Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology

Area of Science:

• Topological Data Analysis
• Statistical Mechanics
• Condensed Matter Physics

Background:

• The two-dimensional XY model is a fundamental system in statistical mechanics, exhibiting rich phase transition phenomena.
• Understanding phase transitions and critical exponents is crucial for characterizing complex physical systems.

Purpose of the Study:

• To apply persistent homology and persistence images as observables for studying phase transitions in three variants of the 2D XY model.
• To develop a robust methodology for estimating critical temperatures and critical exponents with quantifiable errors.

Main Methods:

• Utilizing persistent homology and persistence images to analyze lattice spin model configurations.
• Employing logistic regression and k-nearest neighbor models trained on persistence images to extract critical parameters.
• Focusing on finite-size scaling analysis for accurate estimations.

Main Results:

• Successfully identified phase transitions for all three XY model variants examined.
• Provided accurate determinations of critical temperatures and critical exponents of the correlation length.
• Demonstrated the efficacy of the developed topological data analysis methodology.

Conclusions:

• Persistent homology offers a powerful new observable for studying phase transitions in statistical physics models.
• The developed methodology allows for accurate and quantifiable estimation of critical parameters.
• This approach opens new avenues for analyzing complex systems using topological data analysis.