Models with symmetry-breaking phase transitions triggered by dumbbell-shaped equipotential surfaces

Recently, a number of sufficiency conditions have been shown for the occurrence of a Z2-symmetry breaking phase transition (Z2-SBPT) starting from geometric-topological concepts of potential energy landscapes. In particular, a Z2-SBPT can be triggered by double-well potentials, or equivalently by dumbbell-shaped equipotential surfaces. In this paper, we introduce two models with a Z2-SBPT that, due to their essential feature, show in the clearest way the generating mechanism of a Z2-SBPT. Although they cannot be considered physical models, they all have the features of such models with the same kind of SBPT. At the end of the paper, the ?4 model is revisited in light of this approach. In particular, the landscape of one of the models introduced here turned out to be equivalent to that of the mean-field ?4 model in a simplified version.