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The Nielsen-Ninomiya Theorem has set up a ground rule for the minimal numberof the topological points in a Brillouin zone. Notably, in the 2D Brillouinzone, chiral symmetry and space-time inversion symmetry can properly definetopological invariants as charges characterizing the stability of the nodalpoints so that the non-zero charges protect these points. Due to the chargeneutralization, the Nielsen-Ninomiya Theorem requires at least two stabletopological points in the entire Brillouin zone. However, additionalcrystalline symmetries might duplicate the points. In this regard, for thewallpaper groups with crystalline symmetries, the minimal number of the nodalpoints in the Brillouin zone might be more than two. In this work, we determinethe minimal numbers of the nodal points for the wallpaper groups inchiral-symmetric and space-time-inversion-symmetric systems separately andprovide examples for new topological materials, such as topological nodaltime-reversal-symmetric superconductors and Dirac semimetals. This generalizedNielsen-Ninomiya Theorem serves as a guide to search for 2D topological nodalmaterials and new platforms for twistronics. Furthermore, we show theNielsen-Ninomiya Theorem can be extended to 2D non-Hermitian systems hostingtopologically protected exceptional points and Fermi points for the 17wallpaper groups and use the violation of the theorem on the surface toclassify 3D Hermitian and non-Hermitian topological bulks.

Generalized Fermion Doubling Theorems: Classification of 2D Nodal Systems in Terms of Wallpaper Groups