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Journal of Physics A Mathematical and Theoretical

Transfer Green's functions in two-fold interaction systems

Ursula Schröter, Elke Scheer

Published Jun 9, 2008• 3 citations
Publisher

Abstract

For systems governed by two kinds of interactions it is shown that these can be built successively into the Green’s functions describing the system’s response. Whereas for the ordinary Green’s function the Dyson equation to solve has the same form in each step, we derive the non-trivial second-step equation for the transfer or coupling function, which on the one hand is closely related to the self-energy and on the other hand of practical relevance in transport calculations. PACS numbers: 02.60.Cb, 02.90.+p, 72.10.Bg 1.

DOI
10.1088/1751-8113/41/26/265202

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For systems governed by two kinds of interactions it is shown that these can be built successively into the Green’s functions describing the system’s response. Whereas for the ordinary Green’s function the Dyson equation to solve has the same form in each step, we derive the non-trivial second-step equation for the transfer or coupling function, which on the one hand is closely related to the self-energy and on the other hand of practical relevance in transport calculations. PACS numbers: 02.60.Cb, 02.90.+p, 72.10.Bg 1.
The Dyson equation represents a well-defined connection between the one-particle Green's function and the self-energy. By means of the Dyson equation, the evaluation of the one-particle Green's function can be reduced to the evaluation of the self-energy. As it is well known, the one-particle Green's function is composed of an advanced and a retarded part, which can be represented in the (N+1)- and (N-1)-particle configuration spaces, respectively. The Dyson equation combines these two spaces into a matrix in the union space. This is difficult to understand in configuration-interaction language, where the Hamiltonian matrices for systems containing different numbers of particles are naturally separated. Starting with the usual matrix representations of the Hamiltonian in the (N+1)- and (N-1)-particle spaces, it is shown that it is possible to define a class of unitary transformations that mixes the two parts and gives rise to an effective Hamiltonian matrix in the union space. This effective Hamiltonian leads to all possible representations of the Dyson equation. The transformations used arise from suitable combinations of the matrices of residues of the advanced and retarded Green's functions. Several specific members of the above class of unitary transformations are discussed in some detail also in connection with possible approximation schemes for the Green's function. Special attention is paid to those unitary transformations that depend on the ground state of the N-particle system as the only unknown quantity. This investigation has given rise to some general conclusions and to an alternative approach to the one-particle Green's function.
This book presents an up-to-date formalism of non-equilibrium Green's functions covering different applications ranging from solid state physics, plasma physics, cold atoms in optical lattices up to relativistic transport and heavy ion collisions. Within the Green's function formalism, the basic sets of equations for these diverse systems are similar, and approximations developed in one field can be adapted to another field. The central object is the self-energy which includes all non-trivial aspects of the system dynamics. The focus is therefore on microscopic processes starting from elementary principles for classical gases and the complementary picture of a single quantum particle in a random potential. This provides an intuitive picture of the interaction of a particle with the medium formed by other particles, on which the Green's function is built on.
Perturbation theory using self-consistent Green's functions is one of the most widely used approaches to study many-body effects in condensed matter. On the basis of general considerations and by performing analytical calculations for the specific example of the Hubbard atom, we discuss some key features of this approach. We show that when the domain of the functionals that are used to realize the map between the noninteracting and the interacting Green's functions is properly defined, there exists a class of self-energy functionals for which the self-consistent Dyson equation has only one solution, which is the physical one. We also show that manipulation of the perturbative expansion of the interacting Green's function may lead to a wrong self-energy as a functional of the interacting Green's function, at least for some regions of the parameter space. These findings confirm and explain numerical results of Kozik et al. for the widely used skeleton series of Luttinger and Ward [Phys. Rev. Lett. 114, 156402 (2015)]. Our study shows that it is important to distinguish between the maps between sets of functions and the functionals that realize those maps. We demonstrate that the self-consistent Green's functions approach itself is not problematic, whereas the functionals that are widely used may have a limited range of validity.
While the use of Green's function techniques has a long tradition in quantum chemistry, the possibility of propagating the Kadanoff-Baym equations has remained largely unexplored. We have implemented the time-propagation for atoms and diatomic molecules, starting from a system in the groundstate. The initial stage of the calculation requires solving the Dyson equation self-consistently for the equilibrium Green's function. This Green's function contains a huge amount of information, and we have found it particularly interesting to compare the self-consistent total energies to the results of variational energy functionals of the Green's function. We also use time-propagation for calculating linear response functions, as a means for obtaining the excitation energies of the system. We have presently implemented the propagation for the second Born approximation, while the GW approximation has now been implemented for the ground state calculations.
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.
In a recent letter [Riva et al., Phys. Rev. Lett. 131, 216401 (2023)] we presented the multichannel Dyson equation (MCDE), in which two or more many-body Green's functions are coupled. In this work we will give further details of the MCDE approach. In particular we will discuss (1) the derivation of the MCDE and the definition of the space in which it is to be solved, (2) the rationale of the approximation to the multichannel self-energy, (3) a diagrammatic analysis of the MCDE, and (4) the recasting of the MCDE on an eigenvalue problem with an effective Hamiltonian that can be solved using standard numerical techniques. This work mainly focuses on the coupling between the one-body Green's function and the three-body Green's function to describe photoemission spectra, but the MCDE method can be generalized to the coupling of other many-body Green's functions and to other spectroscopies.