Necessary and sufficient conditions for ${\mathbb{Z}}_2$-symmetry-breaking phase transitions

Abstract

In a recent paper a toy model (hypercubic model) undergoing a first-order ${\mathbb{Z}}_2$-symmetry-breaking phase transition (${\mathbb{Z}}_2$-SBPT) was 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 surfaces (${\Sigma }_v$'s) of configuration space. In this paper we show that at the origin of a ${\mathbb{Z}}_2$-SBPT there is a geometric property of the ${\Sigma }_v$'s, i.e., dumbbell-shaped ${\Sigma }_v$'s suitably defined, which includes a topological change as a limiting case. This property is necessary and sufficient condition to entail a ${\mathbb{Z}}_2$-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 ${\mathbb{Z}}_2$-SBPT belonging to several universality classes, and finally to a physical models, i.e., the mean-field ${\varphi }^4$ model and a simplified version of it.

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Cited by in corpus (3)

• Models with symmetry-breaking phase transitions triggered by dumbbell-shaped equipotential surfaces
• Phase transitions triggered by dumbbell equipotential hypersurfaces
• Simplified energy landscape of the ${\varphi }^4$ model and the phase transition