Proof of the universality of mode symmetries in creating photonic Dirac cones

We formulate a degenerate perturbation theory for the vector electromagnetic field of periodic structures and apply it to the problem of the creation of Dirac cones in the Brillouin-zone center by accidental degeneracy of two modes. We derive a necessary condition by which we can easily select candidates of mode combinations that enable the creation of the Dirac cone. We analyze the structure of a matrix that determines the first-order correction to eigen frequencies by examining its transformation by symmetry operations. Thus, we can obtain the analytical solution of dispersion curves in the vicinity of the zone center and can judge the presence of the Dirac cone. All these findings clearly show that the presence or absence of the Dirac cone in the zone center is solely determined by the spatial symmetry of the two modes.