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Physical review. B, Condensed matter

Two-dimensional Ising-like systems: Corrections to scaling in the Klauder and double-Gaussian models

Mustansir Barma, Michael Ellis Fisher

Published May 1, 1985• 53 citations
Publisher

Abstract

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 ${x}^{21}$ where x\ensuremath{\propto}J/${k}_{B}$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}${\mathrm{cphi}}^{4}$ model.

DOI
10.1103/physrevb.31.5954

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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, \ensuremathγeff\ensuremath≃1.9{\ensuremath{\gamma}}_{\mathrm{eff}\mathrm{\ensuremath{\simeq}}1.9}\ensuremathγeff\ensuremath≃1.9​ to 2.0, is discussed and related to work on the \ensuremath{\lambda}cphi4{\mathrm{cphi}}^{4}cphi4 model.
A two-series approach to partial differential approximant analysis of power series is presented. Instead of double series, $f(x,y)=\ensuremath{\Sigma}{c}_{\mathrm{ij}}{x}^{i}{y}^{j}$, our approach uses two one-variable series in xxx, fff and $\frac{\ensuremath{\partial}f}{\ensuremath{\partial}y}$, and has the efficiency and stability of one-variable methods. 21-term high-temperature series are analyzed for the susceptibility and correlation length squared for double-Gaussian Ising models on the bcc lattice. Critical exponents are $\ensuremath{\gamma}=1.2378(6)$, $2\ensuremath{\nu}=1.2623(6)$, and $\ensuremath{\eta}=0.0375(5)$; correction-to-scaling exponents are ${\ensuremath{\theta}}_{\ensuremath{\chi}}=0.52(3)$ and ${\ensuremath{\theta}}_{{\ensuremath{\xi}}^{2}}=0.49(4)$; and the subdominant critical amplitude ratio is $\frac{{a}_{\ensuremath{\xi}}}{{a}_{\ensuremath{\chi}}}=0.83(5)$.
Although the two-dimensional spin- 1 / 2 Ising model was solved in zero field in 1944, no exact results are yet available for the spin-1 model. This model should have the same dominant critical exponents as the spin- 1 / 2 model, but appears to exhibit non-analytic corrections to scaling as expected for a phi 4 model and unlike the spin- 1 / 2 model. In this paper new 45-term low temperature series for the magnetisation, specific heat and susceptibility of the spin-1 model on the square lattice are presented and analysed. A new 24-term staggered susceptibility series for the hard square model is presented and the extant order parameter series for this model (which is also in the phi 4 universality class) are also considered. For both models a non-analytic confluent correction with exponent 1.0< Delta 1 <1.3 is found. The validity of this result is enhanced by a comparison with the S= 1 / 2 case.
High-temperature series for two bcc lattice models which interpolate between the Gaussian (or free-field) model and the S=12S=\frac{1}{2}S=21​ Ising model are analyzed by partial differential approximants. Series to order 21 in both $x\ensuremath{\propto}\frac{1}{T}$ and the interpolation parameter, yyy, 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.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
If scaling variables and scaling expressions are chosen judiciously, ``critical'' scaling analyses do not need to be restricted to a narrow range of temperatures near the critical point. The standard scaling variable $(T\ensuremath{-}{T}_{c})/{T}_{c}$ is inadapted to wide ranges of temperature because it diverges at high temperatures. With the variable $\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})/T$, in the high-dimension (mean-field) limit, the reduced susceptibility and the second-moment correlation length of the Ising ferromagnet depend on temperature as $\ensuremath{\chi}(T)={\ensuremath{\tau}}^{\ensuremath{-}1}$ and $\ensuremath{\xi}(T)={T}^{\ensuremath{-}1/2}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$ exactly over the entire temperature range above the critical temperature Tc{T}_{c}Tc​. For the canonical two-dimensional square lattice near-neighbor Ising ferromagnet, it is shown that compact ``extended scaling'' expressions analogous to the high-dimensional limit form but with the appropriate exponents give accurate approximations to the true temperature dependencies, again over the entire temperature range from Tc{T}_{c}Tc​ to infinity. Within this approach, for near-neighbor interaction systems, there is no crossover temperature above which mean-field-like behavior sets in.
It is argued that the leading corrections to scaling for planar Ising models, which occur as analytic factors, arise from the quadratic terms of the nonlinear thermal and ordering fields (rather than from irrelevant variables). This yields ${{\ensuremath{\alpha}}_{\mathrm{eff}}}^{\ensuremath{'}}+2{\ensuremath{\beta}}_{\mathrm{eff}}+{{\ensuremath{\gamma}}_{\mathrm{eff}}}^{\ensuremath{'}}=2$ for the effective critical exponents, with no\mathrm{no}no leading corrections, and is confirmed by exact square-lattice results for arbitrary anisotropy J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​. For isotropic lattices the ratios of correction amplitudes and quadratic nonlinear-field terms appear universal.