The generalized Nielsen-Ninomiya Theorem of non-Hermitian lattices for the 17 wallpaper groups

In a 2D Brillouin zone of a non-Hermitian lattice, Fermi points and exceptional points can be topologically protected and immune from any perturbations. These two types of topological points separately obey the Nielsen-Ninomiya doubling theorem since the total charges of the points must be neutralized. However, in the presence of crystalline symmetry, the additional symmetries limit the possible number of the topological points so that the minimal number of the topological points might be more than two; thus, the doubling theorem does not indicate doubling of the points. In this talk, we show the minimal numbers of the non-Hermitian topological points for the 17 wallpaper groups and use this generalized Nielsen-Ninomiya theorem to study the topology of the 3D non-Hermitian bulks.