Topological origin of the phase transition in a mean-field model

TL;DR

This paper links the phase transition in a mean-field XY model to a topological change in its configuration space, using Morse theory to analyze critical points and their growth with system size.

Contribution

It introduces a topological perspective on phase transitions, showing the connection between configuration space topology and thermodynamic behavior in the mean-field XY model.

Findings

Topological transition coincides with the thermodynamic phase transition point.

Number of critical points increases rapidly with system size N.

The thermodynamic limit's approach affects the topological analysis.

Abstract

We argue that the phase transition in the mean-field XY model is related to a particular change in the topology of its configuration space. The nature of this topological transition can be discussed on the basis of elementary Morse theory using the potential energy per particle V as a Morse function. The value of V where such a topological transition occurs equals the thermodynamic value of V at the phase transition and the number of (Morse) critical points grows very fast with the number of particles N. Furthermore, as in statistical mechanics, also in topology the way the thermodynamic limit is taken is crucial.