# Toward a refining of the topological theory of phase transitions

## TL;DR

This paper investigates the topological theory of phase transitions, showing that in the 2D lattice -model, phase transitions are linked to asymptotic topological changes of energy level sets, despite the absence of critical points.

## Contribution

It refines the topological theory of phase transitions by demonstrating the role of asymptotic topological changes in models lacking potential critical points.

## Key findings

- Phase transition linked to asymptotic topology change in the -model.
- Counterexample to previous theorems showing no critical points at transition.
- Numerical evidence supporting topology change as a transition mechanism.

## Abstract

The topological theory of phase transitions was proposed on the basis of different arguments, the most important of which are: a direct evidence of the relation between topology and phase transitions for some exactly solvable models; an explicit relation between entropy and topological invariants of certain submanifolds of configuration space, and, finally, two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topological changes of some submanifolds of configuration space. It has been recently shown that the $2D$ lattice $\phi^4$-model provides a counterexample that falsifies the mentioned theorems. On the basis of a numerical investigation, the present work indicates the way to overcome this difficulty: in spite of the absence of critical points of the potential in correspondence of the transition energy, the phase transition of this model stems from an asymptotic ($N\to\infty$) change of topology of the energy level sets.