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Physical review. B, Condensed matter

Coulomb screening in the two-time Keldysh-Green-function formalism

P. Gärtner, L. Bányai, H. Haug

Published Sep 15, 2000• 21 citations
Publisher

Abstract

The correct formulation of the two-time self-consistent Coulomb quantum kinetics with screening requires a polarization loop obeying the Ward identity stemming from the charge conservation. Otherwise an infrared divergence occurs. For a self-consistent self-energy without vertex part, a polarization loop which includes a vertex part calculated in the ladder approximation, is shown to satisfy the required Ward identity.

DOI
10.1103/physrevb.62.7116

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The correct formulation of the two-time self-consistent Coulomb quantum kinetics with screening requires a polarization loop obeying the Ward identity stemming from the charge conservation. Otherwise an infrared divergence occurs. For a self-consistent self-energy without vertex part, a polarization loop which includes a vertex part calculated in the ladder approximation, is shown to satisfy the required Ward identity.
We consider the fulfillment of conservation laws and Ward identities in the one- and two-loop functional renormalization group approach. It is shown that in a one-particle irreducible scheme of this approach Ward identities are fulfilled only with the accuracy of the neglected two-loop terms $O({V}_{\ensuremath{\Lambda}}^{3})$ at one-loop order, and with the accuracy $O({V}_{\ensuremath{\Lambda}}^{4})$ at two-loop order (${V}_{\ensuremath{\Lambda}}$ is the effective interaction vertex at scale $\ensuremath{\Lambda}$). The one-particle self-consistent version of the two-loop renormalization group equations which leads to smaller errors in Ward identities due to the absence of the terms with nonoverlapping loops, is proposed. In particular, these modified equations exactly satisfy Ward identities in the ladder approximation.
Besides the proof of a modified Luttinger–Ward–Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. L. Hedin, Phys. Rev . 139, A796–823 (1965) Summary In this chapter a set of equations is formulated that determine the self-energy and the one-body Green's function in terms of the screened Coulomb interaction between classical charges. The equations contain a correction to the classical picture in terms of a vertex function. The physical meaning of the various contributions is discussed. The simplest approximation for the vertex yields the random phase approximation for the polarizability and the GW approximation for the self-energy. Various aspects of the GWA are analyzed, with a focus on the physics that is added beyond Hartree– Fock. A brief summary of model cases illustrates the domain of validity and the limits of the GWA. In this chapter we elaborate in more detail on the question of how to calculate the onebody Green's function from a Dyson equation with a self-energy kernel. In the previous chapter a scheme was introduced to design approximations to the self-energy. However, the question of where to stop, which pieces of physics to include and which to neglect, is not yet settled. Of course, there is no unique answer, besides the exact solution, but different strategies can be more or less advantageous in practice. In a system with a few electrons, for example, different aspects will be important than in a system with many electrons. Here we are mostly interested in solids, or more generally in extended systems. In such systems, screening plays an essential role: the interaction between two charges is strongly modified, in general reduced, by the rearrangement of all the other charges. It is therefore most convenient to reformulate the equations such that screening appears explicitly. Some steps in this direction can be found in earlier chapters, in particular in Sec. 8.3, the formulation of the Ψ [ G , W ]-functional of the screened interaction W instead of the Ф[ G , v c ]-functional of the bare v c . In Sec. 10.5 the screened interaction approximation for the self-energy is derived, with the self-energy as a product of the one-body Green's function and the screened interaction W .
We study the effects of local vertex corrections to the self energy of the electron gas. We find that a vertex derived from time-dependent density-functional theory can give accurate self energies without including the explicit time dependence of the exchange-correlation potential provided, however, that a proper decay at large momentum transfer (large q) is built into the vertex function. (The local-density approximation for the vertex fails badly.) Total energies are calculated from the Galitskii-Migdal formula and it is shown that a proper large-q behavior, results in a close consistency between the chemical potentials derived from these energies and those obtained directly from the self energy. We show that this internal consistency depends critically on including the same vertex correction in both the self-energy and the screening function. In addition the total energies become almost as accurate as those from elaborate quantum Monte-Carlo (QMC) calculations. We also study the accuracy and utility of the functional for the total energy proposed by Luttinger and Ward and a generalization by Almbladh, von Barth, and van Leeuwen. For the electron gas, even the simplest and readily evaluated approximations to these functionals yield total energies of similar quality as those of QMC calculations. The functionals depend on the one-electron Green's function and the screened Coulomb interaction and already rather crude approximations to these quantities produce accurate energies thus demonstrating the insensitivity of the functionals to their arguments. Different ways of incorporating vertex corrections beyond the GWGWGW level are studied in simple, exactly soluble polaron-like models. We study models of a structureless core electron coupled to valence electrons and a local polaron model by Cini, Hewson and Newns. Our model results indicate that the first vertex correction alone will in general not suffice to improve the spectrum away from the quasi-particle peak. By including a subsequence of Mahan's fractal vertex series, however, we obtain results with correct physical properties which agree better with exact model results.
This paper gives an introduction to the Keldysh formalism, with emphasis on its usefulness in time-dependent density functional theory. In the first part we introduce the Keldysh contour and the one-particle Green function defined on this contour. We then discuss how to combine and manipulate functions with time-arguments on the contour. The effects of electron-electron interaction can be taken systematically into account, as we illustrate by propagating the Kadanoff-Baym equations for the second-order self-energy approximation. It is important to use conserving approximations such that the evolution of the electron density, momentum, and total energy agrees with the macroscopic conservation laws. One of the main topics in this paper is the non-interacting Green function, which is the relevant quantity for time-dependent density functional theory. We discuss this Green function in detail, and show how the Keldysh contour in a simple way allows us to derive the time-dependent Kohn-Sham potential from an action functional. The formalism in a similar way leads to response functions that obey the causality principle. To illustrate these points, we discuss the time-dependent optimized effective potential equations.