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Physical Review

Irreversibility in Heisenberg Spin Systems. II. Approximate Solution of the High-Temperature Kinetic Equations

Michel De Leener, Pierre M. V. Résibois

Published Dec 2, 1966• 33 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.152.318

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Time-dependent spin correlations in the Heisenberg magnet at infinite temperature

Stephen William Lovesey, E Engdahl, Alessandro Cuccoli +2 more

A coupled-mode theory of spin fluctuations in the d-dimensional Heisenberg magnet at infinite temperature is used to predict the time dependence of various spin correlation functions. The real-space spin autocorrelation function is shown to have a long-time behaviour approximately (1/t) d theta / where theta =(4+d)/2. Properties at intermediate values of the time are extracted from the theory by numerical analysis. In this time window, the reciprocal-lattice spin autocorrelation function, G(q, t), is, to a good approximation, an exponential function of time. The decay rate is proportional to q alpha , where q is the wavevector. Analysis of our numerical data indicates that the exponent alpha depends weakly on d, and it is significantly different from the value 2 which is compatible with a spin diffusion model. In the asymptotic limit, defined by q to 0, t to infinity , and q 2 t to 0, G(q,t) is a function of a single variable=(tq theta ). This result rules against the validity of a diffusion model also in the asymptotic limit.

Journal of Physics Condensed Matter•Aug 29, 1994•15 citations
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 \ensuremathΓab(t){\ensuremath{\Gamma}}_{\mathrm{ab}}(t)\ensuremathΓab​(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 \ensuremathΓab(t){\ensuremath{\Gamma}}_{\mathrm{ab}}(t)\ensuremathΓab​(t). We also compare this theory with previous work on the subject.
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.
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.
Numerical calculations have been performed to obtain the exact infinite-temperature time-dependent spin autocorrelation function G(t)G(t)G(t) for a linear chain of NNN spins (S=12S=\frac{1}{2}S=21​) interacting by nearest-neighbor Heisenberg exchange for N=5,7,9and11N=5, 7, 9 \mathrm{and} 11N=5,7,9and11 by a method different from that of Carboni and Richards. Exact results for the first 20 moments and estimates for M22{M}_{22}M22​ to M30{M}_{30}M30​ of the frequency autocorrelation function for the infinite chain are provided. Excellent agreement is obtained with some recent results obtained by Morita, who, however, gives terms only up to M10{M}_{10}M10​. G(t)G(t)G(t) does not show a simple monotonic diffusive ($\ensuremath{\sim}{t}^{\ensuremath{-}\frac{1}{2}}$) behavior within $\frac{4\ensuremath{\hbar}}{J}$, the time domain up to which we believe our result for N=11N=11N=11 to be equivalent to that of the infinite chain although it is reduced to about 10% of its value at t=0t=0t=0.
The two-time spin-pair correlation function σ(R,t) is expressed as a product of a Gaussian distribution function and a power series with respect to time. The width of the Gaussian distribution function is determined by the second derivative with respect to time of the autocorrelation function σ(0,t). The coefficients of the power series are determined up to terms of order t8 for the square and s.c. isotropic Heisenberg and XY magnets of spin ½ at infinite temperature, and up to terms of t10 for the linear magnets. The expressions obtained by truncating the power series to the exactly known terms give a very good fit at short times to the exact expression for the linear XY model, and to the results of the computer simulation due to Windsor for the s.c. Heisenberg ferromagnet. The Fourier-time transforms of the expressions thus obtained are shown to be the Gram-Charlier expansions. Thus we conclude that the Gram-Charlier expansion of the Fourier-time transform of σ(R,t),σ̃(R,ω), and that of its Fourier-space transform I(k,t),S(k,ω), is especially useful in the cases when the short-time behavior of σ(R,t) or I(k,t) is considered important; generally speaking, this is the case at large k or at ω which is not very small. When the width of the Gaussian distribution function is determined by the second derivative of I(k,t), a convergent result is not obtained for small values of k. For larger k, the convergence is as good as for the case when the width is determined by the autocorrelation function.
A coupled-mode theory of spin fluctuations in the d-dimensional Heisenberg magnet at infinite temperature is used to predict the time dependence of various spin correlation functions. The real-space spin autocorrelation function is shown to have a long-time behaviour approximately (1/t) d theta / where theta =(4+d)/2. Properties at intermediate values of the time are extracted from the theory by numerical analysis. In this time window, the reciprocal-lattice spin autocorrelation function, G(q, t), is, to a good approximation, an exponential function of time. The decay rate is proportional to q alpha , where q is the wavevector. Analysis of our numerical data indicates that the exponent alpha depends weakly on d, and it is significantly different from the value 2 which is compatible with a spin diffusion model. In the asymptotic limit, defined by q to 0, t to infinity , and q 2 t to 0, G(q,t) is a function of a single variable=(tq theta ). This result rules against the validity of a diffusion model also in the asymptotic limit.