Topological properties of the mean field phi^4 model.

We study the thermodynamics and the properties of the stationary points (saddles and minima) of the potential energy for a φ4 mean field model. We compare the critical energy vc (i.e. the potential energy v(T) evaluated at the phase transition temperature Tc) with the energy vθ at which the saddle energy distribution show a discontinuity in its derivative. We find that, in this model, vc ≫ vθ, at variance to what has been found in the literature for different mean field and short ranged systems. By direct calculation of the energy vs(T) of the “inherent saddles”, i.e. the saddles visited by the equilibrated system at temperature T, we find that vs(Tc) ∼ vθ. Thus, we argue that the thermodynamic phase transition is related to a change in the properties of the inherent saddles rather then to a change of the topology of the potential energy surface at T = Tc. Finally, we discuss the approximation involved in our analysis and the generality of our method.