Restricted variation after projection and the Lipkin-Nogami methods

We analyze the ability of a restricted variation after projection method to achieve the full variation after projection solution in a pairing Hamiltonian. The study of the projected potential energy surfaces defined along the fluctuations of the particle number $(\Delta N){}^2$ and $(\Delta N){}^4$ allows the enlargement of the variational space to take into account more correlations within the projection after the variation framework. The results show the equivalence of the full variational projected solution to a restricted projected solution with an educated choice of the restricted variational space. Finally, we show that the Lipkin-Nogami approach can be derived, under certain conditions, in an variational context and that the projected Lipkin-Nogami approach represents an approximation to a restricted variation after projection method with the $(\Delta N){}^2$ as degree of freedom.

Figure 1

(a) Condensation energy (in units of d) as a function of the pairing interaction strength [given in terms of the parameter $\chi =G(N_0-1)/(2d)$] for the BCS (dotted line), VAP (solid line), PAV (dashed line), RVAP${}_1$ (dashed-dotted line), LN (open circles), and RVAP${}_2$ (filled circles) approximations. The vertical lines mark the values of χ where RVAP${}_1$ and RVAP${}_2$ methods have been analyzed in detail. In the inset the exact solution and the VAP approach are plotted. (b) Normalized deviations from the VAP solution for different approaches.

Figure 2

Condensation energy as a function of the particle number fluctuations for the CBCS (dotted line) and particle number projected (solid line) approximations, PCBCS, for two different values of χ, namely χ = 1.09 in (a) and χ = 1.91 in (b). The horizontal lines mark the values of the VAP solutions and the vertical lines the minimum of the PES of the CBCS approach. The minimum of the PCBCS curve corresponds to the RVAP${}_1$ result and is represented by a full circle. In addition, the approximate condensation energy computed in the spirit of the Lipkin-Nogami method is plotted as dashed lines and the LN selfconsistent solution (LN${}_{\mathrm{sc}}$) is represented as an open circle.

Figure 3

The CBCS (dotted line) and particle number projection (solid line) condensation (PCBCS) energies as a function of $(\Delta N){}^4$ for χ = 1.09 and $(\Delta N){}^2$ = 1.6 (a) and χ = 1.91 and $(\Delta N){}^2=34$ (b).

Figure 4

The PCBCS condensation energy as a function of $(\Delta N){}^2$ (short dashed curve) as well as the same quantity obtained by minimizing for each value of $(\Delta N){}^2$ along the $(\Delta N){}^4$ direction (full curve). Panel (a) corresponds to the pairing interaction strength parameter χ = 1.09, whereas panel (b) corresponds to the χ = 1.91 value. The minima of each curve are marked by symbols. The horizontal dashed line marks the value of the VAP solution.

Figure 5

Same as described in the legend to Fig. (1) but for the two-level pairing model.

Figure 6

Same as described in the legend to Fig. (2) but for the two-level model and a pairing interaction strength $\chi =08$.

Figure 7

The $h_2$ parameter of the Lipkin-Nogami method as well as the chemical potential of the $(\Delta N){}^2$ constraint (${\lambda }_{\Delta N^2}$) are plotted as a function of the particle number fluctuation for the two pairing Hamiltonians considered: in panel (a) for the two-level Hamiltonian and $\chi =08$, and in panel (b) for the picket fence Hamiltonian and $\chi =109$.