Particle-hole densities

We give the expression for particle density $\rho$, kinetic density $\tau$, spin density $\bbox{s}$, spin-kinetic density $\bbox{T}$, current density $\bbox{j}$, and spin-current density ${\sf J}$. Tensor-kinetic density $\bbox{F}$ is not used and its expression is omitted.

(a)

Scalar particle density

$$\rho_{m}(\bbox{r}) = \rho_{m}(\bbox{r},\bbox{r}')\biggl\vert _{\bbox{r} = \bbox{r}'}$$ (21)

where $m$ takes values 0, 1, 2 and 3. Suffix 0 represents isoscalar component of the density and 1 to 3 are the isovector components. Expressing the density in terms of HFB wavefunctions (20), we obtain

$$\rho_{m}(\bbox{r})= \sum_{tt'} \rho^{tt'}(rz) \tau_{t't}^{m}$$ (22)

where

$$\rho^{tt'}(rz) = \sum_{k}\biggl[ V^{+\ast}_{kt} V^{+}_{kt'}+V^{-\ast}_{kt} V^{-}_{kt'}\biggr],$$ (23)

and we use the abbreviation of HFB wave functions

$$V^{\pm}_{kt} \equiv V^{\pm}_k(rzt).$$ (24)

Isospin components of the particle density are given by

$$\rho_{0}(\bbox{r}) = \rho^{nn}(rz) +\rho^{pp}(rz)$$

$$= \frac {1} {2} \biggr(\rho^{nn}(rz) +\rho^{pp}(rz) + c.c. \biggr),$$ (25)

$$\rho_{1}(\bbox{r}) = \rho^{np}(rz) +\rho^{pn}(rz)$$

$$= \rho^{np}(rz) + c.c.,$$ (26)

$$\rho_{2}(\bbox{r}) = i[\rho^{np}(rz) -\rho^{pn}(rz) ]$$

$$= i\rho^{np}(rz) + c.c.,$$ (27)

$$\rho_{3}(\bbox{r}) = \rho^{nn}(rz) -\rho^{pp}(rz)$$

$$= \frac {1} {2} \biggr(\rho^{nn}(rz) - \rho^{pp}(rz) + c.c. \biggr).$$ (28)

The isospin structure of the particle density given by Eqs. (25)-(28) is identical for all the following particle-hole densities, and these expressions shall not be repeated in the following.

(b)

Kinetic density

$$\tau_m (\bbox{r}) = (\bbox{\nabla} \cdot \bbox{\nabla} ')\rho_m(\... ...\biggr\vert _{ \bbox{r} = \bbox{r}'} = \sum_{tt'}\tau^{tt'}(rz) \tau_{t't}^{m},$$ (29)

where,

$$\tau^{tt'}(rz) = \sum_{k}\biggr\{ [\partial_{r} V^{+\ast}_{kt}][\partial_{r} V^{+}_{kt'}] +[\partial_{r} V^{-\ast}_{kt}][ \partial_{r}V^{-}_{kt'}]$$

$$+ \frac{(\Lambda^{-})^{2}}{r^2} V^{+\ast}_{kt}V^{+}_{kt'} + \frac{ (\Lambda^{+})^{2}}{r^2} V^{-\ast}_{kt}V^{-}_{kt'}$$

$$+ [\partial_{z} V^{+\ast}_{kt}][\partial_{z} V^{+}_{kt'}] +[\partial_{z} V^{-\ast}_{kt}][ \partial_{z} V^{-}_{kt'}]\biggr\} .$$ (30)

(c)

Pseudovector spin density

$$\bbox{s}_{m}(\bbox{r}) = \bbox{s}_{m}(\bbox{r},\bbox{r}') \biggl\vert _{\bbox{r} = \bbox{r}'} = \sum_{tt'} \bbox{s}^{tt'}(rz) \tau_{t't}^{m},$$ (31)

where

$$\bbox{s}^{tt'}(rz)= \sum_k [ \bbox{e}_r s_r^{tt'}(rz) + \bbox{e}_\phi s_\phi^{tt'}(rz) + \bbox{e}_z s_z^{tt'}(rz) ]$$

$$= \sum_{k}\biggr\{ \bbox{e}_{r}[V^{-\ast}_{kt} V^{+}_{kt'} +V^{+\ast}_{kt} V^{-}_{kt'}]$$

$$+ \bbox{e}_{\phi}i[V^{+\ast}_{kt} V^{-}_{kt'}-V^{-\ast}_{kt} V^{+}_{kt'}]$$

$$+ \bbox{e}_{z}[ V^{+\ast}_{kt} V^{+}_{kt'} -V^{-\ast}_{kt} V^{-}_{kt'}]\biggr\}.$$ (32)

(d)

Pseudovector spin-kinetic density

$$\bbox{T}_m(\bbox{r}) = \biggr[(\bbox{\nabla} \cdot \bbox{\nabla} ')\bbox{s}_m(\bbox{r}, \bbox{r}') \biggr]_{ \bbox{r} = \bbox{r}'}$$

$$= \sum_{tt'}\bbox{T}^{tt'}(rz) \tau_{t't}^m,$$ (33)

where

$$\bbox{T}^{tt'}(rz) = \sum_{k}\biggr\{ \bbox{e}_{r}\biggr( [\partial_{r}V^{+\ast}_{kt}]... ...al_{r} V^{-}_{kt'}]+\frac{\Lambda^{+}\Lambda^{-}}{r^2}V^{+\ast}_{kt}V^{-}_{kt'}$$

$$+[\partial_{z} V^{+\ast}_{kt}][ \partial_{z} V^{-}_{kt'}]+[\partial_{r} V^{-\ast}_{kt}][ \partial_{r} V^{+}_{kt'}]$$

$$+ \frac{\Lambda^{-}\Lambda^{+}}{r^2}V^{-\ast}_{kt}V^{+}_{kt'}+[\partial_{z} V^{-\ast}_{kt}][ \partial_{z} V^{+}_{kt'}]\biggr)$$

$$+i\bbox{e}_{\phi}\biggr( [\partial_{r} V^{+\ast}_{kt}][\partial_{r} V^{-}_{kt'}]+\frac{\Lambda^{-}\Lambda^{+}}{r^2}V^{+\ast}_{kt}V^{-}_{kt'}$$

$$+[\partial_{z} V^{+\ast}_{kt}][ \partial_{z} V^{-}_{kt'}]-[\partial_{r} V^{-\ast}_{kt}][ \partial_{r} V^{+}_{kt'}]$$

$$- \frac{\Lambda^{-}\Lambda^{+}}{r^2}V^{-\ast}_{kt}V^{+}_{kt'} -[\partial_{z} V^{-\ast}_{kt}][ \partial_{z} V^{+}_{kt'}]\biggr)$$

$$+\bbox{e}_z\biggr([\partial_{r} V^{+\ast}_{kt}][\partial_{r} V^{+}_{kt'}]+\frac{\Lambda^{-2}}{r^2}V^{+\ast}_{kt}V^{+}_{kt'}$$

$$+[\partial_{z} V^{+\ast}_{kt}][\partial_{z} V^{+}_{kt}]-[\partial_{r} V^{-\ast}_{kt}][ \partial_{r} V^{-}_{kt'}]$$

$$- \frac{\Lambda^{+2}}{r^2}V^{-\ast}_{kt}V^{-}_{kt'}-[\partial_{z} V^{-\ast}_{kt}][ \partial_{z}V^{-}_{kt'}]\biggr) \biggr\}.$$ (34)

(e)

Vector current density

$$\bbox{j}_{m}(\bbox{r}) = \frac{1}{2i}\biggr[(\bbox{\nabla}-\bbox{\nabla}')\rho_{m}(\bbox{r}, \bbox{r}')\biggr]_{\bbox{r}=\bbox{r}'}$$

$$= \sum_{tt'}\bbox{j}^{tt'}(rz) \tau_{t't}^m,$$ (35)

where

$$\bbox{j}^{tt'}(rz) = \frac{1}{2i}\sum_{k}\biggr\{ \bbox{e}_r\biggr([\partial_{r} V^{+\ast}_{kt}]V^{+}_{kt'}+[\partial_{r} V^{-\ast}_{kt}] V^{-}_{kt'}$$

$$- V^{+\ast}_{kt}[\partial_{r}V^{+}_{kt'}]-V^{-\ast}_{kt} [\partial_{r}V^{-}_{kt'}]\biggr)$$

$$-2i\bbox{e}_{\phi}\biggr( \frac{\Lambda^{-}}{r}V^{+\ast}_{kt}V^{+}_{kt'}+ \frac{\Lambda^{+}}{r}V^{-\ast}_{kt} V^{-}_{kt'}\biggr)$$

$$+\bbox{e}_z\biggr( [\partial_{z}V^{+\ast}_{kt}]V^{+}_{kt'}+[\partial_{z} V^{-\ast}_{kt}] V^{-}_{kt'}$$

$$- V^{+\ast}_{kt}[\partial_{z} V^{+}_{kt'}]-V^{-\ast}_{kt} [\partial_{z} V^{-}_{kt'}]\biggr) \biggr\}.$$ (36)

(f)

Tensor spin current density

$${\sf J}_{m}^{ij}=\frac{1}{2i}(\nabla_i-\nabla'_i)s_m^j(\bbox{r}, ... ...gr\vert _{\bbox{r}=\bbox{r}'} =\sum_{tt'}{\sf J}^{tt'}_{ij}(rz) \tau^m_{t't} .$$ (37)

Explicit expressions of the components are

$${\sf J}_{r\phi}^{tt'}(rz) = \frac{1}{2}\sum_{k}\biggr\{[\partial_{r} V^{+\ast}_{kt}]V^{-}_{kt'}-V^{+\ast}_{kt}[\partial_{r} V^{-}_{kt'}]$$

$$-[\partial_{r} V^{-\ast}_{kt}]V^{+}_{kt'}+V^{-\ast}_{kt}[\partial_{r} V^{+}_{kt'}]\biggr\},$$ (38)

$${\sf J}_{rz}^{tt'}(rz) = \frac{1}{2i}\sum_{k}\biggr\{[\partial_{r} V^{+\ast}_{kt}]V^{+}_{kt'}-V^{+\ast}_{kt}[\partial_{r} V^{+}_{kt'}]$$

$$-[\partial_{r} V^{-\ast}_{kt}]V^{-}_{kt'}+V^{-\ast}_{kt}[\partial_{r} V^{-}_{kt'}]\biggr\},$$ (39)

$${\sf J}_{\phi z}^{tt'}(rz) = -\sum_{k}\biggr\{ \frac{\Lambda^{-}}{r}V^{+\ast}_{kt}V^{+}_{kt'}-\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}V^{-}_{kt'}\biggr\},$$ (40)

$${\sf J}_{z\phi}^{tt'}(rz) = \frac{1}{2}\sum_{k}\biggr\{[\partial_{z} V^{+\ast}_{kt}]V^{-}_{kt'}-V^{+\ast}_{kt}[\partial_{z} V^{-}_{kt'}]$$

$$-[\partial_{z} V^{-\ast}_{kt}]V^{+}_{kt'}+V^{-\ast}_{kt}[\partial_{z} V^{+}_{kt'}]\biggr\},$$ (41)

$${\sf J}_{zr}^{tt'}(rz) = \frac{1}{2i}\sum_{k}\biggr\{[\partial_{z} V^{+\ast}_{kt}]V^{-}_{kt'}-V^{+\ast}_{kt}[\partial_{z} V^{-}_{kt'}]$$

$$+[\partial_{z} V^{-\ast}_{kt}]V^{+}_{kt'}-V^{-\ast}_{kt}[ \partial_{z}V^{+}_{kt'}]\biggr\},$$ (42)

$${\sf J}_{\phi r}^{tt'}(rz)= -\frac{1}{2}\sum_{k}\biggr\{ \frac{\Lambda^{-}}{r}V^{+\ast}_{kt}V^{-}_{kt'}+\frac{\Lambda^{-}}{r}V^{-\ast}_{kt}V^{+}_{kt'}$$

$$+\frac{\Lambda^{+}}{r}V^{+\ast}_{kt}V^{-}_{kt'}+\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}V^{+}_{kt'}\biggr\},$$ (43)

$${\sf J}_{zz}^{tt'}(rz) = \frac{1}{2i}\sum_{k}\biggr\{[\partial_{z} V^{+\ast}_{kt}]V^{+}_{kt'}-V^{+\ast}_{kt}[ \partial_{z}V^{+}_{kt'}]$$

$$-[\partial_{z} V^{-\ast}_{kt}]V^{-}_{kt'}+V^{-\ast}_{kt}[\partial_{z} V^{-}_{kt'}]\biggr\},$$ (44)

$${\sf J}_{\phi \phi}^{tt'}(rz)= -\frac{1}{2i}\sum_{k}\biggr\{ \frac{\Lambda^{-}}{r}V^{-\ast}_{kt}V^{+}_{kt'}+\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}V^{+}_{kt'}$$

$$-\frac{\Lambda^{+}}{r}V^{+\ast}_{kt}V^{-}_{kt'}-\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}V^{-}_{kt'}\biggr\},$$ (45)

$${\sf J}_{rr}^{tt'}(rz) = \frac{1}{2i}\sum_{k}\biggr\{[\partial_{r} V^{+\ast}_{kt}]V^{-}_{kt'}-V^{+\ast}_{kt}[\partial_{r} V^{-}_{kt'}]$$

$$+[\partial_{r} V^{-\ast}_{kt}]V^{+}_{kt'}-V^{-\ast}_{kt}[\partial_{r} V^{+}_{kt'}]\biggr\}.$$ (46)

The trace, antisymmetric, and symmetric parts of the tensor spin-current density are given by

$$J_k(\bbox{r}) = \sum_{a=x,y,z} {\sf J}_{kaa}(\bbox{r}),$$ (47)

$$\bbox{J}_{ka}(\bbox{r}) = \sum_{b,c=x,y,z} \bbox{\epsilon}_{abc} {\sf J}_{kbc}(\bbox{r}),$$ (48)

$$\underline{\sf J}_{kab}(\bbox{r}) = \frac{1}{2}{\sf J}_{kab}(\bbox{r}) + \frac{1}{2}{\sf J}_{kba}(\bbox{r}) - \frac{1}{3} J_k(\bbox{r}) \delta_{ab}.$$ (49)