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Physical review. B, Condensed matter

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

Peter Olsson

Published Aug 1, 1995• 51 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.52.4511

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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.
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).
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.
The thermodynamics of the two-dimensional (2D) XY model is formulated by a transfer-matrix method and analyzed by a density-matrix renormalization group. The finite-size scaling and the beta function of the model are studied by the Roomany-Wyld renormalization-group theory. It is found that the 2D XY model has an essential finite-size effect and the Berezinskii-Kosterlitz-Thouless transition with the critical temperature TBKT=0.892{T}_{\mathrm{BKT}}=0.892TBKT​=0.892 appears in a finite system of 2000--3000 spins as a massless to massive transition with the effective critical temperature ${T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.$
A new logarithmic term in the finite-size helicity modulus has been predicted recently for the three-dimensional classical XY-model universality class under nonperiodic boundary conditions. These theoretical ideas have important implications for the analysis of superfluid-density measurements in confined geometries. We present numerical results from Monte Carlo simulations of the helicity modulus for the three-dimensional classical XY model on a finite simple-cubic lattice, with pinned-surface-spin boundary conditions. Our data are consistent with the presence of an additional logarithmic term and an estimate for the amplitude is given.