Crossover between the Gaussian orthogonal ensemble, the Gaussian unitary ensemble, and Poissonian statistics

Until now only for specific crossovers between Poissonian statistics (P), the statistics of a Gaussian orthogonal ensemble (GOE), or the statistics of a Gaussian unitary ensemble (GUE) have analytical formulas for the level spacing distribution function been derived within random matrix theory. We investigate arbitrary crossovers in the triangle between all three statistics. To this aim we propose an according formula for the level spacing distribution function depending on two parameters. Comparing the behavior of our formula for the special cases of $\mathrm{P}\to \mathrm{GUE}$, $\mathrm{P}\to \mathrm{GOE}$, and $\mathrm{GOE}\to \mathrm{GUE}$ with the results from random matrix theory, we prove that these crossovers are described reasonably. Recent investigations by F. Schweiner et al. [Phys. Rev. E 95, 062205 (2017)] have shown that the Hamiltonian of magnetoexcitons in cubic semiconductors can exhibit all three statistics in dependence on the system parameters. Evaluating the numerical results for magnetoexcitons in dependence on the excitation energy and on a parameter connected with the cubic valence band structure and comparing the results with the formula proposed allows us to distinguish between regular and chaotic behavior as well as between existent or broken antiunitary symmetries. Increasing one of the two parameters, transitions between different crossovers, e.g., from the $\mathrm{P}\to \mathrm{GOE}$ to the $\mathrm{P}\to \mathrm{GUE}$ crossover, are observed and discussed.

Figure 1

The crossover function $P_{▵}(s;\alpha ,\lambda )$ of Eq. (10) for different combinations of the parameters $\alpha$ and $\lambda$. The values of these parameters are given according to the linear equations in each panel with $x=0.02,0.10,0.25,0.5,1.0$ (from dark to bright or left to right).

Figure 2

The optimum values of the parameter $\lambda$ when fitting $P_{\mathrm{GOE}\to \mathrm{GUE}}(s;\lambda )$ to $P_{▵}(s;\alpha ,10)$ for given values of $\alpha$. With these values the distance ${\mathrm{\Delta }}_2$ has been calculated according to Eq. (9).

Figure 3

The triangle of the different statistics with Poissonian $(\lambda =0)$, GOE $(\alpha =0,\lambda =1)$, and GUE statistics $(\alpha =1,\lambda =1)$ located at the corners. The green area shows the domain of the function $P_{▵}(s;\alpha ,\lambda )$.

Figure 4

Cumulative distribution function for ${\delta }^{\prime }=-0.04$ and $\widehat{E}=-0.6$. The numerical data (red lines points) is fitted by the cumulative distribution function $F_{▵}(s;\alpha ,\lambda )$ corresponding to the level spacing distribution of Eq. (10). The optimum fit parameters are here $\alpha =0.65$ and $\lambda =0.261$. Hence, the statistics is in the middle between Poissonian, GOE, and GUE statistics.

Figure 5

Resulting values for the parameters $\alpha \lambda$ and $\lambda$ when fitting the function $F_{▵}(s;\alpha ,\lambda )$ corresponding to the level spacing distribution of Eq. (10) to the cumulative distribution function of the magnetoexciton. Here we show the behavior of the two fit parameters when keeping the value ${\delta }^{\prime }$ fixed (see label in the panels) and increasing the scaled energy $\widehat{E}$ (color scale).

Figure 6

Same results as in Fig. 5 but shown for fixed values of the scaled energy $\widehat{E}$ (see label in the panels) and decreasing values of ${\delta }^{\prime }$ (color scale).