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Physical Review

Calculation of Some Fourth-Order Diagrams in Nuclear Matter

A. G. Petschek

Published Mar 23, 1964• 2 citations
Publisher

Abstract

It is well known that in the Goldstone theory of many-fermion systems certain higher-order diagrams are included in the third-order diagrams if the hole self-energy is evaluated on the energy shell. In this paper a similar generalization of fourth-order diagrams is carried out. With this generalization, the Hartree-Fock potential $U$ which cancels the third-order diagrams no longer cancels the fourth-order diagrams as well as before. The lack of cancellation is evaluated in two simple cases using the reference approximation, and found to be large in one case, small in the other.

DOI
10.1103/physrev.133.b1401

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It is well known that in the Goldstone theory of many-fermion systems certain higher-order diagrams are included in the third-order diagrams if the hole self-energy is evaluated on the energy shell. In this paper a similar generalization of fourth-order diagrams is carried out. With this generalization, the Hartree-Fock potential UUU which cancels the third-order diagrams no longer cancels the fourth-order diagrams as well as before. The lack of cancellation is evaluated in two simple cases using the reference approximation, and found to be large in one case, small in the other.
A generalization of many-fermion Feynman-Goldstone diagrams is derived based on a time-dependent unperturbed Hamiltonian along with a time-dependent reference state. In lowest order, the time-dependent Hartree-Fock equations follow immediately from the same arguments which lead to static Hartree-Fock when the basis is time independent. Systematic corrections to the time-dependent Hartree-Fock mean-field approximation are obtained through higher order diagrams.[NUCLEAR STRUCTURE Generalization of Feynman-Goldstone diagrams to a time-dependent reference state. Diagrammatic derivation of the time-dependent Hartree-Fock equations and their systematic corrections.]
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.
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We present calculations of the energy per particle of pure neutron and symmetric nuclear matter with simplified Argonne nucleon-nucleon potentials for different many-body theories. We compare critically the Brueckner-Hartree-Fock results to other formalisms, such as the Brueckner-Bethe-Goldstone expansion up to third order, self-consistent Green's functions, auxiliary field diffusion Monte Carlo, and Fermi hypernetted chain. We evaluate the importance of spin-orbit and tensor correlations in the equation of state and find these to be important in a wide range of densities.
We advance the many-body perturbation theory (MBPT) calculations of the ground-state energy and radius of closed-shell nuclei beyond third order. Using automated diagram generation and evaluation up to fifth order, we present ground-state properties of selected closed-shell nuclei up to 78^{78}78Ni with two- and three-nucleon interactions derived from chiral effective field theory. A clear convergence trend is observed for the ground-state energy enabling calculations at improved accuracy. We further investigate in detail the decomposition of the fourth-order contributions. For the ground-state energy, the magnitude of the fourth-order contribution is typically less than half of the third order, and a typical cancellation among different classes of diagrams is observed. Finally, we perform a comprehensive comparison between MBPT and non-perturbative in-medium similarity renormalization group (IMSRG) calculations, with the goal to provide insight into many-body uncertainties associated with the IMSRG(2) truncation.