Finite-size scaling method for the Berezinskii-Kosterlitz-Thouless transition

Yun-Da Hsieh, Ying-Jer Kao, A. W. Sandvik
Center of Theoretical Sciences and Department of Physics, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10607, Taiwan
arXiv:1302.2900 [cond-mat.stat-mech], (12 Feb 2013)


   author={Hsieh}, Y.-D. and {Kao}, Y.-J. and {Sandvik}, A.~W.},

   title={"{Finite-size scaling method for the Berezinskii-Kosterlitz-Thouless transition}"},

   journal={ArXiv e-prints},




   keywords={Condensed Matter – Statistical Mechanics, Condensed Matter – Soft Condensed Matter},




   adsnote={Provided by the SAO/NASA Astrophysics Data System}


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We present an improved finite-size scaling method for reliably extracting the critical temperature T_BKT of a Berezinskii-Kosterlitz-Thouless (BKT) transition. Using the known Weber-Minhagen multiplicative logarithmic correction to the spin stiffness rho_s at T_BKT and the Kosterlitz-Nelson relation between the transition temperature and the stiffness, rho_s(T_BKT)=2T_BKT/pi, we define a size dependent transition temperature T_ BKT(L_1,L_2) based on a pair of system sizes L1, L2, e.g., L_2=2L1. We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved, rapidly convergent, and can be reliably extrapolated to the thermodynamic limit, L_1,L_2 -> infinity. Using GPU (graphical processing unit) computing, we obtain high-precision data for L up to 512 and extract a transition temperature T_BKT=0.89274(1), where the statistical error, +/- 1, in the last digit is about 6 times smaller than that of the best previous estimate.
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