Topological Theory of Phase Transitions

The investigation of the Hamiltonian dynamical counterpart of phase transitions, combined with the Riemannian geometrization of Hamiltonian dynamics, has led to a preliminary formulation of a differential-topological theory of phase transitions. In fact, in correspondence of a phase transition there are peculiar geometrical changes of the mechanical manifolds that are found to stem from changes of their topology. These findings, together with two theorems, have suggested that a topological theory of phase transitions can be formulated to go beyond the limits of the existing theories. Among other advantages, the new theory applies to phase transitions in small $N$ systems (that is, at nanoscopic and mesoscopic scales), and in the absence of symmetry-breaking. However, the preliminary version of the theory was incomplete and still falsifiable by counterexamples. The present work provides a relevant leap forward leading to an accomplished development of the topological theory of phase transitions paving the way to further developments and applications of the theory that can be no longer hampered.