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Progress of Theoretical and Experimental Physics

Berezinskii–Kosterlitz–Thouless Transition of the Two-Dimensional XY Model on the Honeycomb Lattice

Fu-Jiun Jiang

Published Sep 26, 2024• 4 citations
Publisher

Abstract

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 ${1}/{\sqrt{2}}$ exactly. Surprisingly, the value of $T_{\text{BKT,H}}$ obtained from our NN calculations is 0.572(3), which deviates significantly from ${1}/{\sqrt{2}}$. In addition, based on the helicity modulus, the $T_{\text{BKT,H}}$ determined is 0.576(4), agreeing well with that resulting from the NN estimation. It will be interesting to carry out a more detailed analytic calculation to obtain a theoretical value consistent with the numerical result reached here.

DOI
10.1093/ptep/ptae147

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

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Frontiers in Physics•Mar 4, 2022•3 citations
Abstract The Berezinskii–Kosterlitz–Thouless (BKT) transition of the two-dimensional XYXYXY 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 TBKT,HT_{\text{BKT,H}}TBKT,H​ is found to be 1/2{1}/{\sqrt{2}}1/2​ exactly. Surprisingly, the value of TBKT,HT_{\text{BKT,H}}TBKT,H​ obtained from our NN calculations is 0.572(3), which deviates significantly from 1/2{1}/{\sqrt{2}}1/2​. In addition, based on the helicity modulus, the TBKT,HT_{\text{BKT,H}}TBKT,H​ determined is 0.576(4), agreeing well with that resulting from the NN estimation. It will be interesting to carry out a more detailed analytic calculation to obtain a theoretical value consistent with the numerical result reached here.
Berezinskii--Kosterlitz--Thouless transition of the two-dimensional XYXYXY model on the honeycomb lattice
The Berezinskii--Kosterlitz--Thouless (BKT) transition of the two-dimensional XYXYXY 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 TBKT,HT_{\text{BKT,H}}TBKT,H​ is found to be 12\frac{1}{\sqrt{2}}2​1​ exactly. Surprisingly, the value of TBKT,HT_{\text{BKT,H}}TBKT,H​ obtained from our NN calculations is 0.560(9) which deviates significantly from 12\frac{1}{\sqrt{2}}2​1​. In addition, based on the helicity modulus, the TBKT,HT_{\text{BKT,H}}TBKT,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.
Abstract In this paper, we thoroughly examined the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the two-dimensional XY model on the honeycomb lattice. To address its thermodynamical behavior, we combined standard numerical Monte Carlo simulations with the simulated annealing (SA) protocol and entropic simulations based on the Wang-Landau (WL) algorithm. The transition temperature was determined using the second (ϒ) and fourth-order (ϒ4) helicity modulus as the order parameter. Our best finite-size scaling analysis suggests T BKT = 0.575(8) from SA and T BKT = 0.576(3) from WL. These values deviate significantly from the expected theoretical value of 1 / 2 . We believe that this discrepancy suggests that the theoretical assumptions regarding the analytical calculation may need to be revisited. Additionally, we calculated the vortex density and the formation energy of the vortex-antivortex pairs, where the obtained vortex formation energy is 2μ = 5.80(12). Upon comparison with the square lattice, our results support the notion of instability of the honeycomb lattice to support the spin long-range order and give additional backing to the critical behavior we found.
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.
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 2D 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.
Using a supervised neural network (NN) trained once on a one-dimensional lattice of 200 sites, we calculate the Berezinskii–Kosterlitz–Thouless phase transitions of the two-dimensional (2D) classical XY and the 2D generalized classical XY models. In particular, both the bulk quantities Binder ratios and the spin states of the studied systems are employed to construct the needed configurations for the NN prediction. By applying semiempirical finite-size scaling to the relevant data, the critical points obtained by the NN approach agree well with the known results established in the literature. This implies that for each of the considered models, the determination of its various phases requires only a little information. The outcomes presented here demonstrate convincingly that the employed universal NN is not only valid for the symmetry breaking related phase transitions, but also works for calculating the critical points of the phase transitions associated with topology. The efficiency of the used NN in the computation is examined by carrying out several detailed benchmark calculations.
Two-dimensional generalized XY spin model on a triangular lattice is studied by means of Monte-Carlo simulations. The critical temperatures of Berezinskii-Kosterlitz-Thouless (BKT) phase transition are obtained by the method of helicity modulus. It is found that the results are consistent with those obtained by other methods. The vortex density and the vortex-antivortex pair formation energy are also obtained. The result shows that the critical temperature decreases with the increase of the generalization parameter q . While the vortex-antivortex pair formation energy increases with the increase of q when q>1 .