*3.4. T-Plot Analysis*

The *T-plot* was introduced in the same Ref. [58], in order to clarify what shapes are more relevant to individual eigenstates of the shell–model calculation. Let us take an example. Figure 12 [64] depicts the PES of 66Ni with the same Hamiltonian as in Figure 10. The small circles on the PES are the T-plot. The T-plot is obtained from MCSM eigenstate. We therefore briefly explain the MCSM eigenstate. An MCSM eigenstate, Ψ, is written, with the ortho-normalization, as

$$\Psi = \sum\_{k} f\_{k} \hat{\mathcal{P}}\_{l}^{n} \,\!\!/ \Psi\_{k} \,\!\!/ \,\tag{16}$$

where *fk* denotes amplitude; <sup>P</sup>ˆ*J<sup>π</sup>* means the projection operator on to the spin/parity *<sup>J</sup><sup>π</sup>* (this part is more complicated in practice); and *φ<sup>k</sup>* stands for a Slater determinant called (*k*-th) MCSM basis vector: *φ<sup>k</sup>* = Π*<sup>i</sup> c* (*k*)† *<sup>i</sup>* |0. Here, |0 is the inert core (closed shell); *c* (*k*)† *i* refers to a superposition of usual single-particle states,

$$c\_i^{(k)\dagger} = \sum\_n D\_{i,n}^{(k)} a\_n^{\dagger} \,. \tag{17}$$

with *a*† *<sup>n</sup>* being the creation operator of a usual single-particle state, for instance, that of the HO potential, and *D*(*k*) *<sup>i</sup>*,*<sup>n</sup>* denoting a matrix element. By choosing an optimum matrix *<sup>D</sup>*(*k*), we can select *φ<sup>k</sup>* so that such *φ<sup>k</sup>* better contributes to the lowering of the corresponding energy eigenvalue. Thus, the determination of *D*(*k*) is the core of the MCSM calculation. The index *k* runs up to 50–100 but sometimes to 300 at maximum. These are much smaller than the dimension of the many-body Hilbert space.

**Figure 12.** PES and T-plot for 66Ni. Taken from Figure 1 of [64].

Each *<sup>φ</sup><sup>k</sup>* has intrinsic quadrupole moments (*φk*|*Q*<sup>ˆ</sup> <sup>0</sup>|*φk* and *φk*|*Q*<sup>ˆ</sup> <sup>2</sup>|*φk*), where *<sup>Q</sup>*<sup>ˆ</sup> 0,2 imply the operators for *Q*0,2 mentioned above. The T-plot circle for *φ<sup>k</sup>* is placed according to those values on the PES with its area proportional to the overlap probability with the corresponding eigenstate, i.e., Ψ in Equation (16). Such T-plot circles are shown in Figure 12. The white circles represent the MCSM basis vectors for the ground state, while the red circles indicate the MCSM basis vectors for the 0<sup>+</sup> <sup>4</sup> state, which is strongly deformed. Although there is no local minimum for oblate shape, the 0<sup>+</sup> <sup>2</sup> state is shown to be moderately oblate deformed. The T-plot can thus give partial labeling to fully correlated eigenstates for mean values as well as fluctuations with respect to their quadrupole shapes. The advantages of mean-field approaches are now nicely incorporated into the shell model.

#### *3.5. Short Summary of This Section*

Type-II shell evolution occurs in various cases, especially in a number of shape coexistence cases, providing deformed states with stronger deformation, lower excitation energies, and more stabilities. It is an appearance of the monopole–quadrupole interplay and plays crucial roles in various phenomena including the first-order quantum phase transition (Zr isotopes [65–67]), the second-order quantum phase transition (Sn isotopes [68]), the multiple even-odd quantum phase transitions (Hg isotopes [69]), as well as the raising of the intruder band due to the suppression of the type-II shell evolution (lighter Ni isotopes [64,70]). As the involvement of the monopole interaction in this manner had not been recognized, type-II shell evolution appears to be among the emerging concepts of nuclear structure. The type-II shell evolution has been clarified by the T-plot in many cases. Including other contributions, the T-plot is undoubtedly one of the emerging concepts of nuclear structure, apart from its impact on the computational methodology.
