows to fulfill higher order sum rules. The momentum conservation also resolves a puzzle about the conductivity which should only be finite in multicomponent systems. 05.30.Fk,21.60.Ev, 24.30.Cz, 24.60.Ky Recently the improvement of the response function in interacting quantum systems has regained much interest [1,2]. This quantity is important in a variety of fields and describes the induced density variation if the system is externally perturbed: δn = χV ext . As an example for an interacting system with potential V the conductivity can be calculated from the response function viaOne of the most fruitful concepts to improve the response functions including correlations are the local field correc-see [1,3,4] and references therein. On the other hand there exists an extremely useful form of the response function when the interactions are abbreviated in the relaxation time approximation τ respecting density conservation [5]. One of the advantages of the resulting Mermin formula (9) is that it leads to the Drude -like form of the dielectric function in the long wavelength limitwith the plasma frequency ω p for the Coulomb potential V from which follows the conductivityHowever one should note that this formula is valid only for the extension to a multicomponent system [6] (at least a two-component system) since it makes no sense to speak of conductivity in a single component system where the conductivity should be infinite. Clearly the Mermin formula does not distinguish these cases and cannot be sufficient to describe the response. Therefore we will show that the inclusion of additional momentum conservation will repair this defect (22) and will lead to a conductivitywhich shows indeed for the static limit a diverging behavior in contrast to (4). There are two distinguishable cases, the single component case where we have to include momentum conservation and obtain divergent conductivity and the multicomponent case where we should expect Mermin-like formulae in order to render the conductivity finite. In order to bring these two extreme cases together the response function for multicomponent systems should be considered [6].In this letter we want to restrict to the one -component situation. In [2] we have derived the density, current and energy response χ, χ J , χ E of an interacting quantum systemto the external perturbation V ext provided the density, momentum and energy are conserved. The interacting system has been described by the quantum kinetic equation for the density operator in r...

Dynamical local field, compressibility, and frequency sum rules for quasiparticles

“…The last formulae corrects a misprint in formula (23) of 46 . This different behavior of the imaginary part is also reflected in different expressions for the third order moment (o(1/ω 4 )) or third order sum rule.…”

“…In fact as found in 46 it is possible to express the density polarization function Π 11 ≡ Π n,j,E including density, current and energy conservation by a simpler one containing density and energy conservation Π n,E as…”

“…where the difference between Π n,j,E and Π 0 has been recasted into a local field contribution 46 derived from (A12)-(A14), * One should not be mislead to the conclusion that this violates the Kramers Kronig relation forfq(ω). A toy example of Imf = sin (aω)/(ω 2 − 4π 2 /a 2 ) and the corresponding real part obtained from the Kramers Kronig relation (22) shows that indeed (37) can hold simultaneously with the Kramers Kronig relation.…”

“…In the former paper 42,46 it was derived formally the density, current and energy response {χ, χ J , χ E } = X {1, 0, 0} T of an interacting quantum system δn q · δJ δE…”

“…In 42,46 was given the polarization function for an interacting quantum system imposing conservation laws on the relaxation time approximation. These polarization functions we have denoted by Π n for density conservation imposed, Π n,j for density and current conservation and Π n,j,E for density, current and energy conservation.…”