Confluent-singularity analysis of high-temperature series for thermodynamic functions of the $S=\frac{1}{2}$ $\mathrm{XY}$ model, the $S=\infty$ $\mathrm{XY}$ model, and the planar-rotator model on the fcc lattice

Leading singularity analysis for the susceptibility index $\gamma$ of these planar models had indicated a nonuniversal variation in the index between the classical spin models and the spin-½ model in three dimensions. Using existing methods of analysis which allow for confluent corrections to the leading singularity, we have reanalyzed the existing susceptibility series for $\gamma$ as well as the second-moment series for the correlation length index $\nu$ for the classical models. Our results are consistent with the following universal values of the indices $\gamma =1.333\mathrm{}\pm 0.010$ and $\nu =0.678\mathrm{}\pm 0.005$. In addition, our analysis suggests that the nonanalytic correction to scaling is absent in the spin-½ model and that the correction to scaling exponent ${\Delta }_1$ may be different for the two classical models investigated, being 0.78±0.08 for the planar rotator model and 0.60±0.08 for the $s=\infty$ $\mathrm{XY}$ model. Analysis of existing series for the second field derivative of the susceptibility to determine the gap index $\Delta$ has indicated a slight difference between the $s=\frac{1}{2}$ and the classical spin values, 1.70±0.02 and 1.67±0.01, respectively. This small difference probably reflects a defect in the analysis, since our analysis clearly indicates that assuming confluent corrections does not satisfactorily account for the scaling corrections in these second-field derivative series.