Phase transitions and topology changes in configuration space

TL;DR

This paper explores the link between phase transitions and topological changes in the configuration space of physical models, providing exact calculations of topological invariants and supporting a topological approach to understanding phase transitions.

Contribution

It presents the first exact analytic computation of the Euler characteristic in configuration space models and compares models with and without phase transitions to support topological theories.

Findings

Topology changes are linked to phase transitions in the mean-field XY model.

No significant topology change occurs in the 1D XY model without phase transition.

A strong topology change correlates with the phase transition in the mean-field model.

Abstract

The relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant (the Euler characteristic) of certain submanifolds in the configuration space of two physical models. The models are the mean-field XY model and the one-dimensional XY model with nearest-neighbor interactions. The former model undergoes a second-order phase transition at a finite critical temperature while the latter has no phase transitions. The computation of this topologic invariant is performed within the framework of Morse theory. In both models topology changes in configuration space are present as the potential energy is varied; however, in the mean-field model there is a particularly "strong" topology change, corresponding to a big jump in the Euler characteristic, connected with the phase transition, which is absent in the one-dimensional model with no phase transition. The comparison between the two models has two major consequences: i) it lends new and strong support to a recently proposed topological approach to the study of phase transitions; ii) it allows us to conjecture which particular topology changes could entail a phase transition in general. We also discuss a simplified illustrative model of the topology changes connected to phase transitions using of two-dimensional surfaces, and a possible direct connection between topological invariants and thermodynamic quantities.