Spin Model with Antiferromagnetic and Ferromagnetic Interactions

The one-dimensional Ising model with nearest-neighbor repulsion and infinitely long-range attraction acting only on even-numbered neighbors is solved exactly, using mean-field theory for the long-range part. Below a certain temperature one gets antiferromagnetic ordering. For temperatures giving antiferromagnetic ordering in zero magnetic field, there will be a phase transition from the antiferromagnetic to the more common ferromagnetic ordering for a finite magnetic field. The transition is of first or second order giving discontinuous magnetization or discontinuous susceptibility, respectively. It is shown by direct calculation that the second-order phase transition coincides with a divergence in the $\gamma$ expansion for this model.