Topology of configuration space of the mean-field phi^4 model by Morse theory

TL;DR

This paper investigates the topology of the configuration space in a mean-field phi^4 model to understand the geometric aspects of phase transitions, using Morse theory and analytical solutions.

Contribution

It provides an analytical study linking the topology of equipotential hypersurfaces to symmetry-breaking phase transitions in a mean-field phi^4 model.

Findings

Topology changes correlate with phase transition points.

Analytical solutions confirm sufficiency conditions for symmetry breaking.

The study advances geometric-topological understanding of critical phenomena.

Abstract

In this paper we present the study of the topology of the equipotential hypersurfaces of configuration space of the mean-field $\phi^4$ model with a $\mathbb{Z}_2$ symmetry. Our purpose is discovering, if any, the relation between the second-order $\mathbb{Z}_2$-symmetry breaking phase transition and the geometric entities mentioned above. The mean-field interaction allows us to solve analytically either the thermodynamic in the canonical ensemble or the topology by means of Morse theory. We have analyzed the results at the light of two theorems on sufficiency conditions for symmetry breaking phase transitions recently proven. This study makes part of a research line based on the general framework of geometric-topological approach to Hamiltonian chaos and critical phenomena.