Topologically Protected Doubling of Tilted Dirac Fermions in Two Dimensions

The doubling of massless Dirac fermions on two-dimensional lattices is theoretically studied. It has been shown that the doubling of massless Dirac fermions on a lattice with broken chiral symmetry is topologically protected even when the Dirac cone is tilted. This is due to the generalized chiral symmetry defined for lattice systems, where such models can be generated by a deformation of the chiral-symmetric lattice models. The present paper shows for two-band lattice models that this is a general way to produce systems with the generalized chiral symmetry in that such systems can always be transformed back to a lattice model with the conventional chiral symmetry. We specifically show that the number of zero modes is an invariant of the transformation, leading to the topological protection \`{a} la Nielsen-Ninomiya of the doubling of tilted and massless Dirac fermions in two dimensions.