Canonical Solution of Classical Magnetic Models with Long-Range Couplings

TL;DR

This paper provides an exact solution for a family of classical long-range spin models on lattices, showing that their thermodynamic behavior is equivalent to a mean-field model regardless of lattice details.

Contribution

It introduces a canonical solution for classical $n$-vector spin models with long-range couplings, demonstrating their equivalence to the $ ext{alpha}=0$ case across all parameters.

Findings

The model's thermodynamics are independent of lattice geometry.

The solution involves spectral analysis of coupling matrices.

The model remains extensive but non-additive due to rescaling.

Abstract

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.