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Canadian Journal of Physics

SOME DYNAMICAL PROPERTIES OF THE ISING FERROMAGNET

Noboru Matsudaira

Published Jun 1, 1967• 28 citations
Publisher

Abstract

Time-dependent statistics of the Ising model proposed by Glauber for the one-dimensional chain are extended to the example of a two-dimensional square lattice. Each spin is assumed to change its state through the interaction with a heat bath. The equations of motion for both the single spin and the spin correlation functions are solved approximately by using a decoupling procedure where the many-body correlation functions are taken as sums of products of pair correlation functions. As a special case, our theory allows the approximate calculation of the equilibrium properties of the system and it turns out that, in this case, our result is an improvement over the Bethe approximation. Both the frequency-dependent magnetic susceptibility and the decay of the magnetic moment to the equilibrium state are calculated above and below the Curie temperature. The fluctuation–dissipation theorem developed by Glauber for the linear chain is shown to hold in the two-dimensional case also.

DOI
10.1139/p67-165

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arXiv:0707.1953•Physical Review B•Aug 6, 2007•14 citations
Time-dependent statistics of the Ising model proposed by Glauber for the one-dimensional chain are extended to the example of a two-dimensional square lattice. Each spin is assumed to change its state through the interaction with a heat bath. The equations of motion for both the single spin and the spin correlation functions are solved approximately by using a decoupling procedure where the many-body correlation functions are taken as sums of products of pair correlation functions. As a special case, our theory allows the approximate calculation of the equilibrium properties of the system and it turns out that, in this case, our result is an improvement over the Bethe approximation. Both the frequency-dependent magnetic susceptibility and the decay of the magnetic moment to the equilibrium state are calculated above and below the Curie temperature. The fluctuation–dissipation theorem developed by Glauber for the linear chain is shown to hold in the two-dimensional case also.
A method to calculate the dynamical as well as the static properties of the Ising ferromagnet, developed in the previous paper for the two-dimensional square lattice, is extended to cubic lattices. The temperature dependence of the frequency-dependent magnetic susceptibility, approach to equilibrium of the magnetization of the system, the fluctuation-dissipation theorem, and some static properties are calculated. It is proved that all these properties are essentially the same for cubic lattices and for two-dimensional model. It is further demonstrated by explicit calculation that the non-linear effect is very important for the critical slowing-down near the Curie point, especially in the ferromagnetic region.
A modified version of Glauber's one-dimensional spin relaxation model is applied to a spin-1 Ising chain in order to study the time dependence of the system in the weak-coupling limit. The individual spin-1 Ising particles are assumed to interact with the heat bath, which causes them to change their states randomly in time. Coupling between the particles is introduced through the assumption that the transition probabilities for any one spin-1 Ising particle depends on the state of the neighboring spin-1 Ising particles. A special assumption about the rate constants is chosen such that the average values of the dipole moment will return to the equilibrium value. We establish the system of rate equations for average values of the dipole and quadruple moments, as well as their coupling. The Ising interaction between the spin-1 particles is assumed to be weak compared to the coupling with the heat reservoir. In this way we can terminate the hierarchy and solve the problem of a linear chain with periodic boundary conditions, using the Fourier transformation. The resulting secular equation determines two sets of relaxation times and two sets of eigenvectors. From this equation both relaxation times are determined by a perturbation method.
Bethe's method of equilibrium statistical mechanics is generalized to treat Glauber's time-dependent Ising models in two and three dimensions. This constitutes the first-order approximation which improves the mean-field theory (the zeroth-order approximation) by incorporating specific short-range correlations into the distribution function. We first calculate the single-spin expectation value. Although the dynamic equation may need to be solved by using a numerical method, its approximate solution in the pretransitional region can be studied in detail. We then calculate the two-spin correlation function. This requires yet another extension of Bethe's method. Again approximate solutions can be obtained in the pretransitional region. Our results for the static pair correlation functions are equivalent to that of Elliott and Marshall, but our derivation is simpler. We also calculate the time-dependent correlation function for the magnetization, and reproduce the fluctuation-dissipation relation.
The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.
A statistical theory of the localized model of ferromagnetism is developed by considering the effective density matrices for a single spin, and two nearest-neighbor spins, respectively. The Ising Model and the Heisenberg model, and even the case of the anisotropic exchange interaction, are treated within a unified mathematical scheme. The result is reduced to that of the Bethe approximation in the case of the Ising model, and the constant coupling approximation in the case of the Heisenberg model. The method is generalized to calculate the static spin-correlation function. It is obtained in Ornstein-Zernike type. In the nonequilibrium state, the expectation value of the spin decays by the homogeneous damping as well as the spin-diffusion process. This relaxation equation is different from that by Van Hove because of the existence of the homogeneous damping. The diffusion constant does not vanish even at the Curie point TC, contrary to Van Hove's result, although the damping constant vanishes at TC. The latter fact causes the critical slowing down of the regression of fluctuation. The solution of this relaxation equation gives the time-dependent spin-correlation function. The method can be applied to the antiferromagnetic case, too.
We study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional Ising models with spin $\ifmmode\pm\else\textpm\fi{}1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest-neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors is assumed to compete with a nearest-neighbor exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent zzz, defined as $1∕\ensuremath{\tau}\ensuremath{\approx}{(1∕\ensuremath{\xi})}^{z}$ (where $\ensuremath{\tau}$ is the longest relaxation time and $\ensuremath{\xi}$ is the correlation length of the chain), suggests that the T=0T=0T=0 single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.