Distribution of the spacing between two adjacent avoided crossings

We consider the frequency at which avoided crossings appear in an energy-level structure when an external field is applied to a quantum chaotic system. The distribution of the spacing in the parameter between two adjacent avoided crossings is investigated. Using a random matrix model, we find that the distribution of these spacings is well fitted by a power-law distribution for small spacings. The powers are 2 and 3 for the Gaussian orthogonal ensemble and Gaussian unitary ensemble, respectively. We also find that the distributions decay exponentially for large spacings. The distributions in concrete quantum chaotic systems agree with those of the random matrix model.

Figure 1

The scaled energy spectrum for ${\mathcal{H}}_{\mathrm{RM}}\left(\Lambda \right)$ of the GOE as a function of $\lambda$. Three examples of avoided-crossing spacings are also shown.

Figure 2

Log-log plots of the cumulative distributions (a) $I_{\mathrm{RM}}^{\left(1\right)}\left(S_{\lambda }\right)$ and (b) $I_{\mathrm{RM}}^{\left(2\right)}\left(S_{\lambda }\right)$.

Figure 3

(a) For $\beta =1$, $P_{\mathrm{RM}}^{\left(\beta \right)}\left(S_{\lambda }\right)$ is compared with $P_{\text{trial}}^{\left(\beta \right)}\left(S_{\lambda }\right)$. (b) The same as (a) except $\beta =2$.

Figure 4

(a) Histogram of $P_{\mathrm{CRT}}\left(S_{\lambda }\right)$ with $P_{\mathrm{RM}}^{\left(1\right)}\left(S_{\lambda }\right)$. (b) $I_{\mathrm{CRT}}\left(S_{\lambda }\right)$ together with $I_{\mathrm{RM}}^{\left(1\right)}\left(S_{\lambda }\right)$ and $I_{\mathrm{RM}}^{\left(2\right)}\left(S_{\lambda }\right)$. The inset shows the magnified figure with a log-log plot near the origin.

Figure 5

Boundaries of the Aharonov-Bohm billiard with $\Lambda =$ (a) 0.15, (b) 0.2, and (c) 0.25.

Figure 6

(a) Histogram of $P_{\mathrm{AB}}\left(S_{\lambda }\right)$ with $P_{\mathrm{RM}}^{\left(2\right)}\left(S_{\lambda }\right)$. (b) $I_{\mathrm{AB}}\left(S_{\lambda }\right)$ together with $I_{\mathrm{RM}}^{\left(1\right)}\left(S_{\lambda }\right)$ and $I_{\mathrm{RM}}^{\left(2\right)}\left(S_{\lambda }\right)$. The inset shows the magnified figure with a log-log plot near the origin.