The static critical exponents of the three dimensional Blume-Capel model

The three-dimensional BC model is simulated on a cellular automaton which improved from the Creutz cellular automaton for simple cubic lattice. The phase diagram characterizing phase transition of the model is obtained for a comparison with those obtained from other calculations. The simulations confirm the existence of a tricritical point at which the phase transition changes from second-order to first-order at D/J =2.82. For the determined of the tricritical point, the thermodynamics quantities are computed using two different procedures which is called as the standard and the cooling algorithm for the anisotropy parameter values in the interval 3≥D/J≥-8. The simulations indicates that the cooling algorithm is a suitable procedure for the calculations near the first-order phase transition region, and the cooling rate is an important parameter in the determining of the phase boundary. The estimated critical temperatures for D/J =0 and 2.82 are compatible with the series expansion results.

The spin-1 Ising model with the dipole–quadrupole interaction (ℓ = L/J) has been simulated using a cellular automaton (CA) algorithm improved from the Creutz cellular automaton (CCA) for a face-centered cubic (fcc) lattice. The simulations have been made for different ℓ values at the reentrant phase transition and the special points such as the tricritical point (k = K/J = 0, d = D/J = 5.7) and the critical end point (k = -0.9, d = 0.7). The simulation results show that the model has the dense ferromagnetic (df, df(+), df(-)) and the ferromagnetic (F, F(+), F(-)) phases with the dipole–quadrupole interaction. The type and the order of the phase transitions change for the nonzero values of ℓ on the special points. Furthermore, the effect of ℓ is similar with the effect of the external magnetic field (h).

The Blume–Capel model in the presence of external magnetic field H has been simulated using a cellular automaton algorithm improved from the Creutz cellular automaton in three-dimension lattice. The field critical exponent δ is estimated using the power law relations and the finite size scaling functions for the magnetization and the susceptibility in the range −0.1 ≤ h = H/J ≤ 0. The estimated value of the field critical exponent δ is in good agreement with the universal value (δ = 5) in three dimensions. The simulations are carried out on a simple cubic lattice under periodic boundary conditions.