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Physical review. B, Solid state

Tests of strong scaling in the three-dimensional Ising model

Michael Ellis Fisher, Howard B. Tarko

Published Feb 1, 1975• 17 citations
Publisher

Abstract

We study numerically the amplitudes ${E}^{\ifmmode\pm\else\textpm\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ of the ${|\frac{\ensuremath{\Delta}T}{{T}_{c}}|}^{1\ensuremath{-}\ensuremath{\alpha}}$ variation in the spin-spin correlation functions $〈{s}_{\stackrel{\ensuremath{\rightarrow}}{0}}{s}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}〉$, of three-dimensional Ising models in zero field above and below ${T}_{c}$. By allowing for reasonable correction terms to the asymptotic behavior ${E}^{\ifmmode\pm\else\textpm\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{E}_{0}^{\ifmmode\pm\else\textpm\fi{}}{(\frac{R}{a})}^{\ensuremath{\zeta}}$, where $a$ is the lattice spacing, we find consistency with the strong-scaling prediction $\ensuremath{\zeta}=\frac{(1\ensuremath{-}\ensuremath{\alpha})}{\ensuremath{\nu}}\ensuremath{-}d+2\ensuremath{-}\ensuremath{\eta}\ensuremath{\simeq}0.31$, and reasonable agreement with universality and $\ensuremath{\epsilon}$-expansion estimates of the amplitudes ${E}_{0}^{\ifmmode\pm\else\textpm\fi{}}$.

DOI
10.1103/physrevb.11.1131

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Physical review. B, Solid state•Nov 1, 1976•91 citations
We study numerically the amplitudes E\ifmmode±\else\textpm\fi(R\ensuremath→){E}^{\ifmmode\pm\else\textpm\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})E\ifmmode±\else\textpm\fi(R\ensuremath→​) of the ∣\ensuremathΔTTc∣1\ensuremath−\ensuremathα{|\frac{\ensuremath{\Delta}T}{{T}_{c}}|}^{1\ensuremath{-}\ensuremath{\alpha}}∣Tc​\ensuremathΔT​∣1\ensuremath−\ensuremathα variation in the spin-spin correlation functions 〈s0\ensuremath→sR\ensuremath→〉〈{s}_{\stackrel{\ensuremath{\rightarrow}}{0}}{s}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}〉〈s0\ensuremath→​​sR\ensuremath→​​〉, of three-dimensional Ising models in zero field above and below Tc{T}_{c}Tc​. By allowing for reasonable correction terms to the asymptotic behavior E\ifmmode±\else\textpm\fi(R\ensuremath→)\ensuremath≈E0\ifmmode±\else\textpm\fi(Ra)\ensuremathζ{E}^{\ifmmode\pm\else\textpm\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{E}_{0}^{\ifmmode\pm\else\textpm\fi{}}{(\frac{R}{a})}^{\ensuremath{\zeta}}E\ifmmode±\else\textpm\fi(R\ensuremath→​)\ensuremath≈E0\ifmmode±\else\textpm\fi​(aR​)\ensuremathζ, where aaa is the lattice spacing, we find consistency with the strong-scaling prediction \ensuremathζ=(1\ensuremath−\ensuremathα)\ensuremathν\ensuremath−d+2\ensuremath−\ensuremathη\ensuremath≃0.31\ensuremath{\zeta}=\frac{(1\ensuremath{-}\ensuremath{\alpha})}{\ensuremath{\nu}}\ensuremath{-}d+2\ensuremath{-}\ensuremath{\eta}\ensuremath{\simeq}0.31\ensuremathζ=\ensuremathν(1\ensuremath−\ensuremathα)​\ensuremath−d+2\ensuremath−\ensuremathη\ensuremath≃0.31, and reasonable agreement with universality and \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-expansion estimates of the amplitudes E0\ifmmode±\else\textpm\fi{E}_{0}^{\ifmmode\pm\else\textpm\fi{}}E0\ifmmode±\else\textpm\fi​.
We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin-SSS Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.
The susceptibility of the spin-sss nearest-neighbor fcc Ising model is studied at zero magnetic field by twelve-term high-temperature series. We find that the data near criticality are consistent with $\ensuremath{\chi}=A(s){t}^{\frac{\ensuremath{-}5}{4}}[1+B(s){t}^{{\ensuremath{\Delta}}_{1}}]$, $t=1\ensuremath{-}\frac{{T}_{c}(s)}{T}$. The correction-to-scaling exponent has the universal value ${\ensuremath{\Delta}}_{1}=0.50\ifmmode\pm\else\textpm\fi{}0.05$ in good agreement with $\ensuremath{\epsilon}$-expansion estimates. The confluent amplitude B(s)B(s)B(s) vanishes at s=12s=\frac{1}{2}s=21​ to within uncertainties. Values of the critical parameters are tabulated for a variety of spins.
High-temperature series for the inverse spin-spin correlation function of the s=12s=\frac{1}{2}s=21​ Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at Tc{T}_{c}Tc​ and in the Ornstein---Zernike region are determined. The Fisher---Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.
We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where ttt is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-ttt behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin- 1 2 Ising model with nearest-neighbor interactions, a spin- 1 2 model with nearest-neighbor and third-neighbor interactions, and a spin-1 model with nearest-neighbor interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behavior reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbor interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), yh =2.4815 (15) and y i = -0.82 (6). The universal ratio Q = #m 2 # 2 /#m 4...
We report a detailed study of twelve-term, high-temperature series for the second moment of spin-pair correlations ${\ensuremath{\mu}}_{2}(t)$ and the specific heat cH(t){c}_{H}(t)cH​(t) of the nearest-neighbor spin-sss Ising model in zero magnetic field on the fcc lattice. Near criticality we find ${\ensuremath{\mu}}_{2}(t)={A}_{2}(s)t{(s)}^{\ensuremath{-}(\ensuremath{\gamma}+2\ensuremath{\nu})}[1 + {B}_{2}(s)t{(s)}^{{\ensuremath{\Delta}}_{1}}+ \ensuremath{\cdots}]$, {$t(s)=\frac{[T\ensuremath{-}{T}_{c}(s)]}{T}$}, showing a confluent correction to the dominant scaling singularity. To within uncertainties the exponents have the universal (i.e., spin-independent) values $\ensuremath{\nu}={0.638}_{\ensuremath{-}0.008}^{+0.002}$ (with $\ensuremath{\gamma}={1.250}_{\ensuremath{-}0.007}^{+0.003}$) and ${\ensuremath{\Delta}}_{1}=0.6\ifmmode\pm\else\textpm\fi{}0.1$. The confluent exponent ${\ensuremath{\Delta}}_{1}$ is in reasonable agreement with the correction-to-scaling index derived from earlier analysis of the susceptibility, as predicted by renormalization-group arguments. A similar analysis of the specific heat cH{c}_{H}cH​ for the same model finds no detectable confluent singularities in rather noisy, high-temperature series and gives $\ensuremath{\alpha}=0.125\ifmmode\pm\else\textpm\fi{}0.020$ in general confirmation of earlier s=12s=\frac{1}{2}s=21​ estimates. With $\ensuremath{\nu}$ as quoted above the hyperscaling relation $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$ at d=3d=3d=3 requires $\ensuremath{\alpha}={0.086}_{\ensuremath{-}0.006}^{+0.024}$ so the validity of hyperscaling remains problematical.