Obtaining critical point and shift exponent for the anisotropic two-layer Ising and Potts models: Cellular automata approach

Using probabilistic cellular automata with the Glauber algorithm, we have precisely calculated the critical points for the anisotropic two-layer Ising and Potts models (K-x not equal K-y not equal K-z) of the square lattice, where K-x and K-y are the nearest-neighbor interactions within each layer in the x and y directions, respectively and Kz is the inter-layer coupling. A general equation is obtained as a function of the inter- and intra-layer interactions (xi, sigma) for both the two-layer Ising and Potts models, separately, where xi = K-z/K-x and sigma = K-y/K-x. Furthermore, the shift exponent for the two-layer Ising and Potts models is calculated. It was demonstrated that in the case of sigma = 1 for the two-layer Ising model, the value of phi = 1.756 (+/- 0.0078) supports the scaling theories' prediction that phi = gamma. However, for the unequal intra-layer couplings for the two-layer Ising model and also in the case of both equal and unequal intra-layer interactions for the two-layer Potts model, our results are different from those obtained from the scaling theories. Finally, an equation is obtained for the shift exponent as a function of intra-layer couplings (a) for the two-layer Ising and Potts models. (c) 2007 Elsevier B.V. All rights reserved.