Relaxation Modes in Random Spin Systems

A method of finding slow relaxation modes in random spin systems is proposed. For stochastic dynamics, an approximate relaxation mode {f_i} and its relaxation rate λ are determined from an eigenvalue problem ∑_j C_i,j(t₀ + t)f_j = exp (−λt)∑_j C_i,j(t₀) f_j, where C_i,j(t) = 〈S_i(t)×Sj(0)〉 is the correlation matrix of spins. The method is applied to the two-dimensional ±J Ising spin glass below the critical temperature of the corresponding nonrandom ferromagnet. It is found through Monte Carlo simulations that the slow relaxation modes obtained by the present method describe the long-time behavior of spins well. The slow relaxation modes are spatially localized and can be regarded as clusters. The distribution of the relaxation rates is consistent with the prediction of the theory which assumes independent motion of clusters.