Local densities

In the HFB theory with the zero-range Skyrme interaction [168,169], or in the local density approximation (LDA) (cf. Refs. [170,167]), the energy functional depends only on local densities, and on local densities built from derivatives up to the second order. These local densities are obtained by setting $\mbox{{\boldmath {$r$}}}'$= $\mbox{{\boldmath {$r$}}}$ in Eqs. (20)-(26) after the derivatives are performed. They will be denoted by having one spatial argument to distinguish them from the non-local densities that have two. Moreover, for local densities the spatial argument will often be omitted in order to lighten the notation.

Following the standard definitions [171,172], a number of local densities are introduced:

$\bullet$ scalar densities:

-

particle and pairing densities:

$${\rho}_k(\mbox{{\boldmath {$r$}}}) = {\rho}_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (38)

$$\vec{\breve {\rho}} (\mbox{{\boldmath {$r$}}}) = \vec{\breve{\rho}} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (39)

-

p-h and p-p kinetic densities:

$${\tau}_k(\mbox{{\boldmath {$r$}}}) = \big[ (\mbox{{\boldmath {$\nabla$}}}\cdot\mbox{{\boldmath {$\nabl... ...{\boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (40)

$$\vec{\breve {\tau}} (\mbox{{\boldmath {$r$}}}) = \big[ (\mbox{{\boldmath {$\nabla$}}}\cdot \mbox{{\boldmath {$\nab... ...{\boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (41)

$\bullet$ vector densities:

-

p-h and p-p spin (pseudovector) densities:

$${\mbox{{\boldmath {$s$}}}}_k(\mbox{{\boldmath {$r$}}}) = {\mbox{{\boldmath {$s$}}}}_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (42)

$$\breve{\mbox{{\boldmath {$s$}}}}_0(\mbox{{\boldmath {$r$}}}) = \breve{\mbox{{\boldmath {$s$}}}}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (43)

-

p-h and p-p spin-kinetic (pseudovector) densities:

$${\mbox{{\boldmath {$T$}}}}_k(\mbox{{\boldmath {$r$}}}) = \big[ (\mbox{{\boldmath {$\nabla$}}}\cdot\mbox{{\boldmath {$\nabl... ...{\boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (44)

$$\breve{\mbox{{\boldmath {$T$}}}}_0(\mbox{{\boldmath {$r$}}}) = \big[ (\mbox{{\boldmath {$\nabla$}}}\cdot\mbox{{\boldmath {$\nabl... ...{\boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (45)

-

p-h and p-p current (vector) densities:

$${\mbox{{\boldmath {$j$}}}}_k(\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2i}}}\big[ (\mbox{{\boldmath {$\nabla$}}} - ... ...{\boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (46)

$$\breve{\mbox{{\boldmath {$j$}}}}_0(\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2i}}}\big[ (\mbox{{\boldmath {$\nabla$}}} - ... ...{\boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (47)

-

p-h and p-p tensor-kinetic (pseudovector) densities:

$${\mbox{{\boldmath {$F$}}}}_k(\mbox{{\boldmath {$r$}}}) =\! {\textstyle{\frac{1}{2}}}\big[ (\mbox{{\boldmath {$\nabla$}}} \!\... ...boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},\!$$ (48)

$$\breve{\mbox{{\boldmath {$F$}}}}_0(\mbox{{\boldmath {$r$}}}) =\! {\textstyle{\frac{1}{2}}}\big[ (\mbox{{\boldmath {$\nabla$}}} \!\... ...boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},\!$$ (49)

$\bullet$ tensor densities:

-

p-h and p-p spin-current (pseudotensor) densities:

$${{\mathsf J}}_k(\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2i}}}\big[ (\mbox{{\boldmath {$\nabla$}}} - ... ...{\boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (50)

$$\vec{\breve{\mathsf J}} (\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2i}}}\big[ (\mbox{{\boldmath {$\nabla$}}} - ... ...{\boldmath {$r$}}}')\big]_{\mbox{{\boldmath {$r$}}}=\mbox{{\boldmath {$r$}}}'},$$ (51)

where $k$=0,1,2,3, and $\otimes$ stands for the tensor product of vectors in the physical space, e.g., $(\mbox{{\boldmath {$v$}}}$$\otimes\,$ $\mbox{{\boldmath {$w$}}})_{ab}$$\,\equiv\,$ $\mbox{{\boldmath {$v$}}}_{a}\mbox{{\boldmath {$w$}}}_{b}$ and $[(\mbox{{\boldmath {$v$}}}$$\otimes\,$ $\mbox{{\boldmath {$w$}}})$$\cdot$ $\mbox{{\boldmath {$z$}}}]_{a}$$\,\equiv\,$ $\mbox{{\boldmath {$v$}}}_{a}(\mbox{{\boldmath {$w$}}}$$\cdot$ $\mbox{{\boldmath {$z$}}})$. Note that for particle, pairing, kinetic, spin, spin-kinetic, and tensor-kinetic densities only the symmetric non-local densities contribute, while for the current and spin-current densities only antisymmetric ones contribute. It is then clear that for each p-h density there exist both isoscalar and isovector component, while for the p-p densities, the isovector component exists only for the pairing, kinetic, and spin-current densities, while the isoscalar one exists only for spin, spin-kinetic, tensor-kinetic, and current densities.

We note here in passing that the complete list of all local densities (up to the derivatives of the second order) also includes the kinetic and spin-kinetic densities in which the two derivatives are coupled to a tensor, i.e., $\mbox{{\boldmath {$\nabla$}}}$$\otimes$ $\mbox{{\boldmath {$\nabla$}}}'$. The resulting local densities are usually disregarded, because they do not have counterparts to form useful terms in the local energy density. There is one set of exceptions, which has been overlooked in the systematic construction presented in Ref. [173], and appears in the averaging of a zero-range tensor force [171], namely, the set of the tensor-kinetic local densities (48). In Sec. 4 we define terms in the energy density that depend on the tensor-kinetic densities.

All tensor densities (50) can be decomposed into trace, antisymmetric, and symmetric parts, giving the standard pseudoscalar, vector, and pseudotensor components that we show here to fix the notation:

$${J}_k(\mbox{{\boldmath {$r$}}}) = \sum_{a=x,y,z} {\mathsf J}_{kaa}(\mbox{{\boldmath {$r$}}}),$$ (52)

$$\vec{\breve{J}} (\mbox{{\boldmath {$r$}}}) = \sum_{a=x,y,z}\vec{\breve{\mathsf J}}_{aa}(\mbox{{\boldmath {$r$}}}),$$ (53)

$${\mbox{{\boldmath {$J$}}}}_{ka}(\mbox{{\boldmath {$r$}}}) = \sum_{b,c=x,y,z}\epsilon_{abc} {\mathsf J}_{kbc}(\mbox{{\boldmath {$r$}}}),$$ (54)

$$\vec{\breve{\mbox{{\boldmath {$J$}}}}}_{a}(\mbox{{\boldmath {$r$}}}) = \sum_{b,c=x,y,z}\epsilon_{abc}\vec{\breve{\mathsf J}}_{bc}(\mbox{{\boldmath {$r$}}}),$$ (55)

$$\underline {{\mathsf J}}_{kab} (\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2}}}{{\mathsf J}}_{kab} (\mbox{{\boldmath {$... ...$r$}}}) - {\textstyle{\frac{1}{3}}}{J}_k (\mbox{{\boldmath {$r$}}})\delta_{ab},$$ (56)

$$\underline{\vec{\breve{{\mathsf J}}}}_{ab}(\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2}}}\vec{\breve{{\mathsf J}}}_{ab} (\mbox{{\... ...\textstyle{\frac{1}{3}}}\vec{\breve{ J}} (\mbox{{\boldmath {$r$}}})\delta_{ab},$$ (57)

where $k$=0,1,2,3.

It follows from Eqs. (28) and (34) that the p-h densities are all real whereas the p-p densities are in general complex and thus the complex-conjugate densities are relevant. The p-p densities become real or imaginary only when the time-reversal symmetry is conserved, see Sec. 7.

Instead of the isoscalar and the third component of isovector p-h density one can always use the neutron and the proton one, e.g.,

$$\rho_n(\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2}}}\left(\rho_0(\mbox{{\boldmath {$r$}}})+\rho_3(\mbox{{\boldmath {$r$}}})\right),$$ (58)

$$\rho_p(\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2}}}\left(\rho_0(\mbox{{\boldmath {$r$}}})-\rho_3(\mbox{{\boldmath {$r$}}})\right),$$ (59)

and just the same for all other p-h densities. Similarly, instead of the $k$=1,2 isovector p-p densities one can use the neutron and proton pairing density, i.e.,

$$\breve{\rho}_n(\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2}}}\left(\breve{\rho}_1(\mbox{{\boldmath {$r$}}})+i\breve{\rho}_2(\mbox{{\boldmath {$r$}}})\right),$$ (60)

$$\breve{\rho}_p(\mbox{{\boldmath {$r$}}}) = {\textstyle{\frac{1}{2}}}\left(\breve{\rho}_1(\mbox{{\boldmath {$r$}}})-i\breve{\rho}_2(\mbox{{\boldmath {$r$}}})\right),$$ (61)

and just the same for all other p-p densities.