Exciton-phonon interaction breaking all antiunitary symmetries in external magnetic fields

Recent experimental investigations by M. Aßmann et al. [Nat. Mater. 15, 741 (2016)] on the spectrum of magnetoexcitons in cuprous oxide revealed the statistics of a Gaussian unitary ensemble (GUE). The model of F. Schweiner et al. [Phys. Rev. Lett. 118, 046401 (2017)], which includes the complete cubic valence band structure of the solid, can explain the appearance of GUE statistics if the magnetic field is not oriented in one of the symmetry planes of the cubic lattice. However, it cannot explain the experimental observation of GUE statistics for all orientations of the field. In this paper we investigate the effect of quasiparticle interactions or especially the exciton-phonon interaction on the level statistics of magnetoexcitons and show that the motional Stark field induced by the exciton-phonon interaction leads to the occurrence of GUE statistics for arbitrary orientations of the magnetic field in agreement with experimental observations. Importantly, the breaking of all antiunitary symmetries can be explained only by considering both the exciton-phonon interaction and the cubic crystal lattice.

Figure 1

Band structure of ${\mathrm{Cu}}_2\mathrm{O}$ [1]. As a consequence of the spin-orbit coupling (8) the valence band splits into a lower lying fourfold-degenerate band (including the hole spin) of symmetry ${\mathrm{\Gamma }}_8^{+}$ of and a higher lying twofold-degenerate band of symmetry ${\mathrm{\Gamma }}_7^{+}$. The lowest lying conduction band of ${\mathrm{Cu}}_2\mathrm{O}$ has ${\mathrm{\Gamma }}_6^{+}$ symmetry. Depending on the bands involved, one distinguishes between the yellow, green, blue, and violet exciton series. Due to the cubic symmetry of ${\mathrm{Cu}}_2\mathrm{O}$, the symmetry of the bands can be assigned by the irreducible representations ${\mathrm{\Gamma }}_i^{\pm }$ of the cubic group $O_{\mathrm{h}}$, where the superscript $\pm$ denotes the parity.

Figure 2

Transition from GOE to GUE statistics when deflecting the field ${\mathit{F}}_{\mathrm{ms}}$ in Eq. (32) from the symmetry plane $y=0$ by an angle ${\phi }_{\mathrm{ms}}$. For the magnetic field we have set $\mathit{B}\left(\phi =0,\vartheta =\pi /6\right)$ with $B=3\mathrm{T}$. To obtain enough eigenvalues for a statistical evaluation, we used the simplified model of Ref. [18], in which the spins of the electron and hole are neglected. To visualize the differences between the cumulative distribution functions more clearly, we subtract $F_{\mathrm{GOE}}\left(s\right)$ from them. The data points (red) were fitted with the analytical function $F_{\mathrm{GOE}\to \mathrm{GUE}}\left(s;\lambda \right)$ defined in Sec. 2c. The optimum values of the fit parameter $\lambda$ are given in each panel and are also shown in Fig. 3. One can observe a good agreement between the numerical data and the analytical function describing the transition between the two statistics in dependence on $\lambda$. For further information see text.

Figure 3

Optimum values of the fit parameter $\lambda$ in dependence on the angle ${\phi }_{\mathrm{ms}}$ for the situation presented in Fig. 2. One can see that the value of $\lambda$ increases very rapidly with increasing ${\phi }_{\mathrm{ms}}$. Already for ${\phi }_{\mathrm{ms}}=5^{\circ }$ the transition to GUE statistics is completed. As regards the value of $\lambda$ for ${\phi }_{\mathrm{ms}}=8^{\circ }$, we have to note that the function $F_{\mathrm{GOE}\to \mathrm{GUE}}\left(s;\lambda \right)$ only slightly varies for $\lambda \ge 0.8$ and hence small fluctuations in the numerical results will lead to a strong change in $\lambda$. For the transition between GOE and GUE statistics only the range of $0.1\le \lambda \le 0.8$ is of importance (green dashed lines) (cf. Ref. [19]). For ${\phi }_{\mathrm{ms}}>8^{\circ }$ it is always $\lambda >0.8$ until ${\phi }_{\mathrm{ms}}\approx {176}^{\circ }$ [cf. Eq. (33)].

Figure 4

Dependence of the energy of specific exciton states on the angle ${\phi }_{\mathrm{ms}}$ of the field ${\mathit{F}}_{\mathrm{ms}}$. For the magnetic field we have set $\mathit{B}\left(\phi =0,\vartheta =\pi /6\right)$ with $B=3\mathrm{T}$. It can be seen that for ${\phi }_{\mathrm{ms}}$ and $\pi +{\phi }_{\mathrm{ms}}$ the exciton energies (blue solid lines) are shifted in the same direction with respect to the energy at ${\phi }_{\mathrm{ms}}=0$ (red dashed lines). The average energy (green solid lines) often clearly differs from the energy at ${\phi }_{\mathrm{ms}}=0$.

Figure 5

(a) Splitting of the $n=5$ exciton states of ${\mathrm{Cu}}_2\mathrm{O}$ in an external magnetic field $\mathit{B}=\mathit{B}\left(\phi =0,\vartheta =\pi /6\right)$ with $F_{\mathrm{ms}}=0$. At $B\approx 1.98\mathrm{T}$ an avoided crossing can be observed (see inset and red box). (b) Energy of the $n=5$ states for $B=1.98\mathrm{T}$ and (i) $F_{\mathrm{ms}}=0$ (blue lines), (ii) $F_{\mathrm{ms}}=9.57\times {10}^3\mathrm{V}/\mathrm{m}$ given by Eq. (32) with ${\phi }_{\mathrm{ms}}=0$ (red lines), and (iii) $F_{\mathrm{ms}}$ given by Eq. (32) but taking the position of the states when averaging over ${\phi }_{\mathrm{ms}}=0$ (green lines). (c) Normalized spacings for the three cases considered. It can be seen that the motional Stark effect further suppresses small spacings. For a comparison, we also show the distribution functions for Poisson statistics (blue dash-dotted line), GOE statistics (red dashed line), and GUE statistics (green solid line).