Particle Number Projection

The advantage of the mean-field approach to the pairing problem lies in its simplicity that allows a straightforward interpretation in terms of pairing fields and deformations (pairing gaps) associated with the spontaneous breaking of gauge symmetry. However, in the intrinsic-system description, the particle-number invariance is internally broken. Therefore, to relate to experiment, the particle number symmetry needs to be restored. This can be done on various levels, including the quasiparticle random phase approximation, Lipkin-Nogami (LN) method, the projected LN method (PLN) [39,22,21], and the particle-number projection before variation (PNP) [40,41,42].

Recently, particle-number restoration before variation has been incorporated for the first time into the Skyrme-DFT framework employing zero-range pairing [43]. It was demonstrated that the resulting projected HFB equations can be expressed in terms of local gauge-angle-dependent densities. In Ref. [43], results of PNP calculations have been compared with those obtained within LN and PLN methods. While the PLN gives results close to PNP for open-shell nuclei, for closed-shell nuclei it breaks down with more than one MeV difference in the total binding energy; see Fig. 5. This pathological behavior of LN and PLN methods around closed-shell nuclei can be partly cured by performing particle-number projection from neighboring open-shell systems [44]. This result is important in the context of large-scale microscopic mass calculations such as those of Ref. [21]. To be on the safe side, however, it is always recommended to apply the complete PNP procedure around closed shells.

Figure 5: The total binding energy (with respect to a linear reference) as a function of the neutron number $N$ for even-even nuclei around doubly magic $^{40}$Ca, $^{48}$Ca, $^{100}$Sn, $^{132}$Sn calculated in LN, PLN, and PNP methods. The crosses for magic nuclei indicate the PLN results obtained by projecting from neighboring nuclei, as indicated by arrows.