  The Nielsen-Ninomiya Theorem has 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 that the non-zero 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 work, 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 new 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 new 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.
