As already discussed in Sec. 3.3, efficient numerical codes that allow for large-scale, self-consistent variational calculations after projecting onto a good particle number have been developed [43]. The particle-number conserving HFB equations [40,41] with Skyrme functionals can be simply obtained from the standard Skyrme-HFB equations in coordinate space by replacing the intrinsic densities and currents by their gauge-angle dependent counterparts. Using the VAP method, one can properly describe transitions between normal and superconducting phases in finite systems, which are inherent in (semi)magic nuclei.
As mentioned above, the restoration of broken symmetries in the framework of DFT causes a number of questions, mainly related to the density dependence of the underlying interaction and to different treatment of particle-hole and particle-particle channels [42,67]. These questions are a matter of ongoing intensive research [65,66].
Related to the particle-number symmetry, but different in origin and
treatment, is the question of the spontaneous isospin breaking. The
isospin-breaking correction is of particular importance around the
line. The isoscalar pairing is believed to contribute to
the additional binding of = nuclei, the so-called Wigner energy
[68]. However, basic questions regarding the collectivity of
such a phase still remain unanswered, and should be part of the future
scientific agenda.
Apart from the presence of charge-dependent terms in the functional, such as the Coulomb term, the isospin symmetry is broken by the quasiparticle mean field (the generalized product wave function is not an eigenstate of isospin). Several techniques have been developed to restore isospin (see the discussion in Refs. [10,69] and references quoted therein). It is fair to say, however, that in spite of many attempts to extend the quasiparticle approach to incorporate the effect of proton-neutron correlations, no symmetry-unrestricted mean-field calculations of proton-neutron pairing, based on realistic effective interaction and the isospin-conserving formalism, have been carried out so far.