Generalized long-range ferromagnetic Ising spin models

Abstract

A class of simply solvable long-range ferromagnetic models is studied in terms of the eigenvalues and eigenvectors of the interaction matrix. The validity of this approach was first systematically studied by Canning (1992). The generalized ferromagnetic models studied in this paper are a ferromagnetic equivalent of Hopfield neural networks and site-disorder spin glass models, although the interactions of the examples studied are chosen in a deterministic way. These ferromagnetic models, in the same way as the separable disordered models, are described by Curie-Weiss mean-field equations of the form (Si)=tanh beta ( Sigma jJij(Sj)), and have a free energy surface with many minima (but finite in number) separated by infinite energy barriers. They have stable states (in the sense that they have an infinite lifetime in the thermodynamic limit) which are non-ferromagnetic, although the ferromagnetic stable states always have the lowest free energy.