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Acta Physica Sinica

ON THE CORRELATION EFFECT IN MANY-BODY SYSTEMS

C Chen

Published Jan 1, 1960• 0 citations
OpenAlex

Abstract

Using the generalized method of self-consistent field developed by one of the authors[4], we discuss various correlation phenomena of many-body systems with a unified viewpoint. The general method of treatment is proposed and compared with the method of perturbation. From the approximation with two-body correlation we obtain, as particular applications, the Bruckner theory of nuclear matter and the Bohm-Pines theory of the collective oscillation of electron,gases. Finally the superconductivity and zero-point oscillations in non-ideal gases are discussed by solving, the bound state problem with the present method and the Gell-mann Low procedure. It is concluded that the method is very effective in treating correlation phenomena.

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Using the generalized method of self-consistent field developed by one of the authors[4], we discuss various correlation phenomena of many-body systems with a unified viewpoint. The general method of treatment is proposed and compared with the method of perturbation. From the approximation with two-body correlation we obtain, as particular applications, the Bruckner theory of nuclear matter and the Bohm-Pines theory of the collective oscillation of electron,gases. Finally the superconductivity and zero-point oscillations in non-ideal gases are discussed by solving, the bound state problem with the present method and the Gell-mann Low procedure. It is concluded that the method is very effective in treating correlation phenomena.
A systematic method of handling a system composed of a number of particles and an intermediary Bose field has been developed. The main idea consists in linearizing the interaction by taking certain average values so that it may be amalgamated to the free Hamiltonian. Only the remaining fluctuational interaction is treated by perturbation theory. The effect of the averaged interaction appears in the form of two generalized Hartree fields associated with the particle and the Bose quantum, respectively. The self-consistent equations for these Hartree fields are derived from an analysis of the SSS matrix, keeping close analogy to the renormalization procedure in quantum electrodynamics. The formalism includes such approximations as the Bohm-Pines, Hartree-Fock, and Thomas-Fermi theories as special cases. General criteria about the validity and plausibility of the picture involved in the present formulation are discussed from various points of view.
The self-consistent mean field in the many-body correlation dynamics is discussed in detail. It is shown that the self-consistent mean field exists in general in many-body correlation dynamics for both stationary and non-stationary cases. The many-body correlations enter the mean field through the two-body correlation c2 and the corresponding collision term I. The role played by I is twofold i.e., dynamical effects and statistical effects on single particle states. The many-body correlations also manifest their effects on the many-body energy and yield an energy expression different from the conventional expression in the HF-Bruc-kner theory.
Many-body Green functions are a very efficient formulation of the many-body problem. We review the application of this method to nuclear physics problems. The formulas which can be derived are of general applicability, e.g., in self-consistent as well as in nonself-consistent calculations. With the help of the Landau renormalization, one obtains relations without any approximations. This allows to apply conservation laws which lead to important general relations. We investigate the one-body and two-body Green functions as well as the three-body Green function and discuss their connection to nuclear observables. The generalization to systems with pair correlations are also presented. Numerical examples are compared with experimental data.
Introducing low-energy effective Hamiltonians is usual to grasp most correlations in quantum many-body problems. For instance, such effective Hamiltonians can be treated at the mean-field level to reproduce some physical properties of interest. Employing effective Hamiltonians that contain many-body correlations renders the use of perturbative many-body techniques difficult because of the overcounting of correlations. In this work, we develop a strategy to apply an extension of the many-body perturbation theory starting from an effective interaction that contains correlations beyond the mean-field level. The goal is to re-organize the many-body calculation to avoid the overcounting of correlations originating from the introduction of correlated effective Hamiltonians in the description. For this purpose, we generalize the formulation of the Rayleigh-Schrödinger perturbation theory by including free parameters adjusted to reproduce the appropriate limits. In particular, the expansion in the bare weak-coupling regime and the strong-coupling limit serves as a valuable input to fix the value of the free parameters appearing in the resulting expression. This method avoids double counting of correlations using beyond-mean-field strategies for the description of many-body systems. The ground state energy of various systems relevant for ultracold atomic, nuclear, and condensed matter physics is reproduced qualitatively beyond the domain of validity of the standard many-body perturbation theory. Finally, our method suggests interpreting the formal results obtained as an effective field theory using the proposed reorganization of the many-body calculation. The results, like ground state energies, are improved systematically by considering higher orders in the extended many-body perturbation theory while maintaining a straightforward polynomial expansion.
The problem of correlations is considered in many-fermion systems, although with some modifications the methods are more general. The many-body problem is formulated in terms of reduced one- and two-particle density matrices, and the equations for these are treated by variational and iterative methods. The formalism is applied to a calculation of the energy per particle for a system interacting with Coulomb, screened Coulomb, and singular potentials. Comparisons with the ideas of Brueckner and others are made, and difficulties are discussed.