Infinite-lattice Critical Temperature Research Articles

The Finite-Size Scaling Relation for the Order-Parameter Probability Distribution of the Six-Dimensional Ising Model

The six dimensional Ising model with nearest-neighbor pair interactions has been simulated and verified numerically on the Creutz Cellular Automaton by using five bit demons near the infinite-lattice critical temperature with the linear dimensions L=4,6,8,10. The order parameter probability distribution for six dimensional Ising model has been calculated at the critical temperature. The constants of the analytical function have been estimated by fitting to probability function obtained numerically at the finite size critical point.

Determination of critical linear lattice size for the four dimensional Ising model on the Creutz cellular automaton

The four-dimensional Ising model is simulated on the Creutz cellular automaton (CCA) near the infinite-lattice critical temperature for the lattice with the linear dimension 4 ⩽ L ⩽ 22. The temperature dependence of Binder parameter (gL) are analyzed for the lattice with the linear dimension 4 ⩽ L ⩽ 22. In this study conducted highly detailed, two different types of behavior were determined as a result of varying linear lattice dimension. The infinite lattice critical temperatures are obtained to be Tc = 6.6845 ± 0.0005 in interval 4 ⩽ L ⩽ 12 and Tc = 6.6807 ± 0.0024 in interval 14 ⩽ L ⩽ 22. The finite and infinite lattice critical exponents for the order parameter, the magnetic susceptibility and the specific heat are computed from the results of simulations by using finite-size scaling relations. Critical linear lattice size have been identified as L = 14.

Finite-size scaling relations for a four-dimensional Ising model on Creutz cellular automatons

The four-dimensional Ising model is simulated on Creutz cellular automatons using finite lattices with linear dimensions 4 ≤ L ≤ 8. The temperature variations and finite-size scaling plots of the specific heat and the Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature for 7, 14, and 21 independent simulations. Approximate values for the critical temperature of the infinite lattice of Tc(∞) = 6.6965(35), 6.6961(30), 6.6960(12), 6.6800(3), 6.6801(2), 6.6802(1) and 6.6925(22) (without the logarithmic factor), 6.6921(22) (without the logarithmic factor), 6.6909(2) (without the logarithmic factor), 6.6822(13) (with the logarithmic factor), 6.6819(11) (with the logarithmic factor), and 6.6808(8) (with the logarithmic factor) are obtained from the intersection points of the specific heat curves, the Binder parameter curves, and straight line fits of specific heat maxima for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the results, 6.6802(1) and 6.6808(8), are in very good agreement with the results of a series expansion of Tc(∞), 6.6817(15) and 6.6802(2), the dynamic Monte Carlo value Tc(∞) = 6.6803(1), the cluster Monte Carlo value Tc(∞) = 6.680(1), and the Monte Carlo value using the Metropolis-Wolff cluster algorithm Tc(∞) = 6.6802632 ± 5 · 10−5. The average values calculated for the critical exponent of the specific heat are α =− 0.0402(15), − 0.0393(12), − 0.0391(11) with 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the result, α =− 0.0391(11), agrees with the series expansions result, α =− 0.12 ± 0.03 and the Monte Carlo result using the Metropolis-Wolff cluster algorithm, α ≥ 0 ± 0.04. However, α =− 0.0391(11) is inconsistent with the renormalization group prediction of α = 0.

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 EFFECT OF THE INCREASE OF LINEAR DIMENSIONS ON EXPONENTS OBTAINED BY FINITE-SIZE SCALING RELATIONS FOR THE FOUR-DIMENSIONAL ISING MODEL ON THE CREUTZ CELLULAR AUTOMATON

The four-dimensional Ising model is simulated on the Creutz cellular automaton using finite-size lattices with linear dimension 4 ≤ L ≤ 22. The exponents in the finite-size scaling relations for the order parameter, the magnetic susceptibility at the finite-lattice critical temperature and the specific heat at the infinite-lattice critical temperature are computed to be β = 0.5072(58), γ = 1.0287(56) and α = -0.096(17), respectively, which are consistent with the renormalization group prediction of β = 0.5, γ = 1 and α = 0. The critical temperatures for the infinite lattice are found to be [Formula: see text] and [Formula: see text], which are also consistent with the precise results.

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.

The effect of the number of simulations on the exponents obtained by finite-size scaling relations for the seven-dimensional Ising model on the Creutz cellular automaton

The seven-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattice with the linear dimension 4⩽ L⩽8. The exponents in the finite-size scaling relations for the magnetic susceptibility, the order parameter and the specific heat at the infinite-lattice critical temperature are computed to be 3.544(24), 3.505(55), 3.501(52), 1.78(64), 1.77(38), 1.76(46) and 0.02(11), 0.01(17), 0.01(16) using 4⩽ L⩽8 for 7,14 and 21 independent simulations, respectively. As the number of independent simulations increase, exponents are in very good agreement with the theoretical predictions of 7/2, 7/4 and 0, respectively. The values for the critical temperature of infinite lattice 12.870(12), 12.870(10), 12.869(3) and 12.871(15), 12.871(5), 12.871(5) are obtained from the straight line fit of the magnetic susceptibility maxima and the specific heat using 4⩽ L⩽8 for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are in very good agreement with the more precise value of 12.86902(33).

The test of the finite-size scaling relations of the Ising models in seven and eight dimensions on the Creutz cellular automaton

The Ising models in seven and eight dimensions are simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4 ⩽ L ⩽ 8 . Three different finite-size exponents for the order parameter near the curie point are computed to be 1.74 (3), 0.95 (2), 2.42 (5), 1.99 (19), 0.96 (1) and 2.95 (11) for d = 7 and d = 8 dimensions, respectively. The obtained results are in good agreement with the theoretical predictions, 7 4 , 1, 5 2 , 2, 1, 3 for d = 7 and 8 dimensions, respectively. The exponent in the finite-size scaling relation for the magnetic suscebtibility at the infinite-lattice critical temperature is computed to be 4.03 (9) using 4 ⩽ L ⩽ 8 , which is in very good agreement with the theoretical prediction of 4 for d = 8 dimension. The finite-size scaling relation for the magnetic susceptibility at the infinite-lattice critical temperature is also valid for the maxima of the magnetic susceptibilities of the finite-size lattices. The finite-size scaling plots of the order parameter and the magnetic susceptibility verify the finite-size scaling relations about the infinite-lattice critical temperature.

The effect of the increase of linear dimensions on exponents obtained by finite-size scaling relations for the six-dimensional Ising model on the Creutz cellular automaton

The six-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4 ⩽ L ⩽ 12. The exponents in the finite-size scaling relations for the magnetic susceptibility, the order parameter and the specific heat at the infinite-lattice critical temperature are computed to be 3.00(12), 1.49(12) and 0.001(68) using 4 ⩽ L ⩽ 10, respectively, which are in very good agreement with the theoretical predictions of 6 2 , 6 4 and 0. The critical temperature for the infinite lattice is found to be 10.8347(52) using 6 ⩽ L ⩽ 12 which is also in very good agreement with the precise results. The finite-size scaling relation for magnetic susceptibility at the infinite-lattice critical temperature is also valid for the maxima of magnetic susceptibilities of the finite-size lattices. The finite-size scaling plots of magnetic susceptibility, the order parameter and the specific heat verify the finite-size scaling relations about the infinite-lattices temperature.

The finite-size scaling study of the specific heat and the Binder parameter for the six-dimensional Ising model

The six-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 , 10 . 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 = 10.838 ( 1 ) , T C = 10.836 ( 20 ) and T C = 10.835 ( 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 more precise value of T C = 10.835 ( 5 ) . The value obtained for the critical exponent of the specific heat, i.e., α = 0.012 ( 2 ) is also in agreement with α = 0 predicted by the theory.

The test of the finite-size scaling relations for the seven-dimensional Ising model on the Creutz cellular automaton

The seven-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4⩽ L⩽8. The exponents in the finite-size scaling relations for the magnetic susceptibility and the order parameter at the infinite-lattice critical temperature are computed to be 3.52(8) and 1.72(7) using 4⩽ L⩽8, respectively, which are in very good agreement with the theoretical predictions of 7/2 and 7/4. The finite-size scaling relation for the magnetic susceptibility at the infinite-lattice critical temperature is also valid for the maxima of the magnetic susceptibilities of the finite-size lattices. The finite-size scaling plots of the magnetic susceptibility and the order parameter verify the finite-size scaling relations about the infinite-lattice critical temperature.

The test of the finite-size scaling relations for the six-dimensional Ising model on the Creutz cellular automaton

The six-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4⩽L⩽10. The exponents in the finite-size scaling relations for the magnetic susceptibility and the order parameter at the infinite-lattice critical temperature are computed to be 3.00(12) and 1.47(10) using 6⩽L⩽10, respectively, which are in very good agreement with the theoretical predictions of 62 and 64. The critical temperature for the infinite lattice is found to be 10.835(5) using 4⩽L⩽10 which is also in very good agreement with the precise results. The finite-size scaling relation for the magnetic susceptibility at the infinite-lattice critical temperature is also valid for the maxima of the magnetic susceptibilities of the finite-size lattices. The finite-size scaling plots of the magnetic susceptibility and the order parameter verify the finite-size scaling relations about the infinite-lattice critical temperature.

THE TEST OF THE FINITE-SIZE SCALING RELATIONS FOR THE FIVE-DIMENSIONAL ISING MODEL ON THE CREUTZ CELLULAR AUTOMATON

The five-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4≤L≤16. The exponents in the finite-size scaling relations for the magnetic susceptibility and the order parameter at the infinite-lattice critical temperature are computed to be 2.52(7) and 1.25(11) using 6≤L≤12, respectively, which are in very good agreement with the Monte Carlo results and with the theoretical predictions of 5/2 and 5/4. The critical temperature for the infinite lattice is found to be 8.779(8) using 8≤L≤16 which is also in very good agreement with the recent precise results.