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Physical review. B, Condensed matter

Hyperscaling in the three-dimensional Ising model

Martin V. Ferer, Michael J. Velgakis

Published Mar 1, 1983• 17 citations
Publisher

Abstract

We analyze the 21st-order series of Nickel for the susceptibility and the correlation length of the spin-$s$ Ising model for the bcc lattice with the use of an unbiased method of confluent singularity analysis tailored to the loose-packed lattices. This modified five-fit method of analysis assumes a parametrization which includes one confluent correction, e.g., for the susceptibility $\ensuremath{\chi}={\mathrm{At}}^{\ensuremath{-}\ensuremath{\gamma}}(1+{\mathrm{Bt}}^{{\ensuremath{\Delta}}_{1}})$, where $t=1\ensuremath{-}\frac{{T}_{c}}{T}$. The method determines sequences for these five parameters, e.g., ${\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}{\ensuremath{\gamma}}_{n}=\ensuremath{\gamma}$, by equating five consecutive even or odd coefficients in the perturbation series with the five corresponding coefficients in the expansion of the assumed parametrization and by solving the five equations for the five parameters numerically. Our analysis of the spin-$s$ Ising-model series shows that the indices $\ensuremath{\gamma}$ and $2\ensuremath{\nu}$ are smaller than indicated by early analysis on shorter series and that the confluent correction is apparently not significant for spin-\textonehalf{}, validating the use of ratio methods for the spin-\textonehalf{} series. Considering the results of both the confluent singularity analysis and the standard analysis of the spin-\textonehalf{} series, we estimate $\ensuremath{\gamma}=1.242 (+0.003, \ensuremath{-}0.005)$ and $2\ensuremath{\nu}=1.268(+0.006, \ensuremath{-}0.008)$, in basic agreement with renormalization-group calculations, hyperscaling, and the biased analysis of Nickel's series. Test-function analysis demonstrates the reliability of the method. Estimates for the leading index represent a significant improvement over older methods, e.g., the ratio method. Estimates for the correction exponent and amplitude are consistently high, presumably because the method must try to have one correction mimic all the corrections to the leading singularity.

DOI
10.1103/physrevb.27.2839

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Confluent singularities and the correction-to-scaling exponent for the d=3 fcc Ising model

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Physical review. B, Solid state•Apr 1, 1975•108 citations
We analyze the 21st-order series of Nickel for the susceptibility and the correlation length of the spin-sss Ising model for the bcc lattice with the use of an unbiased method of confluent singularity analysis tailored to the loose-packed lattices. This modified five-fit method of analysis assumes a parametrization which includes one confluent correction, e.g., for the susceptibility \ensuremathχ=At\ensuremath−\ensuremathγ(1+Bt\ensuremathΔ1)\ensuremath{\chi}={\mathrm{At}}^{\ensuremath{-}\ensuremath{\gamma}}(1+{\mathrm{Bt}}^{{\ensuremath{\Delta}}_{1}})\ensuremathχ=At\ensuremath−\ensuremathγ(1+Bt\ensuremathΔ1​), where t=1\ensuremath−TcTt=1\ensuremath{-}\frac{{T}_{c}}{T}t=1\ensuremath−TTc​​. The method determines sequences for these five parameters, e.g., limn\ensuremath→\ensuremath∞\ensuremathγn=\ensuremathγ{\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}{\ensuremath{\gamma}}_{n}=\ensuremath{\gamma}limn\ensuremath→\ensuremath∞​\ensuremathγn​=\ensuremathγ, by equating five consecutive even or odd coefficients in the perturbation series with the five corresponding coefficients in the expansion of the assumed parametrization and by solving the five equations for the five parameters numerically. Our analysis of the spin-sss Ising-model series shows that the indices \ensuremathγ\ensuremath{\gamma}\ensuremathγ and 2\ensuremathν2\ensuremath{\nu}2\ensuremathν are smaller than indicated by early analysis on shorter series and that the confluent correction is apparently not significant for spin-\textonehalf{}, validating the use of ratio methods for the spin-\textonehalf{} series. Considering the results of both the confluent singularity analysis and the standard analysis of the spin-\textonehalf{} series, we estimate \ensuremathγ=1.242(+0.003,\ensuremath−0.005)\ensuremath{\gamma}=1.242 (+0.003, \ensuremath{-}0.005)\ensuremathγ=1.242(+0.003,\ensuremath−0.005) and 2\ensuremathν=1.268(+0.006,\ensuremath−0.008)2\ensuremath{\nu}=1.268(+0.006, \ensuremath{-}0.008)2\ensuremathν=1.268(+0.006,\ensuremath−0.008), in basic agreement with renormalization-group calculations, hyperscaling, and the biased analysis of Nickel's series. Test-function analysis demonstrates the reliability of the method. Estimates for the leading index represent a significant improvement over older methods, e.g., the ratio method. Estimates for the correction exponent and amplitude are consistently high, presumably because the method must try to have one correction mimic all the corrections to the leading singularity.
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.
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.
Nickel has generated 21 term series for the susceptibility and the correlation length on the B.C.C. lattice, for various values of the spin. We analyse them here using a variant of the ratio method. Our results fully support Nickel's conclusions i.e. that is no apparent disagreement any longer for the exponents γ and ν between the renormalization group and high temperature series estimates. We believe that these results shed serious doubts about any claim, based on shorter series, of evidence for hyperscaling violations.
Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$
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.