The Finite-Size Scaling Study of the Specific Heat and the Binder Parameter for the Five-Dimensional Ising Model

Abstract

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, Tc = 8.8063, Tc = 8.7825 and Tc = 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 Tc = 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 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…”

“…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]…”

High-temperature expansions of the higher susceptibilities for the Ising model in general dimensiond

“…What is interesting for our aims is the fact that the lattice-embedding number of a star graph with l edges is a polynomial in d of degree [l/2] at most. The higher susceptibilities are simply related to the quantities 8 − 56 10 − 120 11 − 126 χ 6 (K; d) 2 χ 2 (K; d) 11 + 4620 12 − 15400 12 − 220 13 − 792…”

“…For example, in discussing [10][11][12][13][14][15][16] how the finite-size-scaling behavior [17] changes for systems above the upper critical dimensionality, accurate estimates of the critical parameters are needed as benchmarks. Our data can also help to assess the accuracy of estimates of physical parameters obtained from approximations of a different nature, such as the the = 4 − d expansion [9,18] of the renormalization group * paolo.butera@mib.infn.it † mario.pernici@mi.infn.it theory, the fixed-dimension renormalization group [19], the 1/d expansion [6,8], the Monte Carlo (MC) simulations, etc.…”

The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field

“…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 [13][14][15][16][17]. The algorithm of approximating finite size behavior of ferromagnetic Ising model is extended to higher dimension [13][14][15][16][17][18][19][20][21][22][23][24]. 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 [13][14][15][16][17][18][19][20][21][22][23][24].…”