Divergences in the Space-Time Correlation Functions for the Heisenberg Magnet in One Dimension

Carboni and Richards have performed exact numerical calculations of the time-dependent two-spin correlation function $〈{S_1}^z\mathrm{}\left(t\right)\mathrm{}{S_1}^z\mathrm{}\left(0\right)\mathrm{}〉$ as $T\to \infty$ for a finite one-dimensional Heisenberg system. The non-Gaussian character of their result was characterized by a steep rise near zero frequency for the Fourier transform. We show here that these characteristics result from the inclusion of a Lorentzian form for $S(k, \omega )$, the paramagnetic scattering function at small wave vectors. We also prove that ${〈{S_1}^z\mathrm{}\left(t\right)\mathrm{}{S_1}^z\mathrm{}\left(0\right)\mathrm{}〉}_{\omega }$, the time Fourier transform of $〈{S_1}^z\mathrm{}\left(t\right)\mathrm{}{S_1}^z\mathrm{}\left(0\right)\mathrm{}〉$ in the $T\to \infty$ limit, obeys the inequality ${〈{S_1}^z\mathrm{}\left(t\right)\mathrm{}{S_1}^z\mathrm{}\left(0\right)\mathrm{}〉}_{\omega }\ge \mathrm{const}\times \ln \left|\frac{1}{\omega }\right|$ as $\omega \to 0$ for a one-dimensional system. We discuss the probable divergence of the same quantity in two dimensions.