Abstract The averaged wave functions allow defining the transport vertex. For the correlation functions this leads to the Generalised Kadanoff and Baym formalism with the Langreth–Wilkins rules. The meaning of the selfenergy is explored. The Green’s function is a secular equation in which the complicated motion of a particle during interaction is represented by the selfenergy. The quasiparticle picture is applicable to long trajectories, where the averaged velocity is of interest. In processes like emission, the Doppler shift depends on the local velocity and one has to return to the microscopic picture. The application scheme how to construct and use the selfenergy is presented.
Abstract The averaged wave functions allow defining the transport vertex. For the correlation functions this leads to the Generalised Kadanoff and Baym formalism with the Langreth–Wilkins rules. The meaning of the selfenergy is explored. The Green’s function is a secular equation in which the complicated motion of a particle during interaction is represented by the selfenergy. The quasiparticle picture is applicable to long trajectories, where the averaged velocity is of interest. In processes like emission, the Doppler shift depends on the local velocity and one has to return to the microscopic picture. The application scheme how to construct and use the selfenergy is presented.
Abstract The method of the equation of motion is used to derive the Martin–Schwinger hierarchy for the nonequilibrium Green’s functions. The formal closure of the hierarchy is reached by using the selfenergy which provides a recipe for how to construct selfenergies from approximations of the two-particle Green’s function. The Langreth–Wilkins rules for a diagrammatic technique are shown to be equivalent to the weakening of initial correlations. The quantum transport equations are derived in the general form of Kadanoff and Baym equations. The information contained in the Green’s function is discussed. In equilibrium this leads to the Matsubara diagrammatic technique.
Abstract In quantum statistics based on many-body Green’s functions, the effective medium is represented by the selfenergy. This book aims to discuss the selfenergy from this point of view. The knowledge of the exact selfenergy is equivalent to the knowledge of the exact correlation function from which one can evaluate any single-particle observable. Complete interpretations of the selfenergy are as rich as the properties of the many-body systems. It will be shown that classical features are helpful to understand the selfenergy, but in many cases we have to include additional aspects describing the internal dynamics of the interaction. The inductive presentation introduces the concept of Ludwig Boltzmann to describe correlations by the scattering of many particles from elementary principles up to refined approximations of many-body quantum systems. The ultimate goal is to contribute to the understanding of the time-dependent formation of correlations. Within this book an up-to-date most simple formalism of nonequilibrium Green’s functions is presented to cover different applications ranging from solid state physics (impurity scattering, semiconductor, superconductivity, Bose–Einstein condensation, spin-orbit coupled systems), plasma physics (screening, transport in magnetic fields), cold atoms in optical lattices up to nuclear reactions (heavy-ion collisions). Both possibilities are provided, to learn the quantum kinetic theory in terms of Green’s functions from the basics using experiences with phenomena, and experienced researchers can find a framework to develop and to apply the quantum many-body theory straight to versatile phenomena.
In a nonequilibrium system, quasiparticles are defined, consistently with the construction of quantum transport equations, as one-electron excitations whose propagator possesses a time-local optical potential. This potential is constructed as a solution of the phase equation, a self-consistency condition reducing the renormalized Dyson equation to a formerly proposed multiplicative composition rule for the propagators. The Dyson equation for the particle correlation function is transformed to a renormalized quasiparticle reconstruction equation, whose leading term represents the quasiparticle Kadanoff-Baym ansatz [Physica E 29, 196 (2005)], an improvement over the standard generalized Kadanoff-Baym ansatz.
Abstract The gradient approximations of the Kadanoff and Baym equations are derived up to first order. The off-shell motions responsible for the satellites are shown to ensure causality. The cancellation of off-shell motions from the drift and correlation part of the reduced density provides a precursor of the kinetic equation for the quasiparticle distribution which leads to a functional between reduced and quasiparticle distribution, named the extended quasiparticle picture. Virial corrections appear as internal gradients in the selfenergy and therefore in the considered processes. With this extended quasiparticle picture, the non-Markovian kinetic equations are transformed into Markovian ones for proper defined quasiparticles without neglect showing the exact cancellation of off-shell parts. Alternative approaches are discussed for comparison.
In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamović-Tomasović (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3. Overview of Green's function approach To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry. The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.
The question of which non-interacting Green's function 'best' describes an interacting many-body electronic system is both of fundamental interest as well as of practical importance in describing electronic properties of materials in a realistic manner. Here, we study this question within the framework of Baym-Kadanoff theory, an approach where one locates the stationary point of a total energy functional of the one-particle Green's function in order to find the total ground-state energy as well as all one-particle properties such as the density matrix, chemical potential, or the quasiparticle energy spectrum and quasiparticle wave functions. For the case of the Klein functional, our basic finding is that minimizing the length of the gradient of the total energy functional over non-interacting Green's functions yields a set of self-consistent equations for quasiparticles that is identical to those of the quasiparticle self-consistent GW (QSGW) (van Schilfgaarde et al 2006 Phys. Rev. Lett. 96 226402-4) approach, thereby providing an a priori justification for such an approach to electronic structure calculations. In fact, this result is general, applies to any self-energy operator, and is not restricted to any particular approximation, e.g., the GW approximation for the self-energy. The approach also shows that, when working in the basis of quasiparticle states, solving the diagonal part of the self-consistent Dyson equation is of primary importance while the off-diagonals are of secondary importance, a common observation in the electronic structure literature of self-energy calculations. Finally, numerical tests and analytical arguments show that when the Dyson equation produces multiple quasiparticle solutions corresponding to a single non-interacting state, minimizing the length of the gradient translates into choosing the solution with largest quasiparticle weight.