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Physical Review

Study of the Perturbation Series for the Ground State of a Many-Fermion System. III

George Allen Baker Jr.

Published Oct 11, 1965• 20 citations
Publisher

Abstract

We derive, through sixth order in the Fermi momentum, the many-fermion energy expansion. This expansion is compared with actual energies previously computed for the square-well potential. We introduce an approximation scheme, valid over a significant range of conditions, which includes, we think, infinite nuclear matter. We show that saturation is almost certainly a singular point of the many-fermion energy curve.

DOI
10.1103/physrev.140.b9

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Physical Review•Feb 15, 1961•16 citations
We derive, through sixth order in the Fermi momentum, the many-fermion energy expansion. This expansion is compared with actual energies previously computed for the square-well potential. We introduce an approximation scheme, valid over a significant range of conditions, which includes, we think, infinite nuclear matter. We show that saturation is almost certainly a singular point of the many-fermion energy curve.
In order to investigate the validity of the Brueckner and the simple sum of the ladder diagrams approximations to the energy of an infinite system of fermions, we have calculated as a function of density all the terms in the perturbation expansion through the fourth order in the strength of the potential. We have done this for a repulsive, square-well, two-body potential spin-\textonehalf{} fermions. We are able to construct rigorous upper and lower bounds from the coefficients for the value of the ladder approximation and find that the standard solution procedures give reasonably accurate results (within a few percent). We find that the error in the solution as obtained in practice to the equations of the Brueckner approximation is large compared to the size of its departure from the ladder approximation. We further find for low-to-moderate densities and for low-to-moderate potential strengths that the Brueckner approximation both as a sum of a certain class of diagrams and as computed in practice lies above the ladder approximation while the complete perturbation theory lies below it. This result arises from the neglect of the ring diagrams by the Brueckner approximation.
We evaluate the fourth-order Hugenholtz diagrams from the Rayleigh-Schr\"odinger many-body perturbation series for a repulsive, square-well potential. We consider a system containing four different equal-mass identically interacting particle species, similar to nuclear matter when only the repulsive core is considered. We use the results of these calculations to extend the L expansion of the ground-state energy of a Fermi fluid by one more term. From this extension, we are able to provide accurate values of the ground-state energy over a density range which includes that for infinite nuclear matter.
Perturbation theoretic equations for several properties of the zero-temperature many-fermion normal system are rederived from general considerations. Though none of the equations are new, some of the derivations are, and taken together they form a brief and simple summary of many-fermion relations and a basis for further investigation of the basic perturbation description of the many-fermion normal system: the Brueckner-Goldstone expansion. A "change-of-parameter" technique is then developed and employed to investigate the possible use of the true momentum density instead of the unperturbed Fermi distribution in the Brueckner-Goldstone expansion and in the Brueckner KKK-matrix approximation. The result is a simpler, approximate perturbation series for the interaction energy, whose accuracy for nuclear matter is estimated to be approximately \ifmmode\pm\else\textpm\fi{}2 MeV. The new approximation is exact to the fourth order.
I am looking at the old problem of the ground-state energy of quantum many-Fermion systems. My tools will be a version of perturbation theory. It is plain that the ordinary, low-density rearrangement can not be used directly to compute the saturation energy and density of a system for which the two-body potential has a strong repulsive core and an attractive portion beyond the core, because the path of continuation from low-density to saturation density passes through the two-phase (vacuum, liquid) region. Likewise it is difficult to supply an expansion directly in lambda for fixed rho (density) because of the difficulty of the strong repulsive core and the tediousness of the calculations. My approach here is to expand the energy in a double series in k/sub F/, the Fermi momentum, and lambda, the strength of the attractive part of the potential. The plan is then to sum up, by the Pade method or a generalization of it, the k/sub F/ expansions of the coefficients of each power of lambda into a function of k/sub F/. The coefficient of lambda/sup 0/ is just the repulsive-core energy itself.
We show that the many-fermion ground-state energy with an attractive potential has a critical singularity. This singularity destroys the validity of ``low-density'' approximations. We also find that the K-matrix formalism is, in principle, not applicable to attractive potentials because of the presence of ``Emery singularities.'' We introduce an R-matrix formalism which is numerically very close to the K matrix and free from manifest ``Emery singularities.'' A model calculation is performed on the lattice gas to try to anticipate what quality of results can be expected from summing an R-matrix expansion with fixed density.
The methods employed in the previous papers of this series for the ground-state energy of a normal Fermion system are extended to finite temperatures. The basic assumption is that the density matrix of the actual system in thermodynamic equilibrium can be derived by adiabatic transformation from the density matrix of system of noninteracting particles at the same temperature, volume, and chemical potential. However, the spectrum of the noninteracting system must be dependent on the intensive parameters. The self-consistency of the scheme is exhibited by demonstrating that the effective potential characterizing this system can be related to the self-energy operator of the actual system. Two equivalent versions of the theory are derived. In one of these the basic equations are precisely those of the Landau theory of Fermi liquids with the difference, however, that all quantities are given explicitly from first principles.