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The ordering of the three-dimensional isotropic {\it XY} spin glass with thenearest-neighbor random Gaussian coupling is studied by extensive Monte Carlosimulations. To investigate the ordering of the spin and the chirality, wecompute several independent physical quantities including the glass orderparameter, the Binder parameter, the correlation-length ratio, the overlapdistribution and the non-self-averageness parameter, {\it etc}, for both thespin-glass (SG) and the chiral-glass (CG) degrees of freedom. Evidence of thespin-chirality decoupling, {\it i.e.}, the CG and the SG order occurring at twoseparated temperatures, $0<T_{SG}<T_{CG}$, is obtained from the glass orderparameter, which is fully corroborated by the Binder parameter. By contrast,the CG correlation-length ratio yields a rather pathological and inconsistentresult in the range of sizes we studied, which may originate from thefinite-size effect associated with a significant short-length drop-off of thespatial CG correlations. Finite-size-scaling analysis yields the CG exponents$\nu_{CG}=1.36^{+0.15}_{-0.37}$ and $\eta_{CG}=0.26^{+0.29}_{-0.26}$, and theSG exponents $\nu_{SG}=1.22^{+0.26}_{-0.06}$ and$\eta_{SG}=-0.54^{+0.24}_{-0.52}$. The obtained exponents are close to those ofthe Heisenberg SG, but are largely different from those of the Ising SG. Thechiral overlap distribution and the chiral Binder parameter exhibit the featureof a continuous one-step replica-symmetry breaking (1RSB), consistently withthe previous reports. Such a 1RSB feature is again in common with that of theHeisenberg SG, but is different from the Ising one, which may be the cause ofthe difference in the CG critical properties from the Ising SG ones despite ofa common $Z_2$ symmetry.

Monte Carlo simulations of the three-dimensional XY spin glass focusing on the chiral and the spin order