Eur. Phys. J. A (2020) 56: 40
https://doi.org/10.1140/epja/s10050-020-00045-8

Regular Article -Theoretical Physics

Normal-ordered k-body approximation in particle-number-breaking theories

¹ CEA, DAM, DIF, 91297, Arpajon, France
² Max-Planck-Institut für Kernphysik, 69117, Heidelberg, Germany
³ IRFU, CEA, Université Paris-Saclay, 91191, Gif-sur-Yvette, France
⁴ Instituut voor Kern- en Stralingsfysica, KU Leuven, 3001, Leuven, Belgium

^* e-mail: alexander.tichai@physik.tu-darmstadt.de

Received: 2 August 2019
Accepted: 19 November 2019
Published online: 5 February 2020

Abstract

The reach of ab initio many-body theories is rapidly extending over the nuclear chart. However, dealing fully with three-nucleon, possibly four-nucleon, interactions makes the solving of the A-body Schrödinger equation particularly cumbersome, if not impossible beyond a certain nuclear mass. Consequently, ab initio calculations of mid-mass nuclei are typically performed on the basis of the so-called normal-ordered two-body (NO2B) approximation that captures dominant effects of three-nucleon forces while effectively working with two-nucleon operators. A powerful idea currently employed to extend ab initio calculations to open-shell nuclei consists of expanding the exact solution of the A-body Schrödinger equation while authorizing the approximate solution to break symmetries of the Hamiltonian. In this context, operators are normal ordered with respect to a symmetry-breaking reference state such that proceeding to a naive truncation may lead to symmetry-breaking approximate operators. The purpose of the present work is to design a normal-ordering approximation of operators that is consistent with the symmetries of the Hamiltonian while working in the context of symmetry broken (and potentially restored) methods. Focusing on many-body formalisms in which U(1) global-gauge symmetry associated with particle-number conservation is broken (and potentially restored), a particle-number-conserving normal-ordered k -body (PNOkB) approximation of an arbitrary N-body operator is designed on the basis of Bogoliubov reference states. A numerical test based on particle-number-projected Hartree–Fock–Bogoliubov calculations permits to check the particle-number conserving/violating character of a given approximation to a particle-number conserving operator. The PNOkB approximation of an arbitrary N-body operator is formulated. Based on this systematic approach, it is demonstrated that naive extensions of the normal-ordered two-body (NO2B) approximation employed so far on the basis of symmetry-conserving reference states lead to particle non-conserving operators. Alternatively, the PNOkB procedure is now available to generate particle-number-conserving approximate operators. The formal analysis is validated numerically. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed. The future formulation of an angular-momentum-conserving normal-ordered k-body approximation based on deformed Slater determinant or Bogoliubov reference states is envisioned.