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arXiv:1309.4963 · Physical Review E

Tensor renormalization group study of classical XY model on the square lattice

Ji-Feng Yu, Z. Y. Xie, Yannick Meurice, Yuzhi Liu A, A. Denbleyker, Haiyuan Zou, Mingpu Qin, J. Chen, Tao Xiang

Published Jan 22, 2014• 96 citations
arXivHTMLPDF

Abstract

Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the thermodynamic properties of the continuous XY model on the square lattice. The temperature dependence of the free energy, the internal energy, and the specific heat agree with the Monte Carlo calculations. From the field dependence of the magnetic susceptibility, we find the Kosterlitz-Thouless transition temperature to be 0.8921(19), consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent δ is estimated as 14.5, close to the analytic value by Kosterlitz.

DOI
10.1103/physreve.89.013308

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We calculate the high-temperature series of the magnetic susceptibility and the second and fourth moments of the correlation function for the XY model on the square lattice to order beta;{33} by applying the improved algorithm of the finite lattice method. The long series allow us to estimate the inverse critical temperature as beta_{c}=1.1200(1) , which is consistent with the most precise value given previously by the Monte Carlo simulation. The critical exponent for the multiplicative logarithmic correction is evaluated to be theta=0.054(10) , which is consistent with the renormalization group prediction of theta=1/16 .
In an attempt to understand the discrepancy between renormalization group predictions of a rounded specific heat peak about 38% above the Kosterlitz–Thouless transition temperature Tc and Monte-Carlo results indicating a sharp peak about 7% above Tc, we have undertaken an analysis of our thirteenth-order, high-temperature series for the free energy of the XY model on the triangular lattice. We have performed N fits on our free energy series in order to estimate the dominant singularity. These results are not consistent with either F≂ξ−2 or a transition at temperatures above Tc. Using these estimates of the leading singularity, we have extrapolated partial sums of our specific heat series for temperatures above Tc, thereby locating the specific heat peak.
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.
Results of Monte Carlo simulations of the ferromagnetic classical XXZ model on a square lattice are reported for interactions of the out of plane spin components \ensuremath{\lambda} varying between 0 and 0.99. For all the value of \ensuremath{\lambda} the signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition was observed, and the corresponding critical temperature was located both by BKT fits and finite-size scaling methods. Data for in-plane and out of plane susceptibility, correlation length, specific heat, and internal energy are given in a broad range of temperature and for lattice sizes ranging from 32\ifmmode\times\else\texttimes\fi{}32 to 256\ifmmode\times\else\texttimes\fi{}256.