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Critical Time-Dependent Fluctuations in Heisenberg Spin Systems

Pierre M. V. Résibois

Published Jan 1, 1969• 0 citations
OpenAlex

Abstract

In the Heisenberg spin system near the critical point, the small wave-number (q) component of the auto-correlation function does not satisfy any diffusion equation but exhibits an oscillatory decay with a characteristic time τq∼q- 5 2. © 1967.

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The dynamical theory of critical fluctuations put forward in a previous work is rephrased and extended so as to be applicable below the transition points as well. The theory is illustrated for the isotropic Heisenberg spin systems where the critical dynamical variables can be chosen in such a way as to reproduce the correct characteristic times governing the dynamics of critical fluctuations which were obtained by the dynamical scaling laws. In this connection the general forms of various equal time correlations of long wave length fluctuations have been obtained with the help of the scaling law equation of state. The theory is useful as a starting point for devising approximate schemes of evaluating time correlation functions of physical quantities of our interest, where at least the correct scaling properties of the characteristic times of the problem are retained. This is illustrated by deriving a new self-consistent equation for the time-displaced spin auto-correlation function, and the truncated moments for describing the part of the spectral line shape that is associated with critical fluctuations are introduced and discussed.
The time-dependent autocorrelation function has been derived for the xxx component of the total spin for the S=12S=\frac{1}{2}S=21​ constant-coupling anisotropic Heisenberg model, i.e., the Van der Waals system. For T>TcT>{T}_{c}T>Tc​, the time correlation is shown to be Gaussian for both XY\mathrm{XY}XY-like and Ising-like regimes of the model. For T<TcT<{T}_{c}T<Tc​, the correlation function is still Gaussian if the model is XY\mathrm{XY}XY-like; but it is oscillatory if the model is Ising-like. Critical slow down appears only with the Ising-like system for this time-correlation function.
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.
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.