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Journal of Applied Physics

Time-Dependent Spin-Correlation Function in Ferromagnet and Antiferromagnet

Takehiko Oguchi, Ikuo Ono

Published Feb 1, 1968• 4 citations
Publisher

Abstract

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.

DOI
10.1063/1.1656298

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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.
The diffusion constant of spins in ferromagnets or of molecules in binary mixtures near the critical point is discussed employing a time-dependent Ising model in which spin interactions are replaced by certain temperature-dependent transition probabilities of spin exchange. The spin diffusion constant is calculated with the single approximation of replacing a reduced spin distribution function by its value for local equilibrium with a given inhomogeneous spin density. The behavior of the diffusion constant near the critical point is dominated by a factor ${\ensuremath{\chi}}^{\ensuremath{-}1}$, where $\ensuremath{\chi}$ is the magnetic susceptibility. This problem is also studied with the use of the Bethe lattice. The effects of surrounding spins on the transition probability for spin exchange are found to be essential for obtaining the critical slowing-down near the critical point. In view of this, Koci\ifmmode \acute{n}\else \'{n}\fi{}ski's calculation of the spin diffusion constant is critically discussed.
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.
A systematic method is developed for studying the time-dependent Ising model proposed by Glauber and extended by others. The graphical technique developed for the equilibrium statistical mechanics has been found to be useful in this nonequilibrium problem as well with some modifications. A hierarchy of equations linking various spin cluster functions is obtained. If we choose as the expansion parameter the reciprocal range of the spin interaction which is taken to be of Kac's type, the hierarchy is terminated, and closed sets of nonlinear kinetic equations satisfied by spin cluster functions follow. The dynamical susceptibility is studied using the hierachy, and the polydispersive nature of the spin relaxation is found to be closely connected to the deviation of the spin pair correlation from the Ornstein-Zernike form via multi-spin cluster functions. A sum rule on the distribution of relaxation times is found, which provides a precise expression for the critical slowing-down at the Curie point. The closed set of kinetic equations involving at most three spin cluster functions is used to evaluate explicitly the approximate dynamical susceptibility above the Curie point, and the characteristic time-scale of the polydispersive relaxation is found to increase indefinitely near the Curie point.
The nature of the local-equilibrium approximation employed in our previous work is examined with the use of the correlation-function expression for the general frequency-dependent transport coefficient for systems described by master equations. The transport coefficient is found to be a nondecreasing function of frequency which in the high-frequency limit reduces to the transport coefficient calculated in the local-equilibrium approximation. Thus the local-equilibrium approximation provides an upper bound to the zero-frequency transport coefficient. For the case of spin diffusion, combined with the result of our previous work, this limit implies that the diffusion constant must vanish at least as fast as the inverse of the magnetic susceptibility near the Curie point, which establishes the existence of the critical slowing down for our model of spin diffusion.
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.