The Energy Density Functional

In the HFB theory the expectation value of Hamiltonian in state $\vert\Psi \rangle$ is a functional of the density matrices, and reads

$$E_{\mbox{\rm\scriptsize {HFB}}} = \langle \Psi \vert{H}\vert... ...+%% \hat{\breve{\Gamma}}\bullet \hat{\breve{\rho}}^{+}\right),$$ (72)

where $\mbox{Tr}$ denotes integration over spatial coordinates and summation over spin and isospin indices. Nuclear many-body Hamiltonian ${H}$,

$${H} = \int\hspace{-1.4em}\sum {\rm d}{x'}{\rm d}{x}\,\hat{T}... ...\hat{V}(x'_1x'_2,x_1x_2)a^+_{x'_1}a^+_{x'_2}a_{x_2} a_{x_1},\!$$ (73)

is composed of one-body kinetic energy ${T}$ and two-body interaction ${V}$, being expressed in (73) by matrix $\hat{T}(x',x)$ and the antisymmetrized matrix elements $\hat{V}(x'_1x'_2,x_1x_2)$, respectively. Matrices $\hat{\Gamma}$ and $\hat{\breve{\Gamma}}$ are the single-particle (p-h) and pairing (p-p) self-consistent potentials, respectively,

$$\hspace*{-2.0em} \hat {\Gamma}(x'_1,x_1) \!\! = \int\hspace{-1.4em}\sum {\rm d}{x'_2}{\rm d}{x_2}\, \hat{V}_{\mbox{\rm\scriptsize {p-h}}}(x'_1x'_2,x_1x_2) \hat{\rho}(x_2,x'_2),$$ (74)

$$\hspace*{-2.0em} \hat{\breve{\Gamma}}(x'_1,x'_2) \!\! = \int\hspace{-1.4em}\sum {\rm d}{x_1}{\rm d}{x_2} F_2\hat{V}_{\mbox{\rm\scriptsize {p-p}}}(x'_1\bar{x}'_2,x_1\bar{x}_2) \hat{\breve{\rho}}(x_1,x_2),$$ (75)

where $F_2$= $8s'_2 s_2 t'_2 t_2$ and $\bar{x}$$\equiv$ $\{\mbox{{\boldmath {$r$}}},$$-s,$$-t\}$. In Eqs. (74) and (75) we have indicated that the p-h and p-p potentials can be determined by different two-body interactions, ${V}_{\mbox{\rm\scriptsize {p-h}}}$ and ${V}_{\mbox{\rm\scriptsize {p-p}}}$, called effective interactions in the p-h and the p-p channel, respectively. This places further derivations in the framework of the energy-density formalism that is not based on a definite Hamiltonian (73). Moreover, effective interactions, ${V}_{\mbox{\rm\scriptsize {p-h}}}$ and ${V}_{\mbox{\rm\scriptsize {p-p}}}$, are supposed to be, in general, density-dependent.

In the case of the Skyrme effective interaction, as well as in the framework of the LDA, the energy functional of Eq. (72) is a three-dimensional spatial integral,

$$\overline{H}=\int {\rm d}^3\mbox{{\boldmath {$r$}}}{\mathcal H}(\mbox{{\boldmath {$r$}}}) ,$$ (76)

of local energy density ${\mathcal H}(\mbox{{\boldmath {$r$}}})$ that is a real, scalar, time-even, and isoscalar function of local densities and their first and second derivatives. (Isospin-breaking terms, like those resulting from different neutron and proton masses and from the Coulomb interaction, can be easily added and, for simplicity, are not considered in the present study.) In the case of no proton-neutron mixing, the construction of the most general energy density that is quadratic in one-body local densities was presented in detail in Ref. [173]. With the proton-neutron mixing included, the construction can be performed analogically by including the additional non-zero local densities derived in Sec. 3. Then the energy density can be written in the following form:

$${\mathcal H}(\mbox{{\boldmath {$r$}}}) = \frac{\hbar^{2}}{2m... ...ath {$r$}}}) +\breve{\chi}_t(\mbox{{\boldmath {$r$}}})\right),$$ (77)

where we assumed that the neutron and proton masses are equal.

The p-h and p-p interaction energy densities, $\chi_t(\mbox{{\boldmath {$r$}}})$ and $\breve{\chi}_t$, for $t$=0 depend quadratically on the isoscalar densities, and for $t$=1 - on the isovector ones. Based on general rules of constructing the energy density, Sec. 3.3, one obtains

$${\chi}_{0}(\mbox{{\boldmath {$r$}}}) = C_{0}^{\rho} \rho_{0}^{2}\ofbboxofr + C_{0}^{\Delta\rho} \rho_{0}... ...oxofr\mbox{{\boldmath {$\nabla$}}}\cdot\mbox{{\boldmath {$J$}}}_{0} \ofbboxofr$$

$$+ C_{0}^{s} \mbox{{\boldmath {$s$}}}_{0}^{2}\ofbboxofr + C_{0}^{\De... ...{\boldmath {$s$}}}_{0} \ofbboxofr\cdot\mbox{{\boldmath {$F$}}}_{0} \ofbboxofr ,$$ (78)

$${\chi}_{1}(\mbox{{\boldmath {$r$}}}) = C_{1}^{\rho} \vec{ \rho}^{\,2}\ofbboxofr + C_{1}^{\Delta\rho} \ve... ...rc\mbox{{\boldmath {$\nabla$}}}\cdot\vec{ \mbox{{\boldmath {$J$}}}} \ofbboxofr$$

$$+ C_{1}^{s} \vec{ \mbox{{\boldmath {$s$}}}}^{\,2}\ofbboxofr + C_{1}... ...th {$s$}}}} \ofbboxofr\cdot \circ~ \vec{ \mbox{{\boldmath {$F$}}}} \ofbboxofr ,$$ (79)

where $\times$ stands for the vector product, and

$$\breve{\chi}_{0}(\mbox{{\boldmath {$r$}}}) = \breve{C}_{0}^{s} \vert\breve{\mbox{{\boldmath {$s$}}}}_0 \ofbbox... ...s$}}}}_0^{\,*}\ofbboxofr\cdot\breve{\mbox{{\boldmath {$T$}}}}_0 \ofbboxofr\big)$$

$$+ \breve{C}_{0}^{j} \vert\breve{\mbox{{\boldmath {$j$}}}}_0 \ofbbox... ...$}}}}_0^{\,*}\ofbboxofr\cdot\breve{\mbox{{\boldmath {$F$}}}}_0 \ofbboxofr\big),$$ (80)

$$\breve{\chi}_1(\mbox{{\boldmath {$r$}}}) = \breve{C}_{1}^{\rho} \vert\vec{\breve {\rho}} \ofbboxofr \vert^2 ... ...big(\vec{\breve {\rho}}^{\,*}\ofbboxofr\circ\vec{\breve {\tau}} \ofbboxofr\big)$$

$$+ \breve{C}_{1}^{J0} \vert\vec{\breve {J}} \ofbboxofr \vert^2 + \br... ...dmath {$\nabla$}}}\cdot\vec{\breve {\mbox{{\boldmath {$J$}}}}} \ofbboxofr\big).$$ (81)

In Eqs. (78) and (80) squares always denote total lengths in space and/or iso-space, for complex densities taken in the complex sense, e.g., $\vert\vec{\breve{\mbox{{\boldmath {$J$}}}}}(\mbox{{\boldmath {$r$}}})\vert^2 = \sum_{ak}\breve{J}^*_{ak}\breve{J}_{ak}$. In the p-p energy density (80) we show only terms in which the products of real parts are added to products of imaginary parts. According to the rules based on the $TC$-symmetry, Sec. 3.3, similar terms with both products subtracted from one another are also allowed. We do not show them explicitly, because they have exactly the form of Eq. (80), but without complex conjugations and with absolute values replaced by real parts of products.

When the effective interaction is density-dependent all coupling constants, $C$'s and $\breve{C}$'s, may also depend on density. If this is the case, however, terms that can be transformed into one another by integration by parts are not anymore equivalent. Then, five more types of terms may appear in the energy density, see Ref. [173]; we do not consider such a possibility in the present study. Note that in the p-h channel all coupling constants appear in two flavors, for $t$=0 and 1, while in the p-p channel each one appears exclusively either for $t$=0, or for $t$=1.

The expression (77) is fairly general. In particular, it is not based on any particular two-body interaction. However, if one assumes that the underlying two-body potential is local and momentum-independent, the form of (77) can be simplified and the number of coupling constants can be reduced. Two particular cases of practical interest are discussed in the following.