Symmetry breaking phase transitions in mean-field models triggered by double-well potentials

We present a general rule to shape a long-range potential of a Hamiltonian system for a DOUBLE-STRUCK CAPITAL Z(2)-symmetry breaking phase transition (DOUBLE-STRUCK CAPITAL Z(2)-SBPT) to occur. The main feature is a double-well potential which competes with the entropy in shaping the free energy accordingly to Landau mean-field theory of SBPTs. Further, we introduce a classical spin model undergoing a first-order DOUBLE-STRUCK CAPITAL Z(2)-SBPT which was born from the effort to find one of the simplest way to generate such a phenomenon by means of a double-well potential. The model may be suitable also for didactic purposes and for numerical investigation of the dynamic near the transition point. Finally, we revisit the Ising model and the spherical model (Berlin-Kac) in mean-field version showing the double-well potential at work.