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We study distributions of the ratios of level spacings of rectangular and Africa-shaped superconducting microwave resonators containing circular scatterers on a triangular grid, so-called Dirac billiards (DBs). The high-precision measurements allowed the determination of, respectively, all 1651 and 1823 eigenfrequencies in the first two bands. The resonance densities are similar to that of graphene. They exhibit two sharp peaks at the van Hove singularities which separate the band structure into regions with a linear and a quadratic dispersion relation, respectively. In the vicinity of the van Hove singularities we observe rapid changes in, e.g., the wave function structure. Accordingly, we question whether the spectral properties are there still determined by the shapes of the DBs. The commonly used statistical measures are no longer applicable; however, we demonstrate in this Letter that the ratio distributions provide suitable measures.

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We report on the experimental realization of a flat, superconducting microwave resonator, a microwave billiard, with partially violated time-reversal (T ) invariance, induced by inserting a ferrite into the cavity and magnetizing it with an external magnetic field perpendicular to the resonator plane. In order to prevent its expulsion caused by the Meissner-Ochsenfeld effect we used a cavity of which the top and bottom plate were made from niobium, a superconductor of type II, and cooled it down to liquid-helium temperature TLHe 4 K. The Cavity had the shape of a chaotic Afrivca billiard. Superconductivity rendered possible the accurate determination of complete sequences of the resonance frequencies and of the widths and strengths of the resonances, an indispensable prerequisite for the unambiguous detection of T invariance violation, especially when it is only partially violated. This allows for the first time the precise specification of the size of T invariance violation from the fluctuation properties of the resonance frequencies and from the strength distribution, which actually depends sensitively on it and thus provides a most suitable measure. For this purpose we derived an analytical expression for the latter which is valid for isolated resonances in the range from no T invariance violation to complete violation.Introduction.-An important aspect of quantum chaos is the understanding of the features of the classical dynamics in terms of the spectral properties of the corresponding quantum system [1][2][3]. Numerous experimental and numerical studies confirmed that for a fully chaotic classical dynamics they coincide with those of random matrices [4] from the Gaussian orthogonal ensemble (GOE) when time-reversal (T ) invariance is preserved, from the Gaussian unitary ensemble (GUE) when it is violated [5][6][7] and from an ensemble interpolating between the GOE and the GUE, when T invariance is only partially violated [8][9][10]. Most suitable for the experimental verification are flat, cylindrical microwave resonators [11,12] and microwave networks [10]. For microwave frequencies below a certain cutoff value f max , the associated Helmholtz equation is mathematically equivalent to the Schrödinger equation of the quantum billiard and the quantum graph of corresponding shape [13,14], respectively. The random-matrix theory (RMT) analysis of the spectral properties of a quantum system and the assignment to one of these ensembles requires complete sequences of several hundreds of eigenvalues [7] or an elaborate procedure to cope with missing levels [15][16][17][18][19][20][21] which hinder or render the unambiguous determination of the strength of T invariance violation unfeasible in cases where it is only partially violated. For T invariant systems complete sequences of up to 5000 eigenvalues [22][23][24] of the corresponding quantum billiard were obtained in high-precision experiments at liquidhelium temperature T LHe = 4 K with niobium and leadcoated microwave resonators which become superconducting at T c...