Canonical solution of classical magnetic models with long-range couplings

We study the canonical solution of a family of classical n-vector spin models on a generic d-dimensional lattice; the couplings between two spins decay as the inverse of their distance raised to the power alpha, with alpha < d. The control of the thermodynamic limit requires the introduction of a rescaling factor in the potential energy, which makes the model extensive but not additive. A detailed analysis of the asymptotic spectral properties of the matrix of couplings was necessary to justify the saddle point method applied to the integration of functions depending on a diverging number of variables. The properties of a class of functions related to the modified Bessel functions had to be investigated. For given n, and for any alpha, d and lattice geometry, the solution is equivalent to that of the alpha = 0 model, where the dimensionality d and the geometry of the lattice are irrelevant.