Overview of Hartree-Fock-Bogoliubov theory and the Lipkin-Nogami method

In this section, we review the basic ingredients of Hartree-Fock-Bogoliubov theory and the Lipkin-Nogami method followed by particle-number projection. Since these are by now standard tools in nuclear structure, we keep the presentation brief and refer the reader to Ref. [30] for further details.

HFB is a variational theory that treats in a unified fashion mean-field and pairing correlations. The HFB equations can be written in matrix form as

$$\left( \begin{array}{cc} h-\lambda & \Delta \\ -\Delta ^{\a... ...\left( \begin{array}{l} U_{n} \\ V_{n} \end{array}\right) \;,$$ (1)

where $E_{n}$ are the quasiparticle energies, $\lambda$ is the chemical potential, $h=t+\Gamma$ and $\Delta$ are the Hartree-Fock (HF) hamiltonian and the pairing potential, respectively, and $U_{n}$ and $V_{n}$ are the upper and lower components of the quasiparticle wave functions. These equations are solved subject to constraints on the average numbers of neutrons and protons in the system, which determine the two corresponding chemical potentials, $\lambda_n$ and $\lambda_p$.

In coordinate representation, the HFB approach consists of solving (1) as a set of integro-differential equations with respect to the amplitudes $U(E_{n},{\bf r})$ and $V(E_{n},{\bf r})$, both of which are functions of the position coordinate ${\bf r}$. The resulting density matrix and pairing tensor then read

$$\rho ({\bf r},{\bf r}^{\prime }) = \sum\limits_{0\leq E_{n}\leq E_{\max }}V^{\ast }(E_{n},{\bf r})V(E_{n},{\bf r}^{\prime })\;,$$ (2)

$$\kappa({\bf r},{\bf r}^{\prime }) = \sum\limits_{0\leq E_{n}\leq E_{\max }}V^{\ast }(E_{n},{\bf r})U(E_{n},{\bf r}^{\prime })\;.$$ (3)

Typically, the HFB continuum is discretized in this approach by putting the system in a large box with appropriate boundary conditions [6].

In the configurational approach, the HFB equations are solved by matrix diagonalization within a chosen single-particle basis { $\psi _{\alpha }$} with appropriate symmetry properties. In this sense, the amplitudes $U_{n}$ and $V_{n}$ entering Eq. (1) may be thought of as expansion coefficients for the quasiparticle states in the assumed basis. The nuclear characteristics of interest are determined from the density matrix and pairing tensor,

$$\rho ({\bf r},{\bf r}^{\prime }) = \sum\limits_{\alpha \beta }\rho _{\alpha \beta }\psi _{\alpha }({\bf r})\psi _{\beta }^{\ast }({\bf r}%% ^{\prime })\;,$$ (4)

$$\kappa({\bf r},{\bf r}^{\prime }) = \sum\limits_{\alpha \beta }\kappa_{\alpha \beta }\psi _{\alpha }({\bf r})\psi _{\beta }({\bf r}^{\prime })\;,$$ (5)

which are expressed in terms of the basis states $\psi _{\alpha }$ and the associated basis matrix elements as

$$\rho _{\alpha \beta }= \sum\limits_{0\leq E_{n}\leq E_{\max }}V_{\alpha n}^{\ast }(E_{n})V_{\beta n}(E_{n})\;,$$ (6)

$$\kappa_{\alpha \beta }= \sum\limits_{0\leq E_{n}\leq E_{\max }}V_{\alpha n}^{\ast }(E_{n})U_{\beta n}(E_{n})\;.$$ (7)

In configuration-space calculations, all quasiparticle states have discrete energies $E_{n}$.

The results from configuration-space HFB calculations should be identical to those from the coordinate-space approach when all the states $\psi _{\alpha }$ from a complete single-particle basis are taken into account. Of course, this is never possible. In the presence of truncation, it is essential that the basis produce rapid convergence, so that reliable results can be obtained within computational limitations on the number of basis states that can be included.

The LN method serves as an efficient method for restoring particle number before variation [24]. 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 [28,31].

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

$$h' = h - 2\lambda_2(1-2\rho),$$ (8)

and by iteratively calculating the constant $\lambda_2$ (separately for neutrons and protons) so as to properly describe the curvature of the total energy as function of particle number. For an arbitrary two-body interaction $\hat{V}$, $\lambda_2$ can be calculated from the particle-number dispersion according to [24],

$$\lambda_{2}=\frac {\langle 0\vert \hat{V} \vert 4\rangle\lan... ...t N^{2}\vert 4 \rangle\langle4\vert\hat N^{2} \vert\rangle} ~,$$ (9)

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 [27]

$$\lambda_{2}=\frac {4{\rm Tr} \Gamma^{\prime} \rho(1-\rho) + ... ...r}\rho (1-\rho )\right]^{2}-16{\rm Tr}\rho^{2}(1-\rho)^{2}} ~,$$ (10)

where the potentials

$$\Gamma^{\prime}_{\mu \mu^{\prime}} = \sum_{\nu \nu^{\prime}}V_{\mu \nu \mu^{\prime} \nu^{\prime}}(\rho(1-\rho))_{\nu^{\prime} \nu},$$ (11)

$$\Delta^{\prime}_{\mu \nu} = \frac{1}{2}\sum_{\mu^{\prime} \nu^{\prime}}V_{\mu \nu \mu^{\prime} \nu^{\prime}}(\rho \kappa)_{\mu^{\prime} \nu^{\prime}},$$ (12)

can be calculated in full analogy to $\Gamma$ and $\Delta$ by replacing the $\rho$ and $\kappa$ in terms of which they are defined by $\rho(1-\rho)$ and $\rho\kappa$, respectively. In the case of the seniority pairing interaction with strength $G$, Eq. (10) simplifies to

$$\lambda_{2}=\frac{G}{4} \frac {{\rm Tr} (1-\rho)\kappa~ {\rm... ... Tr}\rho (1-\rho )\right]^{2}-2~{\rm Tr}\rho^{2}(1-\rho)^{2}}.$$ (13)

An explicit calculation of $\lambda_{2}$ from Eq. (10) requires calculating new sets of fields Eq. (11), which is rather cumbersome. However, we have found [32] that Eq. (10) can be well approximated by the seniority-pairing expression Eq. (13) with the effective strength

$$G=G_{\mbox{\rm\scriptsize {eff}}} = -\frac{\bar{\Delta}^2}{E_{\mbox{\rm\scriptsize {pair}}}}\,$$ (14)

determined from the pairing energy

$$E_{\mbox{\rm\scriptsize {pair}}} = -\frac{1}{2}{\rm Tr}\Delta \kappa \,$$ (15)

and the average pairing gap

$$\bar{\Delta} = \frac{{\rm Tr}\Delta \rho}{{\rm Tr}\rho} \, .$$ (16)

The use of the LN method in HFB theory requires special consideration of the asymptotic properties of quasiparticle states [4,5], of essential importance for weakly-bound systems. Because of the modified HF hamiltonian (8), new terms appear in the HFB+LN equation, which are non-local in coordinate representation and thus can modify the asymptotic conditions. Effectively, this means that the standard Fermi energy $\lambda$ has to be replaced by

$$\lambda' = \lambda + 2\lambda_2(1-2n_{\mbox{\rm\scriptsize {min}}})$$ (17)

or by

$$\lambda'' = \lambda + 2\lambda_2,$$ (18)

where $n_{\mbox{\rm\scriptsize {min}}}$ is the norm of the lower HFB component $V(E_{\mbox{\rm\scriptsize {min}}},\mbox{{\boldmath {$r$}}})$ corresponding to the smallest quasiparticle energy $E_{\mbox{\rm\scriptsize {min}}}$.

The first expression (17) assumes that the asymptotic properties can be inferred from the HFB equation in the canonical basis, in which $\rho$ is diagonal and has eigenvalues that can be estimated by norms of the second HFB components. The second expression (18) pertains to the HFB equation in coordinate representation, in which the integral kernel $\rho(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ vanishes at large distances. Neither of these expressions can be rigorously justified, thereby demonstrating limitations of using the LN method to analyze spatial properties of wave functions. These ambiguities are enhanced by the fact that the LN method overestimates the curvature $\lambda_2$ near magic numbers [28,31].

Note that in the exact projection before variation method, the Fermi energy is entirely irrelevant, and hence one should not attribute too much importance to the choice between $\lambda'$ and $\lambda''$. Nevertheless, since the PNP affects only occupation numbers, leaving the canonical wave functions unchanged, in what follows we use the modified Fermi energy $\lambda'$ in modelling the asymptotic behavior needed to implement the THO method.

Finally, we should note that the HFB machinery detailed above can be readily implemented with a quadrupole constraint [30], as is the case for some of the calculations we will be reporting.