Simple spin models with non-concave entropies

Two simple spin models are studied to show that the microcanonical entropy can be a non-concave function of the energy, and that the microcanonical and canonical ensembles can give non-equivalent descriptions of the same system in the thermodynamic limit. The two models are simple variations of the classical paramagnetic spin model of non-interacting spins and are solved as easily as the latter model.