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Communications in Theoretical Physics

Effective-Field Theory for Kinetic Ising Model on Honeycomb Lattice

Shi Xiao-ling, Guozhu Wei

Published May 1, 2009• 7 citations
Publisher

Abstract

As an analytical method, the effective-field theory (EFT) is used to study the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field. The effective-field equations of motion of the average magnetization are given for the honeycomb lattice (Z = 3). The Liapunov exponent λ is calculated for discussing the stability of the magnetization and it is used to determine the phase boundary. In the field amplitude h 0 /ZJ-temperature T/ZJ plane, the phase boundary separating the dynamic ordered and the disordered phase has been drawn. In contrast to previous analytical results that predicted a tricritical point separating a dynamic phase boundary line of continuous and discontinuous transitions, we find that the transition is always continuous. There is inconsistency between our results and previous analytical results, because they do not introduce sufficiently strong fluctuations.

DOI
10.1088/0253-6102/51/5/34

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As an analytical method, the effective-field theory (EFT) is used to study the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field. The effective-field equations of motion of the average magnetization are given for the honeycomb lattice (Z = 3). The Liapunov exponent λ is calculated for discussing the stability of the magnetization and it is used to determine the phase boundary. In the field amplitude h 0 /ZJ-temperature T/ZJ plane, the phase boundary separating the dynamic ordered and the disordered phase has been drawn. In contrast to previous analytical results that predicted a tricritical point separating a dynamic phase boundary line of continuous and discontinuous transitions, we find that the transition is always continuous. There is inconsistency between our results and previous analytical results, because they do not introduce sufficiently strong fluctuations.
A transverse Ising spin system, in the presence of time-dependent longitudinal field, is studied by the effective-field theory (EFT). The effective-field equations of motion of the average magnetization are given for the simple cubic lattice (Z = 6) and the honeycomb lattice (Z = 3). The Liapunov exponent λ is calculated for discussing the stability of the magnetization and it is used to determine the phase boundary. The dynamic phase transition diagrams in h 0 /ZJ — Γ/ZJ plane and in h 0 /ZJ — T/ZJ plane have been drawn, and there is no dynamical tricritical point on the dynamic phase transition boundary. The effect of the thermal fluctuations upon the dynamic phase boundary has been discussed.
The non-equilibrium phase transition in a ferromagnetic Ising model is investigated by use of a new type of effective field theory (EFT) which correctly accounts for all the single-site kinematic relations by differential operator technique. In the presence of a time dependent oscillating external field, with decrease of the temperature the system undergoes a dynamic phase transition, which is characterized by the period averaged magnetization Q , from a dynamically disordered state Q = 0 to the dynamically ordered state Q ≠ 0. The results of the dynamic phase transition point T c determined from the behavior of the dynamic magnetization and the Liapunov exponent provided by EFT are improved than that of the standard mean field theory (MFT), especially for the one dimensional lattice where the standard MFT gives incorrect result of T c = 0 even in the case of zero external field.
The dynamical phase transitions of the kinetic Ising model in the presence of a random magnetic field with a bimodal probability distribution is studied by using effective-field theory (EFT) with correlations. We have used a Glauber-type stochastic dynamic to describe the time evolution of the system, where the system strongly depends on the H≡√<H(i)(2)>(c) root mean square deviation of the magnetic field. The EFT dynamic equation is given for the simple cubic lattice (z=6), and the dynamic order parameter is calculated. The system presents ferromagnetic and paramagnetic states for low and high temperatures, respectively. Our results predict first-order transitions at low temperatures and large disorder strengths, which corresponds to the existence of a nonequilibrium tricritical point (TCP) in a phase diagram in the T-H plane. We compare the results with the equilibrium phase diagram, where only the first-order line is different. Our qualitative results are compatible with recent Monte Carlo simulations.
The dynamic phase transitions are studied, within a mean-field approach, in the kinetic Blume–Emery–Griffiths model under the presence of a time varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. The behaviour of the time-dependence of the order parameters and the behaviour of the average order parameters in a period, which is also called the dynamic order parameters, as a function of reduced temperature, are investigated. The nature (continuous and discontinuous) of transition is characterized by studying the average order parameters in a period. The dynamic phase transition points are obtained and the phase diagrams are presented in the reduced magnetic field amplitude and reduced temperature plane. The phase diagrams exhibit one, two, or three dynamic tricritical points and a dynamic double critical end point, and besides a disordered and two ordered phases, seven coexistence phase regions exist, which strongly depend on interaction parameters. We also calculate the Liapunov exponent to verify the stability of solutions and the dynamic phase transition points.
Using Monte Carlo simulations based on the Metropolis algorithm, we investigate the dynamic phase transition properties of kinetic Ising model driven by a sinusoidally oscillating magnetic field in the presence of additive white noise. We calculate equilibrium and dynamic properties such as the temperature dependence of average magnetization and magnetic specific heat, as well as the period dependence of dynamic order parameter and scaled variance. After determining the critical period at which order-disorder transition takes place, we perform finite size scaling analysis to extract the exponent ratios, and discuss the variation of these properties in the presence of noisy magnetic field. As a general result, we show that for a noisy system, DPT does not fall into a universality class of the conventional dynamic (and also equilibrium) universality class of the Ising model.
I consider the dynamics of a soft-spin Ising magnet, subject to the time-varying external magnetic field h(t)=h cos\ensuremath{\Omega}t. The system is modeled with a time-dependent Ginzburg-Landau equation and is studied at a mean-field level. The time-averaged magnetization (scrM) acts as the order parameter and divides the temperature-h plane into two phases. In contrast to a previous result that predicted a dynamical tricritical point separating a line of continuous and discontinuous transitions in scrM, I find that the transition is always continuous. The previous work utilized Glauber dynamics which, it is here argued, becomes inapplicable near the phase boundary line due to a critical slowing down.