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Low Temperature Physics

Different forms of the Kadanoff–Baym equations in quantum statistical mechanics

Alexander S. Kondratyev, Nazish Shahid

Published Oct 1, 2011• 2 citations
Publisher

Abstract

A new form of the Kadanoff–Baym equations for a system of interacting particles is offered on the basis of the retarded and advanced quantum Green’s functions. The comparison of the traditional and the offered forms of the equations allows to analyze the question to what extent Landau–Silin kinetic equations for the neutral Fermi-liquid and for the electron liquid of normal metals take into account quickly varying in space and time disturbances.

DOI
10.1063/1.3665796

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•Jan 1, 2002•8 citations
A new form of the Kadanoff–Baym equations for a system of interacting particles is offered on the basis of the retarded and advanced quantum Green’s functions. The comparison of the traditional and the offered forms of the equations allows to analyze the question to what extent Landau–Silin kinetic equations for the neutral Fermi-liquid and for the electron liquid of normal metals take into account quickly varying in space and time disturbances.
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 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.
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).
A Landau-type Fermi-liquid theory for interacting electrons moving in a fixed periodic potential is derived. Two methods are used: the Green's-function method, which gives a more physical description of the system but does not give all the desired results, and the many-body perturbation-theory approach, which is complete but is more complicated. The Green's-function formalism of Kadanoff and Baym is adapted to inhomogeneous systems by rewriting it in operator form. A kinetic equation for an energy-dependent, single-particle density matrix is then derived. The quasiparticle concept is introduced and a formal solution of the kinetic equation is given and discussed. These developments are specialized to the periodic potential and a transport equation of the Landau-Boltzmann type is derived for quasiparticles moving within a single energy band. Expressions for the density and current density are then identified from a continuity equation. Although the present work does not treat real transitions between energy bands, the Green's-function method suggests a plausible form for a Landau-like theory which includes such transitions. The perturbation-theory treatment follows the work of Luttinger and Nozi\`eres, who derived Landau's theory for systems without a lattice potential. When the perturbation theory is expressed in terms of Bloch functions, the quantities involved become matrices between different energy bands. However, it is shown that many of the key results of Luttinger and Nozi\`eres can be suitably generalized. The perturbation-theory calculation shows that when the periodic potential is present the change in quasiparticle energy can still be expressed in terms of the variation of the quasiparticle distribution function, and that the results obtained from the Green's-function approach are exact in the case of slowly varying external fields.