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Particle-Number-Projected Skyrme-HFB Method
The particle-number-projected HFB state can be written as:
![\begin{displaymath}
\displaystyle
\vert\Psi\rangle \equiv P^N \vert\Phi\rangle...
...0}^{2\pi} d\phi\,e^{\imath\phi(\hat{N}-N)}\vert\Phi\rangle ,
\end{displaymath}](img11.png) |
(1) |
where
is the number operator,
is the particle number,
and
is the HFB wavefunction which does not have a
well-defined particle number. As shown in Ref. [3], the PNP
HFB energy
![\begin{displaymath}
\textsf{E}^{N}[\rho,\bar{\rho}]=\frac{\left\langle \Phi \ver...
...langle \Phi \vert e^{i\phi ({\hat{N}}-N)}\vert\Phi
\rangle },
\end{displaymath}](img15.png) |
(2) |
is an energy functional of the unprojected particle-hole
and pairing densities
and
, respectively.
In the case of the Skyrme force, the projected energy (2)
reads:
![\begin{displaymath}
\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),
\end{displaymath}](img18.png) |
(3) |
where
![\begin{displaymath}
\begin{array}{l}
\begin{array}{lllrl}
x(\phi ) &=&\frac{1}{2...
...^{\prime }\,x(\phi^{\prime })}, \\
\end{array} \\
\end{array}\end{displaymath}](img19.png) |
(4) |
is the unit matrix, and the gauge-angle-dependent energy
densities
and
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
and
lead to the PNP Skyrme-HFB equations
![\begin{displaymath}
\left(
\begin{array}{cc}
h^{N} & \tilde{h}^{N} \\
\tilde{h}...
...) ={ E}^N\left(
\begin{array}{c}
U \\
V
\end{array}\right) ,
\end{displaymath}](img25.png) |
(5) |
where
![\begin{displaymath}
\begin{array}{lll}
h^{N} &=&\int d\phi y(\phi ) \left[Y(\phi...
...}(\phi )C(\phi )+ (\tilde{h}(\phi )C(\phi ))^T \}
,
\end{array}\end{displaymath}](img26.png) |
(6) |
and
and
. The gauge-angle-dependent field
matrices
and
are obtained by simply
replacing the particle and pairing local densities in the
unprojected fields with their gauge-angle-dependent counterparts.
Next: Results
Up: enam04mario-02w
Previous: Introduction
Jacek Dobaczewski
2005-01-24