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Journal of Mathematical Physics

Critical-Point Singularities of the Perturbation Series for the Ground State of a Many-Fermion System

George Allen Baker Jr., Joseph Kahane

Published Sep 1, 1969• 19 citations
Publisher

Abstract

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.

DOI
10.1063/1.1665012

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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.

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Hard-core square-well fermions

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Based on the well-known low-density expansion for the ground-state energy per particle of a many-fermion system interacting pairwise via a central potential we develop a perturbation scheme in which the zero-order state is that of the purely-repulsive-spheres fluid. For the hard-core plus square-well pair-interaction case we have deduced, for two-species fermion matter, perturbation formulas up to sixth order in the attractive coupling parameter. We explicitly report results up to fourth order here. Pad\'e-approximant analyses in the density are carried out in each order with the intent of providing the basis necessary for extensive equation-of-state calculations of various many-fermion systems. In particular, for zero order (the fermion hard-sphere fluid) we predict an uncertainty-principle divergence in the energy, and hence pressure, corresponding to the random close packing of spheres, at a density of about 0.5 times the recently obtained value for boson hard spheres and about 0.2 times the well-known empirical value for classical hard spheres.

Physical Review A•Dec 1, 1982•30 citations
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.
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.
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.
The RRR-matrix expansion for the ground-state energy of a many-fermion system is carried through fourth order. We evaluate this expansion for a potential modeled after the nucleon-nucleon one. The calculation is described in detail. We find that the "hole-line" approximation seems to underestimate the attraction so that a hard-core force may well be consistent with the experimental binding energy of large nuclei.
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.
Based on the well-known low-density expansion for the ground-state energy per particle of a many-fermion system interacting pairwise via a central potential we develop a perturbation scheme in which the zero-order state is that of the purely-repulsive-spheres fluid. For the hard-core plus square-well pair-interaction case we have deduced, for two-species fermion matter, perturbation formulas up to sixth order in the attractive coupling parameter. We explicitly report results up to fourth order here. Pad\'e-approximant analyses in the density are carried out in each order with the intent of providing the basis necessary for extensive equation-of-state calculations of various many-fermion systems. In particular, for zero order (the fermion hard-sphere fluid) we predict an uncertainty-principle divergence in the energy, and hence pressure, corresponding to the random close packing of spheres, at a density of about 0.5 times the recently obtained value for boson hard spheres and about 0.2 times the well-known empirical value for classical hard spheres.