Topology and phase transitions: a paradigmatic evidence

TL;DR

This paper demonstrates that topology changes in the configuration space are directly linked to phase transitions, providing numerical evidence using the Euler characteristic in a two-dimensional lattice ^4 model.

Contribution

It introduces a general numerical method to connect topological invariants with phase transitions, applicable across different models.

Findings

Euler characteristic  of equipotential hypersurfaces signals topology changes

Major topology change correlates with phase transition in the ^4 model

Method can be applied to other models to study topology-phase transition link

Abstract

We report upon the numerical computation of the Euler characteristic \chi (a topologic invariant) of the equipotential hypersurfaces \Sigma_v of the configuration space of the two-dimensional lattice $\phi^4$ model. The pattern \chi(\Sigma_v) vs. v (potential energy) reveals that a major topology change in the family {\Sigma_v}_{v\in R} is at the origin of the phase transition in the model considered. The direct evidence given here - of the relevance of topology for phase transitions - is obtained through a general method that can be applied to any other model.