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.

Ground-state phase transitions in odd nuclei are examined. Especially, the phase transitions in odd-odd nuclei are revealed. It is found that odd-even effects may reach their maximal or minimal values around the critical point at $N=90$; the signals of phase transition in odd nuclei are greatly enhanced relative to those in the adjacent even-even species. Moreover, a pairing model analysis of the phase transition in Sm isotopes indicates that the critical behaviors related to pairing may be driven by the decrease in the pair strength $G$ with a sudden flattening in $$\Delta${}G$ around the critical point.

We propose an algebraic approach to elucidate the dynamic characteristics of triaxial rotor modes in nuclei by mapping a triaxial rotor Hamiltonian to the interacting boson model one within a finite-$N$ framework. Our method unveils striking features not observed in conventional modes, exemplified by the $B\left(E2\right)$ anomaly, characterized by $B(E2;4_1^{+}\to 2_1^{+})/B(E2;2_1^{+}\to 0_1^{+})<1$. Using specific examples, we demonstrate that the peculiar properties of low-lying states in both neutron-rich and neutron-deficient Os nuclei can be comprehensively understood through the proposed Hamiltonian, which incorporates both rigid and soft triaxial rotor modes. This algebraic method not only offers fresh insights into triaxial dynamics but also showcases its capability in uncovering emergent exotic collective modes in nuclear structure.

The spectral fluctuations and transition intensity fluctuations in the excited-state quantum phase transitions (ESQPTs) have been investigated within the framework of the interacting boson model (IBM) by adopting three statistical measures, including the nearest neighbor level spacing distribution P ( S ) measuring the chaoticity (regularity) in energy spectra, the Δ ₃ ( L ) statistics of Dyson and Mehta measuring the spectral rigidity and the intensity distribution P ( y ) measuring the chaoticity (regularity) in B ( E 0) transitions. The results indicate that that the ESQPT as a function of the excitation energy may occur as a transition from regular (or semiregular) to highly chaotic if only the associated whole spectrum is chaotic, which fits most of the deformed situations in the IBM including those in the U(5)–SU(3) and SU(3)–O(6) transitional regions. Otherwise, the ESQPT will appear as a transition from regular (or semiregular) to regular such as the cases in the U(5)–O(6) transitional region or those on the ‘Alhassid–Whelan arc’, which represents a nearly regular parameter region connecting the U(5) and SU(3) limits in the IBM.