Phase transitions triggered by dumbbell equipotential hypersurfaces

In a recent paper a toy model (called hypercubic model) undergoing a first-order $\mathbb{Z}_2$ symmetry breaking phase transition (SBPT) has been introduced. The hypercubic model was inspired by the \emph{topological hypothesis}, according to which a phase transition may be entailed by suitable topological changes of the equipotential hypersurfaces $\Sigma_v$ of configuration space. The $\Sigma_v$'s of the hypercubic model have a single topological change, which, under further particular hypotheses of geometric nature, entails the $\mathbb{Z}_2$-SBPT. In this paper we introduce an extended version of the hypercubic model in which no topological change in the $\Sigma_v$'s is present anymore, but nevertheless the $\mathbb{Z}_2$-SBPT occurs the same. We introduce a geometric property of the $\Sigma_v$'s (i.e. dumbbell $\Sigma_v$'s suitably defined) that is sufficient to entail a $\mathbb{Z}_2$-SBPT regardless their topology. The paper ends by applying the picture of the dumbbell $\Sigma_v$'s to a physical model, i.e. the mean-field $\phi^4$ model.