  We investigate the nature of the magnetic excitations of a bilayer
single-orbital Hubbard model in the intermediate-coupling regime. This model
exhibits a quantum phase transition (QPT) between a paramagnetic (PM) and an
insulating antiferromagnetic (AFM) phase at a critical value of the coupling
strength. By using the random phase approximation, we show that the QPT is
continuous when the PM state is a band insulator and that the corresponding
quantum critical point (QCP) arises from the condensation of preformed
excitons. These low-energy excitons reemerge on the other side of the QCP as
the transverse and longitudinal modes of the AFM state. In particular, the
longitudinal mode remains sharp for the model parameters relevant to
Sr$_{3}$Ir$_{2}$O$_{7}$ because of the strong easy-axis anisotropy of this
material.
