Unattainability of a purely topological criterion for the existence of a phase transition for nonconfining potentials

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 and 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 to guarantee the occurrence of a phase transition in systems with nonconfining potentials.