Effective Spin Hamiltonian, Magnetic Ground States and Low-Energy Excitations of Double Perovskites*

We investigate the T=0 properties of the generalized double exchange model for double perovskites (DP) A$_{2}$BB$^{\prime}$O$_{6}$. We present exact analytical results for the effective spin Hamiltonian of a system consisting of two unit cells. This is a generalization of the classic Anderson-Hasegawa analysis for manganites to DPs. Using a variational approach for the core spins, together with an exact diagonalization of the conduction electrons, we determine the magnetic phase boundaries as a function of electron concentration. We find that the wide region of stability for the FM phase depends on the magnitude and the sign of direct $B^\prime$-$B^\prime$ hopping and a metallic AFM phase at larger doping. Coulomb correlations on the $B^\prime$ site in a self-consistent Hartree-Fock theory, shows that these are unable to stabilize the FM state beyond a critical filling. We compare our results with experiments on La$_{x}$Sr$_{(2-x)}$FeMoO$_{6}$. We compute the spin wave spectrum, and determine the doping dependence of the effective ferromagnetic exchange and compare with exact results on small clusters.