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Poincare Journal of Analysis and Applications

SOLUTION OF THE KADANOFF-BAYM EQUATIONS BY ITERATED EXPANSIONS

Cenap Özel

Published Jun 26, 2021• 0 citations
Publisher

Abstract

In this paper, we will show how the general nonlinear Kadanoff-Baym equations can be solved with iterated series expansion.In this regard, we will neglect vertex corrections.Further, we will obtain a formal solution in terms of colored tree graphs.The iteration procedure will show that after proper discretization, wellstructured matrix equations would be obtained which are easy to implement in numerical simulation.

DOI
10.46753/pjaa.2021.v08i01(i).021

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In this paper, we will show how the general nonlinear Kadanoff-Baym equations can be solved with iterated series expansion.In this regard, we will neglect vertex corrections.Further, we will obtain a formal solution in terms of colored tree graphs.The iteration procedure will show that after proper discretization, wellstructured matrix equations would be obtained which are easy to implement in numerical simulation.
This chapter gives a brief survey about the “time-stepping procedure” and the “predictor-corrector scheme” for solving the Kadanoff–Baym equations with two ( t and t ′) time variables. In this respect, the solutions of the Kadanoff–Baym equations for the greater and lesser Green’s functions are combined with each other in the positive quadrant of the t − t ′ plane, together with the solution of the lesser Green’s function along the time diagonal where t = t ′. The “generalized Kadanoff–Baym ansatz”, which aims at somewhat simplifying the solution of the Kadanoff–Baym equations themselves, is also introduced and derived in detail.
Abstract An approximation of the Green function based on the Generalized Kadanoff–Baym Ansatz and its correction is tested on the electron tunneling through a single atom, incorporating the on-site second-order Born interaction. Time evolution resulting from the corresponding kinetic equation for the density matrix is compared with the direct solution of the non-equilibrium Green functions. It is demonstrated that the short-time nature of corrections beyond the Generalized Kadanoff–Baym Ansatz enables their effective numerical approximation. Graphical abstract
The Kadanoff-Baym equations (KBE) are usually derived under the assumption of the weakening of initial correlations (Bogolyubov's condition) and, therefore, fail to correctly describe the short time behavior. We demonstrate that this assumption is not necessary. Using functional derivatives techniques, we present a straightforward generalization of the KBE which allows to include arbitrary initial correlations and which is more general than previous derivations. As a result, an additional collision integral is obtained which is being damped out after a few collisions. Our results are complemented with numerical investigations showing the effect of initial correlations.
Gradient expansions in quantum transport equations of a Kadanoff–Baym form have been reexamined. We have realized that in a consistent approach the expansion should be performed also inside of the self–energy in the scattering integrals of these equations. In the first perturbation order this internal expansion gives new correction terms to the generalized Boltzman equation. These correction terms are found here for several typical systems. Possible corrections to the theory of a linear response to weak electric fields are also discussed. 1 1
A quantum kinetic equation for the density matrix is derived from nonequilibrium Green functions using the generalized Kadanoff-Baym ansatz and corrections beyond. The Markovian reconstruction of the Green function and the quasistationary approximation are used to approximate the corrections, making their implementation in numerical simulations faster. On electron tunneling under conditions when the prediction of the uncorrected kinetic equation seriously departs from the exact solution, we show that the corrections reduce the error by an order of magnitude.
A systematic and unambiguous method of deriving generalized transport equations, i.e., equations for distribution functions having a single-time structure, on the basis of the nonequilibrium Green function is obtained, if the common Kadanoff-Baym (KB) ansatz is replaced by a modification which we call the generalized KB ansatz. This new ansatz is fully consistent with the dynamical structure of the theory and is independent of any specific representation. The resulting equations appear to be the zeroth-order approximation of a systematic expansion in terms of the collision duration. In the case of the electron transport in a strong homogeneous electric field, the generalized ansatz is shown to be in agreement with the superoperator methods, whereas the KB ansatz is known to fail in this case.