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Journal of Applied Physics

Dynamic Behavior of a Near-Ising System Near the Critical Point

Paul H. E. Meijer, John C. Edwards

Published Mar 1, 1969• 2 citations
Publisher

Abstract

Applying the master equation to an Ising model, equations of motion are obtained for the average of a single spin and for the average of a pair of spins. These equations relax to equilibrium values that predict a self-consistency relationship among the internal field, temperature, and exchange energy. The critical temperature obtained is that of the Bethe approximation, and is extended to include a static external field in the magnetization direction. With this, the relaxation of an Ising model near the Curie temperature has been investigated in a self-consistent manner. The results are similar to those of Suzuki and Kubo, but in the Bethe approximation instead of the Bragg-Williams approximation, and hence predict the relaxation of the correlation as well. In particular we studied a nearly Ising spin system that is a system with relatively small isotropic transverse coupling. An oscillating, linear, and external magnetic field applied transverse to the strongly coupled direction of the spins leads to lowering of the critical temperature.

DOI
10.1063/1.1657754

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Applying the master equation to an Ising model, equations of motion are obtained for the average of a single spin and for the average of a pair of spins. These equations relax to equilibrium values that predict a self-consistency relationship among the internal field, temperature, and exchange energy. The critical temperature obtained is that of the Bethe approximation, and is extended to include a static external field in the magnetization direction. With this, the relaxation of an Ising model near the Curie temperature has been investigated in a self-consistent manner. The results are similar to those of Suzuki and Kubo, but in the Bethe approximation instead of the Bragg-Williams approximation, and hence predict the relaxation of the correlation as well. In particular we studied a nearly Ising spin system that is a system with relatively small isotropic transverse coupling. An oscillating, linear, and external magnetic field applied transverse to the strongly coupled direction of the spins leads to lowering of the critical temperature.
Dynamics of a spin-3/2 Ising system Hamiltonian with bilinear and biquadratic nearest-neighbour exchange interactions is studied by a simple method in which the statistical equilibrium theory is combined with the Onsager's theory of irreversible thermodynamics. First, the equilibrium behaviour of the model in the molecular-field approximation is given briefly in order to obtain the phase transition temperatures, i.e. the first- and second-order and the tricritical points. Then, the Onsager theory is applied to the model and the kinetic or rate equations are obtained. By solving these equations three relaxation times are calculated and their behaviours are examined for temperatures near the phase transition points. Moreover, the z dynamic critical exponent is calculated and compared with the z values obtained for different systems experimentally and theoretically, and they are found to be in good agrement.
A semiphenomenological cluster theory is developed for dynamic critical properties, which is not limited to small deviations from equilibrium. Explicit numerical expressions are derived for linear and nonlinear response functions of the kinetic Ising model, which are compatible with dynamic scaling and recent calculations of dynamic critical exponents. An uniaxial magnet (or Ising system) and its critical fluctuations are approximated by a "droplet" or "cluster" model, the critical part of the free energy being the sum of the contributions of clusters each containing lll spins. These clusters are assumed to grow and shrink with a phenomenological rate $\ensuremath{\propto}{l}^{r}$, $r\ensuremath{\approx}1$. These rates are compatible with direct Monte Carlo simulations in two and three dimensions. The resulting cluster reaction and diffusion equation is an approximation to the master equation of the Glauber kinetic Ising model (in which the magnetization is not conserved), and has the same structure as used in nucleation theories. In linear response to space- and time-dependent fields the relaxation times of energy and magnetization are expressed as triple integrals. All relaxation times diverge as ${|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}$. This treatment is consistent with dynamic scaling and universality, and is more general than the conventional (Van Hove) theory of critical slowing down. Also the (smaller) exponents found for the response to localized variations ("autocorrelation functions") agree with dynamic scaling. The wave-vector dependence of the relaxation times arises from the static correlation function only, and not from the cluster diffusion term. If the equilibrium cluster distribution is assumed to be that of the Fisher droplet model, and for special values of rrr, the (discrete) eigenvalue spectrum and eigenfunctions (generalized Laguerre polynomials) of the cluster reaction equation are found explicitly. Relaxation functions and frequency-dependent susceptibilities are expressed by hypergeometric series, even in the case of nonlinear response. At the critical point, the spectrum becomes continuous and the exponential decay for large times is replaced by power-law behavior. Similarly, it is found that dynamic scaling applies also to the nucleation rate for this model, and the scaling exponent of the nucleation rate is $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.
The dynamic phase transition (DPT) temperatures and the dynamic phase diagrams are obtained for spin-1/2 Ising system under the presence of a time-dependent oscillating external magnetic field using the method that was purposed by Meijer and Edwards [ANNALS of PHYSICS, 54, 240 (1969)] within the framework of the Bethe or constant coupling theory. The time variation of magnetization is investigated to find the phases in the system. Thermal behavior of the average magnetization is studied to characterize the nature (continuous and discontinuous) of transitions and to obtain the dynamic phase transition temperatures. The DPT temperatures are found always a second-order; hence there can be no tricritical point separating lines of first- and second-order dynamic phase transitions. This result agrees with the dynamic Monte-Carlo (MC) simulations of a two-dimensional kinetic spin-1/2 Ising model in an oscillating external magnetic field.
We have studied an Ising spin system in a transverse field, at zero temperature, under a time oscillating longitudinal field by means of a mean-field approximation and a Monte Carlo algorithm, appropriate to study the ground-state properties of quantum spin chains. For large values of the transverse field Γ or large amplitude h0 of the oscillating field, the magnetization oscillates around a zero value with the same frequency but delayed. Decreasing Γ, at small a h0, the system undergoes a phase transition to a state in which the magnetization oscillates around a nonzero value. We have also obtained the behavior of the hysteresis loop area near the Curie point.
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.
A markoffian stochastic model of interacting Ising spins is discussed near its critical point. The long time behavior of relaxation of its magnetization and energy is investigated near T c by the use of high-temperature series expansion and the ratio method. The numerical estimate of the critical exponent Δ M M of slowing down of magnetization is made, the results of which show Δ M M ≃2 for the simple square and triangular lattices, and Δ M M ≃1.4 for the simple cubic lattice. The exponent Δ E E of slowing down of energy is also estimated, which gives the result Δ E E ≃2 for the simple square and triangular lattices. The method to derive high-temperature expansion of an n -spin correlation function <σ 1 σ 2 …σ n > for an Ising model of spin 1/2 by the use of an equation of detailed balance is presented and applied to these estimates.