Finite-size scaling relations of the four-dimensional Ising model on the Creutz cellular automaton

The four-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4 ≤ L ≤ 8. 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 for the 7, 14, and 21 independent simulations. The approximate values for the critical temperature of the infinite lattice, Tc(∞) = 6.6965(35), 6.6961(30), 6.6960(12), 6.6800(3), 6.6801(2), 6.6802(1) and 6.6925(22) (without logarithmic factor), 6.6921(22) (without logarithmic factor), 6.6909(2) (without logarithmic factor), 6.6822(13) (with logarithmic factor), 6.6819(11) (with logarithmic factor), 6.6808(8) (with logarithmic factor) are obtained from the intersection points of specific heat curves, the Binder parameter curves and the straight line fit of specific heat maxima for the 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results, 6.6802(1) and 6.6808(8), are in very good agreement with the series expansion results of Tc(∞) = 6.6817(15), 6.6802(2), the dynamic Monte Carlo result of Tc(∞) = 6.6803(1), the cluster Monte Carlo result of Tc(∞) = 6.680(1) and the Monte Carlo using Metropolis and Wolff-cluster algorithm of Tc(∞) = 6.6802632 ± 5⋅10⁻⁵. The average values obtained for the critical exponent of the specific heat are calculated as α = –0.0402(15), –0.0393(12), –0.0391(11) for the 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained result, α = –0.0391(11), is agreement with the series expansions results of α = –0.12 ± 0.03 and the Monte Carlo using Metropolis and Wolff-cluster algorithm of α ≥ 0±0.04. However, α = –0.0391(11) isn’t consistent with the renormalization group prediction of α = 0

Finite-size scaling above the upper critical dimension in Ising models with long-range interactions

The correlation length plays a pivotal role in finite-size scaling and hyperscaling at continuous phase transitions. Below the upper critical dimension, where the correlation length is proportional to the system length, both finite-size scaling and hyperscaling take conventional forms. Above the upper critical dimension these forms break down and a new scaling scenario appears. Here we investigate this scaling behaviour in one-dimensional Ising ferromagnets with long-range interactions. We show that the correlation length scales as a non-trivial power of the linear system size and investigate the scaling forms. For interactions of sufficiently long range, the disparity between the correlation length and the system length can be made arbitrarily large, while maintaining the new scaling scenarios. We also investigate the behavior of the correlation function above the upper critical dimension and the modifications imposed by the new scaling scenario onto the associated Fisher relation.Comment: 16 pages, 5 figure

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

WOS: 000293794700004The four-dimensional Ising model is simulated on Creutz cellular automatons using finite lattices with linear dimensions 4 = 0+/-0.04. However, alpha=-0.0391(11) is inconsistent with the renormalization group prediction of alpha=0. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3610180

Finite-size scaling relations of the four-dimensional Ising model on the Creutz cellular automaton

The four-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4 ? L ? 8. 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 for the 7, 14, and 21 independent simulations. The approximate values for the critical temperature of the infinite lattice, Tc(?) = 6.6965(35), 6.6961(30), 6.6960(12), 6.6800(3), 6.6801(2), 6.6802(1) and 6.6925(22) (without logarithmic factor), 6.6921(22) (without logarithmic factor), 6.6909(2) (without logarithmic factor), 6.6822(13) (with logarithmic factor), 6.6819(11) (with logarithmic factor), 6.6808(8) (with logarithmic factor) are obtained from the intersection points of specific heat curves, the Binder parameter curves and the straight line fit of specific heat maxima for the 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results, 6.6802(1) and 6.6808(8), are in very good agreement with the series expansion results of Tc(?) = 6.6817(15), 6.6802(2), the dynamic Monte Carlo result of Tc(?) = 6.6803(1), the cluster Monte Carlo result of Tc(?) = 6.680(1) and the Monte Carlo using Metropolis and Wolff-cluster algorithm of Tc(?) = 6.6802632 ± 5-10-5 The average values obtained for the critical exponent of the specific heat are calculated as ? =-0.0402(15),-0.0393(12),-0.0391(11) for the 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained result, ? =-0.0391(11), is agreement with the series expansions results of ? =-0.12 ± 0.03 and the Monte Carlo using Metropolis and Wolff-cluster algorithm of a > 0+0.04. However, ? =-0.0391(11) isn't consistent with the renormalization group prediction of ? = 0. © Z. Merdan and E. Güzelsoy, 2011

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. alpha=0.009, is also in agreement with alpha = 0 predicted by the theory.C1 [Merdan, Z.] Gaziosmanpasa Univ, Dept Phys, TR-60250 Tokat, Turkey.[Kalay, M.] Pamukkale Univ, Dept Phys, TR-21100 Denizli, Turkey

The Effect of the Number of Simulations on the Exponents Obtained by Finite-Size Scaling Relations of the Order Parameter and the Magnetic Susceptibility for the Four-Dimensional Ising Model on the Creutz Cellular Automaton

WOS: 000302693400031The four-dimensional Ising model is simulated on the Creutz cellular automaton using finite-size lattices with linear dimension 4a parts per thousand currency signLa parts per thousand currency sign8. The exponents in the finite-size scaling relations for the order parameter and the magnetic susceptibility at the finite-lattice critical temperature are computed to be beta=0.49(7), beta=0.49(5), beta=0.50(1) and gamma=1.04(4), gamma=1.03(4), gamma=1.02(4) for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are consistent with the renormalization group predictions of beta=0.5 and gamma=1. The values for the critical temperature of the infinite lattice T (c) (a)=6.6788(65), T (c) (a)=6.6798(69), T (c) (a)=6.6802(70) are obtained from the straight-line fit of the magnetic susceptibility maxima using 4a parts per thousand currency signLa parts per thousand currency sign8 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 series expansion results of T (c) (a)=6.6817(15), T (c) (a)=6.6802(2), the dynamic Monte Carlo result of T (c) (a)=6.6803(1), the cluster Monte Carlo result of T (c) (a)=6.680(1) and the Monte Carlo using Metropolis and Wolff-cluster algorithm result of T (c) (a)=6.6802632 +/- 5x10(-5)

The test of a new critical exponent by using ising model on the creutz cellular automaton

Above the upper critical dimension d_{c} the Ising model is simulated on the Creutz cellular automaton. The values of a new critical exponent Ϙ are obtained by using the simulations for the order parameter and the magnetic susceptibility. At d=4,5,6,7,8, the values of the new critical exponent Ϙ are 0.9904(16), 1.2721(2), 1.4806(24), 1.7626(17), 1.9997(50) for the order parameter, respectively, while those 1.0415(13), 1.2987(27), 1.5133(1), 1.7741(1), 2.0133(28) are for the magnetic susceptibility in the same order. The computed values of the new critical exponent Ϙ are in agreement with theoretical values

The Finite-Size Scaling Study of Five-Dimensional Ising Model

WOS: 000379821800005The five-dimensional ferromagnetic Ising model is simulated on the Creutz cellular automaton algorithm using finite-size lattices with linear dimension 4 <= L <= 8. The critical temperature value of infinite lattice is found to be T-chi(infinity) = 8.7811 (1) using 4 <= L <= 8 which is also in very good agreement with the precise result. The value of the field critical exponent (delta = 3.0067 (2)) is good agreement with delta = 3 which is obtained from scaling law of Widom. 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 2.5080 (1), 2.5005 (3) and 1.2501 (1) using 4 <= L <= 8, respectively, which are in very good agreement with the theoretical predictions of 5/2 and 5/4. The finite-size scaling plots of magnetic susceptibility and the order parameter verify the finite-size scaling relations about the infinitelattice temperature

The Test of a New Critical Exponent (sic) by Using Ising Model on the Creutz Cellular Automaton

Above the upper critical dimension d(c) the Ising model is simulated on the Creutz cellular automaton. The values of a new critical exponent (sic) are obtained by using the simulations for the order parameter and the magnetic susceptibility. At d = 4; 5; 6; 7; 8, the values of the new critical exponent (sic) are 0.9904(16), 1.2721(2), 1.4806(24), 1.7626(17), 1.9997(50) for the order parameter, respectively, while those 1.0415(13), 1.2987(27), 1.5133(1), 1.7741(1), 2.0133(28) are for the magnetic susceptibility in the same order. The computed values of the new critical exponent (sic) are in agreement with theoretical values