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

Abstract

Carboni and Richards have performed exact numerical calculations of the time-dependent two-spin correlation function $〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$ 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 $S(k, \ensuremath{\omega})$, the paramagnetic scattering function at small wave vectors. We also prove that ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}$, the time Fourier transform of $〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉$ in the $T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, obeys the inequality ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}\ensuremath{\ge}\mathrm{const}\ifmmode\times\else\texttimes\fi{}\mathrm{ln}|\frac{1}{\ensuremath{\omega}}|$ as $\ensuremath{\omega}\ensuremath{\rightarrow}0$ for a one-dimensional system. We discuss the probable divergence of the same quantity in two dimensions.