Density matrix approach to the orbital ordering in the spinel vanadates: A case study

In this work we apply the density matrices approach to orbital ordering (OO) in order to study the OO of the spinel vanadates AV$_{2}$O$_{4}$ (A $\equiv$ Zn, Cd and Mg), which is normally believed to be responsible for the structural transition from cubic to tetragonal phase observed in these compounds. The density matrices of vanadium atoms are obtained by using {\it state-of-the-art} full-potential linearized augmented plane wave method based GGA+U calculations. In the absence of spin-orbit coupling, the present study shows the existence of anti-ferro OO in the global (local octahedral) coordinate system where $d_{xz}$ and $d_{yz}$ ($d_{xz}$+$d_{yz}$ and $d_{xz}$-$d_{yz}$) orbitals are mainly occupied at the neighboring V sites for all the compounds.