Regularities of low-lying states with random interactions in the fermion dynamical symmetry model

In this paper we study low-lying states under random interactions in the framework of the fermion dynamical symmetry model (FDSM), regardless of the ground state spin. Very strong correlations are found for $R_6$ versus $R_4$ (where $R_I\equiv E_{I_1^{+}}/E_{2_1^{+}})$ for the entire ensemble. We present arguments on the origin of these regular patterns in terms of the dynamical symmetries of the FDSM. The regular patterns of $B(E2;4_1^{+}\to 2_1^{+})$ versus $B(E2;2_1^{+}\to 0_1^{+})$ are found.

Figure 1

Distribution of $(R_6,R_4)$ of the random Hamiltonian in the FDSM. Panels (a)–(f) are based on random samples with spin-zero ground states, and $\left({\mathrm{a}}^{\prime }\right)$–$\left({\mathrm{f}}^{\prime }\right)$ are based on those with spin-nonzero ground states. Correlations between $R_6$ and $R_4$ are labeled by using $\alpha ,\beta ,\gamma ,\delta$, and $\xi$, respectively. Refer to the text for the definitions of these correlations.

Figure 2

Distribution of $(R_6,R_4)$ for the SP(6) symmetry, with the focus of $R<2$ for $N=3$ and of $R>2$ for $N=4,5$, respectively. Calculated results of the entire random ensemble, regardless of their ground state spins, are included. One sees that two correlations in each panel, i.e., $R_6=\frac{57}{28}{R}_4+\frac{3}{14}$ and $\beta$ correlation for $N=3,R_6=2R_4+\frac{1}{3}$ and $\gamma$ correlation for $N=4,R_6=2R_4+\frac{1}{3}$ and $\beta$ correlation for $N=5$, are very close to each other.

Figure 3

Distribution of $(R_6,R_4)$ for random samplings which do not well follow dynamical symmetry limits in the FDSM. Calculated results of the random samplings, regardless of their ground state spins, are considered here. One sees that correlations $\alpha ,\beta ,\gamma ,\delta$, and $\xi$ are as remarkable as in Fig. 1.

Figure 4

Distribution of $\left[B\right(E2;4_1^{+}\to 2_1^{+}),B(E2;2_1^{+}\to 0_1^{+})]$ for the random ensemble of the FDSM. The solid trajectories in black are given by the ensemble with the Hamiltonian of Eq. (1); squares in red correspond to random samplings with the SO(5)$\times \mathrm{SU}$(2) or the SU(2) limit; triangles in green correspond to those with the SO(7) limit; and inverse triangles in blue correspond to those with the SO(6) or the SU(3) limit.