Series analysis of corrections to scaling for the spin-pair correlations of the spin-sIsing model: Confluent singularities, universality, and hyperscaling

Abstract

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 ${c}_{H}(t)$ of the nearest-neighbor spin-$s$ 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 ${c}_{H}$ 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 $s=\frac{1}{2}$ estimates. With $\ensuremath{\nu}$ as quoted above the hyperscaling relation $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$ at $d=3$ requires $\ensuremath{\alpha}={0.086}_{\ensuremath{-}0.006}^{+0.024}$ so the validity of hyperscaling remains problematical.