Re-entrant phase transitions of the Blume–Emery–Griffiths model for a simple cubic lattice on the cellular automaton

Abstract

. IntroductionThe Blume-Emery-Griffiths (BEG) model 1 was originally introduced in order to explain the phase separation and superfluidity in the He 3 -He 4 mixtures. Subsequently, the model was used in the description of a variety of different physical phenomena such as multicomponent fluids 2 , microemulsions 3 , and semiconductors alloys 4 , etc.The Hamiltonian of the BEG model is given by,where s i = ±1, 0 and < ij > denotes summation over all nearest-neighboring (nn) spin pairs on a simple cubic lattice. The parameters of J and K are the bilinear and biquadratic interaction energies, respectively and D is the single-ion anisotropy constant. The three-dimensional BEG model has been extensively studied by different techniques, using the mean-field approximation (MFA) (1,5−7) , effective-field theory (8−11) , two-particle cluster approximation (TPCA) 12 , Bethe approximation 13 , high-temperature series expansion 14 , renormalization group theory 15 , Monte Carlo simulations (13,16−17) , linear chain approximation (18,19) and cellular automaton (20,21) In this paper we studied the three-dimensional BEG model using an improved heating algorithm from the Creutz Cellular Automaton (CCA) for simple cubic lattice. The CCA algorithm is a microcanonical algorithm interpolating between the canonical Monte Carlo and molecular dynamics techniques on a cellular automaton, and it was first introduced by Creutz 22 .In the previous papers (20,21,23−26) , the CCA algorithm and improved algo-2 rithms from CCA were used to study the critical behavior of the different Ising model Hamiltonians on the two and three-dimensions. It was shown that they have successfully produced the critical behavior of the models.The BEG model has a complicated phase diagrams and has several kinds of phase transitions, such as re-entrant and double re-entrant transitions for some values of the model parameters on the three-dimensional lattices.However, there exists the differences between the phase diagrams in theter values (6,12,13,16) , where z is the coordination number. Sec.2, the results are discussed in Sec.3 and a conclusion is given in Sec.4. ModelThree variables are associated with each site of the lattice. The value of 3 each sites are determined from its value and those of its nearest-neighbors at the previous time step. The updating rule, which defines a deterministic cellular automaton, is as follows: Of the three variables on each site, the first one is Ising spin B i . Its value may be 0 or 1 or 2. The Ising spin energy for the model is given by Eq.1. In Eq.1, Si = Bi − 1. The second variable is for momentum variable conjugate to the spin (the demon). The kinetic energy associated with the demon, H k , is an integer, which equal to the changing in the Ising spin energy for the any spin flip and its values lie in the interval (0, m). The upper limit of the interval, m, is equal to 24J. The total energyis conserved.The third variable provides a checkerboard style updating, and so it allows the simulation of the Ising model on a cellu...

Cu–Au TYPE STRUCTURES IN THE STAGGERED QUADRUPOLAR REGION OF THE fcc BLUME–EMERY–GRIFFITHS MODEL

“…In the previous papers, the Creutz and AB 3 ) type stoichiometric structures on the fcc BEG model using the heating algorithm [15,16] The CA results for the 5% heating rate nearly agree with the CVM and MFA results for the critical lines. In addition, it is seen that the fcc BEG model on the cellular automaton exhibits the modulated AB (type-II) phase as indicated by the theoretical studies [31,36,37] and the experimental results [39, 41 − 47] for Cu-Au type structures.…”

“…This algorithm is realized by increasing of 5% in the kinetic energy (H k ) of each site [15]. Therefore, the increasing value per site of H k is obtained from the integer part of the 0.05H k .…”

The Study of the Three-dimensional Spin-3/2 Ising Model on a Cellular Automaton

“…The CCA algorithm is a microcanonical algorithm interpolating between the canonical Monte Carlo and molecular dynamics techniques on a cellular automaton, and it was first introduced by Creutz [14]. In the previous papers [14][15][16][17][18][19][20][21], the CCA algorithm and improved algorithms from CCA were used to study the critical behavior of the different Ising model Hamiltonians in two and three dimensions. It was shown that they have successfully produced the critical behavior of the models.…”

The ferrimagnetic phase of the Blume-Emery-Griffiths model on the cellular automaton

“…It has been analyzed using mean-field approximation (MFA) [2][3][4][5][6], renormalization techniques [7,8], cluster variation method (CVM) [9][10][11][12][13], effective field theory [14,15], Monte Carlo renormalization method (MCRG) [16], Monte Carlo simulations [17][18][19][20] and cellular automaton simulations [21][22][23]. The BEG model is described by the Hamiltonian…”

“…Therefore, in this study, our interest is focused to study the ferrimagnetic phase of the BEG model in the case of K /J < 0. Recently, we studied the re-entrant behavior of the BEG model in the case of K /J < 0, using an improved heating algorithm [21][22][23] from the Creutz cellular automaton (CCA) [24][25][26][27][28][29][30][31][32]. It was obtained that the model can show the re-entrant and double re-entrant phase transitions (PTs) for some values of the D/J and K /J parameters.…”