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Journal of Mathematical Physics

Summability of the Many-Fermion Perturbation Series with Attractive Forces

George Allen Baker Jr.

Published Aug 1, 1970• 7 citations
Publisher

Abstract

The summability of the many-fermion perturbation series in the presence of attractive forces is considered. We find that, although it is an asymptotic series, the potential-strength perturbation series is summable. We find that some rearrangements to handle the hard-core potentials also lead to the correct sum; however, the ``hole-line'' summation procedure using Brandow's choice for intermediate state energies is either wrong or deceptive in its convergence.

DOI
10.1063/1.1665397

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The summability of the many-fermion perturbation series in the presence of attractive forces is considered. We find that, although it is an asymptotic series, the potential-strength perturbation series is summable. We find that some rearrangements to handle the hard-core potentials also lead to the correct sum; however, the ``hole-line'' summation procedure using Brandow's choice for intermediate state energies is either wrong or deceptive in its convergence.
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.
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.
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.
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.
The convergence problem of perturbation expansions in the many-body problem is discussed by employing some simple models. The expectation value of rhe resolvent operator is analytic around the origin of the coupling constant so long as the system is enclosed in a finite volume, but as the volume tends to infinity with the density of particles kept constant, the question arises whether this limit is allowed to be taken prior to the sum of the perturbation series. Even if this is the case, the resulting series turns out to be usually divergent, but in some models we consider we can find a function whose asymptotic expansion coincides with this series. It is shown that for the repulsive interactions the function thus obtained gives a correct answer, while for the attractive interactions which lead to a superconducting state it is far from the true one. The asymptotic behavior of the perturbation series in the latter case is examined in some detail.