Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Self-Consistent Boundary Conditions in the 2 D XY Model

Peter Olsson

Published Dec 19, 1994• 49 citations
Publisher

Abstract

A generalization of periodic boundary conditions to incorporate fluctuations with wavelength larger than the system size is discussed. These boundary conditions are applied to Monte Carlo calculations on the two-dimensional $\mathrm{XY}$ model. The finite-size effect on the vortex-vortex correlation function, and thereby the dielectric function, is shown to be virtually eliminated. These Monte Carlo data are shown to be in excellent agreement with the Kosterlitz renormalization group equations, and the critical temperature is found to be ${T}_{c}=0.8922(2)$.

DOI
10.1103/physrevlett.73.3339

Related papers

Monte Carlo analysis of the two-dimensionalXYmodel. I. Self-consistent boundary conditions

Peter Olsson

A finite-size effect in the two-dimensional XY model is analytically obtained in the low-temperature limit. It is then shown that the sign of this effect may be changed by generalizing the boundary conditions to include twists over the system, and also that the finite-size effect may be entirely eliminated by adjusting the amplitude of these twists. At finite temperatures, up to ${\mathit{T}}_{\mathit{c}}$, the finite-size effect is shown to be virtually eliminated through the application of a self-consistent condition for the twist amplitude. This self-consistent condition is given a simple motivation. The use of the dielectric function from Monte Carlo simulations on the two-dimensional Coulomb gas is also critically discussed.

Physical review. B, Condensed matter•Aug 1, 1995•51 citations

Monte Carlo analysis of the two-dimensionalXYmodel. II. Comparison with the Kosterlitz renormalization-group equations

Peter Olsson

The temperature scale of relevance for the vortex excitations in the two-dimensional XY model is identified. The size dependence of the helicity modulus at constant vortex temperature is then obtained by Monte Carlo simulations. The Monte Carlo data are then compared with the results for the length-dependent dielectric function from the Kosterlitz renormalization-group equations. The agreement is excellent for temperatures up to and slightly above ${\mathit{T}}_{\mathit{c}}$. The critical temperature is determined to be ${\mathit{T}}_{\mathit{c}}$=0.892 13(10). The temperature dependence of the characteristic length right above ${\mathit{T}}_{\mathit{c}}$ is also obtained and found to be in agreement with recent results from high-temperature series expansions.

Physical review. B, Condensed matter•Aug 1, 1995•110 citations

Critical behavior of the two-dimensionalXYmodel

Rajan Gupta, CLIVE F. BAILLIE

We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat ${\mathit{C}}_{\mathit{v}}$ for the XY model. The simulations are done on ${64}^{2}$, ${128}^{2}$, ${256}^{2}$, and ${512}^{2}$ lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, ${\mathit{T}}_{\mathit{c}}$=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.

Physical review. B, Condensed matter•Feb 1, 1992•196 citations

Alternative boundary conditions for Monte Carlo simulations based on self-consistent correlations: Application to the two- and three-dimensional Ising models

I. Etxebarria, Cesar Capillas, Luis Elcoro

An alternative to periodic boundary conditions is developed and tested in Monte Carlo simulations of the two- and three-dimensional Ising models. The boundary conditions are based on a mean-field approach that incorporates consistency constraints for the magnetization and correlations between nearest neighbors by means of an effective field and an extra coupling between nearest neighbors at the boundary of the simulation box. During the simulation the self-consistent equations are solved, and statistics are accumulated to obtain thermodynamic averages. In comparison with the standard periodic boundary conditions the method gives a more accurate estimation of nonuniversal magnitudes, such as the transition temperature and the behavior of the magnetization, but it cannot compete with the accuracy of other strategies such as finite-size scaling theory or Monte Carlo renormalization group to obtain critical exponents.

Physical Review E•Mar 12, 2010•0 citations

On the helicity modulus, the critical temperature and Monte Carlo simulations for the two-dimensional XY-model

Peter Olsson, Petter Minnhagen

A lattice size dependence of the helicity modulus for the two-dimensional XY-model is inferred by aid of the Coulomb gas-vortex analogy and an explicit form of this dependence is obtained from a set of self-consistent equations for the Coulomb gas. High precision Monte Carlo simulations in conjunction with this inferred lattice size dependence is used to determine the critical temperature. Some other recent results in the literature are discussed in the light of the present investigation.

Physica Scripta•Feb 1, 1991•38 citations

Monte Carlo simulations of finite-size effects in Kosterlitz Thouless systems

F. Falo, L. M. Florı́a, R Navarro

The two-dimensional planar rotator and XY ferromagnetic models for finite lattice sizes (triangular network), with free and periodic boundary conditions, have been studied using Monte Carlo simulations. Thermodynamic quantities, suitable for comparison with experimental data, have been obtained. Heat capacity data are almost insensitive to the lattice size and to the type of boundary conditions, whereas susceptibility data are sensitive to both the lattice size and the boundary conditions. For the Kosterlitz-Thouless phase transition non-universal critical parameters are inferred. They depend on the lattice, the model and the boundary conditions. It is conjectured that the main differences between free and periodic boundary conditions are caused by the influence of the free surface on vortex energetics.

Journal of Physics Condensed Matter•Aug 7, 1989•6 citations
A generalization of periodic boundary conditions to incorporate fluctuations with wavelength larger than the system size is discussed. These boundary conditions are applied to Monte Carlo calculations on the two-dimensional XY\mathrm{XY}XY model. The finite-size effect on the vortex-vortex correlation function, and thereby the dielectric function, is shown to be virtually eliminated. These Monte Carlo data are shown to be in excellent agreement with the Kosterlitz renormalization group equations, and the critical temperature is found to be Tc=0.8922(2){T}_{c}=0.8922(2)Tc​=0.8922(2).
A finite-size effect in the two-dimensional XY model is analytically obtained in the low-temperature limit. It is then shown that the sign of this effect may be changed by generalizing the boundary conditions to include twists over the system, and also that the finite-size effect may be entirely eliminated by adjusting the amplitude of these twists. At finite temperatures, up to Tc{\mathit{T}}_{\mathit{c}}Tc​, the finite-size effect is shown to be virtually eliminated through the application of a self-consistent condition for the twist amplitude. This self-consistent condition is given a simple motivation. The use of the dielectric function from Monte Carlo simulations on the two-dimensional Coulomb gas is also critically discussed.
The temperature scale of relevance for the vortex excitations in the two-dimensional XY model is identified. The size dependence of the helicity modulus at constant vortex temperature is then obtained by Monte Carlo simulations. The Monte Carlo data are then compared with the results for the length-dependent dielectric function from the Kosterlitz renormalization-group equations. The agreement is excellent for temperatures up to and slightly above Tc{\mathit{T}}_{\mathit{c}}Tc​. The critical temperature is determined to be Tc{\mathit{T}}_{\mathit{c}}Tc​=0.892 13(10). The temperature dependence of the characteristic length right above Tc{\mathit{T}}_{\mathit{c}}Tc​ is also obtained and found to be in agreement with recent results from high-temperature series expansions.
We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat Cv{\mathit{C}}_{\mathit{v}}Cv​ for the XY model. The simulations are done on 642{64}^{2}642, 1282{128}^{2}1282, 2562{256}^{2}2562, and 5122{512}^{2}5122 lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, Tc{\mathit{T}}_{\mathit{c}}Tc​=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.
An alternative to periodic boundary conditions is developed and tested in Monte Carlo simulations of the two- and three-dimensional Ising models. The boundary conditions are based on a mean-field approach that incorporates consistency constraints for the magnetization and correlations between nearest neighbors by means of an effective field and an extra coupling between nearest neighbors at the boundary of the simulation box. During the simulation the self-consistent equations are solved, and statistics are accumulated to obtain thermodynamic averages. In comparison with the standard periodic boundary conditions the method gives a more accurate estimation of nonuniversal magnitudes, such as the transition temperature and the behavior of the magnetization, but it cannot compete with the accuracy of other strategies such as finite-size scaling theory or Monte Carlo renormalization group to obtain critical exponents.
The two-dimensional planar rotator and XY ferromagnetic models for finite lattice sizes (triangular network), with free and periodic boundary conditions, have been studied using Monte Carlo simulations. Thermodynamic quantities, suitable for comparison with experimental data, have been obtained. Heat capacity data are almost insensitive to the lattice size and to the type of boundary conditions, whereas susceptibility data are sensitive to both the lattice size and the boundary conditions. For the Kosterlitz-Thouless phase transition non-universal critical parameters are inferred. They depend on the lattice, the model and the boundary conditions. It is conjectured that the main differences between free and periodic boundary conditions are caused by the influence of the free surface on vortex energetics.