  We present a microscopic theory of carrier-induced ferrimagnetism in metallic
double perovskite compounds such as ${\rm Sr}_{2}{\rm FeMoO}_{6}$ and ${\rm
Sr}_{2}{\rm FeReO}_{6}$ which have recently attracted intense interest for
their possible applications to magnetotransport devices. The theory is based on
an effective "Kondo-like" Hamiltonian treated here within the large-$S$
expansion. We find that depending on the value of the carrier density the
ground state is either a ferrimagnet or a layered antiferromagnet. The
ferrimagnetic state has a robust half-metallic electronic structure. The
transition to antiferromagnetic phase is first order accompanied with the
regime of phase separation. We study spin wave spectrum including quantum
corrections and find strongly enhanced quantum effects in the vicinity of
zero-temperature phase transition.
