Existence of Dirac cones in Brillouin zone of diperiodic atomic crystals according to group theory

We have considered non-magnetic materials with weak spin-orbit coupling, that are periodic in two non-collinear directions, and finite in third, orthogonal direction. In some cases, combined time-reversal and crystal symmetry of such systems, allows the existence of Dirac cones at certain points in the reciprocal space. We have investigated in a systematic way, all points of Brillouin zone of all 80 diperiodic groups and have found sufficient conditions for the existence of s=1/2 Dirac fermions, with symmetry-provided band touching at the vertex of the Dirac cones. Conversely, complete linear dispersion is forbidden for orbital wave-functions belonging to two-dimensional irreducible representations (irreps) of little groups that do not satisfy certain group-theoretical conditions given in this paper. Our results are illustrated by a tight-binding example.