Frontiers in Physics

Numerical Studies of Vortices and Helicity Modulus in the Two-Dimensional Generalized XY Model

Yunzhou Sun, Qin Wu, Yang Xiao-li, Yan Zhou, Lanyan Zhu, Quan Chen, Qing An

Published Mar 4, 2022• 3 citations

Abstract

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 .

DOI: 10.3389/fphy.2022.851322

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Frontiers in Physics

Numerical Studies of Vortices and Helicity Modulus in the Two-Dimensional Generalized XY Model

Yunzhou Sun, Qin Wu, Yang Xiao-li, Yan Zhou, Lanyan Zhu, Quan Chen, Qing An

Published Mar 4, 2022• 3 citations

Publisher

Abstract

DOI: 10.3389/fphy.2022.851322

Related papers

Thermodynamic quantities and phase transition of a generalized XY model on triangular lattice

Yunzhou Sun, Qin Wu, Jingyan Li +1 more

IOP Conference Series Earth and Environmental ScienceMar 1, 20181 citation

Berezinskii-Kosterlitz-Thouless transition in the XY model on the honeycomb lattice: a comprehensive Monte Carlo analysis

Felipe E. F. de Andrade, L. N. Jorge, Cláudio José DaSilva

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.

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David M. Hübscher, Stefan Weßel

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Wolfgang Bietenholz A, Urs Gerber, Fernando G. Rejón-Barrera

The 2D XY model exhibits an essential phase transition, which was predicted long ago—by Berezinskii, Kosterlitz and Thouless (BKT)—to be driven by the (un)binding of vortex–anti-vortex pairs. This transition has been confirmed for the standard lattice action, and for actions with distinct couplings, in agreement with universality. Here we study a highly unconventional formulation of this model, which belongs to the class of topological lattice actions: it does not have any couplings at all, but just a constraint for the relative angles between nearest neighbour spins. By means of dynamical boundary conditions we measure the helicity modulus ϒ, which shows that this formulation performs a BKT phase transition as well. Its finite size effects are amazingly mild, in contrast to other lattice actions. This provides one of the most precise numerical confirmations ever of a BKT transition in this model. On the other hand, up to the lattice sizes that we explored, there are deviations from the spin wave approximation, for instance for the Binder cumulant U 4 and for the leading finite size correction to ϒ. Finally we observe that the (un)binding mechanism follows the usual pattern, although free vortices do not require any energy in this formulation. Due to this observation, one should reconsider an aspect of the established picture, which estimates the critical temperature based on this energy requirement.The 2D XY model exhibits an essential phase transition, which was predicted long ago—by Berezinskii, Kosterlitz and Thouless (BKT)—to be driven by the (un)binding of vortex–anti-vortex pairs. This transition has been confirmed for the standard lattice action, and for actions with distinct couplings, in agreement with universality. Here we study a highly unconventional formulation of this model, which belongs to the class of topological lattice actions: it does not have any couplings at all, but just a constraint for the relative angles between nearest neighbour spins. By means of dynamical boundary conditions we measure the helicity modulus ϒ, which shows that this formulation performs a BKT phase transition as well. Its finite size effects are amazingly mild, in contrast to other lattice actions. This provides one of the most precise numerical confirmations ever of a BKT transition in this model. On the other hand, up to the lattice sizes that we explored, there are deviations from the spin wave approximation, for instance for the Binder cumulant U 4 and for the leading finite size correction to ϒ. Finally we observe that the (un)binding mechanism follows the usual pattern, although free vortices do not require any energy in this formulation. Due to this observation, one should reconsider an aspect of the established picture, which estimates the critical temperature based on this energy requirement.

arXiv:1307.0485Journal of Statistical Mechanics Theory and ExperimentDec 20, 201313 citations

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

Fu-Jiun Jiang

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.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.

arXiv:2406.14812Jun 21, 20240 citations

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.

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

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.