The Finite-Size Scaling Study of the Specific Heat and the Binder Parameter of the Two-Dimensional Ising Model for the Fractals Obtained by Using the Model of Diffusion-Limited Aggregation

The two-dimensional Ising model with nearest-neighbour pair interactions is simulated on the Creutz cellular automaton by using the finite-size lattices with the linear dimensions L = 80, 120, 160, and 200. The temperature variations and the finite-size scaling plots of the specific heat and the Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature. The approximate values for the critical temperature of the infinite lattice Tc = 2.287(6), Tc = 2.269(3), and Tc =2.271(1) are obtained from the intersection points of specific heat curves, Binder parameter curves, and the straight line fit of specific heat maxima, respectively. These results are in agreement with the theoretical value (Tc =2.269) within the error limits. The values obtained for the critical exponent of the specific heat, α = 0.04(25) and α = 0.03(1), are in agreement with α = 0 predicted by the theory. The values for the Binder parameter by using the finite-size lattices with the linear dimension L = 80, 120, 160, and 200 at Tc = 2.269(3) are calculated as gL(Tc) = −1.833(5), gL(Tc) = −1.834(3), gL(Tc) = −1.832(2), and gL(Tc) = −1.833(2), respectively. The value of the infinite lattice for the Binder parameter, gL(Tc) = −1.834(11), is obtained from the straight line fit of gL(Tc) = −1.833(5), gL(Tc) = −1.834(3), gL(Tc) = −1.832(2), and gL(Tc) = −1.833(2) versus L = 80, 120, 160, and 200, respectively

THE FINITE-SIZE SCALING STUDY OF THE SPECIFIC HEAT AND THE BINDER PARAMETER FOR THE FIVE-DIMENSIONAL ISING MODEL

The five-dimensional Ising model with nearest-neighbor pair interactions is simulated on the Creutz cellular automaton by using finite-size lattices with the linear dimensions L=4, 6, 8, 10, 12, 14, and 16. The temperature variations and the finite-size scaling plots of the specific heat and Binder parameter verify the theoretically-predicted expression near the infinite-lattice critical temperature. The approximate values for the critical temperature of the infinite-lattice, T c =8.8063, T c =8.7825 and T c =8.7572, are obtained from the intersection points of specific heat curves, Binder parameter curves and the straight line fit of specific heat maxima, respectively. These results are in agreement with the more precise value of T c =8.7787. The value obtained for the critical exponent of the specific heat, i.e. α=0.009, is also in agreement with α=0 predicted by the theory.

THE FINITE-SIZE SCALING STUDY OF THE SPECIFIC HEAT AND THE BINDER PARAMETER FOR THE 7-DIMENSIONAL ISING MODEL

The 7-dimensional Ising model with nearest-neighbor pair interactions is simulated on the Creutz cellular automaton by using the finite-size lattices with the linear dimensions L = 4, 6, 8. The temperature variations and the finite-size scaling plots of the specific heat and Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature. The approximate values for the critical temperature of the infinite lattice T c = 12.866(2), T c = 12.871(2) and T c = 12.871(49) are obtained from the intersection points of specific heat curves, Binder parameter curves and the straight line fit of specific heat maxima, respectively. These results are in agreement with the more precise values of the Creutz cellular automaton results of T c = 12.8700(42), the 1/d-expansion result of T c = 12.8712, the series expansion result of T c = 12.86902(33), the dynamic Monte Carlo result of T c = 12.8667(50). The values obtained for the critical exponent of the specific heat, i.e., α = 0.011(76), α = -0.002, α = 0.011(5) and α = 0.082(32) corresponding to the above T c values, respectively, are in agreement with α = 0 predicted by the theory. Moreover the values for the Binder parameter are calculated as gL(T c ) = -1.022(28), gL(T c ) = -1.09, gL(T c ) = -1.01(12) and gL(T c ) = -1.34(55) corresponding to the above T c values, respectively.