Phases of Mott-Hubbard bilayers
  A phase diagram of two Mott-Hubbard planes interacting with a short-range Coulomb repulsion is presented. Considering the case of equal amount of doping by holes in one layer as electrons in the other, a holon-doublon inter-layer exciton formation is shown to be a natural consequence of Coulomb attraction. Quasiparticle spectrum is gapped and incoherent below a critical doping $\delta_c$ due to the formation of excitons. A spin liquid insulator (SLI) phase is thus realized without the lattice frustration. The critical value $\delta_{c}$ sensitively depends on the inter-layer interaction strength. In the $tJ$ model description of each layer with the d-wave pairing, $\delta_{c}$ marks the crossover between SLI and d-wave superconductor. The SLI phase, despite being non-superconducting and charge-gapped, still shows electromagnetic response similar to that of a superfluid due to the exciton transport. Including antiferromagnetic order in the $tJ$ model introduces magnetically ordered phases at low doping and pushes the spin liquid phase to a larger inter-layer interaction strength and higher doping concentrations. 
Jung Hoon Han and Chenglong Jia