Toward a global microscopic theory for nuclear structure: Mean field plus random phase approximation vs. shell model

Our understanding of nuclear structure is built upon mean-field theories such as Hartree-Fock and time-dependent Hartree-Fock. The small-amplitude limit of the latter is the random phase approximation (RPA), which is widely used to model giant resonances in nuclei. Despite this popularity, RPA has been mostly validated against toy models; tests against complex models are scarce in the literature. We perform a thorough test of the RPA against full 0ħω shell model (SM) calculations, including in our investigation binding energies, scalar ground-state observables, for which we develop a new method, and transition strengths. We allow deformed Hartree-Fock solutions and compare results for spherical and deformed nuclei. We obtain reasonable agreement between RPA and SM, albeit with some significant failures. Particularly, we find that the low-lying collectivity is poorly described for deformed mean-field solutions, which we interpret as incomplete symmetry restoration in RPA. Results for observables, and in particular for J2, also point out toward the same conclusion regarding the symmetries of the ground state. We also prove, both analytically and numerically, that a long-standing theorem regarding RPA is violated in the case of deformation. The worse violation appears for low-lying transitions, such as isoscalar E2, which we consider as a third argument for an incomplete symmetry restoration.