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Phase Transitions

A simulation of the mixed spin 3–spin 3/2 ferrimagnetic Ising model

Aycan Özkan

Published Sep 11, 2015• 16 citations
Publisher

Abstract

The mixed spin 3–spin 3/2 ferrimagnetic Ising model was simulated using cooling algorithm on cellular automaton (CA). The simulations were carried out in the intervals −4 ≤ DA/J ≤ 8 and −4 ≤ DB/J ≤ 8 for the square lattices with periodic boundary conditions. The ground-state phase diagram of the model has different types of ferrimagnetic phases. Although only the antiferromagnetic nearest-neighbor interaction was contained in the Hamiltonian, the compensation points emerged through DA/J = 2 at kT/J = 0. The values of the critical exponents (ν, α , β and γ) were estimated within the framework of the finite-size scaling theory and power-law relations for the selected DA/J values (−2, 0, 1, 2, and 4). The estimated critical exponent values were in good agreement with the universal values of the two-dimensional Ising model (ν = 1, α = α′ = 0, β = 0.125, β′ = 0.875 and γ = γ′ = 1.75).

DOI
10.1080/01411594.2015.1067702

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The mixed spin 3–spin 3/2 ferrimagnetic Ising model was simulated using cooling algorithm on cellular automaton (CA). The simulations were carried out in the intervals −4 ≤ DA/J ≤ 8 and −4 ≤ DB/J ≤ 8 for the square lattices with periodic boundary conditions. The ground-state phase diagram of the model has different types of ferrimagnetic phases. Although only the antiferromagnetic nearest-neighbor interaction was contained in the Hamiltonian, the compensation points emerged through DA/J = 2 at kT/J = 0. The values of the critical exponents (ν, α , β and γ) were estimated within the framework of the finite-size scaling theory and power-law relations for the selected DA/J values (−2, 0, 1, 2, and 4). The estimated critical exponent values were in good agreement with the universal values of the two-dimensional Ising model (ν = 1, α = α′ = 0, β = 0.125, β′ = 0.875 and γ = γ′ = 1.75).
The mixed spin 3- spin 3/2 Ising model has been simulated using cooling algorithm on cellular automaton (CA). The simulations have been made in the interval -6<=D<=6 for J=1 for the square lattices with periodic boundary conditions. The ground state phase diagram of the model has different type ferrimagnetic orderings. Through D/J=2 line, compensation points occurs at kT/J=0. The values of the critical exponents ( ν, α, β and γ) are estimated within the framework of the finite-size scaling theory and power law relations for selected D/J values (-2, 0, 1, 2 and 4). The estimated critical exponent values are in a good agreement with their universal values of the two dimensional Ising model.
The spin-3/2 Ising model on the simple cubic lattice with nearest-neighbour ferromagnetic bilinear interaction (J > 0) is simulated on a cellular automaton by using the cooling algorithm improved from the Creutz cellular automaton. The phase diagrams of the model are constructed in the (D/J, kT/J) and (K/J, kT/J) plane. Comparison of the results are made with those of other methods. The temperature dependence of the order parameters and associated fluctuations are calculated at various of the model parameters and the static critical exponents are estimated within the framework of the finite-size scaling. The results are compatible with the universal Ising critical behavior except for D/J =−3 and K/J =−2.3.
We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.
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.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
In this work, we simulated a quenched random mixed-spin Ising model on the square lattice. The model system consists of two different particles with spins σ = 1/2 (states ±1/2) and S = 1 (states ±1, 0). These particles are randomly distributed on the lattice, and we considered only nearest-neighbor interactions. This model can represent a random magnetic binary alloy AxB1-x, obtained from the high-temperature quenching of a liquid mixture. We performed Monte Carlo simulations for several lattice sizes and temperatures, and we found its critical temperature through the reduced fourth-order cumulant. We also determined the magnetization, the susceptibility, and the specific heat as a function of temperature. We used finite-size scaling arguments to estimate the critical exponents β, γ, and ν of the model. We showed that the quenched model is in the same universality class of the two-dimensional pure Ising model. We also investigated the sample to sample fluctuations that occur in the values of the thermodynamic quantities in order to describe the self-averaging properties at the critical point.