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Physical Review

Irreversibility in Heisenberg Spin Systems. IV. Time-Dependent Fluctuations at the Curie Point

Michel De Leener, Pierre M. V. Résibois

Published Feb 10, 1969• 13 citations
Publisher

Abstract

We propose an approximate solution of the kinetic equations, recently derived by the authors, for the spin autocorrelation function ${\ensuremath{\Gamma}}_{q}(t)$ of ferromagnets at the critical point. It is shown that, even for small wave numbers and long times, ${\ensuremath{\Gamma}}_{q}(t)$ never reaches a diffusion regime, but decays to zero through damped oscillations. We also compare this theory with previous work on the subject.

DOI
10.1103/physrev.178.819

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A systematic approximation scheme is proposed that permits solution of the kinetic equations, recently derived by the authors, describing the high-temperature behavior of the spin autocorrelation function ${\ensuremath{\Gamma}}_{\mathrm{ab}}(t)$. A numerical solution and approximate analytical expressions are given in the first approximation and the importance of the neglected terms is discussed. It is shown that the direct autocorrelation function has an approximate Gaussian behavior for short times and decays then to zero through damped oscillations. A diffusion equation is derived for the short-wave-number Fourier components of ${\ensuremath{\Gamma}}_{\mathrm{ab}}(t)$. We also compare this theory with previous work on the subject.

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Physical Review•Aug 5, 1966•132 citations
We propose an approximate solution of the kinetic equations, recently derived by the authors, for the spin autocorrelation function \ensuremathΓq(t){\ensuremath{\Gamma}}_{q}(t)\ensuremathΓq​(t) of ferromagnets at the critical point. It is shown that, even for small wave numbers and long times, \ensuremathΓq(t){\ensuremath{\Gamma}}_{q}(t)\ensuremathΓq​(t) never reaches a diffusion regime, but decays to zero through damped oscillations. We also compare this theory with previous work on the subject.
A systematic approximation scheme is proposed that permits solution of the kinetic equations, recently derived by the authors, describing the high-temperature behavior of the spin autocorrelation function ${\ensuremath{\Gamma}}_{\mathrm{ab}}(t)$. A numerical solution and approximate analytical expressions are given in the first approximation and the importance of the neglected terms is discussed. It is shown that the direct autocorrelation function has an approximate Gaussian behavior for short times and decays then to zero through damped oscillations. A diffusion equation is derived for the short-wave-number Fourier components of ${\ensuremath{\Gamma}}_{\mathrm{ab}}(t)$. We also compare this theory with previous work on the subject.
Recently the authors have derived kinetic equations describing the behavior of the spin autocorrelation function ${\ensuremath{\Gamma}}_{q}(t)$ in a Heisenberg system at infinite temperature. In the present paper, this derivation is extended to finite temperatures (above the critical point). It is shown that, in the Weiss limit where the number of neighbors $Z\ensuremath{\rightarrow}\ensuremath{\infty}$, the effects of the equilibrium (Ornstein-Zernicke) correlations present in the system at the initial time can be entirely incorporated within an effective temperature-dependent interaction which governs the temporal behavior of the autocorrelation function (af). A non-Markoffian kinetic equation is obtained in which the kernel is highly nonlinear in the complete af ${\ensuremath{\Gamma}}_{q}(t)$; this contrasts with the infinite-temperature case, where the kernel was a functional of the direct af only. This new feature leads to simple approximations near the critical point, as will be discussed in detail in the next paper of this series.
Starting from the von Neumann equation for the spin density matrix of a Heisenberg system, we analyze the perturbation expansion of the spin autocorrelation function by the diagrammatic technique previously applied to quantum gases. We demonstrate a number of theorems which allow us to express this perturbation series in terms of renormalized graphs only; we then derive a kinetic equation for the autocorrelation function. The main feature of this equation is that the kernel, which is highly nonlinear in the autocorrelation function itself, tends to zero in the limit of long times. The results, which are exact in the high-temperature region and in the Weiss limit (number of neighbors $Z\ensuremath{\rightarrow}\ensuremath{\infty}$), allow us to consider the behavior of the autocorrelation function for times both short and long. This model is a typical example of a system with a discrete unperturbed spectrum showing an irreversible behavior.
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 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.