Dirac cones in two-dimensional artificial crystals for classical waves

We present a study on the existence of Dirac cones in two-dimensional phononic crystals. Specifically, we start from the phononic crystals made of rotatable regular hexagonal or triangular rods arranged in a hexagonal lattice. The additional freedom of the rotation enables the crystals with various symmetries. The numerical and experimental band structures manifest consistently that part of the symmetries can support Dirac cones stably, yet the others cannot. The phenomena are explained successfully by the k . p perturbation theory combined with symmetry analysis. Based on this theoretical framework, a systemic exploration on symmetry is further performed, which yields a complete summary on the existence of Dirac cones. The conclusion also remains in the other types of artificial crystals for classical waves, either scalar or vectorial ones.