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.
High-temperature series for the inverse spin-spin correlation function of the
s=21 Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at
Tc and in the Ornstein---Zernike region are determined. The Fisher---Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.
Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.
The bivariate high-temperature expansion of the spin-spin correlation function of the three-dimensional classical
XY (planar rotator) model, with spatially anisotropic nearest-neighbor couplings, is extended from the tenth through the 21st order. The computation is carried out for the simple cubic lattice, in the absence of magnetic field, in the case in which the coupling strength along the
z axis of the lattice is different from those along the
x and the
y axes. It is then possible to determine accurately the critical temperature as function of the parameter
R which characterizes the coupling anisotropy and to check numerically the universality, with respect to
R, of the critical exponents of the three-dimensional anisotropic system. The analysis of our data also shows that the main predictions of the generalized scaling theory for the crossover from the three-dimensional to the two-dimensional critical behavior are compatible with the series extrapolations.
The energy-density-energy-density correlation function is studied in the three-dimensional Ising model in zero external field, for various spins on the fcc lattice. An extension of the linked-cluster technique is used to obtain high-temperature series expansions for the correlation function and its associated functions, to at least eighth order in the high-temperature interaction variable. Analysis of the series provides values for their critical indices and amplitudes. These values are used to verify the predictions of several theories of critical behavior. The critical index of the coherence length of the energy density correlation function is found to be equal to that of the spin-spin correlation function for all spins within the accuracy obtained here. For spins higher than ~ this critical index appears to obey strong scaling. An analysis of universality indicates that only one scale factor is needed to transform the energy density correlation function from one spin to another. This prediction is checked here and found to be good within our present accuracy. The current-algebra hypothesis predicts definite relations between the leading singularities of various correlation functions at the critical point. This prediction is checked directly and consistent values of the leading coefficients are obtained. The shape of the energy density correlation function at large distances from the origin, both at and above the critical point, is determined. The results are sufficiently accurate to rule out the Ornstein-Zernike form in zero field for T > T , but not sufficient to eliminate the possibility c of anisotropy at T = T .
High-temperature bivariate expansions have been derived for the two-spin-correlation function in a variety of classical lattice
XY (planar rotator) models in which spatially isotropic interactions among first-neighbor spins compete with spatially isotropic or anisotropic (in particular uniaxial) interactions among next-to-nearest-neighbor spins. The expansions, calculated for cubic lattices of dimensions
d=1, 2, and 3, are expressed in terms of the two variables
K1=J1/kT and
K2=J2/kT, where
J1 and
J2 are the nearest-neighbor and the next-to-nearest-neighbor exchange couplings, respectively. This paper deals in particular with the properties of the
d=3 uniaxial
XY model (ANNNXY model) for which the bivariate expansions have been computed through the 18th order, thus extending by 12 orders the results so far available and making a study of this model possible over a wide range of values of the competition parameter
R=J2/J1. Universality with respect to
R on the critical line separating the paramagnetic and the ferromagnetic phases can be verified, and at the same time the very accurate determination $\ensuremath{\gamma}=1.3177(5)$ and $\ensuremath{\nu}=0.6726(8)$ of the critical exponents of the susceptibility and of the correlation length, in the three-dimensional
XY universality class, can be achieved. For the exponents at the multicritical
(m,d,N)=(1,3,2) Lifshitz point the estimates ${\ensuremath{\gamma}}_{l}=1.535(25)$, ${\ensuremath{\nu}}_{\ensuremath{\perp}}=0.805(15)$, and ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=0.40(3)$ are obtained. Finally, the susceptibility exponent is estimated along the boundary between the disordered and the modulated phases.