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Physical Review Letters

Unbiased Estimation of Corrections to Scaling by Partial Differential Approximants

Jing-Huei Chen, Michael Ellis Fisher, Bernie G. Nickel

Published Mar 1, 1982• 178 citations
Publisher

Abstract

High-temperature series for two bcc lattice models which interpolate between the Gaussian (or free-field) model and the $S=\frac{1}{2}$ Ising model are analyzed by partial differential approximants. Series to order 21 in both $x\ensuremath{\propto}\frac{1}{T}$ and the interpolation parameter, $y$, yield unbiased estimates for the correction-to-scaling exponent, $\ensuremath{\theta}=0.54\ifmmode\pm\else\textpm\fi{}5$, and the susceptibility exponent, $\ensuremath{\gamma}=1.2385\ifmmode\pm\else\textpm\fi{}15$. The results are universal and agree tolerably with field-theoretic estimates and well with biased, one-variable analyses of general spin Ising models.

DOI
10.1103/physrevlett.48.630

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A recently proposed approach---partial differential approximants---for analyzing power series in two (or more) variables for functions which exhibit singularities of multicritical, scaling character is tested by analyzing the susceptibility series for the bicritical point describing the Ising-Heisenberg-XY\mathrm{XY}XY crossover in a three-dimensional classical ferromagnet. Encouraging results are obtained for the unbiased estimation of the bicritical point, of the multicritical exponents $\ensuremath{\gamma}$ and $\ensuremath{\varphi}$, and of the slopes of the scaling axes.
Partial-differential approximants are used to study the critical behavior of the susceptibility, \ensuremath{\chi}(x,y), of the Klauder and double-Gaussian scalar spin, or O(1) models on a square lattice using two-variable series to order x21{x}^{21}x21 where x\ensuremath{\propto}J/kB{k}_{B}kB​T while y serves to interpolate analytically from the Gaussian or free-field model at y=0 to the standard spin-(1/2) Ising model at y=1. The pure Ising critical point at y=1 appears to be the only non-Gaussian multisingularity in the range 0<y\ensuremath{\le}1. It is concluded that the exponent \ensuremath{\theta} characterizing the leading irrelevant corrections to scaling lies in the range \ensuremath{\theta}=1.35\ifmmode\pm\else\textpm\fi{}0.25. This supports the validity of Nienhuis's conjecture \ensuremath{\theta}=(4/3) but it is argued that, contrary to normal expectations, this (rational) value entails only logarithmic corrections to pure Ising critical behavior. The existence of strong crossover effects for 0.1\ensuremath{\lesssim}y\ensuremath{\lesssim}0.6 and the appearance of an effective exponent, ${\ensuremath{\gamma}}_{\mathrm{eff}\mathrm{\ensuremath{\simeq}}1.9}$ to 2.0, is discussed and related to work on the \ensuremath{\lambda}cphi4{\mathrm{cphi}}^{4}cphi4 model.
Much attention has been paid to the d=3, s= 1 / 2 Ising model on the BCC lattice in recent years because the best high-temperature series expansions were available for this lattice. However, in order to compare series estimates of critical exponents and temperatures with independent evaluations via the Monte Carlo method, lattices of lower coordination number must be considered, since Monte Carlo studies are usually made on these. A study of extant susceptibility series on the FCC, SC, diamond and tetrahedron lattices is reported on and some comments are made on aspects of recent Monte Carlo analyses of the d=3, s= 1 / 2 Ising model on the SC and diamond lattices.
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We have extended through ${\ensuremath{\beta}}^{23}$ the high-temperature expansion of the second field derivative of the susceptibility for Ising models of general spin, with nearest-neighbor interactions, on the simple cubic and the body-centered cubic lattices. Moreover, the expansions for the nearest-neighbor correlation function, the susceptibility, and the second correlation moment have been extended up to ${\ensuremath{\beta}}^{25}.$ Taking advantage of these new data, we can improve the accuracy of direct estimates of critical exponents and of hyperuniversal combinations of critical amplitudes such as the renormalized four-point coupling gr{g}_{r}gr​ or the quantity usually denoted by ${R}_{\ensuremath{\xi}}^{+}.$ In particular, we obtain $\ensuremath{\gamma}=1.2371(1),$ $\ensuremath{\nu}=0.6299(2),$ ${\ensuremath{\gamma}}_{4}=4.3647(20),$ gr=1.404(3),{g}_{r}=1.404(3),gr​=1.404(3), and ${R}_{\ensuremath{\xi}}^{+}=0.2668(5).$ We have used a variety of series extrapolation procedures and, in some of the analyses, we have assumed that the leading correction-to-scaling exponent $\ensuremath{\theta}$ is universal and roughly known. We have also verified, to high precision, the validity of the hyperscaling relation and of the universality property both with regard to the lattice structure and to the value of the spin.