  We studied the possibility of exciton condensation in Mott insulating
bilayers. In these strongly correlated systems an exciton is the bound state of
a double occupied and empty site. In the strong coupling limit the exciton acts
as a hard-core boson. Its physics are captured by the exciton t-J model,
containing an effective XXZ model describing the exciton dynamics only. Using
numerical simulations and analytical mean field theory we constructed the
ground state phase diagram. Three homogeneous phases can be distinguished: the
antiferromagnet, the exciton checkerboard crystal and the exciton superfluid.
For most model parameters, however, we predict macroscopic phase separation
between these phases. The exciton superfluid exists only for large exciton
hopping energy. Additionally we studied the collective modes and
susceptibilities of the three phases. In the superfluid phase we find the
striking feature that the bandwidth of the spin-triplet excitations,
potentially detectable by resonant inelastic x-ray scattering (RIXS), is
proportional to the superfluid density. The superfluid phase mode is visible in
the charge susceptibility, measurable by RIXS or electron energy loss
spectroscopy (EELS).
