The Test of the Finite-Size Scaling Relations for the Five-Dimensional Ising Model on the Creutz Cellular Automaton

Abstract

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.

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

“…The dependence of the critical temperatures T χ c (L) obtained from the magnetic susceptibility maxima of the finite-size lattices on linear dimension L is given by the following expression [14,26,[33][34][35][36][37]: agrees with the results obtained previously using different methods [14,26,[37][38][39][40][41][42][43][44][45][46]. For a lattice linear dimension L and very small h at T = T c (L), the order parameter is given by…”

“…By considering different approximate methods, the approximations of the solutions of twodimensional ferromagnetic Ising model are presented [9][10][11][12][13]. In addition, the four-dimensional ferromagnetic Ising model solution is approximated by using Creutz cellular automaton algorithm with nearest neighbor interactions and near the critical region [14][15][16][17][18][19][20][21][22][23]. The algorithm of approximating finite size behavior of ferromagnetic Ising model is extended to higher dimensions [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

“…In addition, the four-dimensional ferromagnetic Ising model solution is approximated by using Creutz cellular automaton algorithm with nearest neighbor interactions and near the critical region [14][15][16][17][18][19][20][21][22][23]. The algorithm of approximating finite size behavior of ferromagnetic Ising model is extended to higher dimensions [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. It is established that the algorithm has been powerful in terms of providing the values of static critical exponents near the critical region in four and higher dimensions with nearest neighbor interactions [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]…”

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

“…It was shown that, when measured on the finitesize system-length scale and properly taking dangerous irrelevant variables into account above the upper critical dimension, the correct scaling form is 5) where η Q is related to η through η Q = ϙη + 2(1 − ϙ). The main focus of this paper is on the new exponent ϙ.…”

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

“…The finite-size lattice critical temperatures obtained from the susceptibility maxima T χ c (L) are listed in Table 1. For d = 4 dimension, the finite-size scaling relations for the order parameter, (M L ), and the magnetic susceptibility, (χ L ), have the following forms [8][9][10][13][14][15][16][17][18][19][20][21][22][23][24][25] …”