Magnetic ordering, orbital ordering, and resonant x-ray scattering in perovskite titanates

The effective Hamiltonian for perovskite titanates is derived by taking into account the threefold degeneracy of $t_{2g}$ orbitals and the strong electron-electron interactions. The magnetic and orbital ordered phases are studied in the mean-field approximation applied to the effective Hamiltonian. A large degeneracy of the orbital states in the ferromagnetic phase is found in contrast to the case of the doubly degenerate $e_g$ orbitals. Lifting of this orbital degeneracy due to lattice distortions and spin-orbit coupling is examined. A general form for the scattering cross section of the resonant x-ray scattering is derived and is applied to the recent experimental results in ${\mathrm{YTiO}}_3.$ The spin wave dispersion relation in the orbital ordered ${\mathrm{YTiO}}_3$ is also studied.