A class of long range Ising spin models described by Curie-Weiss mean field theory

We show that Ising spin models where the interaction matrix has eigenvalues whose number and values are finite and invariant with respect to N (the system size) in the limit N → ∞ are exactly solvable (subject also to a few other weaker conditions on the eigenvectors). Systems of this type are described by Curie-Weiss mean field type equations (〈Si〉 = tanh β (ΣiJij 〈Sj〉 )), which we shall refer to as the naive mean field type equations (NMFE), to distinguish them from other mean field equations which have an extra “reaction term”. Some models for which NMFE are valid will briefly be reviewed. It is found that, for certain choices of the interaction architecture, ferromagnetic systems (Jij ⩾ 0) exhibit weak spin-glass-like behaviour in the sense that they have many stable states and not just those associated with ferromagnetism. These extra stable states appear discontinuously. The application of these techniques to structured models of spin glasses and neural networks will also be briefly reviewed.