Zero Temperature Dynamics of the Weakly-disordered Ising Model

The Glauber dynamics of the pure and weakly-disordered random-bond 2d Ising model is studied at zero-temperature. The coarsening length scale, L(t), is extracted from the equal time correlation function. In the pure case, the persistence probability decreases algebraically with L(t). In the disordered case, three distinct regimes are identified: a short time regime where the behaviour is pure-like; an intermediate regime where the persistence probability decays non-algebraically with time; and a long time regime where the domains freeze and there is a cessation of growth. In the intermediate regime, we find that P (t) ∼ L(t) −θ ′ , where θ ′ = 0.420 ± 0.009. The value of θ ′ is consistent with that found for the pure 2d Ising model at zero-temperature. Our results in the intermediate regime are consistent with a logarithmic decay of the persistence probability with time, P (t) ∼ (ln t) −θ d , where θ d = 0.63 ± 0.01. 2 The 'persistence' problem is concerned with the determination of the fraction of space which persists in the same phase up to some later time. So, for spin systems we are interested in the fraction of spins that have not flipped in some time t. This problem has been studied extensively over the last few years [1-12] and, somewhat surprisingly, the persistence exponent (θ) has been found to be highly non-trivial even for simple one-dimensional models, such as the q−state Potts model at zero temperature [6-7], of non-equilibrium coarsening dynamics. Although Stauffer [3] has performed Monte Carlo simulations in up to 5d, most of the work in higher dimensions has been largely limited to 2d. Numerical studies [1,3] estimate that θ = 0.22±0.03 for the 2d Ising model with Glauber dynamics at T = 0. The analogous exponent for non-equilibrium critical dynamics has also come under intensive investigation [9-12]. Very recently, the persistence problem has been generalised to partial survivors [8]. There has, however, been relatively little published in the literature to date on the persistence problem in systems containing disorder. Here we present the results of a numerical study of an Ising model containing quenched impurities. In this work we study domain growth [13] in a weakly disordered random-bond 2d Ising model and restrict ourselves to zero temperature. The model we work with is given by H = − J ij S i S j (1) 3 where the Ising spins …