Study of the Berezinskii-Kosterlitz-Thouless transition: an unsupervised machine learning approach

Area of Science:

• Condensed Matter Physics
• Statistical Mechanics
• Machine Learning Applications

Background:

• The Berezinskii-Kosterlitz-Thouless (BKT) transition is a key phenomenon in 2D systems, but accurately determining its transition temperature remains a challenge.
• Previous attempts using principal component analysis (PCA) with vorticity data for BKT temperature estimation yielded inconclusive results.

Purpose of the Study:

• To develop a reliable method for estimating the BKT transition temperature in anisotropic classical Heisenberg models.
• To investigate the efficacy of PCA in analyzing magnetic phase transitions in both square and triangular lattices.

Main Methods:

• Application of unsupervised machine learning, specifically principal component analysis (PCA), to anisotropic classical Heisenberg XY and XXZ models.
• Analysis of the first principal component-temperature curve to identify and estimate the BKT transition temperature (TBKT).
• Exploration of models with ferromagnetic exchange on a square lattice and antiferromagnetic exchange on a triangular lattice.

Main Results:

• The study demonstrates that analyzing the first principal component-temperature curve using PCA can accurately estimate TBKT, consistent with existing literature.
• This PCA-based method proves effective for anisotropic Heisenberg models with ferromagnetic exchange on a square lattice.
• The approach successfully estimates TBKT for frustrated antiferromagnetic exchange on a triangular lattice, even when transitions are close.

Conclusions:

• Principal component analysis offers a robust method for estimating BKT transition temperatures in complex magnetic systems.
• PCA facilitates the separation and determination of distinct phase transition points, such as the BKT transition and Ising-like transitions, in frustrated magnetic models.