When topology triggers a phase transition

TL;DR

This paper investigates the role of topology changes in the configuration space as a mechanism for phase transitions, identifying conditions under which topology change is necessary or not for thermodynamic singularities.

Contribution

It identifies topology change as the sole mechanism for phase transitions in certain short-range confining potentials and discusses exceptions involving long-range interactions and non-confining potentials.

Findings

Topology change correlates with phase transitions in specific systems.

Long-range interactions can induce phase transitions without topology change.

Non-confining potentials can also lead to phase transitions via maximization processes.

Abstract

Two mathematical mechanisms, responsible for the generation of a thermodynamic singularity, are individuated. For a class of short-range, confining potentials, a topology change in some family of configuration space submanifolds is the only possible such mechanism. Two examples of systems in which the phase transition is not accompanied by a such topology change are discussed. The first one is a model with long-range interactions, namely the mean-field phi^4-model, the second example is a one-dimensional system with a non-confining potential energy function. For both these systems, the thermodynamic singularity is generated by a maximization over one variable (or one discrete index) of a smooth function, although the context in which the maximization occurs is very different.