Collective modes of low-lying states in the interacting boson model with random interactions

In this paper we study yrast states of $sd$ bosons under random interactions by revisiting the collective motions and the linear correlations in the Mallmann plot. Both the SU(3) and the conjugate $\overline{\mathrm{SU}\left(3\right)}$ limits are found to play an important role in the rotational motion in the random ensemble. We point out that the seniority-like correlation is represented by the U(5) limit with $s$-boson dominance for yrast low-spin states and $d$-boson condensation for yrast higher-spin states. The so-called $\alpha$ correlation is interpreted in terms of the $s$-boson dominance of the U(5) limit for the vibrational side, the $\mathrm{SU}\left(3\right)/\overline{\mathrm{SU}\left(3\right)}$ limit for the rotational side, and transitions between dynamical symmetry limits for the intermediate region. The $\alpha$ correlation is also reproduced using the consistent-$Q$ Hamiltonian in the Casten's symmetry triangle.

Figure 1

Distribution of the largest overlap between the wave function obtained under the TBRE and U(5)-limit states for the random samplings of the vibrational peak ($R_4\approx 2$). The solid curve in black, the dashed curve in red, and the dotted curve in blue are for the $0_1^{+},2_1^{+},4_1^{+}$ states, respectively.

Figure 2

Distribution of the largest overlap between the wave function obtained under the TBRE and $\mathrm{SU}\left(3\right)/\overline{\mathrm{SU}\left(3\right)}$-limit states for the $0_1^{+}$ state of the random samplings of the rotational peak ($R_4\approx 3.33$).

Figure 3

Mallmann plot ($R_6$ versus $R_4$) for the random ensemble of the $sd$ bosons. In addition to the vibrational and rotational peaks (denoted by open circles), linear correlations between $R_6$ and $R_4$ are noticed. The colors represent the largest overlap between the wave function obtained under the TBRE and dynamical-symmetry-limit states. In the insert of panels (c) and (f) we show the fine structure of the Mallmann plot around the rotational peak, $(R_4,R_6)=(3.33,7)$, by magnifying the scales.

Figure 4

Expectation value of the O(5)-symmetry quantum number $\tau$ for random samplings following the $\alpha$ correlation with $R_4<2.9$.

Figure 5

(a) The solid dots represent the excitation energies of yrast even-$I$ states obtained by the consistent-$Q$ Hamiltonian [see Eq. (10)] for the $n=6$ system with different sets of the control parameters $\zeta$ and $\mathcal{g}$; the curves represent the Ejiri formula. (b) Mallmann plot ($R_6$ versus $R_4$) obtained by the consistent-$Q$ Hamiltonian with random parameters $\zeta$ and $\mathcal{g}$; the straight line represents the $\alpha$ correlation $R_6=3R_4-3$.