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Spin Frustration and Orbital Order in Vanadium Spinels

We present the results of our theoretical study on the effects of geometrical frustration and the interplay between spin and orbital degrees of freedom in vanadium spinel oxides $A$V$_2$O$_4$ ($A$ = Zn, Mg or Cd). Introducing an effective spin-orbital-lattice coupled model in the strong correlation limit and performing Monte Carlo simulation for the model, we propose a reduced spin Hamiltonian in the orbital ordered phase to capture the stabilization mechanism of the antiferromagnetic order. Orbital order drastically reduces spin frustration by introducing spatial anisotropy in the spin exchange interactions, and the reduced spin model can be regarded as weakly-coupled one-dimensional antiferromagnetic chains. The critical exponent estimated by finite-size scaling analysis shows that the magnetic transition belongs to the three-dimensional Heisenberg universality class. Frustration remaining in the mean-field level is reduced by thermal fluctuations to stabilize a collinear ordering.