Circle Theorem for Hard-Core Binary Lattice Gases

Two-componen t lattice gases o f the general type introduced by Widom and Rowlinsonr and their con t inuum analog have been o f considerable interest in the study of phase transitions and cooperat ive phenomena. ~2-7) For the simplest case o f two symmetric components A and B with repulsive hard cores between unlike particles and a smaller (point) hard core between like particles, a demixing phase transit ion was proven using the Peierls method on the lattice ~2) and a very clever generalization o f it for the con t inuum case. ~3) These were proven for the case where the fugacities of the two components are the same, ZA = ZB, and large.While it has been presumed that no phase transition is possible for such a system away from the symmetry line o f equal fugacities for the two components , it has only recently been proven for high values o f the fugacities. ~8) There are conceptual similarities between these two-component lattice gases and ferromagnetic Ising systems. ~2'6) For the latter we have the wellknown Lee -Yang "circle theorem ''(9) that says that nonanalyt ic thermodynamics is possible only ife ~ (where H = fi • magnetic field) is o f magni tude one. Hence for the logar i thm of the part i t ion function of the ferromagnetic Ising system to be nonanalytic, it is necessary that H be pure imaginary; if it is