Hyperscaling in the three-dimensional Ising model

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 $\chi ={\mathrm{At}}^{-\gamma }(1+{\mathrm{Bt}}^{{\Delta }_1})$, where $t=1-\frac{T_c}{T}$. The method determines sequences for these five parameters, e.g., ${\lim }_{n\to \infty }{\gamma }_n=\gamma$, 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 $\gamma$ and $2\nu$ are smaller than indicated by early analysis on shorter series and that the confluent correction is apparently not significant for spin-½, validating the use of ratio methods for the spin-½ series. Considering the results of both the confluent singularity analysis and the standard analysis of the spin-½ series, we estimate $\gamma =1.242\mathrm{} (+0.003, -0.005)$ and $2\nu =1.268(+0.006, -0.008)$, 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.