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Journal of Mathematical Physics

Two-Time Spin-Pair Correlation Function of the Heisenberg Magnet at Infinite Temperature. II

Tohru Morita

Published May 1, 1972• 24 citations
Publisher

Abstract

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.

DOI
10.1063/1.1666041

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Fernando Carboni, Peter M. Richards

Numerical calculations of quantum-mechanical time correlation functions are reported for finite Heisenberg linear chains containing up to 10 spin-\textonehalf{} particles. The results, obtained by manipulation of eigenvalues and eigenfunctions, give Fourier transforms in histogram form. Two-spin correlation functions show a sharp rise near zero frequency and highly non-Gaussian behavior at infinite temperature. The short-time behavior is estimated and compared with classical calculations and other theories. Four-spin functions are calculated and compared with results of a simple decoupling approximation. Spatial Fourier transforms of two-spin functions as needed for neutron scattering cross sections are computed. These are limited to fairly short wavelengths by the finiteness of the system. The temperature dependence of two-spin functions shows a decrease in the near-zero frequency component with decreasing temperature. Computed correlation functions are used to predict the frequency dependence of the ESR linewidth in the linear-chain salt ${\mathrm{C}\mathrm{u}(\mathrm{N}{\mathrm{H}}_{3})}_{4}$S${\mathrm{O}}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}${\mathrm{H}}_{2}$O, and comparison is made with experiment. Very good agreement is found using the same exchange constant $J$ as inferred from specific-heat and magnetic-susceptibility data. The Gaussian approximation, on the other hand, is in extremely poor agreement.

Physical Review•Jan 10, 1969•104 citations
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.
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 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.
The exact short-time expansion coefficients of the two-time spin-pair correlation function σ(Rif,t)=≪siz(t)sfz(0)≫c at infinite temperature are used in order to estimate the spin diffusion constants of the diluted Heisenberg and XY magnets of spin 1/2. The used coefficients are those up to the term of t10 for the Heisenberg magnet on the linear chain, and those up to the term of t8 for the Heisenberg and XY magnets on the square, sc and bcc lattices. Our calculations are based on the “linear extrapolation approximation”, in which unknown coefficients in Mori's continued fraction representation for ~Is(k), the Laplace-time Fourier-space transform of σ(Rif,t), are linearly extrapolated from the last two of the known coefficients. The obtained values of the diffusion constant and their concentration dependences show a reasonable trend of convergence for successive approximations in a range of concentration not close to the percolation limit. Discussions are also given on the diffusion constants obtained by the “two-parameter Gaussian approximation in time” applied to the memory function.
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.
A theoretical criterion for the occurrence of the spin diffusion is given in terms of the "friction function" or "memory function" which occurs in the kinetic equation of the Fourierspace transform $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, t)$ of the two-time spin-pair correlation function. The friction function in the limit of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{\rightarrow}0$ is determined with the aid of the short-time expansion of the two-time spin-pair correlation function for the square, sc, and bcc isotropic Heisenberg magnets at infinite temperature, and the criterion is checked. The spin diffusion constants for spin \textonehalf{} are 0.860Ja20.860J{a}^{2}0.860Ja2, 0.619Ja20.619J{a}^{2}0.619Ja2, and 0.509Ja20.509J{a}^{2}0.509Ja2, respectively, for these lattices, where JJJ is the exchange integral and aaa is the lattice constant. The spin diffusion constants for larger spins are also provided. It should be noted that 0.509Ja20.509J{a}^{2}0.509Ja2 is in complete agreement with the experimental value $(0.525\ifmmode\pm\else\textpm\fi{}0.06)J{a}^{2}$ for the bcc solid He3{\mathrm{He}}^{3}He3. For the one-dimensional isotropic Heisenberg magnet and for the square, sc, and bcc XY\mathrm{XY}XY magnets, the friction function could not be determined in the present calculation. Suggested values of the spin diffusion constant are given for these lattices. Comparison is made with the results of Windsor's computer-simulation calculation. There exists no spin diffusion for the one-dimensional XY\mathrm{XY}XY model.
Numerical calculations of quantum-mechanical time correlation functions are reported for finite Heisenberg linear chains containing up to 10 spin-\textonehalf{} particles. The results, obtained by manipulation of eigenvalues and eigenfunctions, give Fourier transforms in histogram form. Two-spin correlation functions show a sharp rise near zero frequency and highly non-Gaussian behavior at infinite temperature. The short-time behavior is estimated and compared with classical calculations and other theories. Four-spin functions are calculated and compared with results of a simple decoupling approximation. Spatial Fourier transforms of two-spin functions as needed for neutron scattering cross sections are computed. These are limited to fairly short wavelengths by the finiteness of the system. The temperature dependence of two-spin functions shows a decrease in the near-zero frequency component with decreasing temperature. Computed correlation functions are used to predict the frequency dependence of the ESR linewidth in the linear-chain salt Cu(NH3)4{\mathrm{C}\mathrm{u}(\mathrm{N}{\mathrm{H}}_{3})}_{4}Cu(NH3​)4​SO4{\mathrm{O}}_{4}O4​\ifmmode\cdot\else\textperiodcentered\fi{}H2{\mathrm{H}}_{2}H2​O, and comparison is made with experiment. Very good agreement is found using the same exchange constant JJJ as inferred from specific-heat and magnetic-susceptibility data. The Gaussian approximation, on the other hand, is in extremely poor agreement.