Effect of spin-orbit coupling on magnetic and orbital order in MgV$_2$O$_4$

Recent measurements on MgV$_2$O$_4$ single crystal have reignited the debate on the role of spin-orbit (SO) coupling in dictating the orbital order in Vanadium spinel systems. Density functional theory calculations were performed using the full-potential linearized augmented-plane-wave method within the local spin density approximation (LSDA), Coulomb correlated LSDA+U, and with SO interaction (LSDA+U+SO) to study the magnetic and orbital ordering in low temperature phase of MgV$_2$O$_4$. It is observed that the spin-orbit coupling in the experimentally observed antiferromagnetic phase, affects the orbital order differently in alternate V-atom chains along c-axis. This observation is found to be consistent with the experimental prediction that the effect of spin-orbit coupling is intermediate between that in case of ZnV$_2$O$_4$ and MnV$_2$O$_4$.