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Physical review. B, Condensed matter

Exact functional differential variant of the generalized Kadanoff-Baym ansatz

Hum Chi Tso, Norman J. M. Horing

Published Jul 15, 1991• 11 citations
Publisher

Abstract

We derive an exact functional differential relation for the physical Green's function ${\mathit{g}}_{\ensuremath{\lessgtr}}$(1;2) in terms of its retarded and advanced counterparts and the equal-time function ${\mathit{g}}_{\ensuremath{\lessgtr}}$(1;${\mathbf{x}}_{2}$,${\mathit{t}}_{1}$). This exact, formal, functional differential relation is a variant of the generalized Kadanoff-Baym GKB ansatz, and we examine the approximations involved in reducing it to the GKB ansatz.

DOI
10.1103/physrevb.44.1451

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We derive an exact functional differential relation for the physical Green's function g\ensuremath≶{\mathit{g}}_{\ensuremath{\lessgtr}}g\ensuremath≶​(1;2) in terms of its retarded and advanced counterparts and the equal-time function g\ensuremath≶{\mathit{g}}_{\ensuremath{\lessgtr}}g\ensuremath≶​(1;x2{\mathbf{x}}_{2}x2​,t1{\mathit{t}}_{1}t1​). This exact, formal, functional differential relation is a variant of the generalized Kadanoff-Baym GKB ansatz, and we examine the approximations involved in reducing it to the GKB ansatz.
The generalized Kadanoff-Baym ansatz (GKBA) is an approximation to the Kadanoff-Baym equations (KBE), that neglects certain memory effects that contribute to the Green's function at non-equal times. Here we present arguments and numerical results to demonstrate the practical insignificance of the quantities neglected when deriving the GKBA at conditions at which KBE and GKBA are appropriate. We provide a mathematical proof that places a scaling bound on the neglected terms, further reinforcing that these terms are typically small in comparison to terms that are kept in the GKBA. We perform calculations in a range of models, including different system sizes and filling fractions, as well as experimentally relevant non-equilibrium excitations. We find that both the GKBA and KBE capture the dynamics of interacting systems with moderate and even strong interactions well. We explicitly compute terms neglected in the GKBA approximation and show, in the scenarios tested here, that they are orders of magnitude smaller than the terms that are accounted for, i.e., they offer only a small correction when included in the full Kadanoff-Baym equations.
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
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.
The generalized Kadanoff-Baym ansatz (GKBA) is an approximation to the Kadanoff-Baym equations (KBEs) that neglects certain memory effects that contribute to the Green's function at nonequal times. Here we present arguments and numerical results to demonstrate the practical insignificance of the quantities neglected when deriving the GKBA at conditions at which KBE and GKBA are appropriate. We provide a mathematical proof that places a scaling bound on the neglected terms, further reinforcing that these terms are typically small in comparison to terms that are kept in the GKBA. We perform calculations in a range of models, including different system sizes and filling fractions, as well as experimentally relevant nonequilibrium excitations. We find that both the GKBA and KBE capture the dynamics of interacting systems with moderate and even strong interactions well. We explicitly compute terms neglected in the GKBA approximation and show, in the scenarios tested here, that they are orders of magnitude smaller than the terms that are accounted for, i.e., they offer only a small correction when included in the full KBEs.
In the non-equilibrium Green function calculations, the use of the generalized Kadanoff–Baym ansatz (GKBA) allows for a simple approximate reconstruction of the two-time Green function from its time-diagonal value. With this, a drastic reduction of the computational needs is achieved in time-dependent calculations, making longer time propagation possible and more complex systems accessible. This paper gives credit to the GKBA that was introduced 25 years ago. After a detailed derivation of the GKBA, we recall its application to homogeneous systems and show how to extend it to strongly correlated, inhomogeneous systems. As a proof of concept, we present the results for a two-electron quantum well, where the correct treatment of the correlated electron dynamics is crucial for a correct description of the equilibrium and dynamic properties.