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Acta Physica Sinica

DISCUSSION ON RANDOM PHASE APPROXIMATION OF HYPERNETTED CHAIN METHOD

WANG KE-LIN, 中国科学技术大学基础物理中心

Published Jan 1, 1988• 0 citations
Publisher

Abstract

In discussion of the many body effect of homogenenus Fermion system with hypernetted chain method, RPA is always applied and original wave function is taken as a Fermi sphere distribution function. In this paper, the validity of this method is investigated and the relation between the valid range of this method and the properties of potential and density is obtained.

DOI
10.7498/aps.37.727

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In discussion of the many body effect of homogenenus Fermion system with hypernetted chain method, RPA is always applied and original wave function is taken as a Fermi sphere distribution function. In this paper, the validity of this method is investigated and the relation between the valid range of this method and the properties of potential and density is obtained.
In this article, we revisit the question of the validity of Hartree-Fock and random-phase approximations. We show that there is a connection between the two and while the RPA as it is known in much of the physics literature is of limited validity, there is a generalised sense in which the random phase approximation is of much wider applicability including to systems that do not possess Fermi surfaces. The main conclusion is that the Hartree-Fock approximation is a mean-field idea applied to the density operator, and the random-phase approximation is a mean-field idea applied to the number operator. The generalised RPA is used to compute single-particle properties such as momentum distribution and spectral functions. It is found that we have to go beyond the generalised RPA and include fluctuations in the momentum distribution in order to recover a nonzero imaginary part of the one-particle self energy, which is also explicitly computed, all this in any number of spatial dimensions and no bosonization is needed.
The approach to the many-fermion problem known as the method of higher random phase approximations (RPA) is given a more rigorous formulation. It is shown that the previous heuristic procedures for evaluation of the second RPA are justified, in that expectation values of plane wave operators with respect to the true ground state may validly be replaced by their values in the unperturbed Fermi state. This property of momentum-space occupation renormalization is conjectured to hold also to higher orders of RPA than the second.
The Fermi-hypernetted-chain (FHNC) theory and the effective hypernetted-chain method are applied to study the composite-fermion (CF) states of the fractional quantum Hall effect. Using this theory we compute, in the thermodynamic limit, the correlation energy, radial distribution function, and static structure factor for all unprojected CF wave functions. The unprojected excitation gaps for $\ensuremath{\nu}=1/3,1/5$ were obtained by adopting in the FHNC a scheme previously used to compute nuclear matter excitation spectra. The results obtained so far are consistent with Monte Carlo simulations and small-number exact diagonalizations.
The hypernetted-chain theory is applied to study the fractional quantum Hall effect with the Laughlin wave functions. A new method is proposed to include the effect of the elementary diagrams, which improves upon the commonly used modified hypernetted-chain approximation. The correlation energy, the pair distribution function, as well as the magnetoroton excitation spectrum have been computed within this method. The results obtained are in very good agreement with the available Monte Carlo estimates. The method is generalizable to treat other wave functions, like those corresponding to the hierarchical states or those of composite fermion type.
The Fermi-hypernetted-chain method for quantum many-body calculations is studied and investigated in some detail by calculations of the ground-state energy for neutron matter. The calculations are done for five different two-body potentials and results are compared with other theoretical results. Our Fermi-hypernetted-chain results are rather close to results obtained by lowest-order constrained variation calculations, and the results are quite dependent on the chosen potential, especially at high densities.