Particle-Number-Projected Skyrme-HFB Method

The particle-number-projected HFB state can be written as:

$$\displaystyle \vert\Psi\rangle \equiv P^N \vert\Phi\rangle... ...0}^{2\pi} d\phi\,e^{\imath\phi(\hat{N}-N)}\vert\Phi\rangle ,$$ (1)

where $\hat{N}$ is the number operator, $N$ is the particle number, and $\vert\Phi\rangle$ is the HFB wavefunction which does not have a well-defined particle number. As shown in Ref. [3], the PNP HFB energy

$$\textsf{E}^{N}[\rho,\bar{\rho}]=\frac{\left\langle \Phi \ver... ...langle \Phi \vert e^{i\phi ({\hat{N}}-N)}\vert\Phi \rangle },$$ (2)

is an energy functional of the unprojected particle-hole and pairing densities $\rho$ and $\bar{\rho}$, respectively. In the case of the Skyrme force, the projected energy (2) reads:

$$\textsf{E}^{N}[\rho,\tilde{\rho}]= \int d\phi ~y(\phi ) \int... ...f r} \left({ H}({\bf r},\phi)+\tilde{ H}({\bf r},\phi)\right),$$ (3)

where

$$\begin{array}{l} \begin{array}{lllrl} x(\phi ) &=&\frac{1}{2... ...^{\prime }\,x(\phi^{\prime })}, \\ \end{array} \\ \end{array}$$ (4)

$I$ is the unit matrix, and the gauge-angle-dependent energy densities ${ H}({\bf r},\phi )$ and $\tilde{ H}({\bf r},\phi)$ are derived from the unprojected ones by simply replacing particle (pairing) local densities by their gauge-angle-dependent counterparts. The latter ones are defined by the gauge-angle-dependent density matrices.

Obviously, the projected energy (3) is a functional of the unprojected density matrices. Its derivatives with respect to $\rho_{n^{\prime }n}$ and $\tilde{\rho}_{n^{\prime }n}$ lead to the PNP Skyrme-HFB equations

$$\left( \begin{array}{cc} h^{N} & \tilde{h}^{N} \\ \tilde{h}... ...) ={ E}^N\left( \begin{array}{c} U \\ V \end{array}\right) ,$$ (5)

where

$$\begin{array}{lll} h^{N} &=&\int d\phi y(\phi ) \left[Y(\phi... ...}(\phi )C(\phi )+ (\tilde{h}(\phi )C(\phi ))^T \} , \end{array}$$ (6)

and $Y(\phi ) =ie^{-i\phi }\sin \phi C(\phi ) -i\int d\phi^{\prime }y(\phi^{\prime })e^{-i\phi^{\prime }} \sin \phi^{\prime } C(\phi^{\prime })$ and $C(\phi ) =e^{2i\phi }\left( 1+\rho (e^{2i\phi }-1)\right)^{-1}$. The gauge-angle-dependent field matrices $h(\phi)$ and $\tilde{h}(\phi )$ are obtained by simply replacing the particle and pairing local densities in the unprojected fields with their gauge-angle-dependent counterparts.