Approximate restoration of translational and rotational symmetries within the Lipkin method

Abstract: Background: Nuclear self-consistent mean-field approaches are rooted in the density functional theory and, through the spontaneous symmetry breaking mechanism, allow for including important correlations, while keeping the simplicity of the approach. Because real ground states should have all symmetries of the nuclear Hamiltonian, these methods require subsequent symmetry restoration. Purpose: We implement and study Lipkin method of approximate variation after projection applied to the restoration of the translational or rotational symmetries. Methods: We use Lipkin operators up to quadratic terms in momenta or angular momenta with self-consistently determined values of the Peierls-Yoccoz translational masses or moments of inertia, respectively. Calculations based on Skyrme energy-density functional are performed for heavy, deformed, and paired nuclei. Results: In deformed nuclei, the Peierls-Yoccoz translational masses along three different principal-axes directions of the intrinsic system can be different, which illustrates different widths of the total-momentum distributions. Numerically, the differences are only of the order of a few per cent. For the rotational-symmetry restoration, the Lipkin method compares favorably with the exact angular-momentum projection, which requires much larger computational effort. Conclusions: The Lipkin method of translational and rotational symmetry restoration is a practical low-cost method of determining the corresponding correlation energies. It allows for a simultaneous restoration of several symmetries and can be relatively easily implemented in the standard self-consistent mean-field calculations, including those required for the parameter adjustments.