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