LAUSR

Shapes and shape evolution in the mass-130 region, including the Te, Xe, and Ba isotopes, have long been a focus of discussion in nuclear physics. This mass region consists of complex many-body systems that can behave in astonishingly simple and regular ways, as classified in the Casten symmetry triangle. By applying the shell model Hamiltonian proposed recently, we carry out calculations using the Hartree-Fock-Bogolyubov plus generator coordinate method, in the large model space containing the (1g_{9/2},1g_{7/2},2d_{5/2},2d_{3/2},3s_{1/2},1h_{11/2},2f_{7/2}) orbits. Based on good reproduction of the experimentally known energy levels, spectroscopic quadrupole moments, and E2 transition probabilities, we identify the quasi-SU(3) couplings across the N=50 and 82 shell gaps, which play a role in driving shape evolution and phase transition discussed in the extended Casten triangle. Specifically, we demonstrate that the quasi-SU(3) coupling mechanism in the proton partner orbits (1g_{9/2}, 2d_{5/2}) tends to drive the system to be more γ soft, and that in the neutron partner orbits (1h_{11/2}, 2f_{7/2}) are responsible for the oblate-to-prolate shape phase transition. With an emphasis on discussing spectroscopic quadrupole moments, our Letter uncovers hidden symmetries from the vast shell-model configurations and adds microscopical insights into the empirical symmetry triangle.