Haldane graphene billiards versus relativistic neutrino billiards

We study fluctuation properties in the energy spectra of finite-size honeycomb lattices, graphene billiards, subject to the Haldane-model onsite potential and next-nearest neighbor interaction at critical points, referred to as Haldane graphene billiards in the following. The billiards had the shapes of a rectangular billiard with integrable dynamics, one with chaotic dynamics, and one whose shape has, in addition, threefold rotational symmetry. It had been shown that the spectral properties of the graphene billiards coincide with those of the nonrelativistic quantum billiard with the corresponding shape, both at the band edges and in the region of low energy excitations around the Dirac points at zero energy. There, the dispersion relation is linear and, accordingly, the spectrum is described by the same relativistic Dirac equation for massless half-spin particles as relativistic neutrino billiards, whose spectral properties agree with those of nonrelativistic quantum billiards with violated time-reversal invariance. Deviations from the expected behavior are attributed to differing boundary conditions and backscattering at the boundary, which leads to a mixing of valley states corresponding to the two Dirac points, that are mapped into each other through time reversal. We employ a Haldane model to introduce a gap at one of the two Dirac points so that backscattering is suppressed in the energy region of the gap and demonstrate that there the correlations in the spectra comply with those of the neutrino billiard of the corresponding shape.