The simplified energy landscape of the φ 4 model and the phase transition

The on-lattice φ 4 model is a paradigmatic example of a continuous real-variable model undergoing a continuous symmetry-breaking phase transition (SBPT). Here, we study the Z 2 -symmetric mean-field case without the quadratic term in the local potential. We show that the Z 2 -SBPT is not affected by the quadratic term and that the potential energy landscape is greatly simplified from a geometric-topological viewpoint. In particular, only three critical points exist to confront, with a number growing as eN (N is the number of degrees of freedom) of the model with a negative quadratic term. We focus on the properties of the equipotential surfaces with the aim to deepen the link between SBPTs and the essential properties of a potential that is capable of entailing them. The results are interpreted in view of of some recent achievements regarding rigorous necessary and sufficient conditions for a Z 2 -SBPT.