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Spectral properties of Dirac billiards at the van Hove singularities

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 The commonly used statistical measures, however, are no longer applicable whereas, as demonstrated in this Letter, the ratio distributions provide most suitable ones.