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We present our theoretical and numerical results on thermodynamic propertiesand the microscopic mechanism of two successive transitions in vanadium spineloxides $A$V$_2$O$_4$ ($A$=Zn, Mg, or Cd) obtained by Monte Carlo calculationsof an effective spin-orbital-lattice model in the strong correlation limit.Geometrical frustration in the pyrochlore lattice structure of V cationssuppresses development of spin and orbital correlations, however, we find thatthe model exhibits two transitions at low temperatures. First, a discontinuoustransition occurs with an orbital ordering assisted by the tetragonalJahn-Teller distortion. The orbital order reduces the frustration in spinexchange interactions, and induces antiferromagnetic correlations inone-dimensional chains lying in the perpendicular planes to the tetragonaldistortion. Secondly, at a lower temperature, a three-dimensionalantiferromagnetic order sets in continuously, which is stabilized by thethird-neighbor interaction among the one-dimensional antiferromagnetic chains.Thermal fluctuations are crucial to stabilize the collinear magnetic state bythe order-by-disorder mechanism. The results well reproduce the experimentaldata such as transition temperatures, temperature dependence of the magneticsusceptibility, changes of the entropy at the transitions, and the magneticordering structure at low temperatures. Quantum fluctuation effect is alsoexamined by the linear spin wave theory at zero temperature. The staggeredmoment in the ground state is found to be considerably reduced from saturatedvalue, and reasonably agrees with the experimental data.Comment: 22 pages, 23 figure

Orbital and magnetic transitions in geometrically frustrated vanadium spinels: Monte Carlo study of an effective spin-orbital-lattice coupled model