  We present our theoretical and numerical results on thermodynamic properties
and the microscopic mechanism of two successive transitions in vanadium spinel
oxides $A$V$_2$O$_4$ ($A$=Zn, Mg, or Cd) obtained by Monte Carlo calculations
of an effective spin-orbital-lattice model in the strong correlation limit.
Geometrical frustration in the pyrochlore lattice structure of V cations
suppresses development of spin and orbital correlations, however, we find that
the model exhibits two transitions at low temperatures. First, a discontinuous
transition occurs with an orbital ordering assisted by the tetragonal
Jahn-Teller distortion. The orbital order reduces the frustration in spin
exchange interactions, and induces antiferromagnetic correlations in
one-dimensional chains lying in the perpendicular planes to the tetragonal
distortion. Secondly, at a lower temperature, a three-dimensional
antiferromagnetic order sets in continuously, which is stabilized by the
third-neighbor interaction among the one-dimensional antiferromagnetic chains.
Thermal fluctuations are crucial to stabilize the collinear magnetic state by
the order-by-disorder mechanism. The results well reproduce the experimental
data such as transition temperatures, temperature dependence of the magnetic
susceptibility, changes of the entropy at the transitions, and the magnetic
ordering structure at low temperatures. Quantum fluctuation effect is also
examined by the linear spin wave theory at zero temperature. The staggered
moment in the ground state is found to be considerably reduced from saturated
value, and reasonably agrees with the experimental data.
