Spectral properties of superconducting microwave photonic crystals modeling Dirac billiards

We determined experimentally the eigenvalues of two rectangular quantum billiards that contain circular scatterers forming a triangular grid, so-called Dirac billiards. For this we performed measurements of unprecedented accuracy using superconducting macroscopic-size microwave billiards that enclose a photonic crystal. The objective was the investigation of the peculiar features of the density of the eigenvalues (DOE), which resemble that of a graphene flake, and of their fluctuation properties. We identified in the measured resonance spectra Dirac points and in their adjacent bands the van Hove singularities (VHSs), that show up as sharp peaks in the DOE. The analysis of the experimental resonance frequencies and of the band structure, which was computed with a tight-binding model, revealed that the VHSs divide the associated band into regions where the system is governed by the nonrelativistic Schr\"odinger equation of the quantum billiard and the Dirac equation of the graphene billiard of corresponding shape, respectively. Furthermore, we demonstrate that Dirac billiards are most suitable for the modeling of idealized graphene. Indeed, the DOEs of both systems are well described by a finite tight-binding model which includes first-, second-, and third-nearest-neighbor couplings.