Time-dependent correlations for spin Van der Waals systems

The time-dependent autocorrelation function has been derived for the $x$ component of the total spin for the $S=\frac{1}{2}$ constant-coupling anisotropic Heisenberg model, i.e., the Van der Waals system. For $T>\mathrm{}{T}_c$, the time correlation is shown to be Gaussian for both $\mathrm{XY}$-like and Ising-like regimes of the model. For $T<\mathrm{}{T}_c$, the correlation function is still Gaussian if the model is $\mathrm{XY}$-like; but it is oscillatory if the model is Ising-like. Critical slow down appears only with the Ising-like system for this time-correlation function.