Superfluidity and Density Order in a Bilayer Extended Hubbard Model
  We use cluster dynamical mean field theory to study the phase diagram of the square lattice bilayer Hubbard model with an interlayer interaction. The layers are populated by two-component fermions, and the densities in both layers and the strength of the interactions are varied. We find that an attractive interlayer interaction can induce a checkerboard density ordered phase and superfluid phases, with either interlayer or intralayer pairing. Remarkably, the latter phase does not require an intralayer interaction to be present: it can be attributed to an induced attractive interaction caused by density fluctuations in the other layer. 
Tuomas I. Vanhala, Jildou E. Baarsma, Miikka O. J. Heikkinen, Matthias   Troyer, Ari Harju and P\"aivi T\"orm\"a
cond-mat.quant-gas