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

Abstract

Using probabilistic cellular automata with the Glauber algorithm, we have precisely calculated the critical points for the anisotropic two-layer Ising and Potts models (Kx≠Ky≠Kz) of the square lattice, where Kx and Ky 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 (ξ,σ) for both the two-layer Ising and Potts models, separately, where ξ=Kz/Kx and σ=Ky/Kx. 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=1.756(±0.0078) supports the scaling theories’ prediction that ϕ=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.