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Finite temperature orbital state in a ferromagnetic Mott insulator withtriply-degenerate $t_{2g}$ orbital is investigated numerically. We employ thequantum Monte Carlo simulation with the loop algorithm. Indications forconventional staggered-type orbital order are not remarkable down to the lowesttemperature to which the present simulation can get access. Physical parametersmonitoring the off-diagonal orbital order, which is characterized by a linearcombination of the $(d_{yz}, d_{zx}, d_{xy})$ orbital-wave functions with equalweights, are not conspicuous. It is found that a orbital gap-like behaviorappears in the uniform orbital susceptibility. This is supported by a thresholdbehavior in the staggered correlation function in a calculation with theadditional Ising-type interaction. Some rigorous remarks for the long-rangeorbital order are also presented.

Numerical study of t2g orbital system with ferromagnetic polarization