**3. Comparison to Experimental Data**

The main objective of this paper is to examine how well the shell evolution described by Equation (7) is supported by experimental data. In Figure 2a,b, the proton shell evolution with neutrons occupying the *p f* shell, and the neutron shell evolution with protons occupying the *sd* shell are plotted, respectively. The former and the latter are examined in Sections 3.1 and 3.2, respectively. In the following, for brevity, the quantum number *n* are omitted and only the other quantum numbers like *d*5/2 are given.

**Figure 2.** Evolution of the ESPEs calculated with the SDPF-MU interaction with the tensor force included (solid lines) and not included (dashed lines). (**a**) Proton orbitals measured from 1*d*3/2 for the atomic number *Z* = 20 isotopes and (**b**) neutron orbitals measured from 1 *f*7/2 for the neutron number *N* = 28 isotones. The ESPEs are obtained by assuming filling configurations whose orders are indicated at the bottom of the figure.

In Figure 2, also the ESPE with the tensor force removed is plotted. One can immediately find that the proton *d*5/2 orbital (Figure 2a) and the neutron *f*5/2 orbital (Figure 2b) have the largest effect from the tensor force. Since the ESPEs shown are measured from the *d*3/2 and *f*7/2 orbitals, respectively, this result is a manifestation of a general property that the tensor force strongly affects the spin–orbit splitting (see Figure 1a of Ref. [12]).

To be more specific, when the proton orbital *j* is filled, the evolution of the neutron spin–orbit splitting between *j*<sup>&</sup>lt; and *j*<sup>&</sup>gt; is expressed, by using Equation (7), as Δ*πj*(*εν<sup>j</sup>*<sup>&</sup>lt; − *εν<sup>j</sup>*<sup>&</sup>gt; )=(2*j* + <sup>1</sup>){*V*<sup>m</sup> *pn*(*j*<, *j* ) − *<sup>V</sup>*<sup>m</sup> *pn*(*j*>, *j* )}. The *<sup>V</sup>*<sup>m</sup> *pn*(*j*<, *j* ) and *V*<sup>m</sup> *pn*(*j*>, *j* ) values for the tensor force are always of the opposite sign due to the identity (2*j*<sup>&</sup>gt; + 1)*V*<sup>m</sup> *<sup>T</sup>* (*j*>, *j* ) + (2*j*<sup>&</sup>lt; + 1)*V*<sup>m</sup> *<sup>T</sup>* (*j*<, *j* ) = 0 (valid for any isospin coupling *T*) [12], thus, magnifying the Δ*πj*(*εν<sup>j</sup>*<sup>&</sup>lt; − *εν<sup>j</sup>*<sup>&</sup>gt; ) value.

In addition to evaluating the ESPE, we conducted large-scale shell-model calculations to more directly compare to the data. The procedure of the calculation was the same as that employed earlier [18,25]. The valence shell consists of the full *sd* and *p f* shells. The basis states considered are truncated to allow only 0*h*¯ *ω* (with *h*¯ being the reduced Planck constant and *ω* the angular frequency) excitations for natural-parity states and to allow 1*h*¯ *ω* excitations for unnatural-parity states. Note that, in the present case, *nh*¯ *ω* excitation is equivalent to *n*-particle-*n*-hole excitation across the *N* = *Z* = 20 shell gap.

Let us stress that this truncation scheme (restricted to the lowest *h*¯ *ω* space) is introduced not only to make numerical computation possible but also to be in accordance with the way how the SDPF-MU interaction is constructed (see also Section 2.2 of Ref. [21]): (i) the central force of the cross-shell interaction in SDPF-MU is fitted to the GXPF1B interaction and (ii) the intra-shell interactions employed in the SDPF-MU interaction are based on USD for the *sd* shell and GXPF1B for the *p f* shell. The USD and the GXPF1B interactions are intended for the use of the 0*h*¯ *ω* model space. As shown in Ref. [18], the binding energies of neutron-rich nuclei in this region are well reproduced in this framework. The Hamiltonian matrices spanned by those basis states are numerically diagonalized by using the KSHELL code [26].
