Results and discussion

We have implemented the Lipkin VAPNP method, presented in Sec. 2, in computer code HFODD (v2.68c) [26,27]. This code solves the HFB equations in a three-dimensional Cartesian harmonic oscillator basis. Within this implementation, we tested the Lipkin VAPNP method using the Skyrme SIII parameterization [28] in the particle-hole channel and volume zero-range pairing interaction in the particle-particle channel. SIII parameterization was selected for this study due to the fact that it contains only integer powers of densities. In the calculation of non-diagonal terms, the density-dependent interaction was treated within the ``mixed density prescription", as discussed in Refs. [29,8,5,30].

The neutron pairing strength, $V_0=-155.45$MeVfm$^3$, was adjusted within the LN method to reproduce the empirical neutron pairing gap of $\Delta_n=1.245$MeV in $^{120}$Sn. In principle, at each given order of the Lipkin VAPNP method, this adjustment should be repeated. However, for the sake of meaningful comparison of the results obtained at different orders, we use the same pairing strength throughout all calculations.

For protons, pairing strength was set to zero, that is, the proton subsystem is described by unpaired states. This setup allows us to test the Lipkin VAPNP method in the neutron paired subsystem, resulting in a clearer interpretation of the obtained results and allowing for a better evaluation of the efficiency of the Lipkin VAPNP method. Because of the used zero-range pairing interaction, we adopted the commonly used equivalent-spectrum cutoff of 60MeV, applied in the quasiparticle configuration space. All calculations were performed in the spherical basis of 14 major harmonic-oscillator shells.

Figure 1: (Color online) Lipkin parameters $k_2$ (a), $k_4$ (b), and $k_6$ (c), Lipkin correction energy $E_{\mbox{\rm\scriptsize{corr}}}$ (16) (d), and Lipkin VAPNP energy $E_{N_0}$ (7) (e), determined in $^{120}$Sn at second, fourth, and sixth orders, as functions of the maximum gauge angle $\phi_{M}$. Note that Lipkin parameters $k_2$, $k_4$, and $k_6$ are shown in units of MeV, keV, and eV, respectively, which illustrates the rapid convergence of the Lipkin expansion.

Figure 2: (Color online) Same as Fig. 1, but for $^{100}$Sn.

To begin with, we first study convergence of the Lipkin VAPNP method when terms up to sixth-order in expansion (8) are incorporated. At present, we limit our analysis to the even powers only, that is, we take into account terms with $m=2$, 4, and 6. This corresponds to a symmetric approximation around the central value of the particle number $N_0$. In Figs. 1 and 2, we show, respectively for $^{120}$Sn and $^{100}$Sn, dependence of the Lipkin parameters on the maximum gauge angle $\phi_{M}$ (see the previous section). The figures also show the total Lipkin VAPNP energy $E_{N_0}$ and Lipkin correction energy $E_{\mbox{\rm\scriptsize {corr}}}$

$$E_{\mbox{\rm\scriptsize {corr}}}=\langle\Phi\vert-\hat{K}... ...t{N}-N_0\}\vert\Phi\rangle = -\sum_{m=1}^M k_{m} n_m(0) \, ,$$ (16)

cf. Eqs. (7) and (9).

At second order, the obtained results show a clear dependence on $\phi_{M}$, indicating insufficient expansion. On the other hand, at fourth and sixth orders, total energy $E_{N_0}$ and correction energy $E_{\mbox{\rm\scriptsize {corr}}}$ are already rather insensitive to $\phi_{M}$. Thus, we can conclude that at sixth order, the expansion is well converged, and at least fourth order is required for sufficiently precise results. We also note that for the magic nucleus $^{100}$Sn, the convergence is slightly slower, and the values of Lipkin parameters are significantly higher than those for $^{120}$Sn.

Figure 3: (Color online) Reduced energy kernel $h(\phi)-h(0)$ (full squares) and reduced kernels of Lipkin operator $\sum_{m=1}^M k_m(n_m(\phi)-n_m(0))$ at orders $M=2$, 4, and 6 (open symbols), as functions of the gauge angle up to $\phi=1$, calculated in $^{100}$Sn and $^{120}$Sn.

Figure 4: (Color online) Same as in Fig. 3 but for the gauge angles up to $\phi=2\pi$.

In what follows, we have used the same maximum gauge angle of $\phi_{M}=\frac{2\pi}{51}\simeq0.123$ in all expansions, regardless of the expansion order. In Fig. 3, convergence of the reduced kernels of Lipkin operator (8) in $^{100}$Sn and $^{120}$Sn is shown. Kernel values at $\phi=0$ were subtracted, in order to illustrate how well the reduced Routhian kernels (9) stay constant, that is, independent of the gauge angle $\phi$. Again, we clearly see that the second-order expansion is insufficient, whereas fourth and sixth orders already give satisfactory description of the energy kernels.

Figure 4 shows the same kernels as those plotted in Fig. 3 for the whole range of gauge angle, up to $\phi=2\pi$. We see that in $^{100}$Sn the energy kernel near $\phi=\pi/2$ is poorly described by the Lipkin expansion. This is directly related to the kink in the particle-number dependence of the projected energies, which appears at the magic shell closure, and which cannot be properly described by a polynomial expansion [19,5]. In this kind of case, a good quality of the Lipkin expansion obtained at small gauge angles is not sufficient enough to guarantee a good convergence at larger gauge angles. The situation is entirely different in the open-shell nucleus $^{120}$Sn, where particle-number dependence of the projected energies is given by a smooth function, which can very well be approximated by a polynomial expansion. Here, for all gauge angles, we obtain a perfectly converging Lipkin expansion of the exact energy kernel, even in the vicinity of the pole related to the nearly half-filled 3s$_{1/2}$ orbital [8].

Figure 5: (Color online) The LN, PLN, and Lipkin VAPNP energies of tin isotopes relative to those obtained within the standard HFB method.

Figure 6: (Color online) Same as in Fig. 5, but for lead isotopes.

In Figs. 5 and 6, we show results of the Lipkin VAPNP method for tin and lead isotopes, respectively. For comparison, figures also show results obtained using the LN method, similarly as in Ref. [31], and projectd LN (PLN) method, as in Ref. [5], where the exact PNP energy is obtained via projection from HFB+LN self-consistent solution.

Away from the closed shells, at fourth and sixth orders, results of the Lipkin VAPNP method are very similar to those of PLN results. As pointed out by the VAPNP calculations in Ref. [5], for open shell nuclei, PLN results are very close to exact VAPNP results. Again, fourth and sixth orders give similar results, signaling the convergence of the Lipkin expansion. We can thus conclude that the fourth-order Lipkin VAPNP method is a good approximation of the exact VAPNP method. Near shell closure, differences between various orders of the Lipkin VAPNP method are large, indicating a non-convergent power series of the Lipkin operator. Once again, this is related to the kink in the particle-number dependence on the projected energies [19,5].

Figure 7: (Color online) Top panels (a) and (e): the PLN energies compared with the exact PNP energies determined for states obtained by solving the Lipkin equations at second, fourth, and sixth orders. Lower panels show comparisons at three different orders of the Lipkin VAPNP and exact PNP energies. All energies are plotted relative to those obtained within the standard HFB method.

In Fig. 7, we show results obtained by projecting good particle numbers from the states obtained either by the Lipkin VAPNP or LN methods. It is very gratifying to see that irrespective on whether one uses the Lipkin VAPNP or LN methods, the projected energies, shown in the two top panels, are very similar. This fact means that all approximate methods analyzed in this study lead to similar pair condensates, whereas they differ in the determination of corrective mean-field energies. The main advantage of using the Lipkin VAPNP method is in the fact that the PNP calculation does not have to be performed at all. Then, as shown in lower panels of Fig. 7, the obtained energies very well approximate the PNP energies. This is particularly true near the middle of the shell, where the influence of closed-shell kinks in the projected energies is weaker.

We also see that at closed shells, the second-order Lipkin VAPNP method, similarly as the LN method - see Figs. 5 and 6 - gives results that are very different from those obtained by the PNP. On the contrary, the fourth- and sixth-order Lipkin VAPNP methods gives results almost identical to the PNP. Finally, at fourth and sixth orders, the non-analytic behavior of the PNP energies at closed shells causes the largest discrepancies for 2 or 4 particles away from the closed shell.

Figure 8: (Color online) Reduced energy kernel $h(\phi_{\nu},\phi_{\pi})-h(0,0)$ (left panel) compared to the reduced kernel of the Lipkin operator $\sum_{m=1}^M k_m(n_m(\phi_{\nu},\phi_{\pi})-n_m(0,0))$ at sixth order (right panel), calculated for $^{124}$Xe.

Our current implementation of the Lipkin method in the computer code HFODD allows us to treat pairing correlations simultaneously for neutrons and protons. However, this has been implemented such that the Lipkin operator is simply a sum of neutron and proton contributions of Eq. (8), with Lipkin parameters determined by independent gauge-angle rotations for neutrons and protons. Although this implementation works perfectly well, we have realized that such a method is insufficient in some cases. This is illustrated in Fig. 8(a), which shows the reduced energy kernel of $^{124}$Xe calculated in two dimensions, as a function of the neutron $\phi_n$ and proton $\phi_p$ gauge angles. We clearly see that energy kernel is tilted with respect to main axes of neutron and proton gauge angle. Evidently, the Lipkin operator, here being a sum of neutron and proton contributions separately, leads to a non-tilted energy kernel, as shown in Fig. 8(b). Therefore, to fully reproduce the true energy kernel, one has to use the Lipkin operator that contains cross terms, which depend on products of neutron and proton particle numbers. Within proton-neutron pairing scheme, combined with PLN, these kind of cross-terms are required [32,33]. Implementations of such more complicated forms of the Lipkin operator will be subject of future study.