Flat bands and Wigner crystallization in the honeycomb optical lattice

We study the ground states of cold atoms in the tight-binding bands built from p-orbitals on a two dimensional honeycomb optical lattice. The band structure includes two completely flat bands. Exact many-body ground states with on-site repulsion can be found at low particle densities, for both fermions and bosons. We find crystalline order at n=1/6 with a $\sqrt{3} \times \sqrt{3}$ structure breaking a number of discrete lattice symmetries. In fermionic systems, if the repulsion is strong enough, we find the bonding strength becomes \emph{dimerized} at n=1/2. Experimental signatures of crystalline order can be detected through the noise correlations in time of flight experiments.