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 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 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 TBKT,H is found to be 1/2 exactly. Surprisingly, the value of TBKT,H obtained from our NN calculations is 0.572(3), which deviates significantly from 1/2. In addition, based on the helicity modulus, the 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 XY model on the honeycomb lattice
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 TBKT,H is found to be 21 exactly. Surprisingly, the value of TBKT,H obtained from our NN calculations is 0.560(9) which deviates significantly from 21. In addition, based on the helicity modulus, the 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 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.
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 .
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.
The Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two-dimensional planar rotator and XY models on a square lattice, diluted by randomly placed vacancies, is studied here using hybrid Monte Carlo simulations that combine single spin flip, cluster, and over-relaxation techniques. The transition temperature Tc is determined as a function of vacancy density ${\ensuremath{\rho}}_{\mathrm{vac}}$ by finite-size scaling of the helicity modulus and the in-plane magnetic susceptibility. The results for Tc are consistent with those from the much less precise fourth-order cumulant of Binder. Tc is found to decrease monotonically with increasing ${\ensuremath{\rho}}_{\mathrm{vac}}$, and falls to zero close to the square lattice percolation limit, ${\ensuremath{\rho}}_{\mathrm{vac}}\ensuremath{\approx}0.41$. The result is physically reasonable: the quasi-long-range orientational order of the low-temperature phase cannot be maintained in the absence of sufficient spin interactions across the lattice.