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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Kadanoff-Baym equations with initial correlations

D. Semkat, D. Kremp, M. Bönitz

Published Feb 1, 1999• 88 citations
Publisher

Abstract

Equilibrium and nonequilibrium properties of nonideal many-particle systems are strongly influenced by correlation effects that are well described by generalized quantum kinetic equations, in particular, the Kadanoff-Baym equations (KBE). However, these equations 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 for the derivation of the KBE. Using functional derivatives techniques, we present a straightforward generalization of the KBE that allows us to include arbitrary initial correlations and that 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.

DOI
10.1103/physreve.59.1557

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Equilibrium and nonequilibrium properties of nonideal many-particle systems are strongly influenced by correlation effects that are well described by generalized quantum kinetic equations, in particular, the Kadanoff-Baym equations (KBE). However, these equations 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 for the derivation of the KBE. Using functional derivatives techniques, we present a straightforward generalization of the KBE that allows us to include arbitrary initial correlations and that 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.
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.
A recently developed method [Semkat et al., Phys. Rev. E 59, 1557 (1999); Kremp et al., in Progress in Nonequilibrium Green’s Functions (World Scientific, Singapore, 2000), p. 34] for incorporating initial binary correlations into the Kadanoff–Baym equations (KBE) is used to derive a generalized T-matrix approximation for the self-energies. It is shown that the T-matrix obtains additional contributions arising from initial correlations. Using these results and taking the time-diagonal limit of the KBE, a generalized quantum kinetic equation in binary collision approximation is derived. This equation is a far-reaching generalization of Boltzmann-type kinetic equations: It self-consistently includes memory effects (retardation, off-shell T-matrices) as well as many-particle effects (damping, in-medium T-matrices) and spin-statistics effects (Pauli-blocking).
The nonequilibrium dynamics of quantum fields is an initial-value problem, which can be described by Kadanoff-Baym equations. Typically, and, in particular, when numerical solutions are demanded, these Kadanoff-Baym equations are restricted to Gaussian initial states. However, physical initial states are non-Gaussian correlated initial states. In particular, renormalizability requires the initial state to feature nnn-point correlations that asymptotically agree with the vacuum correlations at short distances. In order to identify physical nonequilibrium initial states, it is therefore a precondition to describe the vacuum correlations of the interacting theory within the nonequilibrium framework. In this paper, Kadanoff-Baym equations for non-Gaussian correlated initial states describing vacuum and thermal equilibrium are derived from the 2PI effective action. A diagrammatic method for the explicit construction of vacuum and thermal initial correlations from the 2PI effective action is provided. We present numerical solutions of Kadanoff-Baym equations for a real scalar ${\ensuremath{\Phi}}^{4}$ quantum field theory, which take the thermal initial four-point correlation as the leading non-Gaussian correction into account. We find that this minimal non-Gaussian initial condition yields an approximation to the complete equilibrium initial state that is quantitatively and qualitatively significantly improved as compared to Gaussian initial states.
We present in detail a method we recently introduced (PRL. 103, 176404 (2009)) to describe finite systems in and out of equilibrium, where the evolution in time is performed via the Kadanoff-Baym Equations (KBE) within Many-Body Perturbation Theory (MBPT). The main property we analyze is the time-dependent density. We also study is the exchange-correlation potential of TDDFT, obtained via reverse engineering from the time-dependent density. Our systems consist of small, strongly correlated clusters, described by a Hubbard Hamiltonian within the Hartree-Fock, second Born, GW and T-matrix approximations. We compare the results from the KBE dynamics to those from exact numerical solutions. The outcome of our comparisons is that, among the many-body schemes considered, the T-matrix approximation is overall superior at all electron densities. Such comparisons permit a general assessment of the whole idea of applying MBPT, in the KBE sense, to finite systems. A striking outcome of our analysis is that when the system evolves under a strong external field, the KBE develop a steady-state solution as a consequence of a correlation-induced damping. This damping is present both in isolated (finite) systems, where it is purely artificial, as well as in clusters contacted to (infinite) macroscopic leads. To illustrate this point we present selected results for a system coupled to contacts within the T-matrix and second Born approximation. The extensive characterization we performed indicates that this behavior is present whenever approximate self energies, based upon infinite partial summations, are used. A second important result is that, for isolated clusters, the steady state reached is not unique but depends on how one switches on the external field. This may also true for clusters connected to leads.
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 pair distribution function (PDF) is a key quantity for the analysis of correlation effects of a quantum system both in equilibrium and far from equilibrium. We derive an expression for the PDF in terms of the single-particle Green functions—the solutions of the Keldysh/Kadanoff-Baym equations in the two-time plane—for a one- or two-component system. The result includes initial correlations and generalizes previous density matrix expressions from single-time quantum kinetic theory. Explicit expressions for the PDF are obtained in second Born approximation.