px,y-orbital counterpart of graphene: Cold atoms in the honeycomb optical lattice

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 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 bandwidth 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 $n=\frac{i}{6}(i=1--6)$. At $n=\frac{1}{6}$, the many-body ground states can be exactly solved as the close-packed hexagon states which can be stabilized even in the weakly interacting regime. The dimerization of bonding strength occurs at both $n=\frac{1}{2}$ and $\frac{5}{6}$, and the latter case is accompanied with the charge-density wave of holes. The trimerization of bonding strength and charge inhomogeneity appear at $n=\frac{1}{3},\frac{2}{3}$. These crystalline orders exhibit themselves in the noise correlations of the time-of-flight spectra.