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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Long-range interactions generated by random Lévy flights: Spin-flip and spin-exchange kinetic Ising model in two dimensions

Huang-Jian Xu, B. Bergersen, Zoltàn Ràcz

Published Mar 1, 1993• 10 citations
Publisher

Abstract

A d=2 dimensional kinetic Ising model that evolves by a combination of spin flips and spin exchanges is investigated. The spin flips satisfy detailed balance for the equilibrium state of the Ising model at temperature T while the spin exchanges are random L\'evy flights of dimension \ensuremath{\sigma}=1.5. Our Monte Carlo (MC) simulations show that the steady state of this system displays a second-order phase transition as T is lowered. Comparing the critical fluctuations of the magnetization to those of an Ising model in which the interaction decays with distance as ${\mathit{r}}^{\mathrm{\ensuremath{-}}3.5}$, we find that, within the resolution of the MC data, the critical exponents and the scaling functions of the two systems coincide. We argue that this coincidence indicates that a recent conjecture about the random L\'evy flights generating long-range interaction of the form ${\mathit{V}}_{\mathrm{eff}}$(r)\ensuremath{\sim}${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathit{d}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$ is valid not only in the spherical limit and in d=1 but also in d=2.

DOI
10.1103/physreve.47.1520

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A d=2 dimensional kinetic Ising model that evolves by a combination of spin flips and spin exchanges is investigated. The spin flips satisfy detailed balance for the equilibrium state of the Ising model at temperature T while the spin exchanges are random L'evy flights of dimension \ensuremath{\sigma}=1.5. Our Monte Carlo (MC) simulations show that the steady state of this system displays a second-order phase transition as T is lowered. Comparing the critical fluctuations of the magnetization to those of an Ising model in which the interaction decays with distance as r\ensuremath−3.5{\mathit{r}}^{\mathrm{\ensuremath{-}}3.5}r\ensuremath−3.5, we find that, within the resolution of the MC data, the critical exponents and the scaling functions of the two systems coincide. We argue that this coincidence indicates that a recent conjecture about the random L'evy flights generating long-range interaction of the form Veff{\mathit{V}}_{\mathrm{eff}}Veff​(r)\ensuremath{\sim}r\ensuremath−d\ensuremath−\ensuremathσ{\mathit{r}}^{\mathrm{\ensuremath{-}}\mathit{d}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}r\ensuremath−d\ensuremath−\ensuremathσ is valid not only in the spherical limit and in d=1 but also in d=2.
Ordering of a one-dimensional stochastic Ising model evolving by a combination of spin flips and spin exchanges is investigated. The spin-flip rates satisfy detailed balance for the equilibrium state of the Ising model at temperature T, while the spin exchanges are random and of arbitrary range. Analytical methods and Monte Carlo simulations are used to show that, depending on the details of the spin-flip rate, finite-temperature phase transition may or may not occur in the system. When ordering occurs, it is of mean-field type and the scaling function describing the finite-size scaling of the magnetization fluctuations is found to be indistinguishable from that of an equilibrium Ising model with infinite-range interaction.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.
Nonequilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and Kawasaki-type spin-exchange kinetics at infinite temperature T are investigated here in one dimension from the point of view of phase transition and critical behaviour. Branching annihilating random walks with an even number of offspring (on the part of the ferromagnetic domain boundaries), is a decisive process in forming the steady state of the system for a range of parameters, in the family of models considered. A wide variety of quantities characterize the critical behaviour of the system. Results of computer simulations and of a generalized mean field theory are presented and discussed.
We have studied the critical phenomena of the Ising model with long-range interaction and the Ising model with spatially correlated random field, both of which are expected to change the effective dimension of the system continuously.The investigation was conducted from the numerical viewpoint by using the Markov chain Monte Carlo method.Second order phase transitions can be classified into the universality classes and the dimensionality of the Ising model changes the universality class: the one-dimensional Ising model has no long-range order at finite temperatures (lower critical dimension), four or higher dimensional model has the mean-field universality (upper critical dimension), two-dimensional Ising model belongs to different universality from either of them, and so on.Although both of the Ising models with long-range interaction and the Ising model with spatially correlated random field are believed to change their critical behavior from its real spatial dimension into different effective dimensions, their detailed behavior has not been known well because of the notorious difficulties in analysis.In this work, we have obtained the detailed behavior of these models by introducing and developing the various methods.To investigate the Ising model with long-range interaction, we introduced the Fukui-Todo method, which drastically reduces the computational cost from O(N 2 ) to O(N ), where N is the number of spins, and developed the generalized Ewald summation, the improved estimator, and the combined Binder ratio technique, which improves the convergence of the estimation.These techniques lead to the successful estimation of the boundary of the universality classes.We also investigated the Ising model with correlated random field by introducing the method for correlated random number generation and the various techniques for the precise analysis.Although the present numerical investigations suggested that the correlated random field causes a shift of effective dimensionality, they also revealed the difficulty for stronger random-field correlations because of the huge finite-size effects.i Prof. Synge
Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.
Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.