Strongly spin-orbit coupled spin-3/2 model for $5d^1$ $AB_2O_4$ spinels

Research on 5d transition metal oxides has been more and more active recently. Unlike in 3d transition metals, the strong spin orbit interaction cannot be treated as a perturbation. The competition between correlation, spin orbit coupling and the kinetic energy of 5d electrons makes the problem nontrivial. We model the $AB_2O_4$ spinel structure with single d electron on atom B as a Hubbard type model. By treating the hopping term perturbatively under large spin orbit coupling we derive an effective spin 3/2 model which is not Heisenberg-like. We further investigate the possible phase diagram of the effective spin 3/2 model.