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 Z_{2}-symmetry breaking phase transition (Z_{2}-SBPT) starting from geometric-topological concepts of potential energy landscapes. In particular, a Z_{2}-SBPT can be triggered by double-well potentials, or equivalently by dumbbell-shaped equipotential surfaces. In this paper, we introduce two models with a Z_{2}-SBPT that, due to their essential feature, show in the clearest way the generating mechanism of a Z_{2}-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 ϕ^{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 ϕ^{4} model in a simplified version.