The HFB+VAPNP method

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

$$E^{N}[\rho ,\kappa] = \frac{\left\langle \Phi \vert HP^{N}\vert\Phi \right\rangle }{ \left\langle \Phi \vert P^{N}\vert\Phi \right\rangle }$$

$$= \frac{\int d\phi \langle \Phi \vert{H}e^{i\phi ({\hat{N}}-N)}\ver... ...le }{ \int d\phi \langle \Phi \vert e^{i\phi ({\hat{N}}-N)}\vert\Phi \rangle },$$ (26)

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

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

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

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

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

where

$${\cal H}^{N}=\left( \begin{array}{cc} \varepsilon^{N}+\Gamma^{N}+... ...st } & -(\varepsilon^{N}+\Gamma^{N}+\Lambda ^{N})^{\ast } \end{array} \right) .$$ (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.,

$$\varepsilon^{N} = \tfrac{1}{2}\int d\phi \,\,y(\phi )\left\{ Y(\phi ){\rm Tr} [e\rho (\phi )]\right.$$

$$+ \left. [1-2ie^{-i\phi }\sin \phi \rho (\phi )]eC(\phi )\right\} +{\rm h.c.},$$ (30)

$$\Gamma^N = \tfrac{1}{4}\int d\phi \,\,y(\phi ) \left\{Y(\phi) {\rm Tr} [\Gamma(\phi)\rho(\phi)] \right.$$

$$+ \left. 2[1-2ie^{-i\phi}\sin\phi\rho(\phi)]\Gamma(\phi)C(\phi)\right\} + {\rm h.c.},$$ (31)

$$\Delta^{N} = \tfrac{1}{2}\int d\phi \;y(\phi )e^{-2i\phi }C\left( \phi \right) \Delta (\phi )-(...)^{T},$$ (32)

$$\Lambda^{N} = -\tfrac{1}{4}\int d\phi \,\,y(\phi )\left\{ Y(\phi ){\rm Tr} [\Delta (\phi )\overline{\kappa }^{\ast }(\phi )]\right.$$

$$- \left. 4ie^{-i\phi }\sin \phi \;C(\phi )\Delta (\phi )\overline{\kappa }^{\ast }(\phi )\right\} + {\rm h.c.},$$ (33)

with

$$\Gamma_{nm}(\phi ) = \sum_{n'm'}V _{nn'mm'}\rho_{m'n'}(\phi ),$$ (34)

$$\Delta_{nn'}(\phi ) = \tfrac{1}{2}\sum_{mm'}V_{nn'mm'}\kappa_{mm'}(\phi ),$$ (35)

where, using the unit matrix $\hat{I}$,

$$\rho (\phi ) = C(\phi )\rho ,$$ (36)

$$\kappa (\phi ) = C(\phi )\kappa,$$ (37)

$$\overline{\kappa }(\phi ) = e^{2i\phi }C^{\dagger }(\phi )\kappa ,$$ (38)

$$C(\phi ) = e^{2i\phi }\left[ 1+\rho (e^{2i\phi }-1)\right]^{-1},$$ (39)

$$x(\phi ) = \frac{1}{2\pi }\frac{e^{-i\phi N}\det (e^{i\phi }I)}{\sqrt{ \det C(\phi )}},$$ (40)

$$y(\phi ) = \frac{x(\phi )}{\int d\phi^{\prime }\,x(\phi^{\prime })},$$ (41)

and

$$Y(\phi ) = \,ie^{-i\phi }\sin \phi \;C(\phi )$$

$$- i\int d\phi^{\prime }y(\phi^{\prime })e^{-i\phi^{\prime }}\sin \phi^{\prime }\;C(\phi^{\prime }).$$ (42)

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

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

Finally, the total HFB+VAPNP energy is given by

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

$$- \int d\phi ~y(\phi )~\tfrac{1}{2}{\rm Tr}\left( \Delta (\phi )\overline{ \kappa }^{\ast }(\phi )\right) .$$ (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).