Narrow Results

The two-dimensional Ising model in a small external magnetic field, is simulated on the Creutz cellular automaton. The values of the static critical exponents for 0.0025 ≤ h ≤ 0.025 are estimated within the framework of the finite size scaling theory. The value of the field critical exponent is in a good agreement with its theoretical value of δ = 15. The results for 0.0025 ≤ h ≤ 0.025 are compatible with Ising critical behavior for T < T_c.

The two-dimensional antiferromagnetic spin-1 Ising model with positive biquadratic interaction is simulated on a cellular automaton which based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transition of the model are presented for a comparison with those obtained from other calculations. We confirm the existence of the intermediate phase observed in previous works for some values of J/K and D/K. The values of the static critical exponents (β, γ and ν) are estimated within the framework of the finite-size scaling theory for D/K<2J/K. Although the results are compatible with the universal Ising critical behavior in the region of D/K<2J/K-4, the model does not exhibit any universal behavior in the interval 2J/K-4<D/K<2J/K.

The two-dimensional BEG model with nearest neighbor bilinear and positive biquadratic interaction is simulated on a cellular automaton, which is based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transitions of the model are presented for comparison with those obtained from other calculations. We confirm the existence of the tricritical points over the phase boundary for D/K>0. The values of static critical exponents (α, β, γ and ν) are estimated within the framework of the finite size scaling theory along D/K=-1 and 1 lines. The results are compatible with the universal Ising critical behavior except the points over phase boundary.

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 (BEG) model has been simulated on a cellular automaton improved from the Creutz cellular automaton (CCA) for a face-centered cubic lattice. The simulations have been made in the 0 ≤ d = D/J ≤ 7 and -1.25 < k = K/J ≤ 0 parameter region. In this region, the ground state diagram (k, d) has ferromagnetic and perfect zero ordering regions. The ferromagnetic ordering region separates into four regions which exhibit different phase transition types as the first order, the second order, the reentrant, the double-reentrant and the successive phase transitions. The simulation results show that the model has the tricritical points, the critical end points and the bicritical points on the (kT_C/zJ, d) and (kT_C/zJ, k) planes as indicated by the Mean Field approximation 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).