Topological properties of the mean-field phi4 model

We study the thermodynamics and the properties of the stationary points (saddles and minima) of the potential energy for a phi4 mean-field model. We compare the critical energy vc [i.e., the potential energy vT evaluated at the phase transition temperature Tc ] with the energy vtheta at which the saddle energy distribution show a discontinuity in its derivative. We find that, in this model, vc >> vtheta, at variance to what has been found in different mean-field and short ranged systems, where the thermodynamic phase transitions take place at vc=vtheta [Phys. Rep. 337, 237 (2000)]]. 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 vsTc approximately vtheta. Thus, we argue that the thermodynamic phase transition is related to a change in the properties of the inherent saddles rather than 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.