The P-H and P-P Mean Fields

By varying the energy functional (72) with respect to the density matrices one obtains the p-h and p-p mean field Hamiltonians,

$$\hat{h}(\mbox{{\boldmath {$r$}}}'s't',\mbox{{\boldmath {$r$}}}st) = \frac{\delta \overline{H}[\hat{\rho},\hat{\breve{\rho}},\hat{\bre... ...rm\scriptsize {r}}}(\mbox{{\boldmath {$r$}}}'s't',\mbox{{\boldmath {$r$}}}st) ,$$ (158)

$$\hat{\breve{h}}(\mbox{{\boldmath {$r$}}}'s't',\mbox{{\boldmath {$r$}}}st) = \frac{\delta \overline{H}[\hat{\rho},\hat{\breve{\rho}},\hat{\bre... ...rm\scriptsize {r}}}(\mbox{{\boldmath {$r$}}}'s't',\mbox{{\boldmath {$r$}}}st) .$$ (159)

The rearrangement potentials $\hat{\Gamma}_{\mbox{\rm\scriptsize {r}}}$ and $\hat{\breve{\Gamma}}_{\mbox{\rm\scriptsize {r}}}$ result from the density dependence of effective interactions on the p-h and p-p densities, respectively. Usually effective interactions are assumed to depend only on the p-h density matrix (most often, only on the isoscalar particle density $\rho_0$). In that case the p-p rearrangement potential vanishes. However, one cannot forget that the dependence of the p-p interaction on the particle density results in a corresponding contribution to the p-h rearrangement potential. In what follows, to simplify the presentation we do not show the rearrangement terms explicitly.

Within the LDA, the mean-field Hamiltonians being originally, like the Skyrme interaction of Eq. (117), either distributions or derivatives of distributions, can, when acting as the integral kernels, be expressed as local, momentum dependent operators, i.e.,

$$\hat {{h}}(\mbox{{\boldmath {$r$}}}'s't',\mbox{{\boldmath {$r$}}}st) = \delta (\mbox{{\boldmath {$r$}}}-\mbox{{\boldmath {$r$}}}')\hat {{h}}(\mbox{{\boldmath {$r$}}};s't',st),$$ (160)

$$\hat{\breve{h}}(\mbox{{\boldmath {$r$}}}'s't',\mbox{{\boldmath {$r$}}}st) = \delta (\mbox{{\boldmath {$r$}}}-\mbox{{\boldmath {$r$}}}')\hat{\breve{h}}(\mbox{{\boldmath {$r$}}};s't',st).$$ (161)

The kinetic energy term in Eq. (158) is already expressed in such a form. The mean-fields Hamiltonians are the second-order operators in momentum and matrices in the spin and isospin spaces. The isospin structure of the local p-h and p-p mean-field Hamiltonians reads

$$\hspace*{-1.5em} \hat{h}(\mbox{{\boldmath {$r$}}};s't',st) = {h}_0(\mbox{{\boldmath {$r$}}};s',s)\delta_{t't}+%% \vec{h}(\mbox{{\boldmath {$r$}}};s',s)\circ \hat{\vec{\tau}}_{t't} ,$$ (162)

$$\hspace*{-1.5em} \hat{\breve{h}}(\mbox{{\boldmath {$r$}}};s't',st) = \breve{h}_0(\mbox{{\boldmath {$r$}}};s',s)\delta_{t't}+%% \vec{\breve{h}}(\mbox{{\boldmath {$r$}}};s',s)\circ \hat{\vec{\tau}}_{t't},$$ (163)

respectively. The isoscalar and isovector parts of the p-h mean-field Hamiltonian can be presented in the compact form

$$h_k(\mbox{{\boldmath {$r$}}};s',s) = - \frac{\hbar^{2}}{2m}\mbox{{\boldmath {$\nabla$}}}^2\delta_{s's}... ... + ({\mathsf B}_k\ofbboxofr\cdot\hat{\mbox{{\boldmath {$\sigma$}}}}_{s's})\big]$$

$$- \mbox{{\boldmath {$\nabla$}}}\cdot\big[M_k\ofbboxofr\delta_{s's... ...r\,\hat{\mbox{{\boldmath {$\sigma$}}}}_{s's}\cdot \mbox{{\boldmath {$\nabla$}}}$$ (164)

for $k=0,1,2,3$, and where

$$({\mathsf B}\cdot\hat{\mbox{{\boldmath {$\sigma$}}}})_a =\sum_b{\mathsf B}_{ab}\hat{\mbox{{\boldmath {$\sigma$}}}}^b ,$$ (165)

for $a=x,y,z$, is the $a$'th component of a space vector. The names of symbols are inspired by those introduced in Ref. [172]. Since the p-h density matrix is hermitian, the p-h mean-field Hamiltonian is also hermitian and, thus, all the potentials, $M_k$, $U_k$, ${\mathsf B}_k$, $\mbox{{\boldmath {$C$}}}_k$, $\mbox{{\boldmath {$D$}}}_k$, $\mbox{{\boldmath {$I$}}}_k$, and $\mbox{{\boldmath {$\Sigma$}}}_k$ are real.

The general form of the mean-field Hamiltonian (164) can be constructed from the momentum $-i\mbox{{\boldmath {$\nabla$}}}$ and spin $\hat{\mbox{{\boldmath {$\sigma$}}}}$ operators, based only on the symmetry properties. Apart from the one-body kinetic energy [the first term in Eq. (164)], the expansion in momentum gives: (i) zero-order terms with scalar ($U_k$) and pseudovector ( $\mbox{{\boldmath {$\Sigma$}}}_k$) potentials, (ii) first-order terms with vector ( $\mbox{{\boldmath {$I$}}}_k$) and pseudotensor (${\mathsf B}_k$) potentials, (iii) second-order-scalar terms with scalar ($M_k$) and pseudoscalar ( $\mbox{{\boldmath {$C$}}}_k$) effective masses, and (iv) second-order-tensor terms. In principle, the most general form of the last category should involve tensor and third-order-pseudotensor potentials. However, in Eq. (164) we show only the particular form of it that corresponds to the energy density (78).

According to Eqs. (158) the p-h mean-field Hamiltonian is the functional derivative of the energy functional over the hermitian p-h density matrix. Functional derivatives of integrals of type:

$$\overline{\langle f\rho \rangle}= \int \delta (\mbox{{\boldm... ...\mbox{{\boldmath {$r$}}}_1{\rm d}^3\mbox{{\boldmath {$r$}}}_2,$$ (166)

where function $f$ is treated as independent of densities and $\rho$ represents a p-h non-local density, can easily be calculated using Eqs. (20) and (24). Bearing in mind that

$$\frac{\delta \hat{\rho}(\mbox{{\boldmath {$r$}}}_1s_1t_1,\mbox{{\... ...2)}{\delta\hat{\rho}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = \delta(\mbox{{\boldmath {$r$}}}_1-\mbox{{\boldmath {$r$}}})\delta... ...box{{\boldmath {$r$}}}')\delta_{s_1s}\delta_{s_2s'}\delta_{t_1t}\delta_{t_2t'},$$

one has

$$\frac{\delta \overline{\langle f\rho _k\rangle}}{\delta\hat{\rho}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = \delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}})f(\mbox{{\boldmath {$r$}}})\delta_{s's}\hat{\tau}^k_{t't},$$ (167)

$$\frac{\delta\overline{\langle f\mbox{{\boldmath {$s$}}}_k\rangle}}{\delta\hat{\rho}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = \delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}})f(\mbox{{\boldmath {$r$}}})\hat{\mbox{{\boldmath {$\sigma$}}}}_{s's}\hat{\tau}^k_{t't}$$ (168)

for $k=0,1,2,3$. The functional derivatives of integrals of local differential densities are obtained from Eqs. (168) through integration by parts. Then, the functional derivatives become dependent on derivatives of the Dirac delta function and thus, in accordance with Eqs. (160), again act as local differential operators. They read:

$$\frac{\delta \overline{\langle f\mbox{{\boldmath {$j$}}}_k\rangle}}{\delta\hat{\rho}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = {\textstyle{\frac{1}{2i}}}\delta(\mbox{{\boldmath {$r$}}}'-\mbox{... ...math {$r$}}})\mbox{{\boldmath {$\nabla$}}}\big) \delta_{s's}\hat{\tau}^k_{t't},$$ (169)

$$\frac{\delta\overline{\langle f\mathsf{J}_{kab}\rangle}}{\delta\hat{\rho}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = {\textstyle{\frac{1}{2i}}}\delta(\mbox{{\boldmath {$r$}}}'-\mbox{... ...nabla$}}}_a\big) \hat{\mbox{{\boldmath {$\sigma$}}}}^b_{s's}\hat{\tau}^k_{t't},$$ (170)

$$\hspace*{-1.5em} \frac{\delta\overline{\langle f\mbox{{\boldmath ... ...e}}{\delta\hat{\rho}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = -\delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}}) \mbox{{\boldmath {$\nabla$}}}_af(\mbox{{\boldmath {$r$}}})\mbox{{\boldmath {$\nabla$}}}_b \delta_{s's}\hat{\tau}^k_{t't},$$ (171)

$$\hspace*{-1.5em} \frac{\delta\overline{\langle f\mbox{{\boldmath ... ...e}}{\delta\hat{\rho}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = -\delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}}) \mbox{{\boldmath {$\nabla$}}}_af(\mbox{{\boldmath {$r$}}})\mbox{{... ...th {$\nabla$}}}_b \hat{\mbox{{\boldmath {$\sigma$}}}}^c_{s's}\hat{\tau}^k_{t't}$$ (172)

for $k=0,1,2,3$ and $a,b,c=x,y,z$. Calculations of the functional derivatives over the density matrix are equivalent to the rules for variations over single-particle wavefunctions given by Engel et al. [172]. Using formulae given above, Eqs. (168)-(172), one obtains the following relations between the potentials defining the p-h mean field (164) and the local p-h densities defining the energy density (78),

$$U_k(\mbox{{\boldmath {$r$}}}) = 2C^{\rho}_t\rho_k\ofbboxofr +2C^{\Delta \rho}_t\Delta \rho_k\ofbb... ...a J}_t\mbox{{\boldmath {$\nabla$}}}\cdot \mbox{{\boldmath {$J$}}}_k\ofbboxofr ,$$ (173)

$$\mbox{{\boldmath {$\Sigma$}}}_k(\mbox{{\boldmath {$r$}}}) = 2C^s_t\mbox{{\boldmath {$s$}}}_k\ofbboxofr + 2(C^{\Delta s}_t-C^{... ... j}_t\mbox{{\boldmath {$\nabla$}}}\times \mbox{{\boldmath {$j$}}}_k\ofbboxofr ,$$ (174)

$$\mbox{{\boldmath {$I$}}}_k(\mbox{{\boldmath {$r$}}}) = 2C^j_t\mbox{{\boldmath {$j$}}}_k\ofbboxofr + C^{\nabla j}_t\mbox{{\boldmath {$\nabla$}}} \times\mbox{{\boldmath {$s$}}}_k\ofbboxofr ,$$ (175)

$${\mathsf B}_k(\mbox{{\boldmath {$r$}}}) = 2C^{J0}_tJ_k\ofbboxofr {\delta } - 2C^{J1}_t\epsilon\cdot\mbox{{\... ...fr + C^{\nabla J}_t\epsilon\cdot\mbox{{\boldmath {$\nabla$}}}\rho_k\ofbboxofr ,$$ (176)

$$M_k(\mbox{{\boldmath {$r$}}}) = C^{\tau}_t\rho_k\ofbboxofr ,$$ (177)

$$\mbox{{\boldmath {$C$}}}_k(\mbox{{\boldmath {$r$}}}) = C^T_t\mbox{{\boldmath {$s$}}}_k\ofbboxofr ,$$ (178)

$$\mbox{{\boldmath {$D$}}}_k(\mbox{{\boldmath {$r$}}}) = C^F_t\mbox{{\boldmath {$s$}}}_k\ofbboxofr ,$$ (179)

for $k=0,1,2,3$. All coupling constants $C_t$ in Eqs. (174) are taken with $t$=0 for $k=0$ (isoscalars), and with $t$=1 for $k=$1,2,3 (isovectors). Symbol ${\delta }$ is the unit space tensor, and $\epsilon\cdot\mbox{{\boldmath {$J$}}}$ stands for the antisymmetric space tensor with components: $(\epsilon\cdot\mbox{{\boldmath {$J$}}})_{ab} = \sum_{c}\epsilon_{acb}\mbox{{\boldmath {$J$}}}_c$, so that, according to Eq. (165), its action on a vector is obviously the vector product: $(\epsilon\cdot\mbox{{\boldmath {$J$}}})\cdot \hat{\mbox{{\boldmath {$\sigma$}}}}=\mbox{{\boldmath {$J$}}}\times \hat{\mbox{{\boldmath {$\sigma$}}}}$.

The p-p mean-field Hamiltonian has the following isoscalar and isovector components:

$$\breve{h}_0(\mbox{{\boldmath {$r$}}};s',s) = \breve{\mbox{{\boldmath {$\Sigma$}}}}_0\ofbboxofr\cdot\hat{\mbox{... ...\,\hat{\mbox{{\boldmath {$\sigma$}}}}_{s's}\cdot \mbox{{\boldmath {$\nabla$}}},$$ (180)

$$\vec{\breve{h}}(\mbox{{\boldmath {$r$}}};s',s) = \vec{\breve{U}}(\mbox{{\boldmath {$r$}}})\delta_{s's} + {\textsty... ...bla$}}}\cdot\vec{\breve{M}}\ofbboxofr\delta_{s's}\mbox{{\boldmath {$\nabla$}}}.$$ (181)

Contrary to the p-h Hamiltonian (164), the p-p Hamiltonian (181) can be non-hermitian, because potentials $\breve{\mbox{{\boldmath {$C$}}}}_0$, $\breve{\mbox{{\boldmath {$D$}}}}_0$, $\breve{\mbox{{\boldmath {$I$}}}}_0$, $\breve{\mbox{{\boldmath {$\Sigma$}}}}_0$, $\vec{\breve{M}}$, $\vec{\breve{U}}$, and $\vec{\breve{\mathsf B}}$ are, in general, complex quantities. This is so, because the p-p density matrix is, in general, not hermitian. Therefore, the energy functional should be treated as a functional of both $\hat{\breve{\rho}}$ and $\hat{\breve{\rho}}^+$.

The p-p mean-field Hamiltonian is the functional derivative of the energy functional over $\hat{\breve{\rho}}^+$, whereas the hermitian conjugate Hamiltonian is the functional derivative over $\hat{\breve{\rho}}$. The p-p densities are, according to Eqs. (22) and (26), functions of $\hat{\breve{\rho}}$, while the complex conjugate densities are functions of $\hat{\breve{\rho}}^+$.

When calculating the p-p functional derivatives, one cannot forget that the p-p density matrix fulfills symmetry condition (12), implying that the p-p densities are either symmetric or antisymmetric functions, Eqs. (34). Therefore, the calculation of functional derivatives over either $\hat{\breve{\rho}}$ or $\hat{\breve{\rho}}^+$ is similar to that leading to Eqs. (168)-(172), however, instead of Eq. (167) one has:

$$\frac{\delta \hat{\breve{\rho}}^+(\mbox{{\boldmath {$r$}}}_1... ...}}})\delta_{s_1-s'}\delta_{s_2-s}\delta_{t_1-t'}\delta_{t_2-t}$$ (182)

In the expressions for functional derivatives, this gives either cancellation or addition of terms coming from the two components of the right-hand side of Eq. (183). Finally, the non-vanishing derivatives are

$$\frac{\delta \overline{\langle f\vec{\breve{\rho}}^{\ast}\rangle}... ...\hat{\breve{\rho}}^+(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = 2\delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}}) f(\mbox{{\boldmath {$r$}}})\delta_{s's}\hat{\vec{\tau}}_{t't},$$ (183)

$$\frac{\delta\overline{\langle f\breve{\mbox{{\boldmath {$s$}}}}^{... ...\hat{\breve{\rho}}^+(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = 2\delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}})f(\mbox{{\boldmath {$r$}}})\hat{\mbox{{\boldmath {$\sigma$}}}}_{s's}\hat{\tau}^0_{t't},$$ (184)

$$\frac{\delta \overline{\langle f\breve{\mbox{{\boldmath {$j$}}}}^... ...\hat{\breve{\rho}}^+(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = -i\delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}})\big(... ...math {$r$}}})\mbox{{\boldmath {$\nabla$}}}\big) \delta_{s's}\hat{\tau}^0_{t't},$$ (185)

$$\frac{\delta\overline{\langle f\vec{\breve{\mathsf{J}}}^{\ast}_{a... ...\hat{\breve{\rho}}^+(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = -i\delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}}) \big... ...a$}}}_a\big) \hat{\mbox{{\boldmath {$\sigma$}}}}^b_{s's}\hat{\vec{\tau}}_{t't},$$ (186)

$$\frac{\delta\overline{\langle f\mbox{{\boldmath {$\nabla$}}}_a\mb... ...\hat{\breve{\rho}}^+(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = -2\delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}}) \mbo... ...h {$r$}}})\mbox{{\boldmath {$\nabla$}}}_b\, \delta_{s's}\hat{\vec{\tau}}_{t't},$$ (187)

$$\frac{\delta\overline{\langle f\mbox{{\boldmath {$\nabla$}}}_a\mb... ...\hat{\breve{\rho}}^+(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't')} = -2\delta(\mbox{{\boldmath {$r$}}}'-\mbox{{\boldmath {$r$}}}) \mbo... ...$\nabla$}}}_b\, \hat{\mbox{{\boldmath {$\sigma$}}}}^c_{s's}\hat{\tau}^0_{t't} ,$$ (188)

for $a,b,c=x,y,z$.

Using Eqs. (184)-(186) one obtains the following relations between the potentials defining the p-p mean-field Hamiltonian (181) and the local p-p densities defining the energy density (80):

$$\breve{\mbox{{\boldmath {$\Sigma$}}}}_0(\mbox{{\boldmath {$r$}}}) = 2\breve{C}^s_0\breve{\mbox{{\boldmath {$s$}}}}_0\ofbboxofr + 2\bi... ...ox{{\boldmath {$\nabla$}}}\times \breve{\mbox{{\boldmath {$j$}}}}_0\ofbboxofr ,$$ (189)

$$\breve{\mbox{{\boldmath {$I$}}}}_0(\mbox{{\boldmath {$r$}}}) = 2\breve{C}^j_0\breve{\mbox{{\boldmath {$j$}}}}_0\ofbboxofr + \bre... ...ox{{\boldmath {$\nabla$}}}\times \breve{\mbox{{\boldmath {$s$}}}}_0\ofbboxofr ,$$ (190)

$$\breve{\mbox{{\boldmath {$C$}}}}_0(\mbox{{\boldmath {$r$}}}) = \breve{C}^T_0\breve{\mbox{{\boldmath {$s$}}}}_0\ofbboxofr ,$$ (191)

$$\breve{\mbox{{\boldmath {$D$}}}}_0(\mbox{{\boldmath {$r$}}}) = \breve{C}^F_0\breve{\mbox{{\boldmath {$s$}}}}_0\ofbboxofr ,$$ (192)

$$\vec{\breve{U}}(\mbox{{\boldmath {$r$}}}) = 2\breve{C}^{\rho}_1\vec{\breve{\rho}}\ofbboxofr + 2\breve{C}^{\De... ...\boldmath {$\nabla$}}}\cdot\vec{\breve{\mbox{{\boldmath {$J$}}}}}_k\ofbboxofr ,$$ (193)

$$\vec{\breve{\mathsf B}}(\mbox{{\boldmath {$r$}}}) = 2\breve{C}^{J0}_1\vec{\breve{J}}\ofbboxofr {\delta } - 2\breve{C}... ...la J}_1\epsilon\cdot\mbox{{\boldmath {$\nabla$}}}\vec{\breve{\rho}}\ofbboxofr ,$$ (194)

$$\vec{\breve{M}}(\mbox{{\boldmath {$r$}}}) = \breve{C}^{\tau}_1\vec{\breve{\rho}}\ofbboxofr .$$ (195)

In the case of the zero-range pairing force (151), the isovector p-p potential is proportional to the p-p isovector density while the isoscalar field has a very different structure, i.e., it is proportional to the scalar product of spin $\hat{\mbox{{\boldmath {$\sigma$}}}}$ and the p-p spin density $\breve{\mbox{{\boldmath {$s$}}}}_0$. This immediately suggests that there exists a connection between the isoscalar pairing and the p-p spin saturation, which is influenced by the spin-orbit splitting. In this context, let us remind the shell-model study [20] which discusses the relation between the magnitude of the $T$=0 pairing and the spin-orbit splitting.