Pairing Regularization Procedure

Using the HFB equations and properties of the Bogoliubov transformation (see appendix A for details), one concludes that the local abnormal density $\tilde\rho$ has a singular behavior when $\epsilon_{\mbox{\rm\scriptsize {cut}}}\rightarrow\infty$. The standard regularization technique is to remove the divergent part and define the regularized local abnormal density $\tilde\rho_r(\mathbf{r})$ as

$$\tilde\rho_r(\mathbf{r})=\lim_{\mathbf{x}\rightarrow 0} \left... ...x}/2,\mathbf{r}+\mathbf{x}/2)-f(\mathbf{r},\mathbf{x})\right],$$ (3)

where $f$ is a regulator which removes the divergence at $\mathbf{x}=0$.

For cutoff energies high enough, one can explicitly identify [17,18,19] components generating divergence in the abnormal density (see, e.g., Eq. (21) of Ref. [18]):

$$f(\mathbf{r},\mathbf{x})=\frac{i\tilde h(\mathbf{r})M^*(\math... ...)}{2}G_{\mu}(\mathbf{r}+\mathbf{x}/2,\mathbf{r}-\mathbf{x}/2),$$ (4)

where $G_{\mu}$ is the s.p. Green's function at the Fermi level $\mu$ in the truncated space, $M^*$ is the effective mass, and the Fermi momentum is

$$k_F(\mathbf{r})=\frac{\sqrt{2M^*(\mathbf{r})}}{\hbar}\sqrt{\mu-U(\mathbf{r})},$$ (5)

with $U$ being the self-consistent mean-field potential.

The first term in Eq. (4) comes from the MacLaurin expansion with respect to $\mathbf{x}$; it guarantees that the regularization procedure does not introduce any constant term to the abnormal density and that $f(\mathbf{r},\mathbf{x})$ solely represents the divergent part of $\tilde\rho$.

Using the Thomas-Fermi approximation, the local s.p. Green's function $G_{\mu}(\mathbf{r}):=G_{\mu}(\mathbf{r},\mathbf{r})$ becomes [18,19]

$$G_{\mu}(\mathbf{r})=\frac{1}{2\pi^2}\lim_{\gamma\rightarrow ... ...mu-\frac{\hbar^2k^2}{2M^*(\mathbf{r})}-U(\mathbf{r})+i\gamma},$$ (6)

where the cutoff momentum is given by:

$$k_{\mbox{\rm\scriptsize {cut}}}(\mathbf{r})=\frac{\sqrt{2M^*(... ...qrt{\epsilon_{\mbox{\rm\scriptsize {cut}}}+\mu-U(\mathbf{r})}.$$ (7)

The regularized pairing Hamiltonian and the pairing energy density may be written, respectively, as [18]:

$$\tilde h(\mathbf{r}) = g(\mathbf{r})\tilde\rho_r(\mathbf{r})=g_{eff}(\mathbf{r})\tilde\rho(\mathbf{r})$$ (8)

$${\mathcal H}_{pair}(\mathbf{r}) = \frac{1}{2}g_{eff}(\mathbf{r})\tilde\rho(\mathbf{r})^2,$$ (9)

where the effective pairing strength [17,18,19],

$$g_{eff}(\mathbf{r})=\left(\frac{1}{g(\mathbf{r})}+\frac{G_{\m... ...rac{iM^*(\mathbf{r})k_F(\mathbf{r})}{4\pi\hbar^2}\right)^{-1},$$ (10)

after calculating integral (6), can be expressed in the form:

$$g_{eff}(\mathbf{r})=\left\{ \begin{array}{lc} \left[\frac{1}... ...)}\right)\right]^{-1}&k_F(\mathbf{r})^2<0 \end{array}\right..$$ (11)

In this regularization scheme, only the Green's function is calculated using the Thomas-Fermi approximation. The densities, potentials, and chemical potential are determined self-consistently within the HFB theory. Consequently, the Fermi momentum (5) depends on microscopic HFB quantities. According to the sign of $k_F^2$, one of the expressions (11) is used.

In Ref. [20] a different regularization scheme has been proposed that involves truncation below and above the Fermi level. However, the HFB calculations in the quasiparticle basis should be performed for a high cutoff energy of 50MeV and higher [8]. Since the magnitude of the self-consistent mean field $U$ is also about 50MeV, for such a high cutoff energy both methods are equivalent. The Thomas-Fermi approximation requires that, in order to obtain results independent of $\epsilon_{\mbox{\rm\scriptsize {cut}}}$, its value should be high enough for $k_{\mbox{\rm\scriptsize {cut}}}$ to be real everywhere.

Through the density dependence of $g_{eff}$, $k_{\mbox{\rm\scriptsize {cut}}}$, and $k_F$, there appear rearrangement terms in the self-consistent mean-field potential:

$${\frac{\delta \mathcal H_{pair}}{\delta\rho}= \frac{\delta g_{eff}}{\delta \rho}\tilde\rho^2=\tilde\rho^2\times}$$

$$\times\left(\frac{\partial g_{eff}}{\partial g}\frac{\delta g}{\d... ... \frac{\delta k_{\mbox{\rm\scriptsize {cut}}}}{\delta \rho}\right).\hspace{8pt}$$ (12)

The first term in Eq. (12) is similar to the usual rearrangement term, while the other two terms associated with the regularization procedure are entirely new. It is easy to check that all the terms appearing in Eq. (12) are continuous at the classical turning point $k_F(\mathbf{r})=0$.

Figure 1: Various contributions to the HFB energy for $^{120}$Sn as a function of $\epsilon_{\mbox{\rm\scriptsize{cut}}}$. Calculations are performed using the SLy4 Skyrme functional and mixed pairing interaction (14).

In Eq. (9), the pairing energy density is divergent with respect to the cutoff energy. However, the pairing energy itself is not an observable, and in order for the energy density functional to be independent of the cutoff, other terms have to cancel out this divergence. As discussed in Refs. [14,15,17,18], the kinetic energy density $\tau$ has the same type of divergence as the abnormal density $\tilde\rho$, and the sum

$$\mathcal{H}_{kin+pair}(\mathbf{r})=-\frac{\hbar^2}{2M^*(\math... ...)}\tau +\frac{1}{2}g_{eff}(\mathbf{r})\tilde\rho^2(\mathbf{r})$$ (13)

does converge.

Various contributions to the total HFB energy as functions of the cutoff energy are shown in Fig. 1. The total energy is stable with respect to $\epsilon_{\mbox{\rm\scriptsize {cut}}}$, although some of the components of the total energy vary significantly. As expected from Eq. (13), two terms exhibiting large fluctuations are the kinetic term (with variations of about 2MeV) and pairing term (with variations of about 1.3MeV). Also, the momentum-dependent spin-orbit term, $E_{S.O.}$, has significant variations of about 1MeV. On the other hand, Skyrme and Coulomb energies are fairly stable with respect to $\epsilon_{\mbox{\rm\scriptsize {cut}}}$.