We report a detailed study of twelve-term, high-temperature series for the second moment of spin-pair correlations ${\mu }_2\mathrm{}\left(t\right)\mathrm{}$ and the specific heat $c_H\mathrm{}\left(t\right)\mathrm{}$ of the nearest-neighbor spin-$s$ Ising model in zero magnetic field on the fcc lattice. Near criticality we find ${\mu }_2\mathrm{}\left(t\right)=A_2\mathrm{}\left(s\right)t{\mathrm{}\left(s\right)\mathrm{}}^{-(\gamma +2\mathrm{}\nu )}[1\mathrm{} + B_2\mathrm{}(s)t{\mathrm{}\left(s\right)\mathrm{}}^{{\Delta }_1}+ \cdots ]$, {$t\left(s\right)=\frac{[T-T_c\mathrm{}(s\left)\right]}{T}$}, showing a confluent correction to the dominant scaling singularity. To within uncertainties the exponents have the universal (i.e., spin-independent) values $\nu ={0.638}_{-0.008\mathrm{}}^{+0.002\mathrm{}}$ (with $\gamma ={1.250}_{-0.007\mathrm{}}^{+0.003\mathrm{}}$) and ${\Delta }_1=0.6\mathrm{}\pm 0.1$. The confluent exponent ${\Delta }_1$ 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 $c_H$ for the same model finds no detectable confluent singularities in rather noisy, high-temperature series and gives $\alpha =0.125\mathrm{}\pm 0.020$ in general confirmation of earlier $s=\frac{1}{2}$ estimates. With $\nu$ as quoted above the hyperscaling relation $d\nu =2-\alpha$ at $d=3\mathrm{}$ requires $\alpha ={0.086}_{-0.006\mathrm{}}^{+0.024\mathrm{}}$ so the validity of hyperscaling remains problematical.

• Received 12 April 1976

DOI:https://doi.org/10.1103/PhysRevB.14.3990