The Skyrme HFB+VAPNP method

Following the VAPNP procedure of Sec. 3.1, one can develop the Skyrme HFB+VAPNP equations by introducing the gauge-angle-dependent transition density matrices:

$$\rho ({\bm r}\sigma ,{\bm r^{\prime }}\sigma^{\prime },\phi ) = \sum_{nn^{\prime }}\rho_{nn^{\prime }}(\phi )~\psi_{n^{\prime }}^{\ast }({\bm r^{\prime }},\sigma^{\prime })\psi_{n}({\bm r},\sigma ),$$ (50)

$$\tilde{\rho}({\bm r}\sigma ,{\bm r^{\prime }}\sigma^{\prime },\phi ) = \sum_{nn^{\prime }}\tilde{\rho}_{nn^{\prime }}(\phi ) ~\psi_{n^{\prime }}^{\ast }({\bm r^{\prime }},\sigma^{\prime }) \psi_{n}({\bm r},\sigma ).$$ (51)

In the above equation, the density matrix $\rho_{nn^{\prime }}(\phi )$ is given by Eq. (36) while

$$\tilde{\rho}(\phi ) =e^{-i\phi }C(\phi )\tilde{\rho}.$$ (52)

The associated gauge-angle-dependent local densities $\rho ({\bm r},\phi )$, $\tau ({\bm r},\phi )$, ${\mathsf J}_{ij}({\bm r },\phi )$, $\tilde{\rho}({\bm r},\phi )$, $\tilde{\tau}({\bm r},\phi )$, and $\tilde{\mathsf J}_{ij}({\bm r},\phi )$ are defined by Eqs. (18) in terms of the density matrices (50) and (51). Using the Wick theorem for matrix elements [2], one can show that the gauge-angle-dependent transition energy density ${\cal H}({\bm r},\phi )$ can be obtained from the intrinsic energy density ${\cal H}({\bm r})$ simply by substituting particle (pairing) local densities with their gauge-angle-dependent counterparts (e.g., $\rho ({\bm r})$ $\rightarrow$ $\rho ({\bm r},\phi )$).

In the case of Skyrme functionals, the HFB+VAPNP energy (26) can be expressed through an integral

$$E^{N}[\rho ,\tilde{\rho}]=\int d\phi ~y(\phi )~E(\phi ),$$ (53)

where the transition energy reads:

$$E(\phi )=\frac{\langle \Phi \vert He^{i\phi \hat{N}}\vert\Phi \ra... ...ert e^{i\phi \hat{N}}\vert\Phi \rangle }=\int d{\bm r}~{\cal H}({\bm r},\phi ).$$ (54)

The projected energy (53) is a functional $E^{N}[\rho ,\tilde{\rho}]$ of the matrix elements of intrinsic (i.e., $\phi$=0) matrices $\rho$ and $\tilde{\rho}$.

In order to compute the derivatives of $E^{N}(\rho ,\tilde{\rho})$ with respect to $\rho$ and $\tilde{\rho}$, one should take first the derivatives of $E^{N}[\rho ,\tilde{\rho}]$ with respect to $\rho (\phi )$ and $\tilde{\rho} (\phi )$, and then the derivatives of $\rho (\phi )$ and $\tilde{\rho} (\phi )$ with respect to the intrinsic densities $\rho$ and $\tilde{\rho}$. For example,

$$\frac{\partial E^{N}[\rho ,\tilde{\rho}]}{\partial \rho _{nn^{\prime }}} = \int d\phi ~y(\phi )~\left[ {\frac{1}{y(\phi )}}\frac{\partial y(\phi )}{ \partial \rho_{nn^{\prime }}}~E(\phi )\right.$$

$$+ \sum_{\alpha \beta }\frac{\partial E(\phi )}{\partial \rho_{\alph... ...hi )}\frac{\partial \rho_{\alpha \beta }(\phi )}{\partial \rho _{nn^{\prime }}}$$

$$+ \sum_{\alpha \beta }\frac{\partial E(\phi )}{\partial \tilde{\rho... ...ial \tilde{\rho}_{\alpha \beta }(\phi )}{ \partial \tilde{\rho}_{nn^{\prime }}}$$

$$+ \left. \sum_{\alpha \beta }\frac{\partial E(\phi )}{\partial \til... ...{\rho}_{\alpha \beta }^{\ast }(-\phi )}{\partial \rho _{nn^{\prime }}}\right] .$$ (55)

With the use of the identity:

$$\delta_{mm^{\prime }}-2ie^{-i\phi }\sin \phi ~\rho_{mm^{\prime }}(\phi )=e^{-2i\phi }C_{mm^{\prime }}(\phi ),$$ (56)

the partial derivatives in Eq. (55) can easily be calculated:

$$\frac{\partial y(\phi )}{\partial \rho_{nn^{\prime }}} = y(\phi )~Y_{nn^{\prime }}(\phi ),$$ (57)

$$\frac{\partial \rho_{mm^{\prime }}(\phi )}{\partial \rho _{nn^{\prime }}} = \delta_{m^{\prime }n^{\prime }}C_{mn}(\phi )$$

$$- 2ie^{-i\phi }\sin(\phi )\rho_{n^{\prime }m^{\prime }}(\phi )C_{mn}(\phi ),$$ (58)

$$\frac{\partial \tilde{\rho}_{mm^{\prime }}(\phi )}{\partial \rho _{nn^{\prime }}} = -2ie^{-i\phi }\sin(\phi )\tilde{\rho}_{n^{\prime }m^{\prime }}(\phi )C_{mn}(\phi ),$$ (59)

$$\frac{\partial \tilde{\rho}_{mm^{\prime }}(\phi )}{\partial \tilde{\rho}_{nn^{\prime }}} = e^{-i\phi }C_{mn}(\phi )\delta _{m^{\prime }n^{\prime }}$$

$$+ e^{-i\phi }C_{m\bar{n}}(\phi )\delta_{\bar{n}m^{\prime }}s_{\bar{n}^{\prime }}s_{\bar{n}}^{\ast },$$ (60)

where $\bar{n}$ and $s_{n}$ ( $s_{n}s_{n}^{\ast }=1$, $s_{\bar{n}}=-s_{n}$) are defined using the time-reversal operator $\hat{T}$, as

$$\hat{T}\psi_{n}({\bm r},\sigma )=s_{n}\psi_{\bar{n}}({\bm r},\sigma ).$$ (61)

By inserting Eqs. (57)-(60) in Eq. (55), the latter reads

$$\frac{\partial E^{N}[\rho ,\tilde{\rho}]}{\partial \rho } =\int d\phi ~y(\phi )~Y(\phi )~E(\phi )$$

$$+\int d\phi ~y(\phi )~e^{-2i\phi }~C(\phi )h(\phi )C(\phi )$$ (62)

$$-\left[ \int d\phi ~y(\phi )~2ie^{-i\phi }\sin(\phi )~\tilde{\rho}(\phi ) \tilde{h}(\phi )C(\phi ) + {\rm h.c.}\right],$$

where

$$h_{nn^{\prime }}(\phi ) = \frac{\partial E(\phi )}{\partial \rho_{n^{\prime }n} (\phi )}$$

$$= \sum_{\sigma \sigma^{\prime }}\int d^{3}{\bm r} ~\psi_{n}^{\ast... ... r,}\sigma ,\sigma^{\prime },\phi )\psi_{n^{\prime }}({\bm r}\sigma^{\prime }),$$ (63)

$$\tilde{h}_{nn^{\prime }}(\phi ) = \frac{\partial E(\phi )}{\partial \tilde{\rho}_{n^{\prime }n}(\phi )}$$

$$= \sum_{\sigma \sigma^{\prime }}\int d^{3} {\bm r}~\psi_{n}^{\ast... ... r},\sigma ,\sigma^{\prime },\phi )\psi_{n^{\prime }}({\bm r}\sigma^{\prime }).$$ (64)

The derivative of $E^{N}(\rho ,\tilde{\rho})$ with respect to $\tilde{\rho}$ can be computed in a similar manner. The $\phi$-dependent fields $h({\bm r},\sigma ,\sigma ^{\prime },\phi )$ and $\tilde{h}({\bm r},\sigma ,\sigma ^{\prime },\phi )$ are obtained by substituting the local particle and pairing densities in the intrinsic fields $h({\bm r},\sigma ,\sigma^{\prime })$ and $\tilde{h}({\bm r} ,\sigma ,\sigma^{\prime })$ with their gauge-angle-dependent counterparts.

The Skyrme HFB+VAPNP equations can finally be written in the form

$$\left( \begin{array}{cc} h^{N} & \tilde{h}^{N} \\ \tilde{h}^{N} &... ...( \begin{array}{c} \varphi^{N}_{1,k} \\ \varphi^{N}_{2,k} \end{array} \right) ,$$ (65)

with particle-hole and particle-particle Hamiltonians

$$h^{N} =\int d\phi y(\phi )Y(\phi )E(\phi )$$

$$+\int d\phi y(\phi )e^{-2i\phi }~C(\phi )h(\phi )C(\phi )$$ (66)

$$-\left[ \int d\phi y(\phi )2ie^{-i\phi }\sin(\phi )\tilde{\rho}(\phi ) \tilde{h}(\phi )C(\phi ) + {\rm h.c.}\right],$$

$$\tilde{h}^{N} = \int d\phi y(\phi )e^{-i\phi }\left[ \tilde{h}(\phi )C(\phi )+(...)^{T}\right].$$ (67)

Finally, solutions of the HFB+VAPNP equations (65) allow for calculating the intrinsic density matrices as,

$$\rho_{nn^{\prime }} = \sum_{E_k>0} \varphi^N_{2,nk} \varphi^{N*}_{2,n^{\prime }k} ,$$ (68)

$$\tilde{\rho}_{nn^{\prime }} = -\sum_{E_k>0} \varphi^N_{2,nk} \varphi^{N*}_{1,n^{\prime }k} .$$ (69)

Let us re-emphasize that the densities and fields that enter the Skyrme HFB+VAPNP equations are immediate generalizations of the analogous quantities that appear in the standard Skyrme HFB formalism. Of course, due to the presence of $C(\phi)$ and integrations over the gauge angle, the Skyrme HFB+VAPNP calculations are appreciably more involved.