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Physical review. B, Condensed matter

Critical behavior of the two-dimensionalXYmodel: An analysis of extended high-temperature series

P. Butera, M. Comi, G. Marchesini

Published Jul 1, 1989• 21 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.40.534

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

Martin V. Ferer, Z. Mo

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.

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High-temperature series are extended through \ensuremathβ17{\ensuremath{\beta}}^{17}\ensuremathβ17 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 \ensuremathβ12{\ensuremath{\beta}}^{12}\ensuremathβ12. 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 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.
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.
High-temperature series expansions of the spin-spin correlation function for the plane rotator (or XY) model on the square lattice are extended by three terms through order ${\mathrm{\ensuremath{\beta}}}^{20}$. Tables of the expansion coefficients are reported for the correlation function spherical moments of order l=0,1,2. The expansion coefficients through ${\mathrm{\ensuremath{\beta}}}^{15}$ for the vorticity are also tabulated. Our analysis of the series supports the Kosterlitz-Thouless predictions on the structure of the critical singularities and leads to fairly accurate estimates of the critical parameters.
We present simple tables of integers from which it is possible to reconstruct the high-temperature series coefficients through ${\mathrm{\ensuremath{\beta}}}^{14}$ for the susceptibility, for the second correlation moment, and for the second field derivative of the susceptibility of the O(N) classical Heisenberg model on a simple (hyper)cubic lattice in dimension d=2, 3, and 4 and for any N. To construct the tables we have used the recent extension of the high-temperature series by M. Luscher and P. Weisz and some analytic properties in N that we have derived from the Schwinger-Dyson equations of the O(N) model. We also present a numerical study of these series in the d=2 case. The main results are: (a) the extended series give further support to the Cardy-Hamber-Nienhuis exact formulas for the critical exponents when -2N2; (b) for N\ensuremath{\ge}3 there are no indications of any critical point at finite \ensuremath{\beta}; (c) the series are consistent with the low-temperature asymptotic forms predicted by the perturbative renormalization group.
High-temperature series expansions of the spin-spin correlation function for the XY (or plane rotator) model on the triangular lattice are extended by two terms up to order ${\mathrm{\ensuremath{\beta}}}^{14}$. Tables of the expansion coefficients are reported for the correlaton function spherical moments of order l=0 and 2. Our analysis of the series supports the Kosterlitz-Thouless predictions on the structure of the critical singularities and leads to fairly accurate estimates of the critical parameters.