Insufficiency of a purely topological approach to phase transitions

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.