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Physical Review

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

George Allen Baker Jr., J. L. Gammel, Bob Hill

Published Nov 1, 1963• 29 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.132.1373

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Reviews of Modern Physics•Oct 1, 1971•117 citations
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 derive through fourth order in the potential strength the many-fermion perturbation series for a velocity-dependent force which exactly simulates, in two-body scattering, a hard core. We use the Pad\'e approximant method to sum the complete series and the ladder and Brueckner approximation series. We compare these results with those of the KKK-matrix procedure and find that the KKK-matrix procedure fails to converge for the Brueckner approximation in the only interesting region. We also investigate near saturation, a system something like He3{\mathrm{He}}^{3}He3. We find that there is not, as some have speculated, a magic cancellation that would cause the ring diagrams to be of little importance there.
The ground state energy of the Bose particle system at low densities is investigated by using the method which is based on summing up the terms of the conventional perturbation series giving rise to the lowest order with respect to the density. The summation is shown to be greatly simplified if the scattering matrix is introduced, and a simple example is demonstrated for the repulsive square well potential by calculating a few terms of the series first directly and secondly with the aid of scattering matrix. The final result is shown to be valid both for weak and strong interactions and the calculation is carried out analytically for some types of potential function. The connection of the present method with the pseudopotential method and with the Brueckner theory is investigated and some problems associated with the attractive interactions are discussed.
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.
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.
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.
Calculational procedures are developed for the ground-state energy of a many-fermion system. The entire theory is developed from the many-body Schr\"odinger equation via perturbation theory. The linked cluster expansion is derived by elementary methods. The analytic singularities of the energy function are analyzed, and procedures designed to avoid or overcome them. Results for the calculation of the energy in the presence of a purely repulsive two-body force are reviewed, and the situation is found to be reasonably satisfactory, though good results are restricted to weak potentials when the system is not dilute. Results for an attractive potential surrounding a strongly repulsive core are reviewed, and procedures discribed for calculations in this case. A sample calculation with an assessment of the probable errors is reviewed.