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Progress of Theoretical Physics

Application of the Dielectric Formulation to the Multi-Component, Many-Particle System with Coulomb Interaction

Mitsuaki Ginoza

Published Dec 1, 1974• 1 citation
Publisher

Abstract

A multi-component, many-particle system with Coulomb interaction is described in terms of a generalized dielectric constant ε(q, ω). The ε(q, ω) is calculated by the use of the equation-of-motion of a correlation function obtained within the random phase approximation (R.P.A.). For the Fermi, Bose, Bose and Bose, and Bose and Fermi systems, the ε(q, ω) obatined above leads directly to the result of Gell-Mann and Brueckner, Foldy, Bassichis, and Ginoza and Kanazawa, respectively. As for the Fermi and Fermi system, it has two collective excitations: plasma wave and acoustic phonon wave. The latter is well-defined only for such a special choice of parameters specifying the system as vF1 ≫s > vF2 and further, for the choice of parameters as vF1 ≫s≫vF2, agrees with the acoustic phonon wave of an electronion system treated by the Thomas-Fermi approximation. For the parameters chosen otherwise, such a wave is unstable or does not exist. In the expressions of the conditions above, s is the velocity of acoustic phonon wave and vF1 and vF2 are Fremi velocities in one component and in the other, respectively. The above results show that in the weak coupling limit, the results obtained by treating Bose components consistently by the Bogoliubov method are equivalent to those obtained with the R.P.A.

DOI
10.1143/ptp.52.1737

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A multi-component, many-particle system with Coulomb interaction is described in terms of a generalized dielectric constant ε(q, ω). The ε(q, ω) is calculated by the use of the equation-of-motion of a correlation function obtained within the random phase approximation (R.P.A.). For the Fermi, Bose, Bose and Bose, and Bose and Fermi systems, the ε(q, ω) obatined above leads directly to the result of Gell-Mann and Brueckner, Foldy, Bassichis, and Ginoza and Kanazawa, respectively. As for the Fermi and Fermi system, it has two collective excitations: plasma wave and acoustic phonon wave. The latter is well-defined only for such a special choice of parameters specifying the system as vF1 ≫s > vF2 and further, for the choice of parameters as vF1 ≫s≫vF2, agrees with the acoustic phonon wave of an electronion system treated by the Thomas-Fermi approximation. For the parameters chosen otherwise, such a wave is unstable or does not exist. In the expressions of the conditions above, s is the velocity of acoustic phonon wave and vF1 and vF2 are Fremi velocities in one component and in the other, respectively. The above results show that in the weak coupling limit, the results obtained by treating Bose components consistently by the Bogoliubov method are equivalent to those obtained with the R.P.A.
By means of a quantum mechanical phase space distribution function introduced by von Roos, the Schroedinger equation for a non-relativistic system of N identical particles with scalar interactions is transformed into a quantum mechanical generalization of the Liouville equation, thereby formulating the problem in terms of a generalized density in phase space, a quantity of primary interest in most treatments of the corresponding classical system (or plasma). This transformation permits a parallel development of the theories of classical and quantum plasmas and thus allows the quantum many-body problem to be discussed virtually completely in classical terms. In particular, a kinetic theory of quantum plasmas is obtained by deriving the quantum analogue of the BBGKY hierarchy, and applying thereto approximation techniques similar to those of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from similar previous studies based on the Wigner distribution function is that the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations, in which case the quantum BBGKY system reduces to a simple quantum generalization of the Vlasov equation. This equation is used to study the response of spatially homogeneous systems to weak external forces, and the associated problems of plasmon and spin-wave excitations. It is also used to derive the quantum and exchange corrected equations of inviscid hydrodynamical transport which are then applied to the problem of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the many-electron atom in the Hartree and Hartree-Fock approximations. The relevant quantum Vlasov equations lead naturally to a statistical theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and exchange corrections to these models are simultaneously generated. The quantum hydrodynamical theory developed earlier is used to determine the influence of these corrections on the boundary conditions of the model, and a theory of the compressed atom is consequently obtained. Considered in somewhat less detail are the effects of non-zero temperature, net orbital angular momentum, relativity and correlations, as well as time dependent processes. The final part deals with the problem of the degenerate electron gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with parallel and anti-parallel spin are obtained by neglecting three particle correlations. From these functions, a quantum-mechanical collision integral is derived which differs from that obtained by Silin and Guernsey and conjectured by Wyld and Pines in that dynamical exchange effects are included. Also obtained from the pair function is an expression for the correlation which reduces in the high density limit to the result of Gell-Mann and Brueckner. At intermediate densities an additional term appears in the energy due to the screening of the exchange interaction by the dielectric properties of the medium. It is evaluated in the high density limit and found to be -0.151 r_s ln r_s Ryd/electron in marked disagreement with the corresponding value obtained by DuBois.
The dispersion equation for acoustic plasmons in a high-density degenerate two-component Fermi gas is given and analysed. The random phase approximation (RPA) dielectric function is used, with exact analytic continuation from the upper to the lower half of the complex frequency plane. For small wavevector the acoustic plasmon frequency and damping depend on only two parameters, the ratio of Fermi velocities and the ratio of Thomas-Fermi wavevectors. A large mass ratio is not necessary for weak damping.
We treat the quantum transport of an interacting system of electrons, impurities, and phonons, in a time-dependent electric field, by using the quantum generalized Langevin equation (GLE), in which the system is shown to be equivalent to a quantum particle in a heat bath. We follow here the philosophy of Ford, Lewis, and O'Connell, who have demonstrated the usefulness of the GLE approach to heat-bath problems. The center of mass of the electrons acts like a quantum particle, while the relative electrons and phonons play the role of heat bath. They are coupled through the electron-impurity and electron-phonon interactions. After eliminating the heat-bath variables, the equation of motion for the quantum particle is written in a form of a quantum generalized Langevin equation, with a memory term which reflects the retarded effects of the heat bath on the quantum particle. The evaluation of the memory term immediately leads to a result for the susceptibility from which we can calculate the conductivity directly, in contrast to Kubo-type calculations which require the evaluation of correlation functions as an intermediate step. As a demonstration of the directness of our approach, we show that the usual random-phase-approximation conductivity results are easily derived. In addition, we derive an expression for the memory function, which incorporates higher-order electron-impurity scattering.
In this paper, we first briefly introduce a two-component anyon model and present unusual commutation relations between the two components. In the Fermi (or Bose) representation, drawing an electrostatic-field analogy, we found that the three-body interaction between particles can be approximately reduced to a logarithmic interaction. In the random-phase approximation, we calculated the effective interaction and dielectric response function. The effect of screening is to make the effective interaction nonsingular at q=0 and to induce a current-current interaction that does not appear in the original interaction. Finally, we found that there exists a restricted class of plasma excitations in the long-wavelength case.
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.
In Chapter 3, we have shown how a Bose condensate leads to the single-particle Green's function G αβ ( Q , ω) appearing as a distinct component of the the dynamic structure factor S ( Q , ω). This feature is a direct consequence of (3.13) and is explicitly shown by the expression for S ( Q , ω) in (3.18). In the present chapter, we review the many-body diagrammatic analysis which specifies the precise relation between G αβ ( Q , ω) and S ( Q , ω) in a Bose-condensed fluid ( n 0 ≠ 0). The power of this “dielectric formalism” lies in showing that both functions mirror each other, as a result of the condensate-induced intermixing of single-particle and density excitations. At the same time, the formalism emphasizes the continuity of these excitations as the fluid passes through T λ, although these excitations have very temperature-dependent weights in G αβ ( Q , ω) and S ( Q , ω). While it is the poles of the single-particle Green's function which describe the elementary excitations, experimentally it is difficult to probe directly this field-fluctuation spectrum. In particular, it is only in a Bose-condensed fluid that one can obtain information about G αβ ( Q , ω) from inelastic neutron-scattering data on S ( Q , ω). In the absence of a condensate, there is little relation between the density-fluctuation modes and the single-particle excitations in a Bose fluid. After setting up the dielectric formalism in Section 5.1, the next two sections consider some model approximations at finite temperatures. These models are most appropriate to a dilute weakly interacting Bose gas (WIDBG), but we believe they also give crucial insight into various aspects of superfluid 4 He.