Monte Carlo static and dynamic simulations of a three-dimensional Ising critical model

The critical dynamics of 'model A" of Hohenberg and Halperin has been studied by the Monte Carlo method. Simulations have been carried out in the three-dimensional (3d) simple cubic Ising model for lattices of sizes L=16 to L=512. Using Wolff's cluster algorithm, the critical temperature is precisely found as β_{c}=0.22165468(5). By Fourier transform of the lattice configurations, the critical scattering intensities I(q[over ⃗]) can be obtained. After circular averaging, the static simulations with L=256 and L=512 provide an estimate of the critical exponent γ/ν=2-η=1.9640(5). The |q[over ⃗]|-dependent distribution of I(q[over ⃗]) showed an exponential distribution, corresponding to a Gaussian distribution of the scattering amplitudes for a large q domain. The time-dependent intensities were then used for the study of the critical dynamics of 3d lattices at the critical point. To simulate results of an x-ray photon correlation spectroscopy experiment, the time-dependent correlation function of the intensities was studied for each |q[over ⃗]|-value. In the q region where I(q[over ⃗]) had an exponential distribution, the time correlations can be fit to a stretched exponential, where the exponent μ=γ/νz≃0.975 provides an estimate of the dynamic exponent z. This corresponds to z=2.0145, in agreement with the observed variations of the characteristic fluctuation time of the intensity: τ(q)∝q^{-z}, which gives z=2.015. These results agree with the ε expansion of field-theoretical methods (2.017). In this paper, the need to take account of the anomalous time behavior (μ<1) in the dynamics is exemplified. This dynamics reflects a nonlinear time behavior of model A, and its large time extension is discussed in detail.