Topological properties of the mean field phi^4 model

TL;DR

This paper investigates the relationship between phase transitions and the properties of stationary points in a mean field phi^4 model, revealing that the transition correlates with saddle properties rather than potential energy surface topology.

Contribution

It demonstrates that in the phi^4 mean field model, the phase transition is linked to saddle properties, challenging previous assumptions about potential energy surface topology.

Findings

Vc is much greater than V{theta} in this model.

Vs(Tc) is approximately equal to V{theta}.

The phase transition relates to saddle properties, not topology changes.

Abstract

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.