Patterns of shape-phase transition in the proton-neutron coupled systems are studied within the $\mathit{SD}$-pair shell model. The results show that some transitional patterns in the $\mathit{SD}$-pair shell model are similar to the U(5)-SU(3) and U(5)-SO(6) transitions with signatures of the critical point symmetry of the interacting boson model.

The exact eigenenergies of the $T_{4c}=[cc][\overline{c}\overline{c}]$, $T_{4b}=[bb][\overline{b}\overline{b}]$, and $T_{2[bc]}=[bc][\overline{b}\overline{c}]$ tetraquarks are calculated within the extended transitional Hamiltonian approach, in which the so-called Bethe ansatz within an infinite-dimensional Lie algebra is used. We fit the parameters appearing in the transitional region from phenomenology associated with potential candidates of tetraquarks. The rotation and vibration transitional theory seems to provide a better description of heavy tetraquarks than other attempts within the same formalism. Our results indicate that the pairing strengths are large enough to provide binding; an extended comparison with the current literature is also performed.

The nucleon pair shell model (NPSM) is cast into the so-called $M$ scheme for the cases with isospin symmetry and without isospin symmetry. The odd system and even system are treated on the same footing. The uncoupled commutators for nucleon pairs, which are suitable for the $M$ scheme, are given. Explicit formula of matrix elements in $M$ scheme for overlap, one-body operators, and two-body operators are obtained. It is found that the CPU time used in calculating the matrix elements in $M$ scheme is much shorter than that in the $J$ scheme of NPSM.

The $\mathit{SD}$-pair shell model is employed to study ${}^{126}\mathrm{Xe}$ and ${}^{128}\mathrm{Ba}$. The results show that the spectra and electromagnetic transition strengths can be nicely described in terms of a three-parameter Hamiltonian. In our previous paper, we got a conclusion that the $\mathit{SD}$-pair approximation improves with the number of $\mathit{SD}$ pairs $N$. This work shows that this conclusion can be extrapolated to the case with $N=5$.

Patterns of the vibrational to rotational phase transition in the SD-pair shell model are studied. The results show that the U(5)-SU(3) transitional patterns of the interacting boson model can be reproduced well in the SD-pair shell model.

The SD-pair shell model (SDPSM) is shown to reproduce approximately typical spectra, $E2$ transition strengths, and wave functions of the $U(5)$, $\mathrm{SO}(6)$, and $\mathrm{SU}(3)$ limits of the interacting boson model (IBM). Consequently, the analysis confirms that the IBM has a sound shell-model foundation; it also demonstrates that the truncation scheme adopted in the SDPSM is reasonable.