Constructing the Critical Curve for the Two-Layer Potts Model Using Cellular Automata

Abstract

The critical points of the 3-states two-layer Potts model on square lattice for different interlayer couplings (K x =K y ≠K z ) are calculated with high precision using probabilistic cellular automata with Glauber algorithm, 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

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

“…In a recent work, Asgari et al [26] showed that probabilistic CA based on the Glauber algorithm [27] is a fast and reliable simulation method for obtaining the critical point of the two-layer Ising and Potts model in the isotropic case (K x = K y = K z ), 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. They have also shown that this approach is useful for the case of different inter-layer coupling of the twolayer Potts model (K x = K y = K z ) and constructed a critical curve for this model [28]. Although most of the works that have been done until now are for qualitative descriptions or for introducing fast methods for solving various Ising models, we have shown in our previous works that the probabilistic CA increases the calculations precision [26,28].…”

“…They have also shown that this approach is useful for the case of different inter-layer coupling of the twolayer Potts model (K x = K y = K z ) and constructed a critical curve for this model [28]. Although most of the works that have been done until now are for qualitative descriptions or for introducing fast methods for solving various Ising models, we have shown in our previous works that the probabilistic CA increases the calculations precision [26,28].…”

“…(2) and (5)) and for each state, it is sufficient to refer to the probability table and use the proper value of the probability. There are several ways to obtain the critical point for these models which were described in detail in reference [28]. Here, we have used the graph of the magnetization per spin versus time to find the critical point.…”

Cellular Automata Simulation of Two-Layer Ising and Potts Models

“…The previous approach could easily be used for calculation critical point of anisotropic twolayer Ising and Potts models which have different interlayer coupling coefficients ( &Ghaemi, 2006 andGhaemi, 2008 Table 1. The results are compared with other numerical methods and it is shown that they are in good agreement.…”

“…It is shown that the critical point is proportional to ξ and in such a way that it increases when the value of ξ or decreases. As we have mentioned in earlier work (Asgari & Ghaemi, 2006), it is possible to increase the precision of the calculation by increasing the number of lattice size in order to make the system to have less fluctuation and so, determination of the critical point will be easier. Also, it should be noted that the number of time steps should be high enough to determine the critical point especially in the case of fourth and more digits after the decimal point.…”