Spin dynamics for the one-dimensionalXYmodel at infinite temperature

The infinite-temperature space- and time-dependent spin-correlation functions $g_r^x\mathrm{}\left(t\right)\mathrm{}$ are studied for the one-dimensional $\mathrm{XY}$ model. Numerical calculations are performed to obtain the exact autocorrelation function $g_0^x\mathrm{}\left(t\right)\mathrm{}$ for chains containing 5, 7, and 9 spins ($S=\frac{1}{2}$). This yields exact results for the first 16 moments of the frequency autocorrelation function of the infinite chain, and estimates for a few of the higher moments as well. The analysis suggests that $g_0^x\mathrm{}\left(t\right)\mathrm{}$ for the infinite chain is identical to $\frac{\exp (-J^2{t}^2)}{4}$. We show that $g_r^x\mathrm{}\left(t\right)\mathrm{}$ for $r\ne 0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for $g_0^x\mathrm{}\left(t\right)\mathrm{}$ is compared with that obtained by Huber for the classical ($S=\infty$) chain.