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arXiv:cond-mat/0502556 · Journal of Physics A Mathematical and General

The two-dimensional XY model at the transition temperature: a high-precision Monte Carlo study

Martin Hasenbusch

Published Jun 15, 2005• 165 citations
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Abstract

We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.

DOI
10.1088/0305-4470/38/26/003

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We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.
Using two sets of high-precision Monte Carlo data for the two-dimensional XY model in the Villain formulation on square L\ifmmode\times\else\texttimes\fi{}L lattices, the scaling behavior of the susceptibility \ensuremath{\chi} and correlation length \ensuremath{\xi} at the Kosterlitz-Thouless phase transition is analyzed with emphasis on multiplicative logarithmic corrections (ln L${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the finite-size scaling region and (ln \ensuremath{\xi}${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the high-temperature phase near criticality, respectively. By analyzing the susceptibility at criticality on lattices of size up to 5122{512}^{2}5122 we obtain r=-0.0270(10), in agreement with recent work of Kenna and Irving on the finite-size scaling of Lee-Yang zeros in the cosine formulation of the XY model. By studying susceptibilities and correlation lengths up to \ensuremath{\xi}\ensuremath{\approx}140 in the high-temperature phase, however, we arrive at quite a different estimate of r=0.0560(17), which is in good agreement with recent analyses of thermodynamic Monte Carlo data and high-temperature series expansions of the cosine formulation.
We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz-Thouless phase transition. We determine the fixed point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single cluster algorithm, we simulate lattices up to L = 4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz-Thouless transition temperature on relatively small lattices. We test the new method at the example of the 2-component φ 4 model on the lattice.
We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz–Thouless phase transition. We determine the fixed-point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single-cluster algorithm, we simulate lattices up to L = 4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz–Thouless transition temperature on relatively small lattices. We test the new method on the example of the two-component ϕ 4 model on the lattice.
Convincing numerical evidence is obtained via extensive quantum Monte Carlo simulations on square lattices as large as 96\ifmmode\times\else\texttimes\fi{}96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless phase transition at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.350(4). Correlation length and in-plane susceptibility diverge at Tc{\mathit{T}}_{\mathit{c}}Tc​ precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​ and exhibits a peak around kT/J=0.45.
We study the two-dimensional classical XY model by the large-scale Monte Carlo simulation of the Swendsen-Wang multi-cluster algorithm using multiple GPUs on the open science supercomputer TSUBAME 2.0. Simulating systems up to the linear system size L=65536, we investigate the Kosterlitz-Thouless (KT) transition. Using the generalized version of the probability-changing cluster algorithm based on the helicity modulus, we locate the KT transition temperature in a self-adapted way. The obtained inverse KT temperature β_{KT} is 1.11996(6). We estimate the exponent to specify the multiplicative logarithmic correction, -2r, and precisely reproduce the theoretical prediction -2r=1/8.
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.