Necessary and sufficient conditions for -symmetry-breaking phase transitions

In a recent paper a toy model (hypercubic model) undergoing a first-order -symmetry-breaking phase transition (-SBPT) was introduced. The hypercubic model was inspired by the topological hypothesis, according to which a phase transition may be entailed by suitable topological changes of the equipotential surfaces (? v 's) of configuration space. In this paper we show that at the origin of a -SBPT there is a geometric property of the ? v 's, i.e. dumbbell-shaped ? v 's suitably defined, which includes a topological change as a limiting case. This property is necessary and sufficient condition to entail a -SBPT. This new approach has been applied to three models: a modified version introduced here of the hypercubic model, a model introduced in a recent paper with a continuous -SBPT belonging to several universality classes, and finally to a physical models, i.e. the mean-field ? 4 model and a simplified version of it.