Magnetic transition and orbital degrees of freedom in vanadium spinels

We propose a scenario for the two phase transitions in $A$V${}_2$O${}_4$ ($A$=Zn, Mg, Cd), based on an effective spin-orbital model on the pyrochlore lattice. At high temperatures, spin correlations are strongly frustrated due to the lattice structure, and the transition at $\sim$50 [K] is an orbital order, supported by Jahn-Teller lattice distortion. This orbital order introduces spatial modulation of spin exchange couplings depending on the bond direction. This partially releases the frustration, and leads to a spin order at $\sim$40 [K]. We also study the stable spin configuration by taking account of third-neighbor exchange couplings and quantum fluctuations. The result is consistent with the experimental results.