Orbital ordering and one-dimensional magnetic correlation in vanadium   spinel oxides AV2O4 (A = Zn, Mg, or Cd)
  We present our theoretical results on the mechanism of two transitions in vanadium spinel oxides $A$V$_2$O$_4$ ($A$=Zn, Mg, or Cd) in which magnetic V cations constitute a geometrically-frustrated pyrochlore structure. We have derived an effective spin-orbital-lattice coupled model in the strong correlation limit of the multiorbital Hubbard model, and applied Monte Carlo simulation to the model. The results reveal that the higher-temperature transition is a layered antiferro-type orbital ordering accompanied by tetragonal Jahn-Teller distortion, and the lower-temperature transition is an antiferromagnetic spin ordering. The orbital order lifts the magnetic frustration partially, and induces spatial anisotropy in magnetic exchange interactions. In the intermediate phase, the system can be considered to consist of weakly-coupled antiferromagnetic chains lying in the perpendicular planes to the tetragonal distortion. 
Yukitoshi Motome and Hirokazu Tsunetsugu