The Skyrme HFB method

For the zero-range Skyrme forces, the HFB formalism can be written directly in the coordinate representation [30,31,32] by introducing particle and pairing densities

$$\rho ({\bm r}\sigma ,{\bm r^{\prime }}\sigma^{\prime }) = \tfrac{1}{2} \rho ({\bm r},{\bm r}^{\prime })\delta_{\sigma \sigma^{\prime }}$$

$$+ \tfrac{1}{2} \sum_{i}(\sigma \vert\sigma_{i}\vert\sigma^{\prime })\rho_{i}({\bm r},{\bm r} ^{\prime }) ,$$ (13)

$$\tilde{\rho}({\bm r}\sigma ,{\bm r^{\prime }}\sigma^{\prime }) = \tfrac{1}{2 } \tilde{\rho}({\bm r},{\bm r}^{\prime })\delta _{\sigma \sigma^{\prime }}$$

$$+ \tfrac{1}{2} \sum_{i}(\sigma \vert\sigma_{i}\vert\sigma^{\prime })\tilde{\rho}_{i}( {\bm r},{\bm r}^{\prime }),$$ (14)

which explicitly depend on spin. The use of the pairing density $\tilde{\rho}$,

$$\tilde{\rho}({\bm r}\sigma ,{\bm r^{\prime }}\sigma^{\prime })=-2\sigma^{\prime }\kappa ({\bm r,}\sigma ,{\bm r^{\prime },}-\sigma^{\prime }),$$ (15)

instead of the pairing tensor $\kappa$ is convenient when restricting to time-even quasiparticle states where both $\rho$ and $\tilde{\rho}$ are hermitian and time-even [31].

The densities $\rho$ and $\tilde\rho$ can be expressed in the single-particle basis:

$$\rho ({\bm r}\sigma ,{\bm r^{\prime }}\sigma^{\prime }) = \sum_{nn^{\prime }}\rho_{nn^{\prime }}~\psi_{n^{\prime }}^{\ast }({\bm r^{\prime }},\sigma ^{\prime })\psi_{n}({\bm r},\sigma ),$$ (16)

$$\tilde{\rho}({\bm r}\sigma ,{\bm r^{\prime }}\sigma^{\prime }) = \sum_{nn^{\prime }}\tilde{\rho}_{nn^{\prime }}~\psi_{n^{\prime }}^{\ast }({\bm r^{\prime }},\sigma^{\prime })\psi_{n}({\bm r},\sigma ),$$ (17)

where $\rho_{n^{\prime }n}$ and $\tilde\rho_{n^{\prime }n}$ are the corresponding density matrices. In this study, we take ${\psi_{n}({\bm r},\sigma )}$ as a set of the HO wave functions.

The building blocks of the Skyrme HFB method are the local densities, namely the particle density $\rho ({\bm r})$, kinetic energy density $\tau ({\bm r})$, and spin-current density ${\mathsf J}_{ij}({\bm r})$:

$$\begin{array}{ccl} \rho ({\bm r}) & = & \rho ({\bm r},{\bm r}... ...prime })\right\vert _{{\bm r}^{\prime }={\bm r}}\;, \end{array}$$ (18)

as well as the corresponding pairing densities $\tilde{\rho}({\bm r})$, $\tilde{\tau}({\bm r})$ and $\tilde{\mathsf J}_{ij}({\bm r})$.

In the coordinate representation, the Skyrme HFB energy (8) can be written as a functional of the local particle and pairing densities:

$$E[\rho,\tilde{\rho}]=\frac{\langle \Phi \vert H\vert\Phi \rangle }{\langle \Phi \vert\Phi \rangle } =\int d{\bm r}~{\cal H}({\bm r}).$$ (19)

The energy density ${\cal H}({\bm r})$ is a sum of the particle $H({\bm r})$ and pairing energy density $\tilde{H}({\bm r})$:

$${\cal H}({\bm r})=H({\bm r})+\tilde{H}({\bm r}).$$ (20)

The derivatives of $E[\rho ,\tilde{\rho}]$ with respect to density matrices $\rho$ and $\tilde{\rho}$ define the self-consistent particle ($h$) and pairing ($\tilde{h}$) fields, respectively. The explicit expressions for $H({\bm r})$, $\tilde{H}({\bm r})$, $h({\bm r},\sigma ,\sigma^{\prime })$, and $\tilde{h}({\bm r} ,\sigma ,\sigma^{\prime })$ have been given in Ref. [31] and will not be repeated here.

The Skyrme HFB equations can be written in the matrix form as:

$$\left( \begin{array}{cc} h-\lambda & \tilde{h} \\ \tilde{h} & -h+... ...E_k \left( \begin{array}{c} \varphi_{1,k} \\ \varphi_{2,k} \end{array} \right),$$ (21)

where

$$h_{nn^{\prime }} = \frac{\partial E[\rho ,\tilde{\rho}]}{\partial \rho _{n^{\prime }n}}$$

$$= \sum_{\sigma \sigma^{\prime }}\int d{\bm r}~\psi_{n}^{\ast }({\bm... ...{\bm r},\sigma ,\sigma^{\prime })\psi_{n^{\prime }}({\bm r} ,\sigma^{\prime }),$$ (22)

and

$$\tilde{h}_{nn^{\prime }} = \frac{\partial E[\rho ,\tilde{\rho}]}{\partial \tilde{\rho}_{n^{\prime }n}}$$

$$= \sum_{\sigma \sigma^{\prime }}\int d{\bm r}~\psi_{n}^{\ast }( {\b... ...{\bm r},\sigma ,\sigma ^{\prime })\psi_{n^{\prime}}({\bm r} ,\sigma^{\prime }),$$ (23)

and $\varphi_{1,k}$ and $\varphi_{2,k}$ are the upper and lower components, respectively, of the quasiparticle wave function corresponding to the positive quasiparticle energy $E_k$. After solving the HFB equations (21), one obtains the density matrices,

$$\rho_{nn^{\prime }} = \sum_{E_k>0} \varphi_{2,nk} \varphi^*_{2,n^{\prime }k} ,$$ (24)

$$\tilde{\rho}_{nn^{\prime }} = -\sum_{E_k>0} \varphi_{2,nk} \varphi^*_{1,n^{\prime }k} ,$$ (25)

which define the spatial densities (16) and (17) .

We note in passing that the derivation of the coordinate-space HFB equations [31] is strictly valid only when the time-reversal symmetry is assumed. When the time-reversal symmetry is broken, one has to introduce additional real vector particle densities ${\bm s}$, ${\bm j}$, ${\bm T}$ [33], while the pairing densities acquire imaginary parts; see Ref. [32] for complete derivations.