Designing Dirac points in two-dimensional lattices

We present a framework to elucidate the existence of accidental contacts of energy bands, particularly those called Dirac points which are the point contacts with linear energy dispersions in their vicinity. A generalized von Neumann-Wigner theorem we propose here gives the number of constraints on the lattice necessary to have contacts without fine tuning of lattice parameters. By counting this number, one could search for the candidate of Dirac systems without solving the secular equation. The constraints can be provided by any kinds of symmetry present in the system. The theory also enables the analytical determination of a k-point having accidental contact by selectively picking up only the degenerate solution of the secular equation. By using these frameworks, we demonstrate that the Dirac points are feasible in various two-dimensional lattices, e.g., the anisotropic Kagom 'e lattice under inversion symmetry is found to have contacts over the whole lattice parameter space. Spin-dependent cases, such as the spin-density-wave state in LaOFeAs with reflection symmetry, are also dealt with in the present scheme.