Representation of second derivatives as matrix element of effective interactions

In this appendix, we discuss interaction matrix elements coming from $E[\rho,\kappa,\kappa^\ast]$, which we take to contain separate Skyrme (i.e. strong-force, $\kappa$-independent), Coulomb, and pairing energy functionals:

$$E[\rho,\kappa,\kappa^\ast] = E_{\rm Skyrme}[\rho] + E_{\rm Coul}[\rho_{\rm p}] +E_{\rm pair}[\rho,\kappa,\kappa^\ast],$$ (25)

where $\rho_{\rm p}$ is the proton density matrix. The most general Skyrme energy functional in common use is given by

$$E_{\rm Skyrme}[\rho] = \sum_{t=0,1} \int d^3r \bigg\{ C_t^{\rho}[\rho_{00}]\, \rho^2_{t0... ...\mbox{\boldmath$j$\unboldmath }^2_{t0}(\mbox{\boldmath$r$\unboldmath }) \right]$$

$$+ C_t^{s}[\rho_{00}]\, \mbox{\boldmath$s$\unboldmath }^2_{t0}(\mb... ...stackrel{\leftrightarrow}{J}{}^2_{t0}(\mbox{\boldmath$r$\unboldmath } ) \right]$$

$$+ C_t^{\nabla J} \left[ \rho_{t0}(\mbox{\boldmath$r$\unboldmath }... ...oldmath$j$\unboldmath }_{t0}(\mbox{\boldmath$r$\unboldmath }) \right] \bigg\} .$$ (26)

(See, e.g., [79,80,63] and references therein for a general discussion.) All densities are labeled by isospin indices $tt_z$, where $t$ takes values zero and one and $t_z$ is always equal to 0. A more general theory could violate isospin at the single-quasiparticle level, leading to additional densities $\rho_{1\pm 1}$ [63]. We do not consider such densities here. The $C_t^{i}$ are the coupling constants for the effective interaction. As usual, two of them are chosen to be density dependent:

$$C_t^\rho [\rho_{00}] = A_t^\rho + B_t^\rho \, \rho_{00}^\alpha(\mbox{\boldmath$r$\unboldmath }) ,$$

$$C_t^s [\rho_{00}] = A_t^s + B_t^s \, \rho_{00}^\alpha(\mbox{\boldmath$r$\unboldmath }) .$$ (27)

Here $\rho_{t0}$, $\mbox{\boldmath$s$\unboldmath }_{t0}$, $\tau_{t0}$, $\mbox{\boldmath$T$\unboldmath }_{t0}$, $\mbox{\boldmath$j$\unboldmath }_{t0}$, $\stackrel{\leftrightarrow}{J}_{t0}$, and $\mbox{\boldmath$J$\unboldmath }_{t0}$ are local densities and currents, which are derived from the general density matrices for protons and neutrons

$$\rho_{00} (\mbox{\boldmath$r$\unboldmath }\sigma,\mbox{\boldmath$r$\unboldmath }'\sigma') = \rho_{\mbox{\footnotesize {n}}} (\mbox{\boldmath$r$\unboldmath }\... ...\mbox{\boldmath$r$\unboldmath }\sigma,\mbox{\boldmath$r$\unboldmath }'\sigma'),$$

$$\rho_{10} (\mbox{\boldmath$r$\unboldmath }\sigma,\mbox{\boldmath$r$\unboldmath }'\sigma') = \rho_{\mbox{\footnotesize {n}}} (\mbox{\boldmath$r$\unboldmath }\... ...\mbox{\boldmath$r$\unboldmath }\sigma,\mbox{\boldmath$r$\unboldmath }'\sigma'),$$ (28)

where

$$\rho_{\mbox{\footnotesize {n}}}(\mbox{\boldmath$r$\unboldmath }\sigma,\mbox{\boldmath$r$\unboldmath }^\prime\sigma^\prime) = \sum_{KK^\prime,\mbox{\footnotesize {neutron}}} \psi^\ast_{K^\pri... ...igma^\prime) \psi_{K}(\mbox{\boldmath$r$\unboldmath }\sigma)\rho_{K K^\prime} ,$$

$$\rho_{\mbox{\footnotesize {p}}}(\mbox{\boldmath$r$\unboldmath }\sigma,\mbox{\boldmath$r$\unboldmath }^\prime\sigma^\prime) = \sum_{KK^\prime,\mbox{\footnotesize {proton}}} \psi^\ast_{K^\prim... ...igma^\prime) \psi_{K}(\mbox{\boldmath$r$\unboldmath }\sigma)\rho_{K K^\prime} ,$$ (29)

and $\sigma = \pm \frac{1}{2}$ labels the spin components so that, e.g., $\psi_K(\mbox{\boldmath$r$\unboldmath }\sigma)$ is a spin component of the single-particle wave function associated with the state $K$. Defining

$$\rho_{t0} (\mbox{\boldmath$r$\unboldmath },\mbox{\boldmath$r$\unboldmath }') = \sum_{\sigma = \pm} \rho_{t0} (\mbox{\boldmath$r$\unboldmath }\sigma,\mbox{\boldmath$r$\unboldmath }'\sigma) ,$$

$$\mbox{\boldmath$s$\unboldmath }_{t0} (\mbox{\boldmath$r$\unboldmath },\mbox{\boldmath$r$\unboldmath }') = \sum_{\sigma,\sigma' = \pm} \rho_{t0} (\mbox{\boldmath$r$\unboldm... ...oldmath }'\sigma') \, {\mbox{\boldmath$\sigma$\unboldmath }_{\sigma' \sigma}} ,$$ (30)

where ${\mbox{\boldmath$\sigma$\unboldmath }_{\sigma' \sigma}}= \langle \sigma^\prime \vert \mbox{\boldmath$\sigma$\unboldmath } \vert \sigma \rangle$ is a matrix element of the vector of Pauli spin matrices, we write the local densities and currents as

$$\rho_{t0}(\mbox{\boldmath$r$\unboldmath }) = \rho_{t0} (\mbox{\boldmath$r$\unboldmath },\mbox{\boldmath$r$\unboldmath }),$$

$$\mbox{\boldmath$s$\unboldmath }_{t0}(\mbox{\boldmath$r$\unboldmath }) = \mbox{\boldmath$s$\unboldmath }_{t0} (\mbox{\boldmath$r$\unboldmath },\mbox{\boldmath$r$\unboldmath }),$$

$$\tau_{t0}(\mbox{\boldmath$r$\unboldmath }) = \mbox{\boldmath$\nabla$\unboldmath } \cdot \mbox{\boldmath$\nabla... ...}') \vert _{\mbox{\boldmath$r$\unboldmath }=\mbox{\boldmath$r$\unboldmath }'} ,$$

$$\mbox{\boldmath$T$\unboldmath }_{t0}(\mbox{\boldmath$r$\unboldmath }) = \mbox{\boldmath$\nabla$\unboldmath } \cdot \mbox{\boldmath$\nabla... ...}') \vert _{\mbox{\boldmath$r$\unboldmath }=\mbox{\boldmath$r$\unboldmath }'} ,$$

$$\mbox{\boldmath$j$\unboldmath }_{t0} (\mbox{\boldmath$r$\unboldmath }) = - {\textstyle{\frac{i}{2}}} (\mbox{\boldmath$\nabla$\unboldmath }... ... }') \vert _{\mbox{\boldmath$r$\unboldmath }=\mbox{\boldmath$r$\unboldmath }'},$$

$$J_{t0,ij}(\mbox{\boldmath$r$\unboldmath }) = - {\textstyle{\frac{i}{2}}} (\mbox{\boldmath$\nabla$\unboldmath }... ... }') \vert _{\mbox{\boldmath$r$\unboldmath }=\mbox{\boldmath$r$\unboldmath }'},$$

$$\stackrel{\leftrightarrow}{J}{\!}^2_{t0}(\mbox{\boldmath$r$\unboldmath }) = \sum_{ij=xyz} J^2_{t0,ij},$$

$$\mbox{\boldmath$J$\unboldmath }_{t0}(\mbox{\boldmath$r$\unboldmath }) = -{\textstyle{\frac{i}{2}}} (\mbox{\boldmath$\nabla$\unboldmath }-... ...) \vert _{ \mbox{\boldmath$r$\unboldmath }=\mbox{\boldmath$r$\unboldmath }' } .$$ (31)

The Coulomb energy functional is given by

$$E_{\mbox{\footnotesize {Coul}}} [\rho_{\rm p}] = \frac{e^2}{... ...{\footnotesize {p}}}^{4/3} (\mbox{\boldmath$r$\unboldmath }) ,$$ (32)

where we make the usual Slater approximation [81] for the exchange term.

For the pairing functional we take the quite general form

$$E_{\mbox{\footnotesize {pair}}}[\rho,\kappa,\kappa^\ast] = E... ...ert\tilde{\rho}_\tau(\mbox{\boldmath$r$\unboldmath })\vert^2 ,$$ (33)

where the density-dependent pairing coupling constant $C^{\tilde\rho}[\rho_{00}(\mbox{\boldmath$r$\unboldmath })]$ is an arbitrary function of $\rho_{00}(\mbox{\boldmath$r$\unboldmath })$. The quantity $\tilde{\rho}_\tau(\mbox{\boldmath$r$\unboldmath })$ is defined as [64]

$$\tilde{\rho}_\tau(\mbox{\boldmath$r$\unboldmath }) = -i \sum_{\sigma\sigma'=\pm} \kappa_{\tau}(\mbox{\boldmath$r$\unbo... ...oldmath }\sigma') \sigma^y_{\sigma\sigma'} ,\ \tau={\rm proton \ or \ neutron},$$ (34)

with

$$\kappa_{\mbox{\footnotesize {n}}}(\mbox{\boldmath$r$\unboldmath }\sigma,\mbox{\boldmath$r$\unboldmath }^\prime\sigma^\prime) = \sum_{KK^\prime,\mbox{\footnotesize {neutron}}} \psi_{K^\prime}(\... ...gma^\prime) \psi_{K}(\mbox{\boldmath$r$\unboldmath }\sigma)\kappa_{KK^\prime} ,$$

$$\kappa_{\mbox{\footnotesize {p}}}(\mbox{\boldmath$r$\unboldmath }\sigma,\mbox{\boldmath$r$\unboldmath }^\prime\sigma^\prime) = \sum_{KK^\prime,\mbox{\footnotesize {proton}}} \psi_{K^\prime}(\m... ...igma^\prime) \psi_{K}(\mbox{\boldmath$r$\unboldmath }\sigma)\kappa_{KK^\prime},$$ (35)

being the standard pairing tensor in the coordinate representation.

The second derivatives of the energy functional in Eq. (13), as the equation indicates and we've already noted, can be written as unsymmetrized matrix elements $\bar{V}^{\rm ph}_{KLK'L'}$ of an effective interaction between uncoupled pairs of single-particle states. The particle-hole matrix elements take the form

$$\bar{V}^{\rm ph}_{KL K' L'} = \langle KL\vert \bar{V}^{\mbo... ...oul}}} + \bar{V}^{\rm eff\ ph}_{\rm pair} \vert K' L'\rangle.$$ (36)

The last term contains the pairing rearrangement discussed at the end of the previous appendix.

The effective Skyrme interaction in Eq. (37) is given by

$$\bar{V}^{\mbox{\footnotesize {eff}}}_{\mbox{\footnotesize {Skyrme}}} = ( a_0 +b_0 \, \mbox{\boldmath$\sigma$\unboldmath } \cdot \mbox{\b... ...' ) \, \delta(\mbox{\boldmath$r$\unboldmath }-\mbox{\boldmath$r$\unboldmath }')$$

$$+ ( a_1 +b_1 \, \mbox{\boldmath$\sigma$\unboldmath } \cdot \mbox{... ...boldmath }-\mbox{\boldmath$r$\unboldmath }')\mbox{\boldmath$k$\unboldmath }^2 )$$

$$+ ( a_2 +b_2 \, \mbox{\boldmath$\sigma$\unboldmath } \cdot \mbox{... ...boldmath }-\mbox{\boldmath$r$\unboldmath }') \, \mbox{\boldmath$k$\unboldmath }$$

$$+ ( a_3 +b_3 \, \mbox{\boldmath$\sigma$\unboldmath } \cdot \mbox{... ... }) \, \delta(\mbox{\boldmath$r$\unboldmath }-\mbox{\boldmath$r$\unboldmath }')$$

$$+ \big[ e_3 \, \rho_{10}(\mbox{\boldmath$r$\unboldmath }) \, (\ta... ... (\mbox{\boldmath$\sigma$\unboldmath } + \mbox{\boldmath$\sigma$\unboldmath }')$$

$$\qquad +m_3 \, \mbox{\boldmath$s$\unboldmath }_{10} (\mbox{\boldm... ... }) \, \delta(\mbox{\boldmath$r$\unboldmath }-\mbox{\boldmath$r$\unboldmath }')$$

$$+ \big[ f_3 \, \rho^2_{10}(\mbox{\boldmath$r$\unboldmath }) +h_3 ... ... }) \, \delta(\mbox{\boldmath$r$\unboldmath }-\mbox{\boldmath$r$\unboldmath }')$$

$$+( a_4 +c_4 \, \vec{\tau} \cdot \vec{\tau}') \, (\mbox{\boldmath$... ...ldmath }-\mbox{\boldmath$r$\unboldmath }') \, \mbox{\boldmath$k$\unboldmath } .$$ (37)

$\vec{\tau}=(\tau^{(\pm 1)},\tau^{(0)})$ is the vector of Pauli matrices in isospin space and

$$\mbox{\boldmath$k$\unboldmath } = - {\textstyle{\frac{i}{2}}} (\mbox{\boldmath$\nabla$\unboldmath }... ...ldmath$\nabla$\unboldmath }') \quad \mbox{\footnotesize {acting to the right,}}$$

$$\mbox{\boldmath$k$\unboldmath }^\dagger = \phantom{-} {\textstyle{\frac{i}{2}}} (\mbox{\boldmath$\nabla$\un... ...oldmath$\nabla$\unboldmath }') \quad \mbox{\footnotesize {acting to the left}}.$$ (38)

The coefficients in Eq. (38) are defined in Tables 3 and 4. Equation (38) contains the usual Skyrme-interaction operators, but, the energy functional (27) does not necessarily correspond to a real (density-dependent) two-body Skyrme interaction because the matrix elements are not antisymmetrized. Compared to the case usually discussed in the literature, the more general functional relaxes relations that would otherwise restrict the spin-isospin structure of the effective interaction in Eq. (38); see, e.g., [16] for a discussion of the increased freedom.

The densities and currents that appear in Eq. (38) come mostly from rearrangement terms and take the values given by the HFB ground state. The isoscalar and isovector spin densities $\mbox{\boldmath$s$\unboldmath }_{t0} (\mbox{\boldmath$r$\unboldmath })$ vanish when the HFB ground state is time-reversal invariant or spherical as assumed here. The terms containing them will therefore not appear in the expressions for the matrix elements of the effective interaction for such states given below.

Table 3: Definitions of $a_i, b_i, c_i, d_i$ $(i=0,\cdots,3)$, $a_4$, and $c_4$ in Eq. (38).

$i$	$a_i$	$b_i$	$c_i$	$d_i$
0	${ }2A_0^{\rho}$	${ }2 A_0^{s}$	${ }2A_1^{\rho}$	${ }2 A_1^{s}$
1	${ }\frac{1}{2}(C_0^{\tau}- 4C_0^{\Delta\rho})$	${ }\frac{1}{2}(C_0^{T} - 4C_0^{\Delta s})$	${ }\frac{1}{2}(C_1^{\tau}- 4C_1^{\Delta\rho})$	${ }\frac{1}{2}(C_1^{T} - 4C_1^{\Delta s})$
2	${ }3C_0^{\tau} + 4C_0^{\Delta\rho}$	${ }3C_0^{T} + 4C_0^{\Delta s}$	${ }3C_1^{\tau} + 4C_1^{\Delta\rho}$	${ }3C_1^{T} + 4C_1^{\Delta s}$
3	${ }B_0^{\rho}(\alpha+2)(\alpha+1)$	${ }2B_0^{s}$	${ }2B_1^{\rho}$	${ }2B_1^{s}$
4	${ }-2iC_0^{\nabla J}$		${ }-2iC_1^{\nabla J}$

Table 4: Definitions of the coefficients appearing in the rearrangement terms in Eq. (38).

$i$	$e_i$	$f_i$	$g_i$	$h_i$	$m_i$	$n_i$
3	${ }2\alpha B_1^{\rho}$	${ }\alpha(\alpha-1) B_1^{\rho}$	${ }2\alpha B_0^{s}$	${ }\alpha(\alpha-1) B_0^{s}$	${ }2\alpha B_1^{s}$	${ }\alpha(\alpha-1) B_1^{s}$

The effective Coulomb interaction in Eq. (37), acting between protons, is given by

$$\bar{V}^{\mbox{\footnotesize {eff}}}_{\rm Coul} = \frac{e^2}... ...{\boldmath$r$\unboldmath }-\mbox{\boldmath$r$\unboldmath }') .$$ (39)

Finally, the ph-type pairing-rearrangement terms in Eq. (37) come from an effective interaction

$$\bar{V}^{\rm eff \ ph}_{\rm pair} = \frac{d^2 C^{\tilde\rho}... ...x{\boldmath$r$\unboldmath }'-\mbox{\boldmath$r$\unboldmath }).$$ (40)

The second derivatives with respect to $\kappa,\kappa^\ast$ also can be written as unsymmetrized matrix elements of effective interactions, this time in the particle-particle channel. The particle-particle effective interaction entering the matrix elements

$$\bar{V}^{\rm pp}_{KK^\prime LL^\prime} = \langle KK^\prime\v... ...ff\ pp}}}_{\mbox{\footnotesize {pair}}} \vert LL^\prime\rangle$$ (41)

is obtained from Eq. (34) through Eq. (14) as

$$\bar{V}^{\mbox{\footnotesize {eff\ pp}}}_{\mbox{\footnotesiz... ...{\boldmath$r$\unboldmath }-\mbox{\boldmath$r$\unboldmath }') .$$ (42)

In the numerical calculations of this paper, we use a volume pairing-energy functional, i.e.,

$$C^{\tilde\rho} = \frac{1}{2}V_0 = {\rm const.}$$ (43)

Last of all are the mixed functional derivatives involving both $\rho$ and $\kappa$ (or $\tilde{\rho}$) in Eq. (15). They also can be written as the unsymmetrized matrix elements of an effective interaction:

$$\bar{V}^{\rm 3p1h}_{K'KL'L} = \langle L'K'\vert \bar{V}^{\rm eff\ 3p1h}_{\rm pair } \vert LT(K)\rangle ,$$ (44)

where $T(K)$ denotes the time-reversed state of $K$, and the 3p1h effective interaction itself is

$$\bar{V}^{\rm eff\ 3p1h}_{\rm pair} = \frac{dC^{\tilde\rho}[... ...{\boldmath$r$\unboldmath }'-\mbox{\boldmath$r$\unboldmath }) ,$$ (45)

where $\tau'_z$ acts on the single-particle states $K'$ and $T(K)$ in Eq. (45), and the eigenvalues 1 and $-1$ are assigned to the neutron and proton, respectively.