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Physical review. B, Condensed matter

Generalized Kadanoff-Baym ansatz for deriving quantum transport equations

P. Lipavský, Václav Špička, Bedřich Velicky´

Published Nov 15, 1986• 424 citations
Publisher

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.

DOI
10.1103/physrevb.34.6933

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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.
In Chapter 4, we introduced the Kadanoff–Baym equations of motion for the imaginary-time nonequilibrium Green's functions for a Bose gas, as given by (4.59) and (4.60). In this chapter, we will use the generalization of these equations of motion to find the equivalent equations of motion for the real-time Green's functions. These can be written in a natural way in the form of a kinetic equation. Using a simple Hartree–Fock approximation, we show how the coupled equations for the condensate and thermal cloud given in Chapter 3 emerge naturally from the Kadanoff–Baym (KB) formalism. This chapter is based on Imamović-Tomasović and Griffin (2001) and Imamović-Tomasović (2001), building on the pioneering work of Kane and Kadanoff (1965). In this chapter and Chapter 7 we review the KB formalism. However, we also encourage the reader to read the original account given by Kadanoff and Baym (1962). The goals and accomplishments of their seminal book are beautifully captured by the following quote from p. 138: Our rather elaborate Green's function arguments provide a means of describing transport phenomena in a self-contained way, starting from a dynamical approximation, i.e. an approximation for G 2 ( U ) in terms of G 1 ( U ). These calculations require no extra assumptions. The theory provides at the same time a description of the transport processes that occur and a determination of the quantities which appear in the transport equations. A closely related way of treating the nonequilibrium dynamics of a Bose-condensed gas is based on the two-particle irreducible (2PI) effective action together with the Schwinger–Keldysh closed-time path formalism. Berges (2004) gives a detailed review of this approach, which allows one to derive the nonequilibrium action on the basis of controllable approximations.
On the basis of the Kadanoff-Baym (KB) version of the time-dependent Green's function method, an Ansatz for the approximation of a spectral function is offered. The Ansatz possesses all the advantages of quasiparticle and extended quasiparticle approximations and satisfies the KB equation for a spectral function in the case of slightly nonequilibrium system when disturbances in space and time are taken into consideration in the gradient approximation. This feature opens opportunities for the microscopic derivation of the Landau kinetic equation for the quasiparticle distribution function of the normal Fermi liquid and provides the widening of these equations' temperature range of validity.
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.
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.
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