Phase transition in zero dimensions: A remark on the spherical model

Abstract

It is shown that the spherical model consisting of N spins with nonzero interaction between two spins only has a phase transition in the limit N → ∞. This is a counterexample to a suggestion of Kac which states that an Ising model will have a transition if the corresponding spherical model has a transition. Possible modifications of Kac's conjecture are suggested and discussed.