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Physical Review

Perturbation Theory Applied to the Nuclear Many-Body Problem

J. S. Levinger, M. Razavy, O. Rojo, N. Webre

Published Mar 15, 1961• 3 citations
Publisher

Abstract

function thus will be slightly changed from the values given in the paper.
DOI
10.1103/physrev.121.1863.5

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The solution of the nuclear A-body problem encounters severe limitations from the size of many-body operators. These limitations are typically related to both the (iterative) storing of the associated tensors and to the computational time related to their multiple contractions in the calculation of various quantities of interest. However, not all the degrees of freedom encapsulated into these tensors equally contribute to the description of many-body observables. Identifying systematic and dominating patterns, a relevant objective is to achieve an \emph{a priori} reduction to the most relevant degrees of freedom via a pre-processing of the A-body problem. The present paper is dedicated to the analysis of two different paradigms to do so. The factorization of tensors in terms of lower-rank ones, whose know-how has been recently transferred to the realm of nuclear structure, is compared to a reduction of the tensors' index size based on an importance truncation. While the objective is to eventually utilize these pre-processing tools in the context of non-perturbative many-body methods, benchmark calculations are presently performed within the frame of perturbation theory. More specifically, we employ the recently introduced Bogoliubov many-body perturbation theory that is systematically applicable to open-shell nuclei displaying strong correlations. This extended perturbation theory serves as a jumpstart for non-perturbative Bogoliubov coupled cluster and Gorkov self-consistent Green's function theories. Results obtained in "small" model spaces are equally encouraging for tensor factorization and importance truncation techniques. While the former requires significant numerical developments to be applied in large model spaces, the latter is presently applied in this context and demonstrates great potential to enable high-accuracy calculations at a much reduced computational cost.
Many-body perturbation theory provides a powerful framework to study the ground state and thermodynamic properties of nuclear matter as well as associated single-particle potentials and response functions within a systematic order-by-order expansion. However, computational challenges can emerge beyond the lowest orders of perturbation theory, especially when computing both single-particle potentials and response functions, which in general are complex-valued and require Cauchy principal value calculations of high-dimensional integrals. We demonstrate that normalizing flows are suitable for Monte Carlo importance sampling of both regular and irregular functions appearing in nuclear many-body calculations. Normalizing flows are a class of machine learning models that can be used to build and sample from complicated distributions through a bijective mapping from a simple base distribution. Furthermore, a well-trained model for a certain target integrand can be efficiently transferred to calculate related integrals with varying physical conditions. These features can enable more efficient tabulations of nuclear physics inputs to numerical simulations of supernovae and neutron star mergers across varying physical conditions and nuclear force models.