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 ≠ K_y ≠ 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 K_z is the inter-layer coupling. A general equation is obtained as a function of the inter- and intra-layer interactions (ξ, σ) for both the two-layer Ising and Potts models, separately, where ξ = K_z / K_x and σ = 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 σ = 1 for the two-layer Ising model, the value of φ{symbol} = 1.756 (± 0.0078) supports the scaling theories' prediction that φ{symbol} = γ. 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 (σ) for the two-layer Ising and Potts models.