Unified description of low-lying states in rare-earth nuclei using the angular-momentum-projected number-conserved BCS method in a shell model framework

In this paper, we present a unified microscopic description of low-lying collective states in even-even rare-earth nuclei using the angular-momentum-projected number-conserved BCS (PNBCS) method within a shell-model space. Systematic calculations for seven isotopic chains from Xe to Dy naturally generate both vibrational and rotational band structures, reproduce level energies, $B(E2)$ transition strengths, gradual shape evolution, sharp shape-phase transitions, and produce results consistent with the empirical ${N}_{p}{N}_{n}$ systematics as well as the $Z=64$ subshell effects. The analysis highlights the importance of the interplay between pairing correlations and proton-neutron quadrupole-quadrupole correlations. Pairing correlations are crucial for reproducing spectra and the observed moments of inertia, but have little influence on nuclear shapes or $B(E2)$ strengths. Proton-neutron quadrupole-quadrupole correlations, together with the $Z=64$ subshell closure and orbital occupations, govern the evolution or transition of collectivity and explain the saturation of the $B(E2)$ strengths in well-deformed nuclei relative to the pseudo- and quasi-SU(3) schemes. Discrepancies in seniority-like nuclei indicate the importance of broken-pair configurations, which are beyond the present framework. Predictions have been made for neutron-rich isotopes with levels or $B(E2)$ strengths that have yet to be measured.