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Background: Nuclear self-consistent mean-field approaches are rooted in thedensity functional theory and, through the spontaneous symmetry breakingmechanism, allow for including important correlations, while keeping thesimplicity of the approach. Because real ground states should have allsymmetries of the nuclear Hamiltonian, these methods require subsequentsymmetry restoration.  Purpose: We implement and study Lipkin method of approximate variation afterprojection applied to the restoration of the translational or rotationalsymmetries.  Methods: We use Lipkin operators up to quadratic terms in momenta or angularmomenta with self-consistently determined values of the Peierls-Yoccoztranslational masses or moments of inertia, respectively. Calculations based onSkyrme energy-density functional are performed for heavy, deformed, and pairednuclei.  Results: In deformed nuclei, the Peierls-Yoccoz translational masses alongthree different principal-axes directions of the intrinsic system can bedifferent, which illustrates different widths of the total-momentumdistributions. Numerically, the differences are only of the order of a few percent. For the rotational-symmetry restoration, the Lipkin method comparesfavorably with the exact angular-momentum projection, which requires muchlarger computational effort.  Conclusions: The Lipkin method of translational and rotational symmetryrestoration is a practical low-cost method of determining the correspondingcorrelation energies. It allows for a simultaneous restoration of severalsymmetries and can be relatively easily implemented in the standardself-consistent mean-field calculations, including those required for theparameter adjustments.

Approximate restoration of translational and rotational symmetries within the Lipkin method