  We demonstrate the existence of a topological disconnection threshold,
recently found in Ref. \cite{JSP}, for generic $1-d$ anisotropic Heisenberg
models interacting with an inter--particle potential $R^{-\alpha}$ when
$0<\alpha < 1$ (here $R$ is the distance among spins). We also show that if
$\alpha $ is greater than the embedding dimension $d$ then the ratio between
the disconnected energy region and the total energy region goes to zero when
the number of spins becomes very large. On the other hand, numerical
simulations in $d=2,3$ for the long-range case $\alpha < d$ support the
conclusion that such a ratio remains finite for large $N$ values. The
disconnection threshold can thus be thought as a distinctive property of
anisotropic long-range interacting systems.
