Abstract
We present results for properties related to the band structure of a microwave photonic crystal, that is, a flat resonator containing metallic cylinders arranged on a triangular grid, referred to as the Dirac (microwave) billiard, with a threefold-symmetric shape. Such systems have been used to investigate finite-size graphene sheets. It was shown recently that the eigenmodes of rectangular Dirac billiards are well described by the tight-binding model of a finite-size honeycomb-kagome lattice of corresponding shape. We compare properties of the eigenstates of the Dirac billiard with those of the associated graphene and honeycomb-kagome billiard and relativistic quantum billiard. We outline how the eigenstates of threefold-symmetric systems can be separated according to their transformation properties under rotation by $\frac{2\ensuremath{\pi}}{3}$ into three subspaces, namely singlets, that are rotationally invariant, and doublets that are noninvariant. We reveal for the doublets in graphene and honeycomb-kagome billiards in quasimomentum space a selective excitation of the valley states associated with the two inequivalent Dirac points. For the understanding of symmetry-related features, we extend known results for nonrelativistic quantum billiards and the associated semiclassical approach to relativistic neutrino billiards.
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