Unattainability of a purely topological criterion for the existence of a phase transition for non-confining potentials

TL;DR

This paper investigates the link between topology changes in configuration space and phase transitions in classical systems, demonstrating that a purely topological criterion cannot universally predict phase transitions in non-confining potentials.

Contribution

It shows that topology changes are necessary but not sufficient for phase transitions, challenging the idea of a purely topological criterion for their occurrence.

Findings

Topology change correlates with phase transitions but is not sufficient.

Differences in topology change strength do not determine phase transition presence.

Purely topological conditions cannot guarantee phase transitions in non-confining systems.

Abstract

The relation between thermodynamic phase transitions in classical systems and topology changes in their configuration space is discussed for a one-dimensional, analytically tractable solid-on-solid model. The topology of a certain family of submanifolds of configuration space is investigated, corroborating the hypothesis that, in general, a change of the topology within this family is a necessary condition in order to observe a phase transition. Considering two slightly differing versions of this solid-on-solid model, one showing a phase transition in the thermodynamic limit, the other not, we find that the difference in the ``quality'' or ``strength'' of this topology change appears to be insignificant. This example indicates the unattainability of a condition of exclusively topological nature which is sufficient as to guarantee the occurrence of a phase transition in systems with non-confining potentials.