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Communications in Theoretical Physics

High-Temperature Cutoff Approximation of the 3D Kinetic Ising Model

Zhu Jian-Yang, Yang Zhan-Ru

Published Apr 15, 2001• 0 citations
Publisher

Abstract

A single-spin transition critical dynamics is used to investigate the three-dimensional kinetic Ising model on an anisotropic cubic lattice. We first derive the fundamental dynamical equations, and then linearize them by a cutoff approximation. We obtain the approximate solutions of the local magnetization and equal-time pair correlation function in zero field. In which the axial-decoupling terms , and as higher infinitesimal quantity are ignored, where . We think that it is reasonable as the temperature of the system is very high. The result of what we obtain in this paper can go back to the one-dimensional Glauber's theory as long as .

DOI
10.1088/0253-6102/35/4/469

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A single-spin transition critical dynamics is used to investigate the three-dimensional kinetic Ising model on an anisotropic cubic lattice. We first derive the fundamental dynamical equations, and then linearize them by a cutoff approximation. We obtain the approximate solutions of the local magnetization and equal-time pair correlation function in zero field. In which the axial-decoupling terms , and as higher infinitesimal quantity are ignored, where . We think that it is reasonable as the temperature of the system is very high. The result of what we obtain in this paper can go back to the one-dimensional Glauber's theory as long as .
A single-spin transition critical dynamics is used to investigate the three-dimensional kinetic Ising model on an anisotropic cubic lattice. We first derive the fundamental dynamical equations, and then linearize them by a cutoff approximation. We obtain the approximate solutions of the local magnetization and equal-time pair correlation function in zero field. In which the axial-decoupling terms γ1γ2, γ2γ3 and γ1γ3 as higher infinitesimal quantity are ignored, where γa = tanh(2kα) = tanh(2Jα/kβT) (α = 1,2,3). We think that it is reasonable as the temperature of the system is very high. The result of what we obtain in this paper can go back to the one-dimensional Glauber's theory as long as k2 = k3= 0.
A single-spin transitional critical dynamics based on the Glauber theory is used to investigate the d-dimensional kinetic S 4 model. We derive the fundamental dynamical equations by a cumulant expansion, and then, a one-dirnensional model is treated. We obtain approximately the local magnetization and equal-time spin-pair correlation function, in which the terms of k 3 and the higher orders are ignored (corresponding to high-temperature cutoff). These are finally extended to d-dimensional case. In addition, we obtain the critical point k c of d-dimensional S 4 model, and give a relation satisfied by the dynamic critical exponent z and the correlation length critical exponent ν .
In this work we study the Glauber dynamics of the one-dimensional Ising model with nearest-neighbor and next-nearest-neighbor interactions, for which an approximate solution of the magnetization per site is obtained. When the dynamical critical exponent z is investigated following the treatment of Cordery, Sarker, and Tobochnik [Phys. Rev. B 24, 5402 (1981)], our observation shows that its upper-bound value is the same as the known value, thus implying that z is independent of the range of the interaction. We also suggest a high-temperature expansion approximation which is then used to solve the two-dimensional Glauber dynamics governed by a master equation; this solution is compared with that of the decoupling method. The time-delayed correlation function is also calculated.
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.
We have studied the nonequilibrium dynamic phase transitions of both three-dimensional (3D) kinetic Ising and Heisenberg spin systems in the presence of a perturbative magnetic field by Monte Carlo simulation. The feature of the phase transition is characterized by studying the distribution of the dynamical order parameter. In the case of anisotropic Ising spin system (ISS), the dynamic transition is discontinuous and continuous under low and high temperatures respectively, which indicates the existence of a tri-critical point (TCP) on the phase \linebreak boundary separating low-temperature order phase and high-temperature disorder phase. The TCP shifts towards the higher temperature region with the decrease of frequency, i.e. T TCP =1.33 × exp(−ω/30.7). In the case of the isotropic Heisenberg spin system (HSS), however, the situation on dynamic phase transition of HSS is quite different from that of ISS in that no stable dynamical phase transition was observed in kinetic HSS after a threshold time. The evolution of magnetization in the HSS driven by a symmetrical external field after a certain duration always tends asymptotically to a disorder state no matter what an initial state the system starts with. The threshold time τ depends upon the amplitude H 0 , reduced temperature T / T C and the frequency ω as τ = C .ω α . H 0 −β .( T / T C ) −γ .
An analytic method for finding the highly accurate values of the critical point for cubic Ising models with arbitrary spin is presented. The thermodynamic quantities of interest are expressed through some higher-order correlations with a reference spin excluded. A reduction scheme for these higher-order correlations is constructed. It leads to two types of resulting equations for correlation functions. The coincidence of two different representations of critical correlations is ensured by the introduced parameter of the reduction scheme. The estimation of critical points for all cubic lattices is compatible with series works to within 1.69%, 0.70% for the limiting values of spin, ½ and ∞, respectively.