Hidden Euclidean dynamical symmetry in the U(n+1) vibron model

Based on the boson realization of the Euclidean algebras, it is found that the E($n$) dynamical symmetry (DS) may emerge at the critical point of the U($n$)-SO($n+1$) quantum phase transition. To justify this finding, we provide a detailed analysis of the critical dynamics in the U($n+1$) vibron model in both quantal and classical ways. It is further shown that the low-lying structure of ${}^{82}$Kr may serve as an excellent empirical realization of the E(5) DS in experiments.