Topological Theory of Phase Transitions

TL;DR

This paper advances a topological theory of phase transitions based on geometric changes in mechanical manifolds, applicable to small systems and those without symmetry-breaking, offering a new perspective beyond traditional theories.

Contribution

It develops a more complete and robust formulation of the topological theory of phase transitions, overcoming previous limitations and enabling broader applications.

Findings

Topological changes in mechanical manifolds correlate with phase transitions.

The theory applies to small N systems and non-symmetry-breaking transitions.

The work provides a more complete framework for the topological approach.

Abstract

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.