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Equilibrium properties of the three-dimensional isotropic Heisenberg spinglass are studied by extensive Monte Carlo simulations, with particularattention to the nature of its phase transition. A finite-size-scaling analysisis performed both for the spin-glass (SG) and the chiral-glass (CG) orders. Ourresults suggest that the model exhibits the CG long-range order at finitetemperatures without accompanying the conventional SG long-range order, incontrast to some of the recent works claiming the simultaneous SG and CGtransition. Typical length and time scales which represent a crossover from thespin-chirality coupling regime at short scales to the spin-chirality decouplingregime at long scales are introduced and examined in order to observe the trueasymptotic transition behavior. On the basis of these crossover scales,discussion is given concerning the cause of the discrepancy between our presentresult and those of other recent numerical works.Comment: 22 pages, 26 figures, minor change

Monte Carlo simulations of the phase transition of the three-dimensional isotropic Heisenberg spin glass