Orbital ordering in the geometrically frustrated MgV$_2$O$_4$: \emph{Ab initio} electronic structure calculations

In the light of recent interesting experimental work on MgV$_2$O$_4$ we employ the density functional theory to investigate the crucial role played by different interaction parameters in deciding its electronic and magnetic properties. The strong Coulomb correlation in presence of antiferromagnetic (AFM) coupling is responsible for the insulating ground state. In the ground state the $d_{xz}$ and $d_{yz}$ orbitals are ordered and intra-chain vanadium ions are antiferromagnetically coupled. The calculation gives small spin-orbit coupling (SOC), which provides a tilt of $\sim11.3^0$ to the magnetic moment from the z-axis. In the presence of weak SOC and strong exchange coupling, the experimentally observed small magnetic moment and low AFM transition temperature appear to arise from spin fluctuation due to activeness of geometrical frustration.