Next: The Lipkin-Nogami method
Up: Variation after particle-number projection
Previous: Variation after particle-number projection
The HFB+VAPNP method
It has been demonstrated [25] that the
HFB+VAPNP energy,
where
is the particle-number projection operator,
![$\displaystyle P^{N}=\frac{1}{2\pi }\int d\phi \ e^{i\phi (\hat{N}-N)},$](img99.png) |
(27) |
can be written as an energy functional of the unprojected densities
(7).
The variation of Eq. (26) results in
the HFB+VAPNP equations:
![$\displaystyle {\cal H}^{N}\left( \begin{array}{c} U_k^N \\ V_k^N \end{array} \right) ={\cal E}_k\left( \begin{array}{c} U_k^N \\ V_k^N \end{array} \right) ,$](img100.png) |
(28) |
where
![$\displaystyle {\cal H}^{N}=\left( \begin{array}{cc} \varepsilon^{N}+\Gamma^{N}+...
...st } & -(\varepsilon^{N}+\Gamma^{N}+\Lambda ^{N})^{\ast } \end{array} \right) .$](img101.png) |
(29) |
Equations (28) and (29) have the same structure as
Eqs. (11) and (12), except that the expressions for
the VAPNP fields are now different [25,27], i.e.,
with
where, using the unit matrix
,
![$\displaystyle \rho (\phi )$](img119.png) |
![$\displaystyle =$](img15.png) |
![$\displaystyle C(\phi )\rho ,$](img120.png) |
(36) |
![$\displaystyle \kappa (\phi )$](img121.png) |
![$\displaystyle =$](img15.png) |
![$\displaystyle C(\phi )\kappa,$](img122.png) |
(37) |
![$\displaystyle \overline{\kappa }(\phi )$](img123.png) |
![$\displaystyle =$](img15.png) |
![$\displaystyle e^{2i\phi
}C^{\dagger }(\phi )\kappa ,$](img124.png) |
(38) |
![$\displaystyle C(\phi )$](img125.png) |
![$\displaystyle =$](img15.png) |
![$\displaystyle e^{2i\phi }\left[ 1+\rho (e^{2i\phi }-1)\right]^{-1},$](img126.png) |
(39) |
![$\displaystyle x(\phi )$](img127.png) |
![$\displaystyle =$](img15.png) |
![$\displaystyle \frac{1}{2\pi }\frac{e^{-i\phi N}\det (e^{i\phi
}I)}{\sqrt{
\det C(\phi )}},$](img128.png) |
(40) |
![$\displaystyle y(\phi )$](img129.png) |
![$\displaystyle =$](img15.png) |
![$\displaystyle \frac{x(\phi )}{\int d\phi^{\prime }\,x(\phi^{\prime
})},$](img130.png) |
(41) |
and
After solving the HFB+VAPNP equations (28), one obtains the
intrinsic density matrix and pairing tensor:
![$\displaystyle \rho =(V^N)^{\ast }(V^N)^{T},\quad \kappa =(V^N)^{\ast }(U^N)^{T}.$](img134.png) |
(43) |
Finally, the total HFB+VAPNP energy is given by
The quantity
plays a role of an
-dependent metric. The
integrands in Eqs. (30)-(32) take the familiar HFB
limit at
=0, while the integrand in (33) vanishes
(
does not appear in the standard HFB approach).
Next: The Lipkin-Nogami method
Up: Variation after particle-number projection
Previous: Variation after particle-number projection
Jacek Dobaczewski
2006-10-13