Central-like form of the pseudopotential

The Skyrme interaction is one of the most important phenomenological effective interaction used in microscopic nuclear structure calculations: such two-body interaction is a short-range expansion up to the second order in derivatives, which contains a certain number of fit parameters adjusted to reproduce the experimental data. In the literature the Skyrme interaction is usually written in cartesian representation, but for our extended pseudopotential we adopt the spherical-tensor representation of operators [25], whose building blocks can be found in [2].

Depending on the specific form of the coupling of the derivative operators with the spin operators, different ways to construct the pseudopotential are possible. A particular form of the pseudopotential, which we call central-like or LS-like, is constructed in the present Section. It is based on coupling together the derivative operators and spin operators, which are then coupled to rotational scalars. An alternative form, called tensor-like or JJ-like, is presented in Section 2.4. There, each derivative operator is coupled with one spin operator, and then they are coupled together to rotational scalars.

In the central-like form, the pseudopotential is a sum of terms,

$$\hat{V}=\sum_{\tilde{n}' \tilde{L}', \tilde{n} \tilde{L},v_... ...at{V}_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'} ,$$ (1)

where the sum runs over the allowed indices of the tensors according to the symmetries discussed below. Each term in the sum is accompanied by the corresponding strength parameter $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}$, and explicitly reads,

$$\hat{V}_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= \frac{1}{2}i^{v_{12}} \left( \left[ \left[K'_{\tilde{n}'\tilde{L}'} K_ {\tilde{n }\tilde{L}}\right]_{S} \hat{S}_{v_{12} S}\right]_{0} \right.$$

$$\left. + (-1)^{v_{12}+S} \left[ \left[K'_{\tilde{n} \tilde{L}} K_ {\tilde{n}'\tilde{L}'}\right]_{S} \hat{S}_{v_{12} S}\right]_{0} \right)$$

$$\times \left(1-\hat{P}^{M}\hat{P}^{\sigma}\hat{P}^{\tau}\right) \hat{\delta}_{12}(\bm{r}'_1\bm{r}'_2;\bm{r}_1\bm{r}_2) .$$ (2)

In Eq. (2), $K_ {\tilde{n }\tilde{L}}$ are the spherical tensor derivatives of order $\tilde{n}$ and rank $\tilde{L}$ built of the spherical representations of the relative momenta $\bm{k}=(\bm{\nabla}_1-\bm{\nabla}_2)/2i$,

$$k_{1,\mu=\left\{-1,0,1\right\}} = -i \left\{{\textstyle{\frac{ 1}{\sqrt{2}}}}\left(k_x-ik_y\right), \right.$$

$$\left. k_z, {\textstyle{\frac{-1}{\sqrt{2}}}}\left(k_x+ik_y\right)\right\} ;$$ (3)

up to sixth order they are listed in Table 1. Similarly, operators $K'_{\tilde{n }\tilde{L}}$ are built of the relative momenta $\bm{k}'=(\bm{\nabla}'_1-\bm{\nabla}'_2)/2i$.

Table 1: Derivative operators $K_{nL}$ up to N$^3$LO as expressed through spherical tensor representation of relative momenta $k$ defined in Eq. (3).

No.	tensor $K_{nL}$	order $n$	rank $L$
1	$1$	0	0
2	${k}$	1	1
3	${[}{k}{k}{]}_{0}$	2	0
4	${[}{k}{k}{]}_{2}$	2	2
5	${[}{k}{k}{]}_{0} {k}$	3	1
6	${[}{k}{[}{k}{k}{]}_{2}{]}_{3}$	3	3
7	${[}{k}{k}{]}_{0} ^{2}$	4	0
8	${[}{k}{k}{]}_{0} {[}{k}{k}{]}_{2}$	4	2
9	${[}{k}{[}{k}{[}{k}{k}{]}_{2}{]}_{3}{]}_{4}$	4	4
10	${[}{k}{k}{]}_{0}^2 {k}$	5	1
11	${[}{k}{k}{]}_{0} {[}{k}{[}{k}{k}{]}_{2}{]}_{3}$	5	3
12	${[}{k}{[}{k}{[}{k}{[}{k}{k}{]}_{2}{]}_{3}{]}_{4}{]}_{5}$	5	5
13	${[}{k}{k}{]}_{0} ^3$	6	0
14	${[}{k}{k}{]}_{0}^2 {[}{k}{k}{]}_{2}$	6	2
15	${[}{k}{k}{]}_{0} {[}{k}{[}{k}{[}{k}{k}{]}_{2}{]}_{3}{]}_{4}$	6	4
16	${[}{k}{[}{k}{[}{k}{[}{k}{[}{k}{k}{]}_{2}{]}_{3}{]}_{4}{]}_{5}{]}_{6}$	6	6

The symmetrized two-body spin operators $\hat{S}_{v_{12} S}$ are defined as,

$$\hat{S}_{v_{12} S} =\left(1-{\textstyle{\frac{1}{2}}}\delta_{v_1,... ..._{v_1}\sigma^{(2)}_{v_2}]_S + [\sigma^{(1)}_{v_2}\sigma^{(2)}_{v_1}]_S \right),$$ (4)

where $v_{12}=v_1+v_2$ and $\sigma^{(i)}_{v\mu}$ are the spherical-tensor components of the rank-$v$ Pauli matrices acting on spin coordinates of particles $i=1$ or 2. They are expressed as

$$\sigma^{(i)}_{00} = \hat{1} ,$$ (5)

$$\sigma^{(i)}_{ 1,\mu=\left\{-1,0,1\right\}} = -i \left\{{\textstyle{\frac{ 1}{\sqrt{2}}}}\left(\sigma^{(i)}_{ x} -i\sigma^{(i)}_{ y}\right), \right.$$

$$\left. \sigma^{(i)}_{ z}, {\textstyle{\frac{-1}{\sqrt{2}}}}\left(\sigma^{(i)}_{ x} +i\sigma^{(i)}_{ y}\right)\right\}$$ (6)

through the spin unity matrix $\hat{1}$ and the standard Cartesian components of the Pauli matrices $\sigma^{(i)}_{ x,y,z}$.

The Dirac delta function,

$$\hspace*{-1.5em} \hat{\delta}_{12}(\bbox{r}'_1\bbox{r}'_2,\bbox{r}_1\bbox{r}_2) = \delta(\bbox{r}'_1\!-\!\bbox{r}_1) \delta(\bbox{r}'_2\!-\!\bbox{r}_2) \delta(\bbox{r} _1\!-\!\bbox{r}_2)$$

$$= \delta(\bbox{r}'_1\!-\!\bbox{r}_2) \delta(\bbox{r}'_2\!-\!\bbox{r}_1) \delta(\bbox{r} _2\!-\!\bbox{r}_1) .$$ (7)

ensures the locality and zero-range character of the pseudopotential. The action of derivatives $K_ {\tilde{n }\tilde{L}}$ and $K'_{\tilde{n }\tilde{L}}$ on $\hat{\delta}_{12}(\bbox{r}'_1\bbox{r}'_2,\bbox{r}_1\bbox{r}_2)$ has to be understood in the standard sense of derivatives of distributions. Whenever the pseudopotential (1) is inserted into integrals to calculate the two-body matrix elements, the integration by parts transfers the derivatives onto appropriate wave functions in the remaining parts of integrands.

The exchange term is explicitly embedded in the pseudopotential through the operator

$$\hat{P}^{M}\hat{P}^{\sigma}\hat{P}^{\tau} = (-1)^{\tilde{n}'}\frac{1}{4}\bigg(1+\sqrt{3}\left[\sigma^{(1)}_1\sigma^{(2)}_1 \right]_0$$

$$\hspace*{-1.4cm} + \sqrt{3}\left[\tau^{(1)}_1\tau^{(2)}_1 \right]... ...1)}_1\sigma^{(2)}_1 \right]_0 \left[\tau^{(1)}_1\tau^{(2)}_1 \right]^0 \bigg) ,$$ (8)

where $\tau^{(i)}_1$ are the standard spherical-tensor isospin Pauli matrices defined analogously as in Eq. (6). The square brackets with superscripts and subscripts denote the coupling of spherical tensors in the isospin space and coordinate space, respectively. The above definitions and conventions exactly correspond to those introduced in Ref. [2].

The zero range of the pseudopotential has an important bearing on the structure of terms in Eq. (2). Indeed, only for the zero-range force, the space-exchange (Majorana) operator $\hat{P}^{M}$ can be replaced, in any individual term, by the phase $(-1)^{\tilde{n}'}$ appearing in Eq. (8). Moreover, apart from the isospin-exchange operator $\hat{P}^{\tau}$, terms of the pseudopotential cannot then depend on isospin. This fact, effectively reduces by half the number of allowed terms of the pseudopotential, as compared to what would have been possible for a finite-range potential. This is at the origin of the numbers of allowed terms of the pseudopotential being equal one half of the numbers of the allowed terms of the EDF, which we discuss below.

The full antisymmetrization of the pseudopotential includes the exchange operator in the isospin space; therefore, in the following we consider the EDF with the isospin degree of freedom included, that is, we discuss both the isoscalar and isovector terms of the N$^3$LO [2], which allows us to fully incorporate the proton-neutron mixing at the level of the energy density [19].

The general form of the pseudopotential and the allowed terms listed below reflect the fact that the fundamental symmetries of the two-body interaction must be respected, see Appendix A. In particular, (i) all terms are scalar operators, that is, they are coupled to the total angular momentum 0, which ensures the rotational invariance, (ii) the total number of derivative operators must be even, namely, $\tilde{n}+\tilde{n}'=0,2,4,6$, which ensures the time-reversal and parity invariances, (iii) the parameters $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}$ of the pseudopotential must be real, to guarantee both the time-reversal invariance and hermiticity, and (iv) the invariance under exchange of the coordinates of particle 1 and 2 is respected by expression (2).