Quantum magnetic phase transitions in a Kugel-Khomskii model including spin-orbit coupling

Using the formalism of pseudospin and isospin operators the Hamiltonian of an effective Kugel-Khomskii model with spin-orbit coupling is derived with an exact account of the $t_{2g}$ multiplet splitting by the crystal field. An analytical solution is obtained for an arbitrary relation between the Hubbard repulsion and crystal field splitting, i.e., interpolating the cases of Mott-Hubbard and charge-transfer insulators. A description of orbital orders is given in terms of octupole moments. The ground-state phase diagram is constructed in the parameter space spanned by spin-orbit coupling, Hund's exchange, and Hubbard interaction. We investigate a quantum phase transition between a state exhibiting hidden magnetic and orbital long-range order and a ferromagnetic state with a reduced magnetic moment accompanied by antiferroorbital order. It is shown that the cooperative effect of Hund's and spin-orbit interactions gives rise to an easy-plane-type anisotropy.