Generalized fermion doubling theorems: Classification of two-dimensional nodal systems in terms of wallpaper groups

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