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Physical Review E

Analytical derivation of critical exponents of the dynamic phase transition in the mean-field approximation

R. A. Gallardo, O. Idigoras, P. Landeros, Andreas Berger

Published Nov 2, 2012• 31 citations
Publisher

Abstract

We have analyzed the dynamic phase transition of the kinetic Ising model in mean-field approximation by means of an analytical approach. Specifically, we study the evolution of the system under the simultaneous influence of time-dependent and time-independent magnetic fields. We demonstrate that within the approximate analytical treatment of our approach, the dynamic phase transition exhibits power-law dependencies for the order parameter that have the same critical exponents as the mean-field equilibrium case. Moreover we have obtained an equation of state, with which we can prove that the time-independent field component is effectively the conjugate field of the order parameter. Our analysis is limited to the parameter range, in which only second-order phase transitions occur, i.e., for small applied field amplitudes and temperatures close to the Curie point. In order to ensure the reliability of our analytical results we have corroborated them by comparison to numerical evaluations of the same model.

DOI
10.1103/physreve.86.051101

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We have analyzed the dynamic phase transition of the kinetic Ising model in mean-field approximation by means of an analytical approach. Specifically, we study the evolution of the system under the simultaneous influence of time-dependent and time-independent magnetic fields. We demonstrate that within the approximate analytical treatment of our approach, the dynamic phase transition exhibits power-law dependencies for the order parameter that have the same critical exponents as the mean-field equilibrium case. Moreover we have obtained an equation of state, with which we can prove that the time-independent field component is effectively the conjugate field of the order parameter. Our analysis is limited to the parameter range, in which only second-order phase transitions occur, i.e., for small applied field amplitudes and temperatures close to the Curie point. In order to ensure the reliability of our analytical results we have corroborated them by comparison to numerical evaluations of the same model.
We studied the dynamical phase transition in kinetic Ising ferromagnets driven by oscillating magnetic field in meanfield approximation. The meanfield differential equation was solved by sixth-order Runge–Kutta–Felberg method. The time-averaged magnetization plays the role of the dynamic order parameter. We studied the relaxation behavior of the dynamic order parameter close to the transition temperature, which depends on the amplitude of the applied magnetic field. We observed the critical slowing down along the dynamic phase boundary. We proposed a power law divergence of the relaxation time and estimated the exponent. We also found its dependence on the field amplitude and compared the result with the exact value in limiting case.
The nonequilibrium dynamic phase transition, in the kinetic Ising model in the presence of an oscillating magnetic field has been studied both by Monte Carlo simulation and by solving numerically the mean-field dynamic equation of motion for the average magnetization. In both cases, the Debye ``relaxation'' behavior of the dynamic order parameter has been observed and the ``relaxation time'' is found to diverge near the dynamic transition point. The Debye relaxation of the dynamic order parameter and the power law divergence of the relaxation time have been obtained from a very approximate solution of the mean-field dynamic equation. The temperature variation of appropriately defined ``specific heat'' is studied by the Monte Carlo simulation near the transition point. The specific heat has been observed to diverge near the dynamic transition point.
The two-dimensional kinetic Ising model, when exposed to an oscillating applied magnetic field, has been shown to exhibit a nonequilibrium, second-order dynamic phase transition (DPT), whose order parameter Q is the period-averaged magnetization. It has been established that this DPT falls in the same universality class as the equilibrium phase transition in the two-dimensional Ising model in zero applied field. Here we study the scaling of the dynamic order parameter with respect to a nonzero, period-averaged, magnetic "bias" field, H(b) for a DPT produced by a square-wave applied field. We find evidence that the scaling exponent, delta(d), of H(b) at the critical period of the DPT is equal to the exponent for the critical isotherm, delta(e), in the equilibrium Ising model. This implies that H(b) is a significant component of the field conjugate to Q. A finite-size scaling analysis of the dynamic order parameter above the critical period provides further support for this result. We also demonstrate numerically that, for a range of periods and values of H(b) in the critical region, a fluctuation-dissipation relation (FDR), with an effective temperature T(eff)(T,P,H0) depending on the period, and possibly the temperature and field amplitude, holds for the variables Q and H(b). This FDR justifies the use of the scaled variance of Q as a proxy for the nonequilibrium susceptibility, partial differential Q/partial differential H(b), in the critical region.
We studied within the framework of a mean-field approach the nonequilibrium dyn amic phase transition of a kinetic Ising spin system subject to a perturbative f ield and temperature simultaneously by comparison between time-dependent Ginzbur g-Landau and Glauber dynamics models. The dynamic phase transition (DPT) boundar ies, separating a symmetry-breaking dynamic ordered phase from its symmetric dyn amic disordered counterpart, were identified through a systematic simulation of the above two models. The dependence of the dynamic order parameter Q and the fo urth order cumulant ratio U upon the temperature t(r0), the frequency ω and amplitude h0 of driving field were also investigated in detai l. A discussion was presented concerning the current controversies on whether bo th a discontinuous dynamic phase transition occurs possibly below a specific low temperature and a tri-critical point exists on the DPT boundary in a kinetic Is ing spin system.
In this work, we explore some interesting details of the time-dependent regime of the long-range systems under mean-field approximation in comparison with the critical dynamics of the short-range systems. First, we discuss some mechanisms of the initial anomalous behavior of the magnetization via two-dimensional Monte Carlo simulations to later compare with results from Mean-field simulations in both: spin 1/2 (Ising) and spin 1 (Blume Capel model) Ising models. The distinction between critical and tricritical points is also investigated. For a complete analysis, we performed short-time simulations in the mean-field regime to determine the critical temperatures optimizing power laws and the critical exponents of the different points, which were independently calculated, i.e. without using previous critical exponents estimates from literature/theory. Our investigations corroborate analytical results here also developed.
The basic equation of a kinetic Ising model under the influence of a time dependent field is rederived applying the Fock space representation of the master equation. The nonequilibrium behaviour of the model in the presence of both a static and an oscillating magnetic field is analyzed by numerical simulations and by scaling arguments. Forcing the system to be initially in the metastable state and imposing an additional oscillating field makes it possible to study the relaxation time of this state. We find a power law behaviour for the waiting time depending on the amplitude and the frequency of the field as well as on the reduced temperature. The critical exponents are determined numerically. Furthermore, the results are supported by scaling arguments.