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Progress of Theoretical Physics

Time Correlation Function in the One-Dimensional Heisenberg System at High Temperatures

N. Nagao, Hiroshi Miyagi

Published Jan 1, 1976• 10 citations
Publisher

Abstract

Moments in the two-time spin-pair correlation function are evaluated up to 16-th order for the one-dimensional Heisenberg system with S = 1/2 at infinite temperature. The method of evaluations with a useful recurrence formula is reported. The current methods of treating the correlation function together with a newly proposed one are examined for both the spatially-Fourier-transformed correlation function and the auto-correlation function.

DOI
10.1143/ptp.55.21

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Moments in the two-time spin-pair correlation function are evaluated up to 16-th order for the one-dimensional Heisenberg system with S = 1/2 at infinite temperature. The method of evaluations with a useful recurrence formula is reported. The current methods of treating the correlation function together with a newly proposed one are examined for both the spatially-Fourier-transformed correlation function and the auto-correlation function.
The two-time spin-pair autocorrelation function $\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{0},t)$ and the Fourier-space transform $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$ of the two-time spin-pair correlation function $\ensuremath{\sigma}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{\mathrm{if}},t)$, are expressed in terms of the "friction function" occurring in the generalized Langevin equation for the spin operator ${s}_{i}^{\ensuremath{\sim}}$ and its Fourier-space transform ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{z}$, respectively. The friction function is determined in the form of a product of a Gaussian distribution function and a power series as a function of time ttt for the isotropic Heisenberg magnet and the XY\mathrm{XY}XY magnet of spin \textonehalf{} at infinite temperature. By truncating the power series to the exactly known term, satisfactory results are obtained for $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$ of the isotropic Heisenberg magnet of the linear chain and sc lattice. Satisfactory results for $\ensuremath{\sigma}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{\mathrm{if}},t)$ are achieved by taking an inverse Fourier-space transform of the thus determined $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$.
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.
Exact coefficients of the short-time expansion of the spin-pair correlations are computed up to order t 30 in the one-dimensional spin-1/2 Heisenberg model with isotropic nearest-neighbour interaction at infinite temperature. The key points of the computational algorithm are sketched. The newly derived coefficients are used to obtain lower and upper bounds on the pair correlations which are graphically indistinguishable up to time nearly twice as large as those up to which previous bounds are reasonably close together. In view of the spin-diffusion hypothesis the spatial spread of the pair correlations is studied as a function of time, but no indication for a diffusive long-time behaviour is found in the extended time region, where the new bounds provide control.
We comment on the recent work of Fernandez and Gersch on the frequency Fourier transform of the selfcorrelation function for a one-dimensional, S=12S=\frac{1}{2}S=21​, Heisenberg spin system with nearest-neighbor exchange. Using a more embellished version of their phenomenological theory, we have computed the frequency Fourier transform of the self-correlation, as well as those for several spatially separated correlations. The results are in good agreement with those calculated by Carboni and Richards with an exact ab initio procedure.
The expansion coefficients in powers of time of the two-time spin-pair correlation function are obtained up to terms of order t8 for the bcc Heisenberg magnet of spin ½ at infinite temperature and up to terms of order t6 for the linear, square, sc, and bcc Heisenberg magnets of an arbitrary spin at infinite temperature. The result is applicable to the isotropic as well as the anisotropic Heisenberg magnet where the exchange integrals in the z direction and in the orthogonal plane are assumed to be different. Analysis of the results are given in separate articles.
The expansion coefficients in powers of time of the two-time spin-pair correlation function are obtained up to terms of order t8 for the Heisenberg magnet of spin ½ at infinite temperature for the square and cubic lattices and up to terms of order t10 for the linear chain. The result is applicable to the isotropic as well as anisotropic Heisenberg magnet where the exchange integrals in the z direction and in the orthogonal plane are assumed to be different. Analysis of the result is left to future work.