Dirac cones protected by chiral symmetry in wallpaper groups

Dirac cones in 2D Brillioun zone (BZ) can be protected by chiral symmetry. The spinless and spin-1/2 systems preserving chiral symmetry can be realized in time-reversal symmetric superconductors and correspond to symmetry class BDI and DIII respectively. Since the total winding number in the entire BZ is zero, there are at least two Diracs cones in the BZ. Furthermore, crystalline symmetries lead to duplicate Dirac cones and additional constraints. The minimal numbers of the Dirac cones in the BZ under different symmetries might be different. In this talk, we classify the minimal numbers of the Dirac cones protected by chiral symmetry in all 17 wallpaper groups for class AIII, BDI, and DIII.