Relationship between phase transitions and topological changes in one-dimensional models

We address the question of the quantitative relationship between thermodynamic phase transitions and topological changes in the potential energy manifold analyzing two classes of one dimensional models, the Burkhardt solid-on-solid model and the Peyrard-Bishop model for DNA thermal denaturation, both in the confining and non-confining version. These models, apparently, do not fit [M. Kastner, Phys. Rev. Lett. 93, 150601 (2004)] in the general idea that the phase transition is signaled by a topological discontinuity. We show that in both models the phase transition energy vc is actually non-coincident with, and always higher than, the energy vθ at which a topological change appears. However, applying a procedure already successfully employed in other cases as the mean field φ 4 model, i. e. introducing a map M: v → vs from levels of the energy hypersurface V to the level of the stationary points ”visited” at temperature T , we find that M(vc)=vθ. This result enhances the relevance of the underlying stationary points in determining the thermodynamics of a system, and extends the validity of the topological approach to the study of phase transition to the elusive one-dimensional systems considered here.