Physical Review B
The phase diagram of the Ising model with antiferromagnetic nearest-neighbor interactions and ferromagnetic equivalent-neighbor interactions on the Cayley tree is determined exactly. A nonuniversal critical line separates the disordered and the ordered phases. A line of first-order transitions separating ferromagnetic order from antiferromagnetic order ends in the midst of the ordered phase at a classical ordered critical point. For a small range of values of the ratio of the two couplings, two transitions occur as the temperature is varied. In this case the uniform magnetization is not a monotonic function of the temperature.
Kaufman, Miron and Kahana, M., "Cayley-Tree Ising Model with Antiferromagnetic Nearest-Neighbor and Ferromagnetic Equivalent-Neighbor Interactions" (1988). Physics Faculty Publications. 81.
Kaufman, Miron and M. Kahana. "Cayley-Tree Ising Model with Antiferromagnetic Nearest-Neighbor and Ferromagnetic Equivalent-Neighbor Interactions." Physical Review B 37 (1988): 7638-7642.
Copyright 1988 American Physical Society. Available on publisher's site at http://link.aps.org/doi/10.1103/PhysRevB.37.7638.