Topological nonconnectivity threshold in long-range spin systems

We demonstrate the existence of a topological disconnection threshold, recently found by Borgonovi [J. Stat. Phys. 116, 1435 (2004)], for generic 1-d anisotropic Heisenberg models interacting with an interparticle potential R-alpha when 0 < 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 of as a distinctive property of anisotropic long-range interacting systems.