Critical temperature of the classical XY model via autoencoder latent space sampling

Area of Science:

• Condensed Matter Physics
• Statistical Mechanics
• Machine Learning Applications

Background:

• The classical XY model is a fundamental system in statistical mechanics.
• The two-dimensional XY model exhibits a topological phase transition known as the Berezinskii-Kosterlitz-Thouless (BKT) transition.
• Understanding the BKT transition is crucial for characterizing topological phenomena in physical systems.

Purpose of the Study:

• To propose a novel machine learning-based method for identifying the BKT phase transition.
• To overcome challenges associated with U(1) symmetry in generating unique states for analysis.
• To accurately determine the critical temperature of the BKT transition.

Main Methods:

• Introduction of an auxiliary field to represent vortex density and mitigate U(1) symmetry.
• Utilizing an autoencoder to map auxiliary fields into a lower-dimensional latent space.
• Sampling from the latent space to compute the thermal average of vortex density.

Main Results:

• The machine learning method successfully identified the emergence of the BKT phase transition.
• The thermal average of the vortex density was accurately computed using samples from the latent space.
• The critical temperature of the phase transition was determined with the proposed methodology.

Conclusions:

• Machine learning offers a powerful tool for analyzing topological phase transitions in physical models.
• The developed auxiliary field and autoencoder approach effectively handles symmetry issues.
• This method provides a robust way to determine critical temperatures in systems like the XY model.