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Physical Review

Divergence and Summability of the Many-Fermion Perturbation Series

George Allen Baker Jr.

Published Aug 15, 1963• 15 citations
Publisher

Abstract

We investigate the convergence of the many-fermion perturbation series and show, for the case of the square-well potential, that it is a divergent series. We bound the rate of divergence and show that, by using appropriate summation procedures, it may be summed to the physically correct sum, provided the density is low enough.
DOI
10.1103/physrev.131.1869

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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 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.
A sudden change in a short-range single-particle potential excites an infinite number of low-energy particlehole pairs in a degenerate Fermion gas. Although each order of perturbation theory diverges, the perturbation series can be summed formally. The sum agrees in the limit of zero temperature with a result due to Nozi\`eres and De Dominicis. At finite temperature, it confirms a conjecture of Anderson's. The excitation spectrum at energies that are small compared to the temperature can be interpreted in terms of a relaxation frequency, which is proportional to the temperature times the square of the phase shift.
In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can be reconstructed formally as a sum over its eigenvalues and eigenstates, this series is typically even more divergent. For the simple cases of the square-well and the harmonic-oscillator potentials, this paper explains how to use the elementary procedure of Euler summation to sum these divergent series and thereby to make sense of the formal statement of the completeness of the formal sum that represents the reconstruction of the Hamiltonian.