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The HFB+VAPNP method

It has been demonstrated [25] that the HFB+VAPNP energy,

$\displaystyle E^{N}[\rho ,\kappa]$ $\displaystyle =$ $\displaystyle \frac{\left\langle \Phi \vert HP^{N}\vert\Phi \right\rangle }{
\left\langle \Phi \vert P^{N}\vert\Phi \right\rangle }$  
  $\displaystyle =$ $\displaystyle \frac{\int d\phi \langle \Phi \vert{H}e^{i\phi ({\hat{N}}-N)}\ver...
...le }{ \int d\phi \langle \Phi \vert e^{i\phi ({\hat{N}}-N)}\vert\Phi
\rangle },$ (26)

where $ P^{N}$ is the particle-number projection operator,

$\displaystyle P^{N}=\frac{1}{2\pi }\int d\phi \ e^{i\phi (\hat{N}-N)},$ (27)

can be written as an energy functional of the unprojected densities (7).

The variation of Eq. (26) results in the HFB+VAPNP equations:

$\displaystyle {\cal H}^{N}\left( \begin{array}{c} U_k^N \\ V_k^N \end{array} \right) ={\cal E}_k\left( \begin{array}{c} U_k^N \\ V_k^N \end{array} \right) ,$ (28)

where

$\displaystyle {\cal H}^{N}=\left( \begin{array}{cc} \varepsilon^{N}+\Gamma^{N}+...
...st } & -(\varepsilon^{N}+\Gamma^{N}+\Lambda ^{N})^{\ast } \end{array} \right) .$ (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.,
$\displaystyle \varepsilon^{N}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2}\int d\phi \,\,y(\phi )\left\{ Y(\phi
){\rm Tr}
[e\rho (\phi )]\right.$  
  $\displaystyle +$ $\displaystyle \left. [1-2ie^{-i\phi }\sin \phi \rho (\phi )]eC(\phi )\right\}
+{\rm h.c.},$ (30)
       
$\displaystyle \Gamma^N$ $\displaystyle =$ $\displaystyle \tfrac{1}{4}\int d\phi \,\,y(\phi ) \left\{Y(\phi) {\rm
Tr}
[\Gamma(\phi)\rho(\phi)] \right.$  
  $\displaystyle +$ $\displaystyle \left. 2[1-2ie^{-i\phi}\sin\phi\rho(\phi)]\Gamma(\phi)C(\phi)\right\} +
{\rm h.c.},$ (31)
       
$\displaystyle \Delta^{N}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2}\int d\phi \;y(\phi )e^{-2i\phi }C\left(
\phi \right) \Delta (\phi )-(...)^{T},$ (32)
       
$\displaystyle \Lambda^{N}$ $\displaystyle =$ $\displaystyle -\tfrac{1}{4}\int d\phi \,\,y(\phi )\left\{ Y(\phi
){\rm Tr}
[\Delta (\phi )\overline{\kappa }^{\ast }(\phi )]\right.$  
  $\displaystyle -$ $\displaystyle \left. 4ie^{-i\phi }\sin \phi \;C(\phi )\Delta (\phi
)\overline{\kappa }^{\ast }(\phi )\right\} + {\rm h.c.},$ (33)

with
$\displaystyle \Gamma_{nm}(\phi )$ $\displaystyle =$ $\displaystyle \sum_{n'm'}V
_{nn'mm'}\rho_{m'n'}(\phi ),$ (34)
$\displaystyle \Delta_{nn'}(\phi )$ $\displaystyle =$ $\displaystyle \tfrac{1}{2}\sum_{mm'}V_{nn'mm'}\kappa_{mm'}(\phi ),$ (35)

where, using the unit matrix $ \hat{I}$,
$\displaystyle \rho (\phi )$ $\displaystyle =$ $\displaystyle C(\phi )\rho ,$ (36)
$\displaystyle \kappa (\phi )$ $\displaystyle =$ $\displaystyle C(\phi )\kappa,$ (37)
$\displaystyle \overline{\kappa }(\phi )$ $\displaystyle =$ $\displaystyle e^{2i\phi
}C^{\dagger }(\phi )\kappa ,$ (38)
$\displaystyle C(\phi )$ $\displaystyle =$ $\displaystyle e^{2i\phi }\left[ 1+\rho (e^{2i\phi }-1)\right]^{-1},$ (39)
$\displaystyle x(\phi )$ $\displaystyle =$ $\displaystyle \frac{1}{2\pi }\frac{e^{-i\phi N}\det (e^{i\phi
}I)}{\sqrt{
\det C(\phi )}},$ (40)
$\displaystyle y(\phi )$ $\displaystyle =$ $\displaystyle \frac{x(\phi )}{\int d\phi^{\prime }\,x(\phi^{\prime
})},$ (41)

and
$\displaystyle Y(\phi )$ $\displaystyle =$ $\displaystyle \,ie^{-i\phi }\sin \phi \;C(\phi )$  
  $\displaystyle -$ $\displaystyle i\int d\phi^{\prime }y(\phi^{\prime })e^{-i\phi^{\prime }}\sin
\phi^{\prime }\;C(\phi^{\prime }).$ (42)

After solving the HFB+VAPNP equations (28), one obtains the intrinsic density matrix and pairing tensor:

$\displaystyle \rho =(V^N)^{\ast }(V^N)^{T},\quad \kappa =(V^N)^{\ast }(U^N)^{T}.$ (43)

Finally, the total HFB+VAPNP energy is given by

$\displaystyle E^N[\rho ,\kappa]$ $\displaystyle =$ $\displaystyle \int d\phi ~y(\phi )~{\rm Tr}\left( e\rho (\phi )+\tfrac{1}{2}
\Gamma (\phi )\rho (\phi )\right)$  
  $\displaystyle -$ $\displaystyle \int d\phi ~y(\phi )~\tfrac{1}{2}{\rm Tr}\left( \Delta (\phi
)\overline{ \kappa }^{\ast }(\phi )\right) .$ (44)

The quantity $ y(\phi )$ plays a role of an $ N$-dependent metric. The integrands in Eqs. (30)-(32) take the familiar HFB limit at $ \phi$=0, while the integrand in (33) vanishes ( $ \Lambda^{N}$ does not appear in the standard HFB approach).


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Next: The Lipkin-Nogami method Up: Variation after particle-number projection Previous: Variation after particle-number projection
Jacek Dobaczewski 2006-10-13