Regularity and chaos in $0^{+}$ states of the interacting boson model using quantum measures

Background: Statistical measures of chaos have long been used in the study of chaotic dynamics in the framework of the interacting boson model. The use of large numbers of bosons renders possible additional studies of chaos that can provide a direct comparison with similar classical studies of chaos.

Purpose: We intend to provide complete quantum chaotic dynamics at zero angular momentum in the vicinity of the arc of regularity and link the results of the study of chaos using statistical measures with those of the study of chaos using classical measures.

Method: Statistical measures of chaos are applied on the spectrum and the transition intensities of $0^{+}$ states in the framework of the interacting boson model.

Results: The energy dependence of chaos is provided for the first time using statistical measures of chaos. The position of the arc of regularity was also found to be stable in the limit of large boson numbers.

Conclusions: The results of the study of chaos using statistical measures are consistent with previous studies using classical measures of chaos, as well as with studies using statistical measures of chaos, but for small number of bosons and states with angular momentum greater than 2.

Figure 1

The IBM symmetry triangle and the points in the vicinity of the arc, where calculations were performed.

Figure 2

Results for the four points shown in Fig. 1, for $N_B=175$ and $J=0$ states, using the fluctuation measures $P\left(S\right)$ of Eq. (2) [33] and ${\mathrm{\Delta }}_3\left(L\right)$ of Eq. (7) [34, 35], and for $N_B=50$ and $J=0$ states, for the fluctuation measure $P\left(y\right)$ of Eq. (8) [36, 37]. The black dotted line shows the regular limit, the red (gray) dashed line describes the chaotic limit, while the blue (gray) solid line is the fit to the distribution. It is evident from the values of $\omega$, $q$, and $\nu$ that the SU(3) vertex and the point on the arc are more regular compared to the other two points above and below the arc.

Figure 3

Results obtained for the nearest spacing distribution $P\left(S\right)$ of Eq. (2) [33] for each of the cases shown in Fig. 1, for $N_B=175$, and when the set of $J=0$ states is divided into 8 sets of 330 states each. The states 1–330 (labeled as interval 1 in Table 2) appear on the top, and the states 2311–2640 (labeled as interval 8 in Table 2) appear at the bottom. The black dotted line shows the regular limit, the red (gray) dashed line describes the chaotic limit, while the blue (gray) solid line is the fit to the distribution.

Figure 4

Results obtained for the spectral rigidity measure ${\mathrm{\Delta }}_3^q\left(L\right)$ of Eq. (7) [34, 35], for each of the cases shown in Fig. 1, for $N_B=175$, and when the set of $J=0$ states is divided into 8 sets of 330 states each. The states 1–330 (labeled as interval 1 in Table 3) appear on the top, and the states 2311–2640 (labeled as interval 8 in Table 3) appear at the bottom. The black dotted line shows the regular limit, the red (gray) dashed line describes the chaotic limit, while the blue (gray) solid line is the fit to the distribution.

Figure 5

Results obtained for the quantum statistical parameters $\omega$ and $q$, for 14 different values of $\chi$ along the $\eta =0.632$ line of the triangle. The calculations were performed for $N_B=175$ and $J=0$.

Figure 6

Results obtained for the quantum statistical parameters $\omega$ and $q$, for various values of $N_B$ at the three points in the vicinity of the arc of regularity.

Figure 7

Results obtained for the statistical measure $P\left(y\right)$ of Eq. (8) [36, 37], for each of the cases shown in Fig. 1, for $N_B=50$ and, when the set of 234 $J=0$ states is divided into 7 sets of 30 states and one set of 24 states. The states 1–30 (labeled as interval 1 in Table 5) appear on the top, and the states 211–234 (labeled as interval 8 in Table 5) appear at the bottom. The blue (gray) solid line is the fit to the distribution.

Figure 8

Results obtained for the quantum statistical parameter $\nu$, for 14 different values of $\chi$ along the $\eta =0.632$ line of the triangle. The calculations were performed for $N_B=50$ and $J=0$.