Lipkin method applied to the two-level pairing model

Figure 9: (Color online) Ground-state energies in the two-level pairing model calculated within the sixth-order Lipkin and exact VAPNP methods. To render the curves symmetric with respect to the closed shell at $N=20$, the appropriate linear term was added. Normalization of $50G+\epsilon=1$MeV was used.

Figure 10: (Color online) Ratios of approximate pairing energies, calculated within the approximate LN (open circles) and Lipkin VAPNP methods, relative to those of the exact VAPNP method. The figure shows results obtained for $\phi_{M}=0.06$ in function of the pairing-strength parameter $x=G/2\epsilon$. Note that panels (a)-(j) are drawn in very different scales, indicating the discrepancies up to 100% for $N=20$ and only 0.2% for $N=2$.

Figure 11: (Color online) Same as in Fig. 10, but for the results obtained for $x=0.03$ (weak pairing) plotted in function of the maximum gauge angle $\phi_{M}$.

In this appendix, we apply the Lipkin method to the standard two-level pairing model, which is characterized by two $\Omega$-fold degenerate levels with the single-particle energy difference $2\epsilon$ and pairing strength $G$. Below we closely follow the notations and definitions presented in Refs. [18,19], where the results obtained within the LN method have been studied.

In Fig. 9, we show particle-number dependence of the ground-state energies obtained for $\Omega=20$ and for three values of the ratio $x=G/2\epsilon$ equal to 0.03 (weak pairing) 0.053 (critical pairing), and 1 (strong pairing). Results show excellent agreement between the sixth-order Lipkin and exact VAPNP methods, which in the absolute scale of energy cannot be distinguished one from another. To compare the approximate and exact VAPNP methods in fine detail, in Fig. 10 we plotted ratios of the respective pairing energies, $R=E^{\mbox{\rm\scriptsize {approx}}}_{\mbox{\rm\scriptsize {pair}}}$/ $E^{\mbox{\rm\scriptsize {VAPNP}}}_{\mbox{\rm\scriptsize {pair}}}$, as functions of $x$. The pairing energies are defined [18,19] as differences between the total and Hartree-Fock energies. Note that the results are exactly symmetric with respect to the mid shell, that is, those for particle numbers of $N$ and $2\Omega-N$ are exactly identical.

The ratio of $R=1$, that is, perfect agreement, is for all particle numbers reached in the strong-pairing regime. For weak pairing, the largest discrepancies appear at mid shell, $N=20$, and they gradually decrease towards smaller (or larger) particle numbers. This is related to the kink in particle-number dependence of ground-state energies [19], cf. Fig. 9, which disappears with increasing pairing correlations.

For $N=20$, with increasing order of the Lipkin expansion, the agreement with exact results gradually increases, and the Lipkin VAPNP method, even at second order, is here visibly superior to the LN method. Note that at $N=20$, the odd orders of expansion (third and fifth) do not bring any improvement - this is owing to the symmetry of the model with respect to the mid shell.

For $N=18$, the Lipkin expansion cannot reproduce the kink appearing at the adjacent particle number of $N=20$ (see Fig. 9), and it does not seem to converge to the exact result, whereas the LN results are clearly superior. For smaller particle numbers, this pattern gradually changes, and for $N\leq12$ the Lipkin expansion does converge to the exact result and at orders higher than four becomes better than the LN method.

We stress here that in the realistic cases discussed in Sec. 3, the pattern of comparison between the LN and Lipkin VAPNP methods pertains to moderately high pairing strengths, certainly beyond the pairing phase transition, which in the two-level model appears at $x_c=1/(\Omega-1)\simeq0.053$.

Finally, in Fig. 11, we show dependence of results on the maximum gauge angle $\phi_{M}$ used in the Lipkin VAPNP method, see Secs. 2 and 3. We see that for all particle numbers, the second-order results do depend on $\phi_{M}$, indicating an insufficient order of expansion. For $N\leq12$ we see that with increasing order of expansion, the results become perfectly independent of $\phi_{M}$, which characterizes a converging expansion. On the other hand, closer to the mid shell, even at sixth order a visible dependence on $\phi_{M}$ still remains.