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Abstract

The Nielsen-Ninomiya theorem set up a ground rule for the minimal number of the topological points in a Brillouin zone. Notably, in the 2D Brillouin zone, chiral symmetry and space-time inversion symmetry can properly define topological invariants as charges characterizing the stability of the nodal points so the nonzero charges protect these points. Due to the charge neutralization, the Nielsen-Ninomiya theorem requires at least two stable topological points in the entire Brillouin zone. However, additional crystalline symmetries might duplicate the points. In this regard, for the wallpaper groups with crystalline symmetries, the minimal number of the nodal points in the Brillouin zone might be more than two. In this paper, we determine the minimal numbers of the nodal points for the wallpaper groups in chiral-symmetric and space-time-inversion-symmetric systems separately and provide examples for topological materials, such as topological nodal time-reversal-symmetric superconductors and Dirac semimetals. This generalized Nielsen-Ninomiya theorem serves as a guide to search for 2D topological nodal materials and platforms for twistronics. Furthermore, we show the Nielsen-Ninomiya theorem can be extended to 2D non-Hermitian systems hosting topologically protected exceptional points and Fermi points for the 17 wallpaper groups and use the violation of the theorem on the surface to classify 3D Hermitian and non-Hermitian topological bulks.