Topology and phase transitions: paradigmatic evidence

We report upon the numerical computation of the Euler characteristic chi (a topologic invariant) of the equipotential hypersurfaces Sigma(v) of the configuration space of the two-dimensional lattice varphi(4) model. The pattern chi(Sigma(v)) versus v (potential energy) reveals that a major topology change in the family Sigma(v)(vinR) is at the origin of the phase transition in the model considered. The direct evidence given here-of the relevance of topology for phase transitions-is obtained through a general method that can be applied to any other model.