We present a method to calculate the excitonic spectra of all direct semiconductors with a complex valence band structure. The Schrödinger equation is solved using a complete basis set with Coulomb Sturmian functions. This method also allows for the computation of oscillator strengths. Here we apply this method to investigate the impact of the valence band structure of cuprous oxide (Cu2O) on the yellow exciton spectrum. Results differ from those of J. Thewes et al. [Phys. Rev. Lett. 115, 027402 (2015)]; the differences are discussed and explained. The difference between the second and third Luttinger parameter can be determined by comparisons with experiments, however, the evaluation of all three Luttinger parameters is not uniquely possible. Our results are consistent with band structure calculations. Considering also a finite momentum K of the center of mass, we show that the large K-dependent line splitting observed for the 1S exciton state by G. Dasbach et al. [Phys. Rev. Lett. 91, 107401 (2003)] is not related to an exchange interaction but rather to the complex valence band structure of Cu2O.

Two of the most striking experimental findings when investigating exciton spectra in cuprous oxide using high-resolution spectroscopy are the observability and the fine structure splitting of F excitons reported by J. Thewes et al. [Phys. Rev. Lett. 115, 027402 (2015)]. These findings show that it is indispensable to account for the complex valence band structure and the cubic symmetry of the solid in the theory of excitons. This is all the more important for magnetoexcitons, where the external magnetic field reduces the symmetry of the system even further. We present the theory of excitons in Cu2O in an external magnetic field and especially discuss the dependence of the spectra on the direction of the external magnetic field, which cannot be understood from a simple hydrogen-like model. Using high-resolution spectroscopy, we also present the corresponding experimental spectra for cuprous oxide in Faraday configuration. The theoretical results and experimental spectra are in excellent agreement as regards not only the energies but also the relative oscillator strengths. Furthermore, this comparison allows for the determination of the fourth Luttinger parameter κ of this semiconductor.

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We present a theoretical calculation of the absorption spectrum of cuprous oxide $\left(\mathrm{Cu_{2}O}\right)$ based on the general theory developed by Y. Toyozawa. An inclusion not only of acoustic phonons but also of optical phonons and of specific properties of the excitons in $\mathrm{Cu_{2}O}$ like the central-cell corrections for the $1S$-exciton allows us to calculate the experimentally observed line widths in experiments by T. Kazimierczuk et al [Nature 514, 343, (2014)] within the same order of magnitude, which demonstrates a clear improvement in comparison to earlier work on this topic. We also discuss a variety of further effects, which explain the still observable discrepancy between theory and experiment but can hardly be included in theoretical calculations

We show analytically and numerically that the application of an external magnetic field to highly excited Rydberg excitons breaks all antiunitary symmetries in the system. Only by considering the complete valence band structure of a direct band gap cubic semiconductor, the Hamiltonian of excitons leads to the statistics of a Gaussian unitary ensemble (GUE) without the need for interactions with other quasi-particles like phonons. Hence, we give theoretical evidence for a spatially homogeneous system breaking all antiunitary symmetries.PACS numbers: 05.30.Ch, 78.40.Fy, For more than 100 years one distinguishes in classical mechanics between two fundamentally different types of motion: regular and chaotic motion. Their appearance strongly depends on the presence of underlying symmetries, which are connected with constants of motion and reduce the degrees of freedom in a given system. If symmetries are broken, the classical dynamics often becomes nonintegrable and chaotic. However, since the description of chaos by trajectories and Lyapunov exponents is not possible in quantum mechanics, it has been unknown for a long time how classical chaos manifests itself in quantum mechanical spectra [1,2].The Bohigas-Giannoni-Schmit conjecture [3] suggests that quantum systems with few degrees of freedom and with a chaotic classical limit can be described by random matrix theory [4,5] and thus show typical level spacings. At the transition to quantum chaos, the level spacing statistics will change from Poissonian statistics to the statistics of a Gaussian orthogonal ensemble (GOE) or a Gaussian unitary ensemble (GUE) as symmetry reduction leads to a correlation of levels and hence to a strong suppression of crossings [1].To which of the two universality classes a given system belongs is determined by remaining antiunitary symmetries in the system. While GOE statistics can be observed in many different systems like, e.g., in atomic [6,7] and molecular spectra [8], for nuclei in external magnetic fields [9][10][11][12] There is no example for a system showing GUE statistics in atomic physics. This is especially true for one of the prime examples when studying quantum chaos: the highly excited hydrogen atom in strong external fields. Even though the applied magnetic field breaks timereversal invariance, at least one antiunitary symmetry, e.g., time reversal and a certain parity, remains and GOE statistics is observed [1,22,23].Excitons are fundamental quasi-particles in semiconductors, which consist of an electron in the conduction band and a positively charged hole in the valence band.Recently, T. Kazimierczuk et al [24] have shown in a remarkable high-resolution absorption experiment an almost perfect hydrogen-like absorption series for the yellow exciton in cuprous oxide (Cu 2 O) up to a principal quantum number of n = 25. This experiment has drawn new interest to the field of excitons experimentally and theoretically [25][26][27][28][29][30][31][32][33][34][35].Since excitons in semiconductors are often treated as the hydro...