Phase transitions triggered by dumbbell equipotential hypersurfaces

TL;DR

This paper demonstrates that phase transitions can occur without topological changes in configuration space, using a geometric property called dumbbell shape, and applies this to the mean-field $$ model.

Contribution

It introduces a new geometric criterion based on dumbbell-shaped equipotential hypersurfaces that can trigger symmetry-breaking phase transitions independently of topology.

Findings

Dumbbell-shaped hypersurfaces are sufficient for $$-SBPT.

Phase transitions can occur without topological changes in $$ models.

The geometric property explains phase transitions in the mean-field $$ model.

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 $\phi^4$ model.