*5.1. Traditional View*

Figure 16a shows the left-lower part of the nuclear chart (Segrè chart) for *Z* ≤ 16. The black squares represent stable nuclei while the orange ones exotic nuclei (see Section 1). An isotopic chain is a horizontal belt, and its neutron-rich end is called neutron dripline. The location of the dripline in the nuclear chart implies the extent of the isotopes and is of fundamental importance to nuclear science. The experimental determination of the dripline is a very difficult task. Very recently, as shown by red squares in Figure 16a, the driplines of F and Ne isotopes and its candidate of Na isotope were reported [94].

The traditional view of the dripline is shown in Figure 16b: all bound single-particle orbits are occupied, and the next neutron goes away. It is an open question whether this view is valid for all nuclei or not. We look into this question now [57].

The structure of neutron-rich exotic isotopes of F, Ne, Na, and Mg can be well described by the shell–model calculation with the full *sd*+*p f* shells and the EEdf1 interaction [95]. This interaction was derived from the chiral EFT interaction of Machleidt and Entem [96], first processed by the Vlow-*<sup>k</sup>* method [97,98] and then processed by the EKK (Extended Krenciglowa-Kuo) method [99–101]. The Vlow-*<sup>k</sup>* method is used to transform the nuclear forces in the free space into a tractable form for further treatments. The Vlow-*<sup>k</sup>* method has been adopted for the derivation of other modern shell–model interactions, for instance, the one by Coraggio et al. for Sn and Cr-Fe regions [102,103].

**Figure 16.** (**a**) Left-lower part of the nuclear chart with stable (black square), exotic (orange) and (confirmed) unbound (blank) nuclei as well as dripline nuclei (red, and purple). (**b**) Schematic illustration of the traditional view of the dripline. Based on Figure 2 of [57].

The present work is unique in the usage of the EKK method, which enlarges the scope of the approaches based on the many-body perturbation theory (MBPT) [29]. The MBPT produced the G-matrix interactions in its early formulations [28], from which many useful shell–model interactions have been constructed (see Section 2.6). However, the resulting G-matrix interaction shows a limitation that if two major shells are merged, the results may diverge [101]. As the gap between two shells often vanishes or becomes smaller in exotic nuclei, this difficulty can be fatal there, although it is irrelevant to one-major-shell calculations. The EKK method nicely avoids this difficulty besides other merits.

Here, I present a very quick sketch of the formal aspect of the EKK method focusing on the logical flow based on Refs. [99–101] particularly the last one. This paragraph is not so relevant for understanding later parts of the article and can be skipped. In this paragraph, the symbol ˆ for operators is omitted for clarity. The EKK method starts from the separation of the Hamiltonian *H* with a parameter *ξ* as

$$H = \begin{pmatrix} \mathfrak{F} & 0\\ 0 & QH\_0Q \end{pmatrix} + \begin{pmatrix} P(H-\mathfrak{F})P & PVQ\\ QVP & QVQ \end{pmatrix} \tag{20}$$

where *P* stands for the projection onto the Hilbert space explicitly treated (called *P* space usually), and *Q* = 1 − *P*. From this equation, we obtain the effective Hamiltonian for the *P* space at the *n*-th stage of the successive process,

$$
\bar{H}\_{\rm eff}^{(n)} = \bar{H}\_{\rm BH}(\xi) + \sum\_{k=1}^{\infty} Q\_k(\xi) \{ \bar{H}\_{\rm eff}^{(n-1)} \}^k,\tag{21}
$$

where *<sup>O</sup>*˜ means *<sup>O</sup>* <sup>−</sup> *<sup>ξ</sup>* for any operator *<sup>O</sup>*, e.g., *<sup>H</sup>*˜ BH(*ξ*) = *<sup>H</sup>*BH(*ξ*) <sup>−</sup> *<sup>ξ</sup>*. Here, the Bloch– Horowitz Hamiltonian is written as,

$$H\_{\rm BH}(\xi) = PHP + PVQ \frac{1}{\overline{\xi} - QHQ} QVP \, \, \, \, \tag{22}$$

where the second term on the r.h.s. is called the *Q*-box. The quantity *Qk* in Equation (21) represents its *k*-th derivative with respect to *ξ*. Provided that *H*˜ (*n*) eff <sup>≈</sup> *<sup>H</sup>*˜ (*n*−1) eff is achieved, we can regard and use them as the effective Hamiltonian, *H*˜ eff. The effective interaction, like the EEdf1 interaction, is obtained as *V*eff = *H*eff − *PH*0*P* with *H*<sup>0</sup> being the unperturbed Hamiltonian (usually the SPEs). The solution of the given many-body problem remains (almost) unchanged within a certain range of *ξ*. In fact, the *ξ* parameter can be interpreted

as the origin point of a Tayler expansion in a generalized sense. The divergence due to the energy denominator does not occur if the adopted *ξ* values are far from the poles causing the divergence. I would like to stress that by construction, this effective Hamiltonian produces the exact solutions, once the convergence is achieved. This sketch is expected to depict that the EKK method is an expansion but not a perturbation one. This can be exemplified by the feature that the final result is independent of the *ξ* parameter, in contrast to the perturbation expansion.

The EEdf1 interaction has thus been derived in an ab initio way by the Vlow-*<sup>k</sup>* and EKK methods from the chiral EFT interaction of Machleidt and Entem [96]. Some effects of 3NF are included in terms of the effective *NN* interaction by averaging over the hole states in the inert core, of which the monopole part is discussed in Section 2.9. While the Fujita–Miyazawa 3NF was used so far, other 3NF can be taken [57]. The EEdf1 interaction describes the properties of the ground and low-lying states of F, Ne, Na, and Mg isotopes quite well [57,95].
