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

Abstract

Recently we proposed a scheme that applies the variational principle to a coherent-pair condensate in the BCS case [1]. This work extends the scheme to the HFB case by allowing variation of the canonical single-particle basis. The result is equivalent to that of the so-called variation after particle-number projection in the HFB case, but now the particle number is always conserved and the time-consuming projection is avoided. Specifically, we derive the analytical expression for the gradient of the average energy with respect to changes of the canonical basis, which is then used in the family of gradient minimizers to iteratively minimize energy. In practice, we find the so-called ADAM minimizer (adaptive moments of gradient), borrowed from the machine-learning field, is very effective. The new algorithm is demonstrated in a semi-realistic example using the realistic V low-k interaction and large model spaces (up to 15 harmonic-oscillator major shells). It easily runs on a laptop, and practically the computer time cost (in the HFB case) is several times that of solving HF by the gradient minimizers. We hope the new algorithm could become a common practice because of its simplicity.

Nuclear states projected from a pair condensate

“…Firstly construct W = p p, then diagonalize W , so that W V = V Λ where V contains all the eigenvectors and the diagonal matrix Λ contains all the eigenvalues. Then p = V pV is the canonical TABLE II: Structure coefficients of the optimized pairs for 32 Si, 18,20 O, in canonical form.…”

“…The pairs optimized for N = Z nucleus 28 Si have N p /2 = N n /2 = 3 non-negligible structure coefficients, when transformed into canonical form, which means the pair condensate approximates one Slater determinant in canonical basis. On the other hand, the pairs optimized for semi-magic nuclei 18,20 O have more scattered coefficients in canonical form, indicating that the optimized pair condensates are mixtures of more than one Slater determinants. matrix in the new basis,…”

“…That explains why PVPC generates results for N = Z nuclei not much better than those from PHF. However for 18,20 O the canonical pair structure coefficients in Tab.…”

“…In low-lying states, such pairs are favored energetically. An example of this is the general seniority model, which focuses on S-pair condensates and their breaking [17][18][19][20], and is especially useful in semi-magic nuclei. Going further, the nucleonpair approximation (NPA) truncates the configuration space with angular-momentum coupled fermion pairs, and has proved to be effective in medium-heavy and heavy nuclei with schematic pairing-plus-quadrupole-quadrupole interactions [21,22].…”

Optimizing the Computation of Many-Pair Density Matrix in VDPC

“…Recently, [18] and [19] has proposed a method that applies the variational principle directly to coherent-pair condensate (VDPC). This method does not require performing the time-consuming numerical projection and also maintains particle number conservation.…”