Canonical solution of classical magnetic models with long-range couplings

Abstract

We study the canonical solution of a family of classical $n-vector$ spin\nmodels on a generic $d$-dimensional lattice; the couplings between two spins\ndecay as the inverse of their distance raised to the power $\\alpha$, with\n$\\alpha<d$. The control of the thermodynamic limit requires the introduction of\na rescaling factor in the potential energy, which makes the model extensive but\nnot additive. A detailed analysis of the asymptotic spectral properties of the\nmatrix of couplings was necessary to justify the saddle point method applied to\nthe integration of functions depending on a diverging number of variables. The\nproperties of a class of functions related to the modified Bessel functions had\nto be investigated. For given $n$, and for any $\\alpha$, $d$ and lattice\ngeometry, the solution is equivalent to that of the $\\alpha=0$ model, where the\ndimensionality $d$ and the geometry of the lattice are irrelevant.\n