GOE-GUE-Poisson transitions in the nearest-neighbor spacing distribution of magnetoexcitons

Recent investigations on the Hamiltonian of excitons by Schweiner et al.. [Phys. Rev. Lett. 118, 046401 (2017)] revealed that the combined presence of a cubic band structure and external fields breaks all antiunitary symmetries. The nearest-neighbor spacing distribution of magnetoexcitons can exhibit Poissonian statistics, the statistics of a Gaussian orthogonal ensemble (GOE), or a Gaussian unitary ensemble (GUE) depending on the system parameters. Hence, magnetoexcitons are an ideal system to investigate the transitions between these statistics. Here we investigate the transitions between GOE and GUE statistics and between Poissonian and GUE statistics by changing the angle of the magnetic field with respect to the crystal lattice and by changing the scaled energy known from the hydrogen atom in external fields. Comparing our results with analytical formulas for these transitions derived with random matrix theory, we obtain a very good agreement and thus confirm the Wigner surmise for the exciton system.

Figure 1

Level spacing probability distribution functions $P\left(s\right)$ (left) and cumulative distribution functions $F\left(s\right)$ (right) for ${\delta }^{\prime }=-0.15$, $B=3\mathrm{T}$, $\vartheta =\pi /6$, and two different values of $\phi$. Besides the numerical data (red boxes or red dots), we also show the corresponding functions of a Poissonian ensemble (black dashed line), GOE (blue dash-dotted line), and GUE (green solid line). Only if the magnetic field is oriented in one of the symmetry planes of the lattice, one antiunitary symmetry is present and GOE statistics can be observed in (a) and (b). In all other cases, all antiunitary symmetries are broken and GUE statistics appears in (c) and (d).

Figure 2

Level spacing probability distribution functions $P\left(s\right)$ (first row) and cumulative distribution functions $F\left(s\right)$ (second row) for the transitions (a) $\mathrm{P}\to \mathrm{GOE}$, (b) $\mathrm{P}\to \mathrm{GUE}$, and (c) $\mathrm{GOE}\to \mathrm{GUE}$. The blue dotted lines show the transition functions of Eqs. (32), (33), and (34) for $\lambda =0.1,0.2,0.4,0.6,0.8$. To visualize the differences between the cumulative distribution functions more clearly, especially for the $\mathrm{GOE}\to \mathrm{GUE}$ transition, we also show in the third row the difference between the distributions for a fixed value of $\lambda$ and the initial distribution, respectively.

Figure 3

Transition from GOE to GUE statistics for fixed values of the magnetic field strength $B$ and increasing values of the angle $\phi$ in $\mathit{B}\left(\phi ,\vartheta =\pi /6\right)$. The results are presented in the same way as in the bottom most panel of Fig. 2 to show the differences between $F_{\mathrm{GOE}}\left(s\right)$ and $F_{\mathrm{GUE}}\left(s\right)$ more clearly. The data points (red) were fitted with the analytical function $F_{\mathrm{GOE}\to \mathrm{GUE}}\left(s;\lambda \right)$. The optimum values of the fit parameter $\lambda$ are given in each panel, but also shown in Fig. 4. One can observe a good agreement between the numerical data and the analytical function describing the transition between the two statistics in dependence on $\lambda$. Only for $\phi =0$ the data shows a slight admixture of Poissonian statistics to the expected GOE statistics. For further information see text.

Figure 4

(a) Optimum values of the fit parameter $\lambda$ in dependence on the angle $\phi$ for the situation presented in Fig. 3. The blue dashed line only serves as a guide to the eye. (b) The function $\sigma \left(\phi \right)$ of Eq. (23) for $\vartheta =\pi /6$. We obtain a qualitatively good agreement between both curves, i.e., as expected, both values $\lambda \left(\phi \right)$ and $\sigma \left(\phi \right)$ increase from zero to a certain value and then decrease for $\phi \gtrsim \pi /8$.

Figure 5

Transition from Poissonian to GUE statistics for fixed values of the angles $\phi =\pi /8$, $\vartheta =\pi /6$, and increasing values of the scaled energy $\widehat{E}$. Except for $\widehat{E}=-1.009$ a good agreement between the numerical data and the analytical function $F_{\mathrm{P}\to \mathrm{GUE}}\left(s;\lambda \right)$ is obtained. For further information see text.

Figure 6

Optimum values of the fit parameter $\lambda$ in dependence on the scaled energy $\widehat{E}$ for the situation presented in Fig. 5. The blue dashed line only serves as a guide to the eye. The value of $\lambda$ increases from a small value at low scaled energies to about $\lambda \approx 0.7$, where the function $F_{\mathrm{P}\to \mathrm{GUE}}\left(s;\lambda \right)$ almost describes GUE statistics.