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Extended Two-Particle Green’s Functions and Optical Potentials for Two-Particle Scattering by Many-Body Targets

Joachim Brand, Lorenz S. Cederbaum

Published Jan 1, 1996• 7 citations
OpenAlex

Abstract

An extension of the fermionic particleparticle propagator is presented that possesses similar algebraic properties to the single-particle Green’s function. In particular, this extended two-particle Green’s function satisfies Dyson’s equation and its self energy has the same analytic structure as the the self energy of the single-particle Green’s function. For the case of a system interacting with one-particle potentials only, the two-particle self energy takes on a particularly simple form, just like the common self energy does. The new two-particle self energy also serves as a well behaved optical potential for the elastic scattering of a two-particle projectile by a many-body target. Due to its analytic structure, the two-particle self energy avoids divergences that appear with effective potentials derived by other means. 1996 Academic Press, Inc. I.

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An extension of the fermionic particleparticle propagator is presented that possesses similar algebraic properties to the single-particle Green’s function. In particular, this extended two-particle Green’s function satisfies Dyson’s equation and its self energy has the same analytic structure as the the self energy of the single-particle Green’s function. For the case of a system interacting with one-particle potentials only, the two-particle self energy takes on a particularly simple form, just like the common self energy does. The new two-particle self energy also serves as a well behaved optical potential for the elastic scattering of a two-particle projectile by a many-body target. Due to its analytic structure, the two-particle self energy avoids divergences that appear with effective potentials derived by other means. 1996 Academic Press, Inc. I.
The standard text book Green's function possesses a self-energy that is known to be an optical potential for elastic scattering. The introduction of an optical potential reduces the complex many-body scattering problem into a tractable one-body problem. In this paper inelastic Green's functions are introduced and discussed which possess self-energies that are optical potentials for inelastic scattering. If the projectile is indistinguishable from particles comprising the target, intriguing aspects arise even for noninteracting particles.
An approximate solution of a Dyson equation satisfied by a single particle Green's function in many-impurity problems in solids is obtained for an extended impurity perturbation potential. We assume that there is no overlapping of any two impurity potentials and that the range of the impurity potential is arbitrary. The coordinate representation of the Dyson equation is solved by transforming an iterative solution of the equation into a form with no repetitive summation indices appearing in any term of given order. By neglecting fluctuations of the impurity distribution, a set of equations are derived to determine the Green's function and the self-energy in a self-consistent manner. A group-theoretic consideration of the self-energy and some sum rules are given.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
Recently we proposed a multichannel Dyson equation, in which we coupled the one- and the three-body Green's functions, to model photoemission spectra. In this work we extend this concept to higher-order Green's functions and to other spectroscopies. We show the general structure of the equations and how one can systematically approximate the corresponding multichannel self-energy. As a particular case, we focus on the coupling of the two-body and the four-body Green's functions in the electron-hole channel to describe neutral excitations. This formulation allows for the description of important many-body effects, such as biexcitons, in a natural way. We illustrate our approach by applying it to a two-level model system, which, in a one-particle picture, exhibits single and double excitations. Our method can correctly describe both kinds of excitation, unlike standard approaches, and in good agreement with the exact results.
The paper addresses the many-body quasiparticle dynamics of the Anderson impurity model at finite temperatures in the framework of the equation-of-motion method. We find a new exact identity relating the one-particle and many-particle Green’s functions. Using this identity we present a consistent and general scheme for a construction of generalized mean fields (elastic scattering corrections) and self-energy (inelastic scattering) in terms of Dyson equation. A new approach for the complex expansion for the single-particle propagator in terms of Coulomb repulsion U and hybridization V is proposed. Using the exact identity, the essentially new many-body dynamical solution of SIAM has been derived. This approach offers a new way for the systematic construction of the approximative interpolating dynamical solutions of the strongly correlated electron systems.