A model with symmetry-breaking phase transition triggered by a double-well potential

In some recent papers some theorems on sufficiency conditions for the occurrence of a $\mathbb{Z}_2$-symmetry breaking phase transition ($\mathbb{Z}_2$-SBPT) have been showed making use of geometric-topological concepts of potential energy landscapes. In particular, a $\mathbb{Z}_2$-SBPT can be triggered by double-well potentials, or in an equivalent way, by dumbbell-shaped equipotential surfaces. In this paper we introduce a model with a classical $\mathbb{Z}_2$-SBPT which, due to its essential feature, shows in the clearest way the generating-mechanism of a $\mathbb{Z}_2$-SBPT above mentioned. Despite the model is not a physical model, it has all the features of such a model with the same kind of SBPT. At the end of the paper a comparison with the $\phi^4$ model is made. The model may be useful for didactic purposes.