**2. Shell Evolution Caused by the SDPF-MU Interaction**

#### *2.1. Monopole Matrix Elements*

The SDPF-MU interaction was constructed in Ref. [18] to describe the structure of neutron-rich nuclei around *N* = 28 whose Fermi surface is located in the *sd* shell for protons and the *p f* shell for neutrons. Hence, the proton–neutron cross-shell interaction, i.e., the part of the interaction that is relevant to both the sd shell and the pf shell is responsible for the shell evolution occurring in this region.

The cross-shell part of the SDPF-MU interaction is provided by a minor modification of the *V*MU interaction [13]. The *V*MU interaction was proposed to give a universal behavior of shell evolution over the nuclear chart, consisting of a Gaussian central force and a *π* + *ρ* meson exchange tensor force. In the SDPF-MU interaction, the following refinements to the original *V*MU interaction are introduced:


The central force of the shell-model effective interaction is subject to complicated renormalization and many-body effects. The basic strategy of *V*MU is to determine the central force so that its monopole matrix elements are close to those of a reliable effective interaction. Here, the monopole matrix element between the orbitals *j*<sup>1</sup> and *j*<sup>2</sup> is defined by

$$V\_{\mathcal{T}}^{\mathfrak{m}}(j\_1, j\_2) = \frac{\sum\_{\mathcal{I}} (2\mathcal{I} + 1) \left< j\_1 j\_2; \mathcal{I} \middle| \begin{array}{c} V \middle| j\_1 j\_2; \mathcal{I} \middle| \mathcal{T} \end{array} \right>}{\sum\_{\mathcal{I}} (2\mathcal{I} + 1)},\tag{1}$$

where *J* runs over all the possible angular-momentum coupling that the Pauli principle allows, and *T* is the isospin coupling. In constructing the original *V*MU interaction, the GXPF1A interaction [19] was used as a reference, and a reasonable but not perfect agreement was achieved. Namely, while most of the monopole matrix elements agree within 0.2 MeV, a few matrix elements differ by 0.5 MeV or more; see Figure 1 of Ref. [13]. To obtain a better result, the central force of the SDPF-MU interaction has the form of

$$N\_{\mathbb{C}}(1,2) = D(R) \sum\_{\mathbb{S},T} f\_{\mathbb{S},T} P\_{\mathbb{S},T} \exp\left(-\left(r/\mu\right)^2\right),\tag{2}$$

where *S* and *T* denote the spin and isospin coupling, respectively, and *PS*,*<sup>T</sup>* is the projection operator onto a given (*S*, *T*). The*r* and *R* are the relative and center-of-mass coordinates, respectively: *<sup>r</sup>* <sup>=</sup> *r*<sup>1</sup> <sup>−</sup> *r*<sup>2</sup> and *R* = (*r*<sup>1</sup> + *r*2)/2. The *D*(*R*) is the density dependent part that was newly introduced in the refined *V*MU, and its form was taken from the FPD6 interaction [20] as

$$D(R) = 1 + A\_d \{ 1 + \exp(\left( R - R\_0 \right) / a) \}^{-1} \tag{3}$$

with *R*<sup>0</sup> = 1.2*A*−1/3 MeV and *a* = 0.6 fm. The interaction, thus, has six free parameters, *fS*,*T*, *μ*, and *Ad*. They were chosen to be *f*0,0 = −140 MeV, *f*1,0 = 0, *f*0,1 = 0.6 *f*0,0, *f*1,1 = −0.6 *f*0,0, *μ* = 1.2 fm, and *Ad* = −0.4. The resulting agreement with the monopole matrix elements of the central force of GXPF1B is quite good, as illustrated in Figure 1 of Ref. [21].

The two-body spin–orbit force in the SDPF-MU interaction was taken from that of the M3Y (Michigan 3-range Yukawa) interaction [22]. The two-body spin-orbit force plays a minor role on shell evolution compared with the central and tensor forces, as far as a restricted region of the nuclear chart is considered: see Table 1 and discussion below. However, some specific evolutions of shell gaps are dominated by the two-body spin–orbit force, thus, included here for completeness.

The overall strength of the SDPF-MU interaction is scaled by a factor *A*−0.3 in the same way as the USD (Universal *sd*) [23] and GXPF1 [24] interactions.

**Table 1.** Proton–neutron monopole matrix elements between the *sd* and *p f* orbitals obtained by the SDPF-MU interaction for the atomic mass number *A* = 42. The second to the fifth columns list the central (C), tensor (T), spin–orbit (LS), and the total values (in MeV), respectively. The sixth to ninth columns indicate the hierarchy of theC+T monopole matrix elements. The texts in red (blue) are to highlight the correspondence between the most attractive matrix elements of the central (tensor) force and Δ*n* = 0 (spin direction). See text for details.


Table 1 presents the proton–neutron cross-shell monopole matrix elements, calculated with the SDPF-MU interaction, for central, tensor, and spin–orbit forces. The proton– neutron monopole matrix element for a pair with (*n*1, *l*1) = (*n*2, *l*2) is given by

$$V\_{pn}^{\rm m}(j\_1, j\_2) = \frac{1}{2} \{ V\_{T=0}^{\rm m}(j\_1, j\_2) + V\_{T=1}^{\rm m}(j\_1, j\_2) \}. \tag{4}$$

The second column of Table 1 indicates that the strengths of the central matrix elements can be grouped into two categories: one has ∼ −1.1 MeV, and the other has much weaker strengths. As explained [1,13], this difference occurs because two orbitals with the difference of the number of nodes, Δ*n* = 0, have a large spatial overlap, thus, gaining much attraction through short-range forces. Comparing the second and the sixth columns, one finds a good correspondence between Δ*n* and the strength of the central matrix elements.

The monopole matrix elements of the tensor force are characterized by the relative spin direction between the two orbitals considered, as pointed out in Ref. [12]. When the spins of two orbitals (with *l* > 0) are parallel, i.e., *j*> - *j* <sup>&</sup>gt; or *j*<sup>&</sup>lt; - *j* <sup>&</sup>lt; (*j*<sup>&</sup>gt; and *j*<sup>&</sup>lt; stand for *j* = *l* + 1/2 and *j* = *l* − 1/2, respectively), the tensor monopole matrix element is positive and otherwise negative. The third and the seventh columns of Table 1 exactly point to this property. This fact is accepted now, [1,12], and quantitative aspects of the tensor monopole matrix elements are as follows.


From the point 2, one concludes that the tensor force plays a role as important as the central force in shell evolution.

On the basis of the above arguments, let us label the orbital pairs to simply estimate the strengths of the monopole matrix elements due to the central and tensor forces without numerical calculations.


These labels are listed in the eighth column of Table 1. The actual sum of the central and tensor monopole matrix elements shown in the ninth column of Table 1 rather well follows this ordering, except for a few cases with Δ*n* = 1 in which the tensor force is less dominant.

Next, the two-body spin–orbit force is examined whose monopole matrix elements are presented in the fourth column of Table 1. The strengths of the elements are usually rather weak (see details in Supplemental Material in Ref. [1]), and the typical order of the monopole matrix elements is ∼ <sup>20</sup>*A*−5/3 MeV ≈ 0.04 MeV at *<sup>A</sup>* = 42. The signs of the elements are determined so that the inner nucleon (usually with lower orbital angular momentum, *l*) produces the normal spin–orbit splitting to the outer orbitals. Namely, when the inner and the outer orbitals are labeled *i* and *j*, respectively, their monopole matrix elements satisfy *V*<sup>m</sup> *pn*(*i*, *j*) < 0 for *j* = *l* + 1/2 and *V*<sup>m</sup> *pn*(*i*, *j*) > 0 for *j* = *l* − 1/2.

More specifically, when monopole matrix elements between the *sd* and 1 *f* orbitals are considered, the *sd* orbitals are located closer to the center and thus can be regarded as the inner orbitals. Hence, this rule causes negative and positive monopole matrix elements for the 1 *f*7/2 and 1 *f*5/2 orbitals, respectively. One can also find that the monopole matrix elements between the 2*s* and 2*p* orbitals are much larger than the others. This is because this pair, having a relative orbital angular momentum *L*rel = 1 alone, gains much energy due to the short-range nature of the two-body spin–orbit force.
