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 phi^4 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 V{theta} at which the saddle energy distribution show a discontinuity in its derivative. We find that, in this model, Vc >> V{theta}, 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) ~ V{theta}. 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.