Topology and phase transitions: a 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 $\phi^4$ model. The pattern \chi(\Sigma_v) vs. v (potential energy) reveals that a major topology change in the family {\Sigma_v}_{v\in R} 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.Comment: 4 pages, 4 figure

Topological conditions for discrete symmetry breaking and phase transitions

In the framework of a recently proposed topological approach to phase transitions, some sufficient conditions ensuring the presence of the spontaneous breaking of a Z_2 symmetry and of a symmetry-breaking phase transition are introduced and discussed. A very simple model, which we refer to as the hypercubic model, is introduced and solved. The main purpose of this model is that of illustrating the content of the sufficient conditions, but it is interesting also in itself due to its simplicity. Then some mean-field models already known in the literature are discussed in the light of the sufficient conditions introduced here