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Recently we proposed a scheme that applies the variational principle to acoherent-pair condensate in the BCS case [Phys. Rev. C 99, 014302 (2019)]. Thiswork extends the scheme to the HFB case by allowing variation of the canonicalsingle-particle basis. The result is equivalent to that of the so-calledvariation after particle-number projection in the HFB case, but now theparticle number is always conserved and the time-consuming projection isavoided. Specifically, we derive the analytical expression for the gradient ofthe average energy with respect to changes of the canonical basis, which isthen used in the family of gradient minimizers to iteratively minimize energy.In practice, we find the so-called ADAM minimizer (adaptive moments ofgradient), borrowed from the machine-learning field, is very effective. The newalgorithm is demonstrated in a semi-realistic example using the realistic$V_{{\rm{low}}{\textrm{-}}k}$ interaction and large model spaces (up to $15$harmonic-oscillator major shells). It easily runs on a laptop, and practicallythe computer time cost (in the HFB case) is several times that of solving HF bythe gradient minimizers. We hope the new algorithm could become a commonpractice because of its simplicity.

Application of the variational principle to a coherent-pair condensate: The HFB case