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Physical review. B, Condensed matter

High-temperature critical behavior of two-dimensional planar models: A series investigation

Martin V. Ferer, Michael J. Velgakis

Published Jan 1, 1983• 21 citations
Publisher

Abstract

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 $n$-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 $n$-fit method is significantly more reliable in treating a number of possible corrections to the leading singularity than is the $D$ log Pad\'e analysis of the logarithm and logarithmic derivative used in earlier series work on tenth-order series. Our $n$-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 $\ensuremath{\eta}=\frac{1}{4}$.

DOI
10.1103/physrevb.27.314

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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 Aexp(bt\ensuremath−\ensuremathν)A\mathrm{exp}(b{t}^{\ensuremath{-}\ensuremath{\nu}})Aexp(bt\ensuremath−\ensuremathν) 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 \ensuremathν=0.5\ifmmode±\else\textpm\fi0.1\ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1\ensuremathν=0.5\ifmmode±\else\textpm\fi0.1 and \ensuremathη=0.27\ifmmode±\else\textpm\fi0.03\ensuremath{\eta}=0.27\ifmmode\pm\else\textpm\fi{}0.03\ensuremathη=0.27\ifmmode±\else\textpm\fi0.03 in good agreement with the Kosterlitz-Thouless predictions \ensuremathν=12\ensuremath{\nu}=\frac{1}{2}\ensuremathν=21​ and \ensuremathη=14\ensuremath{\eta}=\frac{1}{4}\ensuremathη=41​.
To allow analysis of square lattice O(2) model series, we have extended our q-fit method of analysis previously applied to triangular lattice models. We present analysis of our twelfth-order high-temperature series and of tenth-order strong-coupling series. The results from the high-temperature series are consistent with our earlier results \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, \ensuremath{\eta}=0.27\ifmmode\pm\else\textpm\fi{}0.03 for the triangular lattice with the reduced precision usually accompanying a reduction in coordination number. However, the analysis of the strong-coupling expansion, significantly smoothed by an Euler transformation, convincingly supports \ensuremath{\nu}=0.50\ifmmode\pm\else\textpm\fi{}0.05.
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.
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 have extended the series for the zero-field susceptibility of the spin-sss Ising model through tenth order in the reduced inverse temperature KKK on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients hn(s){h}_{n}(s)hn​(s) are expressed as simple polynomials in X=s(s+1)X=s(s+1)X=s(s+1). Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of sss. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For s=12s=\frac{1}{2}s=21​ only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of sss, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
We describe a new algorithm to compute high-temperature expansions for lattice-spin models. The method, based on a recently proposed iterative solution of the Schwinger-Dyson equations, may be efficiently implemented by a computer program. As an application we calculate the (zero-field) two-point correlation functions and their moments up to 15th order for the planar rotator model on a two-dimensional square lattice. The high-temperature series previously known up to tenth order are thus substantially improved and are analyzed within the framework of the Kosterlitz-Thouless theory.
We present new twelfth-order high-temperature series for the susceptibility, correlation length, and free energy of the nearest-neighbor classical Heisenberg ferromagnet in zero field on a fcc lattice. Checks corroborating the correctness of the series are discussed. Standard analysis of these series produces results in essential agreement with the renormalization-group calculations and several other series analyses. However, a more reliable confluent singularity analysis suggests slightly larger values, \ensuremath{\gamma}${=1.40}_{\mathrm{\ensuremath{-}}0.01}^{+0.03}$ and \ensuremath{\nu}=0.72\ifmmode\pm\else\textpm\fi{}0.01, as well as a confluent correction of just the size to cause the observed standard analysis underestimate.