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arXiv:cond-mat/9901314 · Physical review. B, Condensed matter

Essential finite-size effect in the two-dimensionalXYmodel

S. G. Chung

Published Oct 15, 1999• 40 citations
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Abstract

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 ${T}_{\mathrm{BKT}}=0.892$ appears in a finite system of 2000--3000 spins as a massless to massive transition with the effective critical temperature ${T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.$

DOI
10.1103/physrevb.60.11761

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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 Tc=1.07\ifmmode±\else\textpm\fi0.01.{T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.Tc​=1.07\ifmmode±\else\textpm\fi0.01.
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.
In recent works [S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998); S. T. Bramwell et al., Phys. Rev. Lett. 84, 3744 (2000)], a generalized universality has been proposed, linking phenomena as dissimilar as two-dimensional (2D) magnetism and turbulence. To test these ideas, we performed Monte Carlo simulations of the 2D XY model. We found that the shape of the probability distribution function for the magnetization M is non-Gaussian and independent of the system size-in the range of the lattice sizes studied-below the Kosterlitz-Thoules temperature. However, our results suggest that in the full 2D XY model the shape of these distributions has a slight dependence on temperature-for finite volume-below the lattice-shifted critical temperature T*(L). This behavior can be explained by using renormalization group arguments and an extended finite-size scaling analysis, and by the existence of bounds for M.
We test an improved finite-size scaling method for reliably extracting the critical temperature T BKT of a Berezinskii–Kosterlitz–Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness ρ s at T BKT in combination with the Kosterlitz–Nelson relation between the transition temperature and the stiffness, ρ s ( T BKT ) = 2 T BKT /π, we define a size-dependent transition temperature T BKT ( L 1 , L 2 ) based on a pair of system sizes L 1 , L 2 , e.g., L 2 = 2 L 1 . We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining T BKT ( L 1 , L 2 ). For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for L up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result T BKT = 0.8935(1) is several error bars above the previously best estimates of the transition temperature, T BKT ≈ 0.8929. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated T BKT because of the neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.
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Research in recent years has shown that combining finite-size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two-dimensional statistical mechanical and one-dimensional quantum mechanical systems. We review finite-size scaling theory from the general point of view of renormalization group theory for both continuous and first-order transitions (both for systems with discrete and continuous symmetries). We review applications where a comparison with exact results can be made. These include the Ising, Baxter, and q-state Potts models and the Ising model with a defect line. Various other applications such as quantum systems, the self-avoiding random walk, percolation, and Kosterlitz-Thouless transitions are briefly reviewed. The Kosterlitz-Thouless transitions and the critical fan in the antiferromagnetic 3-state Potts model are discussed at somewhat greater length.