Fluctuation and susceptibility for the spin van der Waals model and the bounds of Falk and Bruch

The transverse components of the susceptibility and fluctuation for the spin van der Waals model are obtained for high- and low-temperature $\mathrm{XY}$- and Ising-like regimes. The difference between the susceptibility and fluctuation, which arises because of noncommutativity, is related to certain correlation functions of $S_z$. In the $\mathrm{XY}$-like regime, the difference vanishes since the principal long-range order is $〈S_x〉$. But in the Ising-like regime, the difference persists since the principal long-range order now is $〈S_z〉$. Our results are compared with the bounds of Falk and Bruch on the ratio of the susceptibility to the fluctuation. In the critical region as well as in the high-temperature region the upper and lower bounds merge. In these regions, the two correlation functions become identical in the manner indicated by Falk and Bruch. In the low-temperature region, the bounds do not merge. Here we find that in the Ising-like regime of the model the susceptibility and fluctuation are properly different, but in the $\mathrm{XY}$-like regime the two functions nevertheless become identical. Thus, the susceptibility and fluctuation can evidently be the same without the bounds necessarily merging.