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In this paper we present the study of the topology of the equipotentialhypersurfaces of configuration space of the mean-field $\phi^4$ model with a$\mathbb{Z}_2$ symmetry. Our purpose is discovering, if any, the relationbetween the second-order $\mathbb{Z}_2$-symmetry breaking phase transition andthe geometric entities mentioned above. The mean-field interaction allows us tosolve analytically either the thermodynamic in the canonical ensemble or thetopology by means of Morse theory. We have analyzed the results at the light oftwo theorems on sufficiency conditions for symmetry breaking phase transitionsrecently proven. This study makes part of a research line based on the generalframework of geometric-topological approach to Hamiltonian chaos and criticalphenomena.

Topology of configuration space of the mean-field phi^4 model by Morse theory