Quasidynamical Symmetry in an Interacting Boson Model Phase Transition

The apparent persistence of symmetry in the face of strong symmetry-breaking interactions is examined in a many-boson model. The model exhibits a transition between two phases associated with U(5) and O(6) symmetries, respectively, as the value of a control parameter progresses from 0 to 1. The remarkable fact is that, in spite of strong mixing of the symmetries for intermediate values of the control parameter, the model continues to exhibit the characteristics of its closest symmetry limit for all but a relatively narrow transition region that becomes progressively narrower as the boson number increases. This phenomenon is explained in terms of quasidynamical symmetry.

Figure 1

The energy function $V_{\alpha }(\beta )$ for different values of $\alpha$.

Figure 2

Excitation energies of the Hamiltonian $\widehat{H}(\alpha )$ as a function of $\alpha$. The continuous lines show precise, numerically computed energies. The dotted lines are results of an RPA calculation (for $\alpha <0.5$) and an SHA calculation (for $\alpha >0.5$). Note that $N$ levels coalesce to form a degenerate ground state, and $N-m$ levels coalesce to the $m$th excited multiplet, as $\alpha \to 1$. Thus, the level density is much higher in the mid-$\alpha$ region than this figure would suggest.

Figure 3

$B(E2)$ transition rates for decay of the first excited $v=1$ energy level to the ground state for various values of $N$ expressed in units such that $B(E2;1\to 0)=100$ in the U(5) $(\alpha =0)$ limit. The continuous lines for $\alpha \le 0.5$ are for the RPA and those for $\alpha \ge 0.5$ are for the SHA. The dots are precise (numerically computed) results.

Figure 4

Wave functions for the $N=60$ ground state and first excited $v=0$ state for $\alpha =1$ and 0.75. The dots give the coefficients of the states in a basis of $d$-boson number $n$. The continuous lines are harmonic oscillator wave functions.