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The relation between saddle points of the potential of a classicalmany-particle system and the analyticity properties of its Boltzmann entropy isstudied. For finite systems, each saddle point is found to cause anonanalyticity in the Boltzmann entropy, and the functional form of thisnonanalytic term is derived for the generic case of potentials having the Morseproperty. With increasing system size the order of the nonanalytic term growsunboundedly, leading to an increasing differentiability of the entropy.Nonetheless, a distribution of an unboundedly growing number of saddle pointsmay cause a phase transition in the thermodynamic limit. Analyzing thecontribution of the saddle points to the density of states in the thermodynamiclimit, conditions on the distribution of saddle points and their curvatures arederived which are necessary for a phase transition to occur. With theseresults, the puzzling absence of topological signatures in the spherical modelis elucidated. As further applications, the phase transitions of the mean-fieldXY model and the mean-field k-trigonometric model are shown to be induced bysaddle points of vanishing curvature.

Nonanalyticities of the entropy induced by saddle points of the potential energy landscape