  In [H. Ammari et al., Honeycomb-lattice Minnaert bubbles. arXiv:1811.03905],
the existence of a Dirac dispersion cone in a bubbly honeycomb phononic crystal
is shown. The aim of this paper is to prove that, near the Dirac points, the
Bloch eigenfunctions is the sum of two eigenmodes. Each eigenmode can be
decomposed into two components: one which is slowly varying and satisfies a
homogenized equation, while the other is periodic across each elementary
crystal cell and is highly oscillating. The slowly oscillating components of
the eigenmodes satisfy a system of Dirac equations. Our results in this paper
proves for the first time a near-zero effective refractive index near the Dirac
points for the plane-wave envelopes of the Bloch eigenfunctions in a
sub-wavelength metamaterial. They are illustrated by a variety of numerical
examples. We also compare and contrast the behaviour of the Bloch
eigenfunctions in the honeycomb crystal with that of their counterparts in a
bubbly square crystal, near the corner of the Brillouin zone, where the maximum
of the first Bloch eigenvalue is attained.
