Berezinskii--Kosterlitz--Thouless transition of the two-dimensional $XY$ model on the honeycomb lattice

TL;DR

This study investigates the BKT transition of the 2D XY model on a honeycomb lattice using neural networks and Monte Carlo simulations, revealing discrepancies in critical temperature estimates and highlighting the need for further analytical work.

Contribution

It applies neural network techniques to analyze the BKT transition on a honeycomb lattice, providing new estimates of the critical temperature and identifying discrepancies with theoretical predictions.

Findings

NN estimates of T_BKT,H are 0.560(9), deviating from the expected 1/√2.

Helicity modulus analysis yields T_BKT,H as 0.571(8), aligning with NN results.

The study highlights the need for detailed analytic calculations to resolve the discrepancy.

Abstract

The Berezinskii--Kosterlitz--Thouless (BKT) transition of the two-dimensional $XY$ model on the honeycomb lattice is investigated using both the techniques of Neural Network (NN) and Monte Carlo simulations. It is demonstrated in the literature that with certain plausible assumptions, the associated critical temperature $T_{\text{BKT,H}}$ is found to be $\frac{1}{\sqrt{2}}$ exactly. Surprisingly, the value of $T_{\text{BKT,H}}$ obtained from our NN calculations is 0.560(9) which deviates significantly from $\frac{1}{\sqrt{2}}$. In addition, based on the helicity modulus, the $T_{\text{BKT,H}}$ determined is 0.571(8) agreeing well with that resulting from the NN estimation. The outcomes presented in this study indicate that a detailed analytic calculation is desirable to solve the found discrepancy.