Long-range interactions and nonextensivity in ferromagnetic spin models

The Ising model with ferromagnetic interactions that decay as 1r^α is analyzed in the nonextensive regime 0<~α<~d, where the thermodynamic limit is not defined. In order to study the asymptotic properties of the model in the N→∞ limit (N being the number of spins) we propose a generalization of the Curie-Weiss model, for which the N→∞ limit is well defined for all α>~0. We conjecture that mean-field theory is exact in the last model for all 0<~α<~d. This conjecture is supported by Monte Carlo heat-bath simulations in the $d=1$ case. Moreover, we confirm a recently conjectured scaling by Tsallis that allows for a unification of extensive (α>d) and nonextensive (0<~α<~d) regimes.