Quantum phase transition in the U(4) vibron model and the E(3) symmetry

Abstract

We study the details of the U(3)--O(4) quantum phase transition in the U(4) vibron model. Both asymptotic analysis in the classical limit and rigorous calculations for finite boson number systems indicate that a second-order phase transition is still there even for the systems with boson number $N$ ranging from tens to hundreds. Two kinds of effective order parameters, including $E1$ transition ratios $B(E1:{2}_{1}\ensuremath{\rightarrow}{1}_{1})/B(E1:{1}_{1}\ensuremath{\rightarrow}{0}_{1})$ and $B(E1:{0}_{2}\ensuremath{\rightarrow}{1}_{1})/B(E1:{1}_{1}\ensuremath{\rightarrow}{0}_{1})$, and the energy ratios ${E}_{{2}_{1}}/{E}_{{0}_{2}}$ and ${E}_{{3}_{1}}/{E}_{{0}_{2}}$ are proposed to identify the second-order phase transition in experiments. We also found that the critical point of phase transition can be approximately described by the E(3) symmetry, which persists even for moderate $N~10$ protected by the scaling behaviors of quantities at the critical point. In addition, a possible empirical example exhibiting roughly the E(3) symmetry is discussed.