Long-range interactions and nonextensivity in ferromagnetic spin models

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