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CRC Press eBooks

Green’s Functions

John W. Negele, Henri Orland

Published Mar 5, 2018• 0 citations
Publisher

Abstract

This chapter describes the salient properties of zero-temperature, real-time finite temperature, and thermal Green's functions in turn. It considers the effect of the energy and momentum dependence of the self-energy on quasiparticle propagation in an interacting Fermi system. The chapter illustrates the major points for the case of a nucleon in translationally invariant nuclear matter. Although Green's functions were introduced because of their physical connection to response functions, they contain sufficient information to evaluate all ground state observables or thermal averages. The real-time Green's function, which describes the physical response of a system, may thereby, be obtained from the thermal Green's function. The essential point in relating the optical potential to the self-energy is the observation that the wave function can be written in terms of the one-particle Green's function. A good approximation to the response may be obtained using the response function for a non-interacting Fermi gas corrected for the medium dependence of the nucleon self-energy.

DOI
10.1201/9780429497926-5

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CRC Press eBooks•Mar 5, 2018•0 citations
This chapter describes the salient properties of zero-temperature, real-time finite temperature, and thermal Green's functions in turn. It considers the effect of the energy and momentum dependence of the self-energy on quasiparticle propagation in an interacting Fermi system. The chapter illustrates the major points for the case of a nucleon in translationally invariant nuclear matter. Although Green's functions were introduced because of their physical connection to response functions, they contain sufficient information to evaluate all ground state observables or thermal averages. The real-time Green's function, which describes the physical response of a system, may thereby, be obtained from the thermal Green's function. The essential point in relating the optical potential to the self-energy is the observation that the wave function can be written in terms of the one-particle Green's function. A good approximation to the response may be obtained using the response function for a non-interacting Fermi gas corrected for the medium dependence of the nucleon self-energy.
Linear density response functions are calculated for symmetric nuclear matter of normal density by time-evolving two-time Green's functions in real time. Of particular interest is the effect of correlations. The system is therefore initially time-evolved with a collision term calculated in a direct Born approximation until fully correlated. An external time-dependent potential is then applied. The ensuing density fluctuations are recorded to calculate the density response. This method was previously used by Kwong and Bonitz for studying plasma oscillations in a correlated electron gas. The energy-weighted sum-rule for the response function is guaranteed by using conserving self-energy insertions as the method then generates the full vertex-functions. These can alternatively be calculated by solving a Bethe -Salpeter equation as done in works by Bozek et al. The (first order) mean field is derived from a momentum-dependent (non-local) interaction while 2 nd order self-energies are calculated using a particle-hole two-body effective (or residual) interaction given by a gaussian local potential. We show results of calculations of the response function S(ɷ, q 0 ) for q 0 = 0.2, 0.4 and 0.8 fm -1 . Comparison is made with the nucleons being un-correlated i.e. with only the first order mean field included. We discuss the relation of our work with the Landau quasi-particle theory as applied to nuclear systems by Babu and Brown and followers.
In this chapter, we discuss the central concept in many-body physics, namely the single-particle Green function. It is shown that many physical observables, such as current and number density, are related to this object. However, this quantity is more important than that since it provides information of the nature of ‘quasiparticles’. If we choose to characterize the atomic constituents of the model being studied minus the mutual interactions among the constituents-‘particles’, then ‘quasiparticles’ would be the effective description of such particles in the presence of such mutual interaction between the constituents. The single-particle Green function contains information about the energy momentum relation of these quasiparticles and also their lifetime. We consider this concept both for systems in thermodynamic equilibrium and for systems out of equilibrium.
In this chapter, a summary of the main properties of the many-electron Green function, which is a central quantity in many-electron theory, is given. The physical interpretation of the Green function and its connection to the experiment are described. Hedin's equations as a set of iterative equations determining the self-energy, and hence the Green function, are derived. Linear response theory and the screened interaction underlying the self-energy formalism are elucidated in detail. Finally, we also describe the thermal (Matsubara) Green function that can be used to describe systems at finite temperatures.
This chapter examines the energy of the resulting quasi-particle using the Green function technique. The field-theoretic Green function technique developed originally by Feynman, Dyson, and others for quantum electrodynamics finds extensive applications to the many body problems encountered in different branches of physics, for its proven efficacy in dealing with the dynamical effects associated with these problems. The Green function formalism is essentially a systematic time-dependent perturbation theory involving certain quantities called the Green functions or propagators. The energy-dependent Green function, G, provides great many physical information about a system, in particular the energy of the quasi-particles. Puff and Whitfield have considered the thermodynamic Green functions for the problem and found at the end of the calculation the zero-temperature limit. In the zero-temperature limit the approximate thermodynamic Green function of Mahan becomes equivalent to the intermediate coupling polaron model of Haeringen and is thus expected to give the same result as given by method of Haeringen.
Abstract Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.
This chapter presents field-theoretic methods and discusses three well-known "pictures" or representations used in discussing quantum mechanical problems. It explores how to build up the true one-electron and one-phonon Green's functions as a perturbation series involving the corresponding functions for the non-interacting system. In elementary discussions of quantum mechanics one usually works in the "Schrodinger picture" in which the dynamical variables are taken to be time independent so that the wave function contains the time dependence of the problem. The one-electron Green's function gives information about the spin and charge densities and the momentum distribution of the electrons as well as information about the excitation spectrum of the system. When the system consists only of fermions interacting via an instantaneous two-body potential, the one-particle Green's function also suffices to determine the ground-state energy of the system. The electrical conductivity, magnetic susceptibility, and many other non-equilibrium properties can be obtained from the two-particle Green's function.