The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K9 for the hyper face centred (HFC) lattice, to order K11 for the hyper body centred cubic (HBCC) lattice and to order K12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t-1 mod In t mod p, predicted by the renormalisation group theory (t=1-Kc/K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.
Extended series expansions are derived for the high-temperature susceptibility of the classical Heisenberg model, on three-dimensional lattices. Series coefficients are presented to twelfth order for the simple cubic (SC) and face centred cubic (FCC) lattices and to eleventh order for the body centred cubic (BCC) lattice. The results are in agreement with earlier calculations apart from a small discrepancy at the tenth order on the FCC lattice. Extrapolation studies on the extended series are used to obtain revised estimates for the critical points (K c ) and the susceptibility exponent ( gamma ). On the FCC lattice, the authors also investigate the possibility of a confluent non-analytic correction to the dominant singularity. While the coefficients are consistent with the presence of such a correction term with an exponent ( Delta 1 ) of 0.55, as predicted by renormalisation group (RG) calculations, the amplitude of the correction term appears to be very small compared with that of the first analytic correction term.
The validity of the hyperscaling relation 2 Delta -dv- gamma =0 is studied for the four-dimensional spin- 1 / 2 Ising model. High-temperature series expansions are derived for the fourth-field derivative chi 0 (2) of the free energy on the four-dimensional hyper-face-centred cubic (HFCC) and hyper-body-centred cubic (HBCC) lattices to order 9 and 11 respectively. These are analysed, together with other series already available for the susceptibility chi 0 and correlation length xi for the HFCC, HBCC and the hyper-simple cubic (HSC) lattices. All these series are found to behave consistently with the asymptotic form t -q mod lnt mod p , where t is a reduced temperature variable and q is the appropriate mean-field exponent (so that hyperscaling is satisfied automatically). The best estimates for p are as follows: p=0.30+or-0.05 (HFCC), 0.32+or-0.05 (HBCC) for xi 0 (with q=1= gamma ) and p=0.33+or-0.05 for xi 2 (with q=1=2v). These estimates are in good agreement with the renormalisation group (RG) prediction of p= 1 / 3 . Results for chi 0 (2) are more slowly convergent, but are still consistent with p= 1 / 3 for q=4=2 Delta + gamma .
We present simple tables of integers from which it is possible to reconstruct the high-temperature series coefficients through ${\mathrm{\ensuremath{\beta}}}^{14}$ for the susceptibility, for the second correlation moment, and for the second field derivative of the susceptibility of the O(N) classical Heisenberg model on a simple (hyper)cubic lattice in dimension d=2, 3, and 4 and for any N. To construct the tables we have used the recent extension of the high-temperature series by M. Luscher and P. Weisz and some analytic properties in N that we have derived from the Schwinger-Dyson equations of the O(N) model. We also present a numerical study of these series in the d=2 case. The main results are: (a) the extended series give further support to the Cardy-Hamber-Nienhuis exact formulas for the critical exponents when -2N2; (b) for N\ensuremath{\ge}3 there are no indications of any critical point at finite \ensuremath{\beta}; (c) the series are consistent with the low-temperature asymptotic forms predicted by the perturbative renormalization group.
For pt. I see ibid., vol. 5, no. 5, 624. Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K for the simple cubic lattice to v 17 , the body-centred cubic lattice to v 15 , and the face-centred cubic lattice to v 12 , inclusive. The coefficients are analysed by the ratio method and it is estimated that, subject to the implicit assumptions thereof, the critical temperatures correspond to v f =0.21813+or-0.00001 (SC), v f =0.15612+or-0.00003 (BCC) and v f =0.10174+or-0.00001 (FCC). The symptotic behaviour of the coefficients is studied in detail; for the loose-packed lattices the decay of the even-odd oscillation in the ratios is found to be consistent with the assumption that the critical index of the high temperature specific heat is very close to 1/8.
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.