Leading singularity analysis for the susceptibility index $\ensuremath{\gamma}$ of these planar models had indicated a nonuniversal variation in the index between the classical spin models and the spin-\textonehalf{} 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 $\ensuremath{\gamma}$ as well as the second-moment series for the correlation length index $\ensuremath{\nu}$ for the classical models. Our results are consistent with the following universal values of the indices $\ensuremath{\gamma}=1.333\ifmmode\pm\else\textpm\fi{}0.010$ and $\ensuremath{\nu}=0.678\ifmmode\pm\else\textpm\fi{}0.005$. In addition, our analysis suggests that the nonanalytic correction to scaling is absent in the spin-\textonehalf{} model and that the correction to scaling exponent ${\ensuremath{\Delta}}_{1}$ may be different for the two classical models investigated, being 0.78\ifmmode\pm\else\textpm\fi{}0.08 for the planar rotator model and 0.60\ifmmode\pm\else\textpm\fi{}0.08 for the $s=\ensuremath{\infty}$
XY model. Analysis of existing series for the second field derivative of the susceptibility to determine the gap index $\ensuremath{\Delta}$ has indicated a slight difference between the
s=21 and the classical spin values, 1.70\ifmmode\pm\else\textpm\fi{}0.02 and 1.67\ifmmode\pm\else\textpm\fi{}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.
The spin-1/2 XY model in three dimensions is a particularly suitable model for the study of dynamical critical properties, both for comparison with experimental systems such as liquid He near Tλ and certain magnetic systems, and for testing of the predictions of dynamic-scaling theory.Sixteen coefficients have been obtained in the high-frequency high-temperature expansion of the perpendicular (order-parameter) susceptibility, [Formula: see text]. Analysis of [Formula: see text] yields estimates of dynamic critical indices not in conflict with experiment or dynamic scaling, but longer series are needed for a definitive comparison.
A new method of deriving high temperature series expansions for the susceptibility of the Ising and classical vector spin models is presented. The method involves generating the expansion for the inverse of the susceptibility, in which case only contributions from star graphs need to be considered. An additional term (the thirteenth) has been added to the Ising high temperature susceptibility series for the FCC lattice. Analysis of the extended series does not indicate that revision of previous critical estimates is required. In a test of the capabilities of the method for other systems, seven terms of the susceptibility series of the FCC Heisenberg and planar vector models have been reproduced. Techniques used in generating the graph data are discussed.