Quantum magnetism and counterflow supersolidity of up-down bosonic dipoles

We study a gas of dipolar bosons confined in a two-dimensional optical lattice. Dipoles are considered to point freely in both up and down directions perpendicular to the lattice plane. This results in a nearest-neighbor repulsive (attractive) interaction for aligned (anti-aligned) dipoles. We find regions of parameters where the ground state of the system exhibits insulating phases with ferromagnetic or anti-ferromagnetic ordering, as well as with rational values of the average magnetization. Evidence for the existence of a novel counterflow supersolid quantum phase is also presented.