Skyrme Hartree-Fock-Bogoliubov Equations

Variation of the energy (9) with respect to $\rho$ and $\tilde{\rho}$ results in the Skyrme HFB equations:

$$\sum_{\sigma'} \left(\begin{array}{cc} h({\bf r},\sigma,\sig... ... (E,{\bf r}\sigma) \\ V (E,{\bf r}\sigma) \end{array}\right),$$ (16)

where local fields $h({\bf r},\sigma ,\sigma^{\prime })$ and $\tilde{h}({\bf r},\sigma ,\sigma^{\prime })$ can be easily calculated in the coordinate space by using the following explicit expressions:

$$\begin{array}{rcl} h_{q}({\bf r},\sigma ,\sigma^{\prime }) &... ...rho}{\rho_0}\right)^\gamma \right)\tilde{\rho}_{q}, \end{array}$$ (17)

where

$$\begin{array}{lcl} M_{q} &=&\frac{\hbar^{2}}{2m}+{\textstyle... ...{\bf J}_{ij}\right. +\left. {\bf J}_{q,ij}\right] . \end{array}$$ (18)

Properties of the HFB equation in the spatial coordinates, Eq. (16), have been discussed in Ref. [2]. In particular, it has been shown that the spectrum of eigenenergies $E$ is continuous for $\vert E\vert$$>$$-\lambda$ and discrete for $\vert E\vert$$<$$-\lambda$. In the present implementation, we solve the HFB equations by expanding quasiparticle wave functions on a finite basis; therefore, the quasiparticle spectrum $E_k$ becomes discretized. Hence in the following we use the notation $V_k({\bf r}\sigma)=V(E_k,{\bf r}\sigma)$ and $U_k({\bf r}\sigma) =U(E_k,{\bf r}\sigma)$. Since for $E_k$$>$0 and $\lambda$$<$0 the lower components $V_k({\bf r}\sigma)$ are localized functions of ${\bf r}$, the density matrices,

$$\rho({\bf r}\sigma,{\bf r}'\sigma') = ~~ \sum_k V_k ({\bf r} \sigma ) V_k^*({\bf r}'\sigma') ,$$ (19)

$$\tilde\rho({\bf r}\sigma,{\bf r}'\sigma') = - \sum_k V_k({\bf r} \sigma ) U_k^*({\bf r}'\sigma') ,$$ (20)

are always localized. The orthogonality relation for the single-quasiparticle HFB wave functions reads

$$\int\mbox{\scriptsize {d}}^3{\bf r} \sum_{\sigma} \left[ U_k... ...f r} \sigma ) V_{k'} ({\bf r}\sigma) \right] = \delta_{k,k'} ,$$ (21)

and the norms of lower components $N_k$,

$$N_k = \int\mbox{\scriptsize {d}}^3{\bf r}\sum_{\sigma}\vert V_k({\bf r} \sigma )\vert^2,$$ (22)

define the total number of particles

$$N=\int\mbox{\scriptsize {d}}^3{\bf r}~\rho({\bf r}) = \sum_{n}N_k.$$ (23)