Abstract
In a 2D Brillouin zone (BZ), chiral symmetry can protect nodal points, which of each is characterized by a quantized winding number with an integral path enclosing the point. The spinless and spin-1/2 systems preserving chiral symmetry can be realized in time-reversal symmetric superconductors and belong to symmetry class BDI and DIII respectively. It is known that Nielsen-Ninomiya Theorem leads to at least two nodal points in the BZ, since the total winding number in the entire BZ must vanish. However, two is not always the minimal number of the nodal points in generic lattice systems. The presence of crystalline symmetries might duplicate or eliminate nodal points. In this regard, for different wallpaper groups, the minimal numbers of the nodal points in the BZ might be different. Providing the interplay of the nodal points with winding numbers and symmetries other than chiral symmetry, we classify the minimal numbers of the nodal points protected by chiral symmetry in all 17 wallpaper groups for symmetry classes possessing chiral symmetry.