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

Abstract

The four-dimensional ferromagnetic Ising model in external magnetic field 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, T c χ (∞) = 6.680(1) obtained for h = 0 agrees well with the values T c (∞) ≈ 6.68 obtained previously using different methods. Moreover, h = 0.00025 in our work also agrees with all the results obtained from h = 0 in the literature. However, there are no works for h ≠ 0 in the literature. The value of the field critical exponent (δ = 3.0136(3)) is in good agreement with δ = 3 which is obtained from scaling law of Widom. In spite of the finitesize scaling relations of |M L (t)| and χ L (t) for 0 ≤ h ≤ 0.001 are verified; however, in the cases of 0.0025 ≤ h ≤ 0.1 they are not verified.

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

“…The order parameter M L (t, h) and the magnetic susceptibility χ L (t, h) are obtained from Eq. (6) as below [13]:…”

“…The slopes of the graphs are −β/ν and Υ/ν, respectively. Since the Ising model in d ≥ 4 has β = 1/2, Υ = 1 and ν = 1/2, the new critical exponent is 0.9904 (16) for the order parameter and 1.0415 (13) for the magnetic susceptibility (Table I, II). Both the order parameter and the magnetic susceptibility are 1.00 because is equal to d/d c where d c = 4.…”

“…(t, h) of a hypercubic finite system L d with periodic boundary conditions is adapted for the Ising model in the critical dimension d C = 4, by proposing the finite-size scaling function Y (x.y), correct to leading logarithms as below[13]:f (S)…”

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

“…As the dimensionality and/or the lattice size increases, the simulation of the ferromagnetic Ising model by the conventional Monte Carlo method becomes impractical and faster algorithms are needed. The Creutz cellular automaton algorithm [2] does not require high-quality random numbers, it is an order of magnitude faster than the conventional Monte Carlo method and compared to the Q2R cellular automaton [3], it has the advantage of fluctuating internal energy from which the specific heat can be computed.…”

“…The four-dimensional Ising model in the presence of external magnetic field is simulated [3]. In two dimensions, the solution of ferromagnetic Ising model is investigated [4][5][6][7][8].…”