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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Connected-kernel multifermion Bethe-Salpeter equations

Kenneth L. Kowalski

Published Nov 15, 1979• 3 citations
Publisher

Abstract

Classes of connected-kernel Bethe-Salpeter integral equations for multifermion Green's functions are developed using the combinatorial techniques of many-particle scattering theory. The kernels and inhomogeneous terms of these equations consist of various subsystem Green's functions, and appropriate well-defined integral equations are found for these auxiliary quantities as well.

DOI
10.1103/physrevd.20.2526

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Classes of connected-kernel Bethe-Salpeter integral equations for multifermion Green's functions are developed using the combinatorial techniques of many-particle scattering theory. The kernels and inhomogeneous terms of these equations consist of various subsystem Green's functions, and appropriate well-defined integral equations are found for these auxiliary quantities as well.
The generalization of the pair-labeled Rosenberg equations for many-particle scattering are found in the case where there are arbitrary multiparticle interactions. These are called interaction-set equations because they involve auxiliary transition operators which are labeled by the same set of partitions which characterizes the various connectivities of the interparticle interactions. The technique which we employ also provides the analogous extension of a recently proposed set of connected-kernel multiple scattering equations for the Watson-type transition operators. Further, the structure of the interaction-set equations leads to the identification of an entire class of interaction-set connected-kernel scattering integral equations, each of which is based upon a distinct choice of unperturbed Green’s function and its associated connectivity structure. The generalized Rosenberg equations and the connected-kernel Watson-type multiple scattering equations, which are limiting members of this class, correspond to the choice of the N-free-particle and two-cluster-channel unperturbed Green’s functions, respectively.
A Bethe-Salpeter equation for the three-particle correlation function is derived, representing a second-order response generalization of the usual Bethe-Salpeter equation. The equation can be solved formally, giving the three-particle correlation function in terms of the two-particle correlation function and many-body interaction kernels. The similarity to second-order response time-dependent density-functional theory can be used to improve the understanding of the higher-order exchange and correlation kernels of this theory. An exact expression for such a kernel is derived.
This dissertation deals with aspects of many-body problem in the electronic structure of solids. More specifically, we study how the introduction of Green’s function of many bodies and their equation of motion leads to information about the interactions between particles. Thus, we get to the GW and Bethe-Salpeter formalisms. We show in specific cases how the GW is able to describe energy gaps at a level that goes beyond the Density Functional Theory. Likewise, we investigate the inclusion of excitonic effects through the Bethe-Salpeter equation, showing how the approach can provide optical spectra in good agreement with experimental results. We exemplify the above points with the discussion of specific cases.