In the present study, I have analyzed the translational-symmetry restoration by the approximate variation after projection based on the Lipkin method. For translational symmetry, the Gaussian Overlap Approximation gives an excellent representation of numerical results obtained in doubly-magic spherical nuclei by using the Skyrme energy density functionals. The Lipkin method is based on subtracting from the Skyrme energy the center-of-mass kinetic energy with the Peierls-Yoccoz mass, and not with the true mass. However, calculations show that the Peierls-Yoccoz mass is only a few percent different than the true mass of the nucleus.
I have also studied properties of the direct part of the center-of-mass kinetic energy and I showed that this direct part does not fulfill Lipkin conditions and thus is not a proper correction within the variation after projection method to restore the translational symmetry. In fact, it seems that there is no good argument in support of neglecting the exchange part of the center-of-mass kinetic energy at all, apart from the fact that it is a cheap method. Nevertheless, shell effects induced by the exchange part appear to be weak, and thus the exchange part may only weakly influence the overall agreement of the calculated masses with experimental data.
Due to the fact that the three Cartesian components of the total momentum operator commute, the Lipkin method to restore the translational symmetry can be easily generalized to deformed nuclei. In this case, one obtains two or three different Peierls-Yoccoz translational masses in axially or triaxially deformed nuclei, respectively. This natural result reflects the fact that in deformed nuclei the momentum fluctuations along the principal axes of the mass distribution are different.
The methods presented in this study may constitute an interesting
alternative to the Lipkin-Nogami expressions [7,18]
for calculating the correcting factor 
, required for the
approximate particle-number-symmetry restoration. Indeed, the
Lipkin-Nogami expressions are difficult to implement in realistic
calculations, and approximate work-around procedures have been used
in practical approaches [19,4]. Expressions analogous
to Eqs. (28) and (29), which are based on the
transition energy and overlap kernels with respect to a simple shift
in the gauge angle, could open here a quite useful new possibility.
The largest impact of the Lipkin method can be expected for the rotational-symmetry restoration. Here, the Lipkin operator corresponds to the total angular momentum squared, which, being included before variation in the functional, may induce deformation in all nuclei, including the magic ones. This may create substantial rotational corrections in these nuclei, in analogy to those that are obtained by minimizing the angular-momentum projected energies after variation over deformation, see e.g. Ref. [20]. However, the Lipkin method may magnify these corrections even more, because it treats them before variation. Since in the case of the rotational symmetry, the Lipkin corrections can be expected to be strongly shell dependent, they may have a strong impact on the agreement of the calculated masses with experimental data.
The same methodology can also be applied to restore the isospin symmetry in nuclei. Here, the Lipkin operator corresponds to the total isospin operator squared, cf. Refs. [21,22], and the Peierls-Yoccoz mass must be calculated by considering kernels of the nuclear energy only. In this way, the dependence of this nuclear energy on the total isospin is flattened, which this leaves to the Coulomb energy the possibility of inducing the correct isospin-mixing effects.
It is also worth noting that the Lipkin method with the Peierls-Yoccoz collective mass is based on properties of the energy kernels for relatively small shifts of the collective coordinates. Therefore, singularities of these kernels, see Ref. [23] and references cited therein, which plague the exact symmetry-restoration methods within the energy-density-functional approaches, are not causing problems. This observation has to be substantiated by analyzing the singularity-free corrected kernels [24,25,26] and checking that the proposed corrections are appropriately small for small shifts of the collective coordinates.
Altogether, the Lipkin method reviewed in this study can be systematically applied to restore, within the mean-field or energy-density-functional theories, all broken symmetries. This method constitutes a practical alternative with respect to the exact projection techniques, which are very costly and thus cannot be applied to several broken symmetries simultaneously. The work on implementing the Lipkin method to symmetries other than the translational one studied in this work is now in progress.
Finally, the Lipkin method can also be used for approximate calculation of average values of observables in projected states. However, this method cannot replace the real projection in cases when the good quantum numbers are necessary to calculate matrix elements and properly account for selection rules. Nevertheless, by projecting good quantum numbers from the state that minimizes the Lipkin projected energy one probably obtains the best viable alternative to the exact variation after projection.
This work was supported in part by the Polish Ministry of
Science, by the Academy of Finland and University of Jyväskylä
within the FIDIPRO programme, and by the UNEDF SciDAC Collaboration
under the U.S. Department of Energy grant No. DE-FC02-07ER41457.