Proposed Orbital Ordering in MnV$_2$O$_4$ from First-principles Calculations*

Based on density functional calculations, we propose a possible orbital ordering in MnV$_2$O$_4$ which consists of orbital chains running along crystallographic $a$ and $b$ directions with orbitals rotated alternatively by about 45$^{\circ}$ within each chain. We show that the consideration of correlation effects as implemented in the local spin density approximation (LSDA)+U approach is crucial for a correct description of the space group symmetry signifying a strong influence of the correlation-driven orbital ordering on the structural transitions in this system. Inclusion of spin-orbit effects does not seem to influence the orbital ordering pattern. We further find that the proposed orbital arrangement favors a non-collinear magnetic ordering of V spins, as observed experimentally. Exchange couplings among V spins are also calculated and discussed.