  We study the ground state properties of the interacting spinless fermions in
the $p_{x,y}$-orbital bands in the two dimensional honeycomb optical lattice,
which exhibit different novel features from those in the $p_z$-orbital system
of graphene. In addition to two dispersive bands with Dirac cones, the
tight-binding band structure exhibits another two completely flat bands over
the entire Brillouin zone. With the realistic sinusoidal optical potential, the
flat bands acquire a finite but much smaller band width compared to the
dispersive bands. The band flatness dramatically enhanced interaction effects
giving rise to various charge and bond ordered states at commensurate fillings
of $n=\frac{i}{6} (i=1 \sim 6)$. At $n=1/6$, the many-body ground states can be
exactly solved as the close packed hexagon states which can be stabilized even
in the weak interacting regime. The dimerization of bonding strength occurs at
both $n=1/2$ and 5/6, and the latter case is accompanied with the charge
density wave of holes. The trimerization of bonding strength and charge
inhomogeneity appear at $n={1/3},{2/3}$. These crystalline orders exhibit
themselves in the noise correlations of the time of flight spectra.

Strongly Correlated Electrons;Materials Science