BKT transitions of the XY and six-state clock models on the various two-dimensional lattices

TL;DR

This paper investigates the Berezinskii-Kosterlitz-Thouless (BKT) transitions in XY and six-state clock models across various 2D lattices using Monte Carlo and machine learning methods, comparing transition temperatures with the 2D Ising model.

Contribution

It provides a comprehensive numerical analysis of BKT transitions on multiple 2D lattices and introduces machine learning for phase classification in these models.

Findings

Identified BKT transition temperatures for XY and six-state models on honeycomb, kagome, and diced lattices.

Compared BKT transition temperatures with 2D Ising model, finding close but non-universal ratios.

Demonstrated effectiveness of machine learning in classifying phases in spin models.

Abstract

In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii-Kosterlitz-Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various two-dimensional lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases -- namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.