The quasi-classical limit of the many-particle perturbation theory is considered for electrons moving in the Coulomb potential of the nucleus. The problem is analyzed using the Green's function formulation of many-particle physics. When corrections involving the gradient of the self-consistent potential are neglected, the results agree to all orders in the interaction strength with the procedure of Lewis for including correlation corrections in the Thomas-Fermi model. The disagreement between the work of Lewis and the recent work of Baraff is shown to be due to the latter's neglect of anomalous contributions to the perturbation theory which arise, because the local Fermi momentum is a function of the interaction strength. The calculation of inhomogeneity corrections is also considered.
The quasi-classical limit of the many-particle perturbation theory is considered for electrons moving in the Coulomb potential of the nucleus. The problem is analyzed using the Green's function formulation of many-particle physics. When corrections involving the gradient of the self-consistent potential are neglected, the results agree to all orders in the interaction strength with the procedure of Lewis for including correlation corrections in the Thomas-Fermi model. The disagreement between the work of Lewis and the recent work of Baraff is shown to be due to the latter's neglect of anomalous contributions to the perturbation theory which arise, because the local Fermi momentum is a function of the interaction strength. The calculation of inhomogeneity corrections is also considered.
Beyond the usual many body perturbation theory, with the eigenfunctional theory, we directly calculate the correlation function produced by the Coulomb interaction of electrons in the equation of one-electron Green function, and give the general expression of the non-local effective interaction potential in a Hartree-type potential, which is absent in all previous many body perturbation theories. This effective interaction potential originates from the quantum many body effect of the system, and it cannot be obtained directly by the usual perturbation expression approach. Moreover, using theoretical models, we demonstrate that this effective interaction potential can be used to characterize the electron correlation strength of the system.
The individual particle theory of many body systems was studied starting from a microscopic point of view. A technique which is equivalent to a generalized thermodynamics was used to show that, under the appropriate conditions of temperature and external field, the quasi-particle approach is correct for three very dissimilar systemsi the normal degenerate Fermi system, the metallic electron-phonon system, and the polaron. The fact that an approach like this works for such dissimilar systems emphasizes the underlying unity of the quasi-particle models. The concept that the real t~me non-equilibrium self-energy and the incoherent part of the density-density correlation function may be generated as functional derivatives of a generalized thermodynamic potential (~), together with local translational and rotational invariance and Galilean relativity, leads to a transformation from the microscopic particle density variables g«p,m,R,T) to those of the quasi-particle theory n (R,T). ~ - p- It was shown that, for zero temperature, such a ~ derivability is exact; at low temperature and for slowly varying fields, this ~ derivability is approximately true. This was shown by a procedure which is eqUivalent to an expansion in terms of the real scattering probability for quasiparticles. At low temperature and for slowly varying fields, such real scattering is very improbable and the expansion parameter is small. Perturbation theory was used only to determine the actual size of terms which, on physical grounds are necessarily small. As such, this approach is non-perturbative, depending only on properties of the exact selfenergies and correlation functions. In addition to demonstrating that the quasi-particle models are correct, information on the regions of validity for ~hese models is obtained. .As by-products of these derivations, microscopic expressions for the parameters of the quasi-particle theory and simple microscopic expressions for the currents in degenerate Fermi systems and the electron-phonon system are obtained.
A systematic method is presented for deriving the Thomas-Fermi equation for an atom and the quantum corrections from the many-body description. The novel feature of the method is that it does not require any a priori assumptions about the assignment of electrons to fully occupied single-particle states or about the distribution of electrons in phase space, but shows instead that the distribution which is usually assumed, or derived from the assumption of fully occupied single particle states, is a direct consequence of specifying that the many particle system is in its ground state. The procedure used in the derivation is the expansion of the mixed position-momentum representation of the Green's function in a series of powers of $\ensuremath{\hbar}$. The lowest order term is found to correspond with the Thomas-Fermi density. The form of the higher order terms, which are to be considered as corrections to zeroth order term, depends on the approximations made in the many-body equations for obtaining the Green's function. This paper deals only with the Hartree-Fock approximation, but the methods presented here allow generalization to other approximations which can include correlation effects.
Art is the elimination of the unnecessary. Pablo Picasso Summary Condensed matter is constituted by a huge number of electrons and nuclei interacting with Coulomb potentials. The topic of this chapter is a way to deal with the full, coupled problem by separating it into a part that is tractable and the rest that is approximated. This naturally leads to the appearance of dynamical fields, and to the concept of “quasi-particles” that have the same quantum numbers as non-interacting electrons. The quasi-particles obey equations where the potential and bare interactions in the hamiltonian are replaced by dynamical self-energies and screened interactions that can describe many of the effects of correlation. In this chapter we discuss the intuitive concepts behind this approach in general terms to motivate the more rigorous formulations and approximations used in the Green's function methods of the following chapter and in Part III. The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches. One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.
The self-consistent theory of the correlation effects in Highly Correlated Systems(HCS) is presented. The novel Irreducible Green's Functions(IGF) method is discused in detail for the Hubbard model and random Hubbard model. The interpolative solution for the quasiparticle spectrum, which is valid for both the atomic and band limit is obtained. The (IGF) method permits to calculate the quasiparticle spectra of many-particle systems with the complicated spectra and strong interaction in a very natural and compact way. The inelastic scattering corrections leads to the damping of the quasiparticles and are the main topic of the present consideration. The calculation of the damping has been done in a self-consistent way for both limits. For the random Hubbard model the weak coupling case has been considered and the self-energy operator has been calculated using the combination of the IGF method and Coherent Potential Approximation (CPA). The other applications of the method to s-f model, Anderson model, Heisenberg antiferromagnet, electron-phonon interaction models are discussed briefly.