Lipkin-Nogami Method

The LN method constitutes an efficient method for approximately restoring the particle numbers before variation [33]. With only a slight modification of the HFB procedure outlined above, it is possible to obtain a very good approximation for the optimal HFB state, on which exact particle number projection then has to be performed [34,35].

In more detail, the LN method is implemented by performing the HFB calculations with an additional term included in the HF Hamiltonian,

$$h' = h - 2\lambda_2(1-2\rho),$$ (60)

and by iteratively calculating the parameter $\lambda_2$ (separately for neutrons and protons) so as to properly describe the curvature of the total energy as a function of particle number. For an arbitrary two-body interaction $\hat{V}$, $\lambda_2$ can be calculated from the particle-number dispersion according to [33],

$$\lambda_{2}=\frac {\langle 0\vert \hat{V} \vert 4\rangle\lan... ...t N^{2}\vert 4 \rangle\langle4\vert\hat N^{2} \vert\rangle} ~,$$ (61)

where $\vert\rangle$ is the quasiparticle vacuum, $\hat{N}$ is the particle number operator, and $\vert 4\rangle\langle4\vert$ is the projection operator onto the 4-quasiparticle space. On evaluating all required matrix elements, one obtains [36]

$$\lambda_{2}=\frac {4{\rm Tr} \Gamma^{\prime} \rho(1-\rho) + ... ...r}\rho (1-\rho )\right]^{2}-16{\rm Tr}\rho^{2}(1-\rho)^{2}} ~,$$ (62)

where the potentials

$$\begin{array}{rcl} \Gamma^{\prime}_{\alpha \alpha^{\prime}} ... ...me}}(\rho \kappa)_{\alpha^{\prime} \beta^{\prime}}, \end{array}$$ (63)

can be calculated in a full analogy to $\Gamma$ and $\Delta$ by replacing $\rho$ and $\kappa$ by $\rho(1-\rho)$ and $\rho\kappa$, respectively. In the case of the seniority-pairing interaction with strength $G$, Eq. (62) simplifies to

$$\lambda_{2}=\frac{G}{4} \frac {{\rm Tr} (1-\rho)\kappa~ {\rm... ... Tr}\rho (1-\rho )\right]^{2}-2~{\rm Tr}\rho^{2}(1-\rho)^{2}}.$$ (64)

An explicit calculation of $\lambda_{2}$ from Eq. (62) requires calculating new sets of fields (63), which is rather cumbersome. However, we have found [25] that Eq. (62) can be well approximated by the seniority-pairing expression (64) with the effective strength

$$G=G_{\mbox{\scriptsize {eff}}} = -\frac{\bar{\Delta}^2}{E_{\mbox{\scriptsize {pair}}}}\,$$ (65)

determined from the pairing energy

$$E_{\mbox{\scriptsize {pair}}} = -{\textstyle{\frac{1}{2}}}{\rm Tr}\Delta \kappa \,$$ (66)

and the average pairing gap

$$\bar{\Delta} = \frac{{\rm Tr}\Delta \rho}{{\rm Tr}\rho} \, .$$ (67)

Such a procedure is implemented in the code HFBTHO (v1.66p).