Critical Exponents for the Re-entrant Phase Transitions in the Three-Dimensional Blume--Emery--Griffiths Model on the Cellular Automaton

The critical behaviour of the three-dimensional Blume--Emery--Griffiths (BEG)
model is investigated at D/J=0, -0.25 and -1 in the range of -1≤K/J≤0 for J=100. The simulations are carried out on a simple cubic lattice using the heating algorithm improved from the Creutz cellular automaton (CCA) under periodic boundary conditions. The universality of the model are obtained for re-entrant and double re-entrant phase transitions which occur at certain D/J and K/J parameters, with J and K representing the nearest-neighbour bilinear and biquadratic interactions, and D being the single-ion anisotropy parameter. The values of static critical exponents β, γ and υ are estimated within the framework of the finite-size scaling theory. The results are compatible with the universal Ising critical behaviour for all continuous phase transitions in these ranges.