  Cold atoms in an optical lattice with brick-wall geometry have been used to
mimic graphene, a two-dimensional material with characteristic Dirac
excitations. Here we propose to bring such artificial graphene into the
proximity of a second atomic layer with a square lattice geometry. For
non-interacting fermions, we find that such bilayer system undergoes a phase
transition from a graphene-like semi-metal phase, characterized by a band
structure with Dirac points, to a gapped band insulator phase. In the presence
of attractive interactions between fermions with pseudospin-1/2 degree of
freedom, a competition between semi-metal and superfluid behavior is found at
the mean-field level. Using the quantum Monte Carlo method, we also investigate
the case of strong repulsive interactions. In the Mott phase, each layer
exhibits a different amount of long-range magnetic order. Upon coupling both
layers, a valence-bond crystal is formed at a critical coupling strength.
Finally, we discuss how these bilayer systems could be realized in existing
cold atom experiments.
