We have extended through ${\ensuremath{\beta}}^{23}$ the high-temperature expansion of the second field derivative of the susceptibility for Ising models of general spin, with nearest-neighbor interactions, on the simple cubic and the body-centered cubic lattices. Moreover, the expansions for the nearest-neighbor correlation function, the susceptibility, and the second correlation moment have been extended up to ${\ensuremath{\beta}}^{25}.$ Taking advantage of these new data, we can improve the accuracy of direct estimates of critical exponents and of hyperuniversal combinations of critical amplitudes such as the renormalized four-point coupling
gr or the quantity usually denoted by ${R}_{\ensuremath{\xi}}^{+}.$ In particular, we obtain $\ensuremath{\gamma}=1.2371(1),$ $\ensuremath{\nu}=0.6299(2),$ ${\ensuremath{\gamma}}_{4}=4.3647(20),$
gr=1.404(3), and ${R}_{\ensuremath{\xi}}^{+}=0.2668(5).$ We have used a variety of series extrapolation procedures and, in some of the analyses, we have assumed that the leading correction-to-scaling exponent $\ensuremath{\theta}$ is universal and roughly known. We have also verified, to high precision, the validity of the hyperscaling relation and of the universality property both with regard to the lattice structure and to the value of the spin.
We have extended the series for the zero-field susceptibility of the spin-
S Ising model to eighth order in the reduced temperature
K, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series
hn(S) are expressed as simple polynomials in
X=S(S+1). For the face-centered-cubic lattice, an accurate polynomial fit to the critical point
Kc(S) is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all
S. However, the large-
S series seems to exhibit more rapid apparent convergence.
High-temperature expansions for the free energy, the susceptibility, and the second correlation moment of the classical
N-vector model [also known as the
O(N) symmetric classical spin-Heisenberg model or as the lattice
O(N) nonlinear $\ensuremath{\sigma}$ model] on the simple-cubic and the body-centered-cubic lattices are extended to order ${\ensuremath{\beta}}^{21}$ for arbitrary
N. The series for the second field derivative of the susceptibility is extended to order ${\ensuremath{\beta}}^{17}$. We report here on the analysis of the computed series for the susceptibility and the (second moment) correlation length which yields updated estimates of the critical parameters for various values of the spin dimensionality
N, including
N=0 (the self-avoiding walk model),
N=1 (the Ising spin-1/2 model),
N=2 (the
XY model), and
N=3 (the classical Heisenberg model). For all values of
N we confirm a good agreement with the present renormalization-group estimates. A study of the series for the other observables will appear in a forthcoming paper.
We have extended the series for the zero-field susceptibility of the spin-
s Ising model through tenth order in the reduced inverse temperature
K on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients
hn(s) are expressed as simple polynomials in
X=s(s+1). Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of
s. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For
s=21 only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of
s, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
The spin- 1 / 2 Ising model high-temperature series are re-examined by an analysis that parallels the method used to analyse the continuum phi 4 spin model. Results of the Ising model in three dimensions do not agree with results in the literature based on more conventional methods of analysis. The authors cannot decide using only the series terms presently available, which of the methods is to be preferred. They conclude that previous error assessments are unduly optimistic and that the three-dimensional, spin- 1 / 2 , Ising model may satisfy hyperscaling in agreement with the continuum results. An attempt is made to estimate the number of additional high-temperature series terms necessary to resolve the hyperscaling question.