Particle-number-projected Hartree–Fock–Bogoliubov study with effective shell model interactions

Abstract

We perform particle-number projected mean-field study using the recently developed symmetryprojected Hartree-Fock-Bogoliubov (HFB) equations. Realistic calculations have been performed in sd-and fp-shell nuclei using the shell model empirical intearctions, USD and GXPFIA. It is demonstrated that the mean-field results for energy surfaces, obtained with these shell model interactions, are quite similar to those obtained using the density functional approaches. Further, it is shown that particle-number projected results, for neutron rich isotopes, can lead to different ground-state shapes in comparison to the bare HFB calculations. PACS numbers: 24.75.+i, 21.60.Jz, 27.90.+b, 24.10.Pa I. INTRODUCTIONOne of the primary research goals in nuclear structure physics is to consider correlations going beyond the mean-field approximation [1]. The mean-field theory in the form of Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) approaches have been quite successful in describing the gross features of atomic nuclei [2][3][4][5][6]. In particular, mean-field study with density dependent effective interactions, which include Skyrme [7-10], Gogny [11][12][13] and relativistic models [14-16] have provided an accurate description of the ground-state properties of atomic nuclei ( binding energies, deformations, radii and etc.). However, it is well recognized that meanfield approximation breaks down, in particular, when approaching the limits of spin, particle stability and isospin. In approaching these limits, the correlations going beyond the mean-field approximation play a pivotal role. The related problem is that the product wavefunction ansatz employed in the mean-field theory breaks the symmetries that original many-body Hamiltonian obeys [1,4]. These broken symmetries, for instance, include gauge-symmetry associated with the particle-number, rotational and isospin symmetries.

Variation after projection with a triaxially deformed nuclear mean field

“…To test the quality of the projected wavefunctions, one can compare them with the exact shell model ones using a common Hamiltonian. HFB and variation after projected HFB calculations with shell model Hamiltonians have been reported by several authors [2][3][4][5]. For those calculations without projection, the HFB vacuum states are often assumed to be axially symmetric [4].…”

Application of the gradient method to Hartree-Fock-Bogoliubov theory

“…Since one has a Hamiltonian for the sd-shell that describes the structure very well, one could test the approximations to introduce correlations, such as projection, the random-phase approximation, etc and compare them with the exact results from the Shell Model. Preliminary results along this line are discussed in [11,12]. As a first step in this program, one needs a robust SCMF code that treats shell-model Hamiltonians.…”

Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states

“…In general, the optimum state |Θ will be different for each irrep. Such a search can be achieved with variational methods for single product states [75,76] and also for superpositions of product states, either formulated as a Generator Coordinate Method [24,77] or as the configuration mixing in a non-orthogonal basis [30,31,78]. In both cases, the evaluation of matrix elements between projected states remains tractable, although non-trivial, 7 while the calculation of the projected wave function is not.…”

“…Mϕ (72), the non-physical contributions to the matrix element for N 0 are eliminated as a result of relation (76) until convergence to the physical value for the irrep N 0 is reached. For convergence of the sum rule, it is obviously necessary that the summation covers all irreps found in the original state |Φ a and that the number of discretization point is sufficient to converge the calculation of the projected matrix elements.…”