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 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.
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
0.860Ja2,
0.619Ja2, and
0.509Ja2, respectively, for these lattices, where
J is the exchange integral and
a is the lattice constant. The spin diffusion constants for larger spins are also provided. It should be noted that
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. For the one-dimensional isotropic Heisenberg magnet and for the square, sc, and bcc
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 model.
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.
Rigorous results are presented for the dynamics of the nearest-neighbour Heisenberg chain, for fixed but arbitrary wavevector and frequency, to lowest non-trivial order in the temperature. The authors obtain the two-spin correlation function, in the antiferromagnet for all wavevectors, and in the ferromagnet for wavelengths shorter than a coherence length. The energy correlation function at T=0 is obtained exactly, and is identical in the two systems. They find a violation of dynamical scaling in the form of the spectral function in the antiferromagnet for a range of wavevectors. An extension of the perturbative results for the ferromagnet to treat wavelengths longer than a coherence length shows that the diffusion coefficient vanishes logarithmically, in contrast to the scaling prediction that it is constant. An approximate value for the coefficient of the logarithmic term is obtained which agrees well with the results of simulations.