  We study the ordering of the spin and the chirality in the fully frustrated
XY model on a square lattice by extensive Monte Carlo simulations. Our results
indicate unambiguously that the spin and the chirality exhibit separate phase
transitions at two distinct temperatures, i. e. , the occurrence of the
spin-chirality decoupling. The chirality exhibits a long-range order at
T_c=0.45324(1) via a second-order phase transition, where the spin remains
disordered with a finite correlation length \xi_s(T_c) \sim 120. The critical
properties of the chiral transition determined from a finite-size scaling
analysis for large enough systems of linear size L > \xi_s(T_c) are well
compatible with the Ising universality. On the other hand, the spin exhibits a
phase transition at a lower temperature T_s=0.4418(5) into the
quasi-long-range-ordered phase. We found \eta(T_s)=0.201(1), suggesting that
the universality of the spin transition is different from that of the
conventional Kosterlitz-Thouless (KT) transition.
