Canonical Solution of Classical Magnetic Models with Long-Range Couplings

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.

Submitted for publication in Journal of Statistical Physics

References in corpus (3)

• Ensemble inequivalence in systems with long-range interactions
• Metastable states in a class of long-range Hamiltonian systems
• Microcanonical solution of lattice models with long range interactions

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