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Physical review. B, Condensed matter

Comment on ‘‘Critical behavior of the two-dimensionalXYmodel: An analysis of extended high-temperature series’’

Martin V. Ferer, Z. Mo

Published Dec 1, 1990• 7 citations
Publisher

Abstract

We have used our unbiased 4-fit method to analyze the high-temperature series of Butera, Comi, and Marchesini [Phys. Rev. B 40, 534 (1989)]. Our results are consistent with theirs. However, our method of analysis has the advantage that it is tailored to detect either a power-law singularity or a singularity like that predicted by Kosterlitz and Thouless, in that the value of one of the parameters determined in our 4-fit procedure signals which type of singularity is dominant. Therefore, our method provides a quantitative indicator of the dominant type of singularity. Our results unambiguously support Kosterlitz-Thouless predictions over power-law forms for the singularities.

DOI
10.1103/physrevb.42.10769

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We have used our unbiased 4-fit method to analyze the high-temperature series of Butera, Comi, and Marchesini [Phys. Rev. B 40, 534 (1989)]. Our results are consistent with theirs. However, our method of analysis has the advantage that it is tailored to detect either a power-law singularity or a singularity like that predicted by Kosterlitz and Thouless, in that the value of one of the parameters determined in our 4-fit procedure signals which type of singularity is dominant. Therefore, our method provides a quantitative indicator of the dominant type of singularity. Our results unambiguously support Kosterlitz-Thouless predictions over power-law forms for the singularities.
High-temperature series are extended through ${\ensuremath{\beta}}^{17}$ for the susceptibility and for the first and second moments of the correlation functions, in the case of the classical two-dimensional O(2) Heisenberg model (also called the XY model). For the ``true mass gap'' the high-temperature expansion is computed through ${\ensuremath{\beta}}^{12}$. The calculations are performed by a new computer program which solves iteratively the Schwinger-Dyson equations of the model. Ratio-extrapolation, Euler-transform and Pad\'e techniques are used to analyze the series and to discriminate between the recently reproposed conventional power-law form of the critical singularity and the form predicted by Kosterlitz and Thouless. The critical parameters are determined for the Kosterlitz-Thouless form, which is clearly favored by our data.
We have analyzed new twelfth-order high-temperature series for the susceptibility and correlation length of classical planar models on the triangular lattice using an nnn-fit method of analysis tailored to the form of the singularity $A\mathrm{exp}(b{t}^{\ensuremath{-}\ensuremath{\nu}})$ predicted by Kosterlitz and Thouless. Test-function analysis shows that the nnn-fit method is significantly more reliable in treating a number of possible corrections to the leading singularity than is the DDD log Pad\'e analysis of the logarithm and logarithmic derivative used in earlier series work on tenth-order series. Our nnn-fit analysis leads to the results $\ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1$ and $\ensuremath{\eta}=0.27\ifmmode\pm\else\textpm\fi{}0.03$ in good agreement with the Kosterlitz-Thouless predictions $\ensuremath{\nu}=\frac{1}{2}$ and $\ensuremath{\eta}=\frac{1}{4}$.
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 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.
Several new general methods of series analysis are introduced and tested on an assortment of known functions. These methods are then applied to several of the series expansions known for the thermodynamic properties of the spin-\textonehalf{} Ising model. We are able to obtain an unbiased analysis of the high-temperature specific-heat singularity, and find $\ensuremath{\alpha}\ensuremath{\approx}0.13\ifmmode\pm\else\textpm\fi{}0.01$. The application of our methods in other cases generally agrees with that of previous workers. Our methods in practice amount to procedures for looking more closely at the dominant singularity, at the expense of accuracy for competing singularities. For certain types of function, thought to occur in the study of critical phenomena, our methods are noticeably more effective than those previously employed.
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.