Statistical mechanics of Z(M) models on Cayley trees

The problem of “small-field” phase transitions for Z(M) models on Cayley trees is solved in detail. Phase diagrams for zero field are obtained for M = 2, 3, 4, 5 and 6. As special cases, Potts models are also considered and all phases (not only those in zero field) are identified. The M → ∈ limit, the planar rotator model, is also solved completely for the zero-field case. The relevance of the results (especially of the phase diagrams) for the problems associated with Z(M) models on real lattices is discussed.