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Physical Review

Dielectric Formulation of Quantum Statistics of Interacting Particles

F. Englert, R. Brout

Published Nov 15, 1960• 56 citations
Publisher

Abstract

Formal relations between the free energy, the "dielectric constant" which expresses the response of the system to an external perturbation, and the two-particle Green's function in temperature space are derived. Connection with perturbation expansion for free energy and for general admittance tensors are established. These results are applied to a general discussion of the random-phase approximation at finite temperature and it is shown that the sum of ring diagrams corresponds to the calculation of the dielectric constant in the approximation of the neglect of local field corrections.

DOI
10.1103/physrev.120.1085

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Physical review. B, Condensed matter•Feb 15, 1979•18 citations
Formal relations between the free energy, the "dielectric constant" which expresses the response of the system to an external perturbation, and the two-particle Green's function in temperature space are derived. Connection with perturbation expansion for free energy and for general admittance tensors are established. These results are applied to a general discussion of the random-phase approximation at finite temperature and it is shown that the sum of ring diagrams corresponds to the calculation of the dielectric constant in the approximation of the neglect of local field corrections.
In a preceding paper [J. Math. Phys. 3, 475 (1962)], some model Hamiltonians were proposed for quantum-mechanical many-body systems with pair forces. For infinite systems in thermal equilibrium, they led to temperature-domain propagator expansions which were formally summable and expressible by closed equations. These expansions were identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models corresponded to ring- and ladder-diagram summations from the true propagator expansion, augmented by infinite classes of self-energy corrections. The model Hamiltonians were called stochastic because they contained parameters whose phases were fixed by random choices. In the present paper, more general models are formulated which yield formally summable propagator expansions for finite systems. The analysis is extended to correlation and Green's functions defined for nonequilibrium ensembles. The nonequilibrium treatment is developed in the Heisenberg representation in such a way that unlinked diagrams do not arise. A basic convergence question associated with the formal closed equations for the model propagators and correlation functions is examined by means of finite-difference integration of the Heisenberg equations of motion. This procedure appears to converge independently of whether the perturbation expansions for the propagators and correlation functions converge. It yields substantial support for the validity of the formal closed model equations.
A generalized self-consistent field (SCF) theory of many-particle systems is developed by modifying the usual functional relationship between energy and wave function. A parameter $\ensuremath{\xi}$, $0\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}1$, is thus introduced into the energy so that for $\ensuremath{\xi}=0$, the particles interact only with a self-consistent field, for $\ensuremath{\xi}=1$ the particles interact dynamically among themselves with the correct field, and for intermediate values of $\ensuremath{\xi}$ both SCF and dynamical interactions occur. A systematic development of the theory is given for time-independent and time-dependent probelms, for finite temperatures, and for both uniform and nonuniform systems. The derivative with respect to $\ensuremath{\xi}$ of the total energy, of free energy, is expressed in terms of the dielectric function and an improved version of the dielectric formulation of the many-body problem thereby obtained. A brief discussion of the advantages of the method, possible applications, and further generalizations or extensions is included.
A perturbation technique useful for computing many-time thermal averages of classical quantities is developed. The canonical distribution function for the system is shown to evolve isothermally from that for a free-particle system as the interaction is switched on slowly. This permits convenient use of an interaction picture in which to perform thermal averaging. The technique is applied to the calculation of the time-dependent pair correlation function in position for a uniform gas. The correlation function is shown to be a sum of two components, one the solution of a kinetic equation and essentially a generalization of the autocorrelation in equilibrium, the other a generalization of the mutual correlation function in equilibrium. The equations arise as sums over diagrams. The equations resulting from the random phase approximation, valid for the short-time behavior, are solved exactly. It is shown directly that the generalized dielectric function given in terms of the correlation function is identical with that found by solution of the kinetic equation in the random phase approximation for all frequencies.
An alternative approach to the quantum statistics of interacting particles is proposed. It consists of calculating the equilibrium thermodynamical quantities of the many-body system via the pair distribution function with the assumption that the particles interact with each other only through pair central forces. The proposed approach has some advantage over the usual treatment via the partition function in that the pair distribution function is easier to deal with than the partition function in certain circumstances particularly when the collective motion description of the system is desirable. This is because only the pair distribution function can be expressed directly in terms of the collective interaction which is closely connected with collective elementary excitation, such as a plasmon in the electron gas and a phonon in the hard-sphere Boson gas. The equation of state as well as the internal energy are obtained in the form of integrals of the pair distribution function. The close analogy between the pair distribution function and the two-body propagator, which appears in the quantum field theory, makes it possible to analyze the former by the use of Feynman diagrams identical with those usually introduced for the latter. The collective interaction, which is defined by the sum of the direct and the indirect interactions, is introduced as a particular partial sum of the perturbation series of the pair distribution function. This is used in rewriting the pair distribution function in terms of the collective interaction.It is shown that, while the simple chain approximation to the collective interaction in the electron gas is responsible for the transfer of a plasmon, the same approximation to the collective pseudo-interaction in the hard-sphere Boson system has a relation similar to that of transfer of a phonon, both cases occuring at low temperatures. The explicit calculation of the pair distribution functions for these systems at the absolute zero temperature is carried out up to the first order in the collective interaction (simple chain approximation.) These results are used to calculate the ground-state energies. For the electron gas the energy thus obtained confirms Gell-Mann-Bruckner's calculation of the correlation energy. For the hard-sphere Bosons the calculated energy reproduces the result of Lee, Huang, and Yang. The extension of the calculation to the finite temperature case is also indicated. In particular the classical Debye-Huckel equation of state for the electron gas is briefly discussed.
For theoretical description of pseudospin systems with essential shortrange and long-range interactions we use the method based on calculations of the free energy functional taking into account the short-range interactions within the reference approach in cluster approximation.We propose a consistent formulation of the cluster expansion method for quantum pseudospin systems.We develop a method allowing one to obtain within the cluster approximation an Ornstein-Zernike type equation for reference cumulant Green function of an arbitrary order.In the two-particle cluster approximation we derived an explicit expression for pair temperature cumulant Green function of the reference system.In the cluster random phase approximation we calculated and studied thermodynamic characteristics, elementary excitation spectrum, and integral intensities of the Ising model in transverse field.
In this paper we derive a formula for the time-dependent Hartree-Fock dielectric function by using Green's-function theory. We develop a set of self-consistent equations for the self energy, polarization, and Green's function which can be iterated to obtain successively more accurate approximations for the polarization. In this procedure the Hartree and Hartree-Fock approximations occur naturally as the first and second steps in the iterations. As the iterations are carried out, we obtain not only the expressions for the polarization, but also the one-electron Hartree-Fock equations, so that there is no question as to which polarization formula corresponds to which equation. In this way we obtain an expression for the dielectric function which clearly and directly corresponds to the Hartree-Fock equation and which is therefore called the time-dependent Hartree-Fock (TDHF) dielectric function. In this TDHF formula for the dielectric function, the first term is the time-dependent Hartree or random-phase approximation dielectric function. We show that in order to be consistent with the theory, one should use Hartree wave functions and energies to compute random-phase approximation dielectric functions and Hartree-Fock wave functions and energies to compute TDHF dielectric functions. Numerical results are presented for the free-electron gas.