Correlations between low-lying yrast states for $sd$ bosons with random interactions

In this paper we study low-lying states of $sd$ bosons under random interactions. We find that the yrast states are highly correlated, regardless of whether the ground state spin is zero or not. Many of the random samplings in the random ensemble well overlap with predictions of the U(5) limit in the interacting boson model.

Figure 1

Correlation of $R_6$ versus $R_4$ of $sd$ bosons with two-body random interactions. $n$ is the boson number. Only random samples with spin-nonzero ground states are considered here. In addition to the peaks at $(R_4,R_6)=(3.3,7.0)$, which corresponds to rotational motion, and $(R_4,R_6)=(1.0,1.0)$, which corresponds to the seniority-type correlation, strong correlations (denoted by $\alpha ,\beta ,\gamma ,\delta$, and $\xi )$ between $R_4$ and $R_6$ are seen. Correlations $\beta$ and $\xi$ are described by the same formula in the $R_6\text{−}{R}_4$ plot, and thus are “mixed” in the case of odd $n$ when $n=3k$. See the text for more details.

Figure 2

Same as in Fig. 1 but $R_6$ is replaced by $R_8$. In the $R_8\text{−}{R}_4$ plot, correlations $\beta$ and $\xi$ are different.

Figure 3

Histogram plots of expected values of $d$ boson number $\left(n_d\right)$ and $d$-boson seniority $\left(\tau \right)$ in yrast states of spin 0, 2, 4, 6, 8, for random samples with spin-$2n$ ground states corresponding to correlations $\beta$ [panels (a) and (a′) with boson number $n=6\text{]}$, $\gamma$ [panels (b) and (b′) with $n=8\text{]}$, $\delta$ [panels (c) and (c′) with $n=7\text{]}$, and $\xi$ [panels (d) and (d′) with $n=7\text{]}$. Both $P\left(n_d\right)$ and $P\left(\tau \right)$ are normalized to 1. One sees that the distributions of $n_d$ and $\tau$ are consistent with those predicted by assuming $d$ boson condensation for these random samples.

Figure 4

Distribution of coefficients $C_1,C_2$ of Eq. (2) for random samples with spin-$2n$ ground states, corresponding to correlations $\beta$ [panel (a): $n=6\text{]}$, $\gamma$ [panel (b): $n=8\text{]}$, $\delta$ [panel (c): $n=7\text{]}$ and $\xi$ [panel (d): $n=7\text{]}$. One sees here that both $C_1$ and $C_2$ are dominantly negative for correlations $\beta$ and $\delta$, and that $C_1<0$ and $C_2>0$ for correlations $\gamma$ and $\xi$. Such $C_1$ and $C_2$ values are consistent with predictions in Table 2 by assuming $d$ boson condensation for these correlations.