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International Journal of Modern Physics C

CRITICAL PROPERTIES OF 3D ISING SYSTEMS WITH NON-HAMILTONIAN DYNAMICS

Jouke R. Heringa, Henk W. J. Blöte, A. Hoogland

Published Jun 1, 1994• 6 citations
Publisher

Abstract

We investigate two three-dimensional Ising models with non-Hamiltonian Glauber dynamics. The transition probabilities of these models can, just as in the case of equilibrium models, be expressed in terms of Boltzmann factors depending only on the interacting spins and the bond strengths. However, the bond strength associated with each lattice edge assumes different values for the two spins involved. The first model has cubic symmetry and consists of two sublattices at different temperatures. In the second model a preferred direction is present. These two models are investigated by Monte Carlo simulations on the Delft Ising System Processor. Both models undergo a phase transition between an ordered and a disordered state. Their critical properties agree with Ising universality. The second model displays magnetization bistability.

DOI
10.1142/s0129183194000763

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We investigate two three-dimensional Ising models with non-Hamiltonian Glauber dynamics. The transition probabilities of these models can, just as in the case of equilibrium models, be expressed in terms of Boltzmann factors depending only on the interacting spins and the bond strengths. However, the bond strength associated with each lattice edge assumes different values for the two spins involved. The first model has cubic symmetry and consists of two sublattices at different temperatures. In the second model a preferred direction is present. These two models are investigated by Monte Carlo simulations on the Delft Ising System Processor. Both models undergo a phase transition between an ordered and a disordered state. Their critical properties agree with Ising universality. The second model displays magnetization bistability.
We investigate two-dimensional Ising models with a non-Hamiltonian Glauber dynamics. The transition probabilities are expressed in terms of Boltzmann factors depending only on the nearest neighbour spins and the associated bond strengths. However, the bond strength on each lattice edge is two-valued; it assumes different values with respect to the spins at either of its ends. In one of the two models, the bond strength pattern is consistent with the four-fold rotational symmetry of the square lattice. In contrast, a preferred direction is present in the second model. This model can be interpreted in terms of inhomogeneous interactions and temperature such that their ratios, i. e. the couplings, are uniform. Monte Carlo simulations of these models show that the phase transition persists when these two types of non-Hamiltonian dynamics are introduced. Furthermore, our results indicate that both models belong to the Ising universality class.
In this work, Markov Chain-Monte Carlo technique was used to study the phase transition in two and three dimensional Ising Model (IM) in a square and cubic lattice. The study of temperature dependence of average magnetization and specific heat in different magnetic fields has been carried out in the 3x3 and 3x3x3 lattice with periodic boundary. Critical temperature point kBTc / J for 2D and 3D Ising Model has been observed at around 2.2 and 4.3 respectively at zero field. Our work satisfies Onsager’s critical value in 2D IM. The simulation suggests bifurcation in average magnetization below critical temperature Tc. Temperature plays the role of increasing randomness of spins. We found that Ising Model in small lattice size still retains interesting features like spontaneous magnetization and symmetry breaking below Tc at B = 0. At a non-zero field, the likelihood of spins to prefer certain alignment depends on the direction of the external field and magnitude of magnetization depends on magnitude of field ±B. Specific heat Cv, which gives us fluctuation of energy at particular temperature, has been found to attain maximum value at critical regions.
By Monte Carlo simulations we study critical properties of the mixed spin-1/2 and spin-1 Ising model on a triangular lattice, considering two different ways of the spin-value distributions on the three sublattices: (1/2,1/2,1) and (1/2,1,1). In the former case, we find standard Ising universality class phase transitions between paramagnetic and magnetic phases but no phase transitions between two distinct magnetic phases (±1/2,±1/2,±1) and (±1/2,±1/2,0) at any finite temperature, except for some interesting non-critical anomalies displayed by response functions. On the other hand, the latter case turns out to be a rare (or perhaps the only) example of a two-dimensional mixed spin-1/2 and spin-1 Ising model on a standard lattice that displays tricritical behavior.
In this work we performed numerical simulations for the Ising model on three dimensional lattices with coordination number equal 5. With Monte Carlo simulations in the static case we evaluated the critical temperature and the static critical exponents ννν, γγγ and βββ. Once that we have the critical temperature value we investigated the dynamical critical behavior with Glauber dynamics, starting from disordered states. From our simulations we obtained the values for the dynamic exponent z=2.037(8)z=2.037(8)z=2.037(8), the exponent for the autocorrelation λ/z=1.364(5)λ/z=1.364(5)λ/z=1.364(5), the exponent of the critical initial increase θ′=0.109(8)θ'=0.109(8)θ′=0.109(8) and the asymptotic value of the fluctuation-dissipation ratio X∞=0.433(1)X^\infty=0.433(1)X∞=0.433(1). All of these results are in good agreement to the previous values reported for the 3D Ising model universality class.
We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.