  A crucial result in quantum chaos, which has been established for a long
time, is that the spectral properties of classically integrable systems
generically are described by Poisson statistics whereas those of time-reversal
symmetric, classically chaotic systems coincide with those of random matrices
from the Gaussian orthogonal ensemble (GOE). Does this result hold for
two-dimensional Dirac material systems? To address this fundamen- tal question,
we investigate the spectral properties in a representative class of graphene
billiards with shapes of classically integrable circular-sector billiards.
Naively one may expect to observe Poisson statistics, which is indeed true for
energies close to the band edges where the quasiparticle obeys the
Schr\"odinger equation. However, for energies near the Dirac point, where the
quasiparticles behave like massless Dirac fermions, Pois- son statistics is
extremely rare in the sense that it emerges only under quite strict symmetry
constraints on the straight boundary parts of the sector. An arbitrarily small
amount of imperfection of the boundary results in GOE statistics. This implies
that, for circular sector confinements with arbitrary angle, the spectral
properties will generically be GOE. These results are corroborated by extensive
numerical computation. Furthermore, we provide a physical understanding for our
results.
