Topological Signature of First Order Phase Transitions

TL;DR

This paper demonstrates that the topology of the potential energy landscape, specifically the Euler characteristic, can reveal the presence and nature of first order phase transitions without relying on traditional thermodynamic measures.

Contribution

It introduces a topological approach to identify phase transitions, linking the Euler characteristic of configuration space submanifolds to thermodynamic behavior in a solvable mean-field model.

Findings

Euler characteristic signals phase transition types

Topological analysis distinguishes between no transition, second order, and first order transitions

Conjectured link between Euler characteristic and thermodynamic entropy

Abstract

We show that the presence and the location of first order phase transitions in a thermodynamic system can be deduced by the study of the topology of the potential energy function, V(q), without introducing any thermodynamic measure. In particular, we present the thermodynamics of an analytically solvable mean-field model with a k-body interaction which -depending on the value of k- displays no transition (k=1), second order (k=2) or first order (k>2) phase transition. This rich behavior is quantitatively retrieved by the investigation of a topological invariant, the Euler characteristic, of some submanifolds of the configuration space. Finally, we conjecture a direct link between the Euler characteristic and the thermodynamic entropy.