*4.3. A Historical Touch and a Short Summary of This Section*

The collective bands in heavy nuclei have traditionally been understood in terms of the ground band with axially symmetric prolate shape and the side bands with the *β* or *γ* vibrational excitations from the ground state. This picture is consistent with the Nilsson model [88] and was confirmed by the Pairing + Quadrupole-Quadrupole (P+QQ) model [89,90], where the monopole interaction is not included, however. It has been shown in this section that the monopole interaction is crucial also for the collective bands in heavy nuclei. We just note that in lighter nuclei, the situation can be different mainly because of small model spaces comprising single or a few active orbits, where the rotational motion has been nicely described by symmetry-based approaches, e.g., SU(3) model of Elliott for the *sd* shell [91,92], and by realistic calculations, e.g., on 48Cr [93].

Regarding heavy nuclei, for individual rotational bands, the monopole interaction contributes differently, and the intrinsic structure is determined not only by the quadrupole interaction but also by the monopole interaction, as verified by the monopole-frozen analyses. Thus, the monopole–quadrupole interplay arises. The monopole interaction does not directly drive the deformation but optimizes the ESPEs so that more binding energy is gained. This gain is state-dependent and even can alter the ordering of bands as mentioned above. The present monopole–quadrupole interplay can be described also from the viewpoint of the *self-organization* [71]: the nucleus is changed from a disorder (original SPEs) to an order (ESPEs tailored to the shape of interest) by activating the monopole interaction. As this occurs "purposely" towards certain shapes with positive feedback, particularly between the monopole and quadrupole effects, the whole picture fits well the (quantal) self-organization [71]. The self-organization for collective bands is among the emerging concepts of nuclear structure, showing novel consequences. For example, the dominant fraction of the ground states of heavy nuclei are expected to show triaxial shapes, as another emerging concept of nuclear structure, in contrast to the traditional view of the prolate shape dominance in those states.

Appendix B presents a possible extension or generalization of the current idea to "many-ingredient" systems outside nuclear physics.

#### **5. Dripline Mechanism**
