Phase transitions triggered by dumbbell equipotential hypersurfaces

Abstract

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 ${\varphi }^4$ model.

8 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:1106.3870