A spin-glass transition has been investigated for a long time but we have not reached a conclusion yet due to difficulties in the simulation studies. They are slow dynamics, strong finite-size effects, and sample-to-sample dependencies. We found that a size of the spin-glass order reaches a lattice boundary within a very short Monte Carlo step. A competition between the spin-glass order and a boundary condition causes these difficulties. Once the boundary effect was removed, physical quantities exhibited quite normal behaviors. They became self-averaging in a limit of large replica numbers. These findings suggest that the nonequilibrium relaxation method is a good choice for solving the difficulties if a lattice size and a replica number are set sufficiently large. A dynamic scaling analysis on nonequilibrium relaxation functions gave a result that the spin-glass transition and the chiral-glass transition occurs at the same temperature in the Heisenberg model in three dimensions. The estimated critical exponent ν agrees with the experimental result.
The finite-size-scaling method in the equilibrium Monte Carlo (MC) simulations and the finite-time-scaling method in the nonequilibrium relaxation simulations are compromised. MC time data of various physical quantities are scaled by the MC time data of the dynamic correlation length. It corresponds to changing the system size in the finite-size-scaling method. This scaling method is tested in the three-dimensional ferromagnetic Ising model and in the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ Ising spin-glass model. The transition temperature and the critical exponents are obtained. We also comment on the definition of the dynamic correlation length in the nonequilibrium relaxation process. The Ornstein-Zernike formula is not always appropriate.