Necessary and sufficient conditions for Z2-symmetry-breaking phase transitions in Hamiltonian systems

Investigation of the geometric, and possibly also topological, characteristics of the potential energy landscape in configuration space capable of giving rise to a phase transition associated with spontaneous Z2-symmetry breaking in the canonical treatment [1]. The recurring feature in infinite-range systems is the presence of a double well with a gap proportional to the number of degrees of freedom (N) of the system [2]. This characteristic causes the formation of so-called 'dumbbell-shaped' equipotential surfaces under a certain critical potential which have been shown to be a necessary and sufficient condition to generate spontaneous Z2-symmetry breaking [3]. This feature was found in the Ising model [2], in the ?4 model [4] and in some other models solvable analytically built for this research specifically [5,6]. Currently, we are trying to extend these results to the most interesting case from the physical point of view of short-range potentials. In this framework, a Z2-symmetry breaking phase transition can also be defined without invoking the thermodynamic limit.