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International Journal of Modern Physics B

STABILITY OF ISING SYSTEMS AGAINST NON-HAMILTONIAN, SYMMETRY BREAKING DYNAMICS

Henk W. J. Blöte, Jouke R. Heringa, A. Hoogland, Royce K. P. Zia

Published Feb 20, 1991• 25 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979291000389

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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.
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 perform a comprehensive numerical study of how variations in microscopic coupling strength affect critical behavior in the two-dimensional Ising model. Using extensive Monte Carlo simulations based on the Wolff cluster algorithm, we investigate the clean Ising model on a square lattice for interaction strengths in the range 0.80 ≤ J ≤ 1.20. While the exact solution at J = 1 has long established the model’s critical properties, a systematic numerical assessment of universality under continuous coupling rescaling has not been performed. Here, we address this gap by analyzing systems of linear size L = 20 − 500, allowing for high-precision finite-size scaling of thermodynamic observables. For each coupling, the critical temperature T_C(J) is determined via finite-size extrapolation of the susceptibility maxima, yielding results in quantitative agreement with Onsager’s exact prediction. At the extrapolated critical points, we extract the critical exponents ν, β, γ, and α through detailed scaling analyses. These exponents remain consistent with the exact Ising benchmarks across the full range of interaction strengths, confirming that the universal aspects of the phase transition are preserved. Our findings demonstrate that modifying the microscopic coupling strength rescales the critical temperature without altering the asymptotic critical behavior, thereby providing a clear and quantitative confirmation of the robustness of the two-dimensional Ising universality class.
Phase transitions of the spin-one Ising model with nearest-neighbour and next-nearest-neighbour interactions are investigated for a square lattice using a molecular-field theory and a Monte Carlo method. Special attention is paid to the effects of the biquadratic next-nearest-neighbour interaction on the phase transitions of the model. Phase diagrams characterizing types of phase transition are obtained for several values of the coupling constants. The existence of the multi-critical points and the successive phase transitions are also discussed in connection with these phase diagrams.
We present an extensive study of the effects of quenched disorder on the dynamic phase transitions of kinetic spin models in two dimensions. We undertake a numerical experiment performing Monte Carlo simulations of the square-lattice random-bond Ising and Blume-Capel models under a periodically oscillating magnetic field. For the case of the Blume-Capel model we analyze the universality principles of the dynamic disordered-induced continuous transition at the low-temperature regime of the phase diagram. A detailed finite-size scaling analysis indicates that both nonequilibrium phase transitions belong to the universality class of the corresponding equilibrium random Ising model.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.