Dirac Cones at k=0 in Phononic Crystals

We show that two-dimensional phononic crystals exhibit Dirac cone dispersion at k=0 by exploiting dipole and quadrupole accidental degeneracy. While the equi-frequency surface of Dirac cone modes is isotropic, such systems exhibit "super-anisotropy", meaning that only transverse waves are allowed along certain directions while only longitudinal waves are allowed along some other directions. Only one mode, not two, is allowed near the Dirac point, and only two effective parameters, not four, are needed to describe the dispersion. Effective medium theory finds that the phononic crystals have effectively zero mass density and zero 1/C44 at the Dirac point. Numerical simulations are used to demonstrate the unusual elastic wave properties near the Dirac point frequency.