Summary

In the framework of the nuclear energy-density-functional theory, we derived the Lipkin method of approximate particle-number-symmetry restoration up to sixth order. The Lipkin parameters were determined from non-diagonal energy kernels, resulting in a more manageable approach as compared to the traditional Lipkin-Nogami approach.

Convergence of the Lipkin VAPNP method was tested by investigating gauge dependence of expansion parameters. Taking $^{120}$Sn as an example, the Lipkin expansion up to the second order was found to have explicit gauge angle dependence. Inclusion of fourth order terms subsequently diminished dependence on the gauge angle significantly. With the inclusion of sixth-order terms of the expansion, overall change is minimal, indicating a converging series. Accuracy of the Lipkin VAPNP method was tested by comparing the reduced energy kernel and Lipkin operator approximated by a power series. It was found that the chosen Lipkin operator describes well the small gauge-angle rotation of the intrinsic wave function. The results obtained for $^{100}$Sn and $^{120}$Sn showed that the second-order Lipkin expansion is typically not sufficiently converged. Within fourth order, the series already mimics the reduced energy kernels rather well. With inclusion of sixth order term, the results stay practically the same, indicating again a well converged series.

In chains of tin and lead isotopes, we compared the Lipkin VAPNP method to LN and PLN methods. As pointed in Ref. [5], for mid-shell nuclei, PLN is a very good approximation to the exact VAP method. Our results show that for the mid-shell nuclei, the Lipkin VAPNP method already at the second order gives rather well converged results. When advancing to higher orders, the results are improved. Near closed shells, because of the kink in the particle-number dependence on the projected energy, the Lipkin VAPNP method is unable to reproduce the exact projected energy. Also, near the closed-shell region, the pairing correlations have a dynamic nature [2]. Within the Lipkin VAPNP method, these kind of features cannot be reproduced with a well converging series expansion. The main features of the results obtained for nuclei were corroborated within the exactly-solvable two-level pairing model.

When neutrons and protons were treated simultaneously within the Lipkin VAPNP method, we observed a necessity to include in the Lipkin operator cross terms, which depend simultaneously on neutron and proton number operators. For the case of $^{124}$Xe, the contour lines of the reduced energy kernel, with respect to neutron's and proton's gauge angles, show tilted shapes. Without the cross terms, this kind of behavior cannot be reproduced. A study of the cross terms will be subject of future work.

The Lipkin VAPNP method presented in this work allows for a computationally inexpensive way to approximate the exact VAPNP energy of the ground state. We need to point out that this method cannot replace the exact projection for the case of, e.g., calculation of transition matrix elements or evaluation of selection rules, where good quantum numbers are mandatory. The Lipkin method can also be applied to approximate the restoration of other symmetries, broken on the mean-field level, with a small or no extra computational cost. This is important, for example, at the stage of adjusting the functionals, where rapid evaluations of penalty functions are required. This can be particularly significant for adjustments of EDFs tailored for beyond mean-field multireference studies [34]. Work towards restoring isospin and rotational symmetries within the framework of the Lipkin method is currently in progress.

This work was supported in part by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme, by the Polish National Science Center under Contract No. 2012/07/B/ST2/03907, by the Academy of Finland under the Centre of Excellence Programme 2012-2017 (Nuclear and Accelerator Based Physics Programme at JYFL), by the European Union's Seventh Framework Programme ENSAR (THEXO) under Grant No. 262010, and by the US Department of Energy under Contract Nos. DE-FC02-09ER41583 (UNEDF SciDAC Collaboration) and DE-FG02-96ER40963 (University of Tennessee). We acknowledge the CSC - IT Center for Science Ltd, Finland, for the allocation of computational resources.