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Reviews of Modern Physics

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

George Allen Baker Jr.

Published Oct 1, 1971• 117 citations
Publisher

Abstract

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.

DOI
10.1103/revmodphys.43.479

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Cluster Expansion Technique in the Many-Fermion Systems. I

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A new cluster expansion technique is developed for the calculation of the energy expectation value of the many-fermion systems with singular potentials.The wave function, which is studied in this paper, is of the formwhich, in a special case, coincides with Jastrow wave function.The formula for the energy expectation value is written in terms of the complex cluster integrals.The method introduced in the derivation of this formula has an analogy to the classical cluster expansion method in the theory 0f an imperfect classical gas of multicomponents.

Progress of Theoretical Physics•Dec 1, 1960•5 citations
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.
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.
We extend to the N-body case previous techniques for solving three-body problems with repulsive interparticle potentials and periodic boundary conditions on each particle. For clarity, we begin with one-dimensional problems, although the techniques are not peculiar to them and can be generalized to three dimensions. We decompose the wavefunction ψ into N(N − 1)/2 parts, according to Ψ= ∑ i<j=1Nψij;from the Schrödinger equation we find a basic equation for a typical ψij. As a test of this equation, we apply it for N = 4 and the case of δ-function interactions, and solve it numerically to find close agreement between the energy per particle of a four-body system and a system of the same density in which N → ∞. The numerical results are consistent with the analytic one that Eg(∞), which is the ground-state energy per particle of the system with an infinite number of particles, is related to Eg(N), which is that energy for the N-body system of the same density, by Eg(N) = Eg(∞)[1 − (1/N)] for weak repulsion and by Eg(N) = Eg(∞)[1 − (1/N2)] for very strong repulsion. We begin a similar comparison for the three-dimensional hard-sphere case by working out the problem of two hard spheres with periodic boundary conditions over a length L2 such that the density is the same as for an N-body problem in volume V=LN3:2/L23=N/LN3. We find that we get the analog of the one-dimensional result, for which case Eg(2) = Eg(∞)/2 for N = 2. That is, we find Eg(2) = πa2h/mv, which is just one-half of the many-body theory result. We also calculate the two-body correlation function and find good agreement with the many-body one, as calculated by Lee, Huang, and Yang.
The ground-state energy and low-energy excitations of single-particle character of an infinite medium of fermions are discussed with the aid of time-dependent Green's functions, which are convenient generalizations of the exact particle correlation functions in the ground state of NNN particles. The power series development for the one- and two-particle functions, under the restriction to two-body forces, is derived and described by means of Feynman diagrams. The derivation of the linked-cluster expansion for the energy then follows immediately. The equivalence to previous versions is established. The one-particle function is examined in particular detail, and it is shown that the poles of its space-time Fourier transform studied as a function of the energy variable, for fixed momentum, determined the (N+1N+1N+1)-particle and ($N\ensuremath{-}1$)-particle excited states which have single-particle character. For a reasonable assumption about the full spectrum of excited states, it is found that for the interacting system, single-particle excitations with a real energy occur only at the Fermi momentum. It is pointed out that the corresponding energy, termed the perturbed Fermi energy, equals the binding energy per particle in the ground state of NNN particles for a saturating system at equilibrium density.It is shown, finally, that the entire structure of the theory may be carried over to the case of finite temperature, requiring only a redefinition of the Green's functions. The analogy is constructed from a discussion of the internal energy.
A new cluster expansion technique is developed for the calculation of the energy expectation value of the many-fermion systems with singular potentials.The wave function, which is studied in this paper, is of the formwhich, in a special case, coincides with Jastrow wave function.The formula for the energy expectation value is written in terms of the complex cluster integrals.The method introduced in the derivation of this formula has an analogy to the classical cluster expansion method in the theory 0f an imperfect classical gas of multicomponents.