THERMALLY DILUTED ISING SYSTEMS

Fractals ◽  
2003 ◽  
Vol 11 (supp01) ◽  
pp. 53-65
Author(s):  
MANUEL I. MARQUÉS ◽  
JULIO A. GONZALO ◽  
JORGE ÍÑIGUEZ
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Disordered Systems ◽  
Short Range ◽  
Finite Size ◽  
Universality Class ◽  
Finite Size Scaling ◽  
Local Distribution ◽  
Ising Systems ◽  
Theoretical Predictions

In this paper finite size scaling techniques are used to study the universality class of thermally diluted Ising systems, in which the realization of the disposition of magnetic atoms and vacancies is taken from the local distribution of spins in the pure original Ising model at criticality. The critical temperature, the critical exponents and therefore the universality class of these thermally diluted Ising systems depart markedly from the ones of short range correlated disordered systems. This result is in agreement with theoretical predictions previously made by Weinrib and Halperin for systems with long range correlated disorder.

THE RESTRICTED SPIN MODEL

1990 ◽  
Vol 04 (16) ◽  
pp. 1029-1041
Author(s):  
H.A. FARACH ◽  
R.J. CRESWICK ◽  
C.P. POOLE
Keyword(s):  
Monte Carlo ◽  
Critical Temperature ◽  
Heisenberg Model ◽  
Anisotropy Parameter ◽  
Finite Size ◽  
Universality Class ◽  
Spin Model ◽  
Scaling Analysis ◽  
Finite Size Scaling ◽  
Monte Carlo Calculations

We present a novel anisotropic Heisenberg model in which the classical spin is restricted to a region of the unit sphere which depends on the value of the anisotropy parameter Δ. In the limit Δ→1, we recover the Ising model, and in the limit Δ→0, the isotopic Heisenberg model. Monte Carlo calculations are used to compare the critical temperature as a function of the anisotropy parameter for the restricted and unrestricted models, and finite-size scaling analysis leads to the conclusion that for all Δ>0 the model belongs to the Ising universality class. For small A the critical behavior is clearly seen in histograms of the transverse and longitudinal (z) components of the magnetization.

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

2009 ◽  
Vol 64 (12) ◽  
pp. 849-854 ◽  
Author(s):  
Ziya Merdan ◽  
Mehmet Bayirli ◽  
Mustafa Kemal Ozturk
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Specific Heat ◽  
Finite Size ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Infinite Lattice ◽  
Straight Line ◽  
Binder Parameter

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 7-DIMENSIONAL ISING MODEL

2007 ◽  
Vol 21 (04) ◽  
pp. 215-224 ◽  
Author(s):  
Z. MERDAN ◽  
D. ATILLE
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Cellular Automaton ◽  
Specific Heat ◽  
Finite Size ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Infinite Lattice ◽  
Creutz Cellular Automaton ◽  
Binder Parameter

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

2008 ◽  
Vol 22 (13) ◽  
pp. 1329-1341 ◽  
Author(s):  
G. MÜLAZIMOGLU ◽  
A. DURAN ◽  
Z. MERDAN ◽  
A. GUNEN
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Cellular Automaton ◽  
Finite Lattice ◽  
Finite Size ◽  
Scaling Relations ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Infinite Lattice ◽  
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

2007 ◽  
Vol 21 (28) ◽  
pp. 1923-1931 ◽  
Author(s):  
M. KALAY ◽  
Z. MERDAN
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Specific Heat ◽  
Nearest Neighbor ◽  
Finite Size ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Infinite Lattice ◽  
Creutz Cellular Automaton ◽  
Binder Parameter

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

1999 ◽  
Vol 10 (07) ◽  
pp. 1237-1245 ◽  
Author(s):  
N. AKTEKIN ◽  
Ş. ERKOÇ ◽  
M. KALAY
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Cellular Automaton ◽  
Finite Size ◽  
Scaling Relations ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Infinite Lattice ◽  
Creutz Cellular Automaton ◽  
Good Agreement

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.

Sign in / Sign up

Export Citation Format

Share Document