Robust correlations between quadrupole moments of low-lying $2^+$ states within random-interaction ensembles

In the random-interaction ensembles, three proportional correlations between quadrupole moments of the first two $I^{\pi}=2^+$ states robustly emerge, including $Q(2^+_1)=\pm Q(2^+_2)$ correlations consistently with realistic nuclear survey, and the $Q(2^+_2)=-\frac{3}{7}Q(2^+_1)$ correlation, which is only observed in the $sd$-boson space. These correlations can be microscopically characterized by the rotational SU(3) symmetry and quadrupole vibrational U(5) limit, respectively, according to the Elliott model and the $sd$-boson mean-field theory. The anharmonic vibration may be another phenomenological interpretation for the $Q(2^+_1)=- Q(2^+_2)$ correlation, whose spectral evidence, however, is insufficient.