Phase Diagram of Ising Systems with Additional Long Range Forces

We consider ferromagnetic Ising systems where the interaction is given by the sum of a fixed reference potential and a Kac potential of intensity λ≥0 and scaling parameter γ>0$. In the Lebowitz Penrose limit γ→0⁺$ the phase diagram in the (T,λ) positive quadrant is described by a critical curve λ^mf(T), which separates the regions with one and two phases, respectively below and above the curve. We prove that if $λ>^mf(T), i.e. above the curve, there are at least two Gibbs states for small values of γ. If instead λ<λ^mf(T) and if the reference Gibbs state (i.e. without the Kac potential) satisfies a mixing condition at the temperature T, then, at the same temperature the full interaction (i.e. with also the Kac potential) satisfies the Dobrushin Shlosman uniqueness condition for small values of γ so that there is a unique Gibbs state. Received: 9 April 1996 / Accepted: 26 November 1996