Sixth Frequency Moment of the Frequency-Wave-Vector-Dependent Correlation Function for Isotropic Heisenberg Paramagnets at Elevated Temperatures

Abstract

We calculate the sixth frequency moment ${〈{\ensuremath{\omega}}^{6}〉}_{\mathrm{K}}$ of the frequency-wave-vector-dependent spectral function for the isotropic Heisenberg paramagnet at elevated temperatures with arbitrary spin and arbitrary range of exchange interaction. These exact results are compared with the approximate sixth moment predicted by the two-parameter Gaussian representation used in the preceding paper for the generalized diffusivity. The general agreement of the approximate and the exact results for ${〈{\ensuremath{\omega}}^{6}〉}_{\mathrm{K}}$ is satisfactory. For the exactly soluble spin-\textonehalf{} nearest-neighbor one-dimensional $\mathrm{XY}$ model, we compare the predictions of the two-parameter Gaussian approximation for ${〈{\ensuremath{\omega}}^{6}〉}_{{\mathrm{K}}^{\mathrm{zz}}}$ and ${〈{\ensuremath{\omega}}^{8}〉}_{{\mathrm{K}}^{\mathrm{zz}}}$ against the exact results for these moments (which are derived by using the corresponding exact spectral function given by Katsura et al. The agreement is again found to be satisfactory. This reinforces our confidence in the qualitative validity of the simple Gaussian approximation for the generalized diffusivity.