Topology and phase transitions: paradigmatic evidence

Area of Science:

• Computational physics
• Topological data analysis
• Lattice field theory

Background:

• The two-dimensional lattice varphi(4) model is a fundamental system in quantum field theory.
• Phase transitions in physical models are often associated with changes in system topology.
• Understanding the relationship between topology and phase transitions is crucial for theoretical physics.

Purpose of the Study:

• To numerically compute the Euler characteristic (a topological invariant) of equipotential hypersurfaces in the 2D lattice varphi(4) model.
• To investigate the relationship between topological changes and phase transitions in this model.
• To demonstrate a generalizable method for applying topological analysis to phase transitions.

Main Methods:

• Numerical computation of the Euler characteristic (chi).
• Analysis of equipotential hypersurfaces (Sigma(v)) in the configuration space.
• Correlation of topological patterns (chi versus potential energy v) with phase transitions.

Main Results:

• The Euler characteristic was computed for the equipotential hypersurfaces Sigma(v).
• A significant change in topology was observed at the origin of the potential energy (v=0).
• This major topology change directly corresponds to the phase transition in the model.

Conclusions:

• Topology plays a critical role in driving phase transitions in the 2D lattice varphi(4) model.
• The study provides direct evidence for the relevance of topology in understanding phase transitions.
• The developed numerical method is applicable to other physical models for topological analysis of phase transitions.