Theory of Critical-Point Scattering and Correlations. II. Heisenberg Models

Abstract

The spin-spin correlation functions and the critical-scattering intensity for Heisenberg models of general spin, $S=\frac{1}{2}$ to $\infty$, on the sc, bcc, and fcc lattices are studied on the basis of high-temperature series expansions along the lines developed in Paper I [M. E. Fisher and R. J. Burford, Phys. Rev. 156, 583 (1967)]. Subject to increased uncertainties for low spin, it is concluded that the exponents $\gamma ={1.375}_{-0.\mathrm{}01}^{+0.\mathrm{}02}$, $2\nu ={1.405}_{-0.\mathrm{}01}^{+0.\mathrm{}02}$, and $\eta =0.\mathrm{}043\pm 0.014$ describe all lattices and all spin. Explicit formulas are presented for the susceptibility/zero-angle scattering ${\chi }_0\mathrm{}\left(T\right)\mathrm{}$, for the inverse correlation length ${\kappa }_1\mathrm{}\left(T\right)\mathrm{}$, for the effective interaction range $r_1\mathrm{}\left(T\right)\mathrm{}$, and using the Fisher-Burford approximant, for the total scattering $\hat{\chi }(\overrightarrow{\mathrm{k}},T)$. The shape parameter ${\phi }_c$ attains the "universal" value ${\phi }_c\simeq 0$. 11 for large spin but shows signs of spin dependence (and lattice dependence) for low spin. At fixed k the scattering is predicted to display a maximum above $T_c$ determined by $\frac{{\kappa }_1\left(T_{max}\right)}{k}\simeq 0.10 \left(\mathrm{for} S\gtrsim 2\right)$ to 0.15. A detailed study is made of the structure dependence of the critical-point correlations ${〈S_0^z{S}_{\overrightarrow{\mathrm{r}}}^z〉}_c$ for various models. This leads to the revised, universal estimate ${\phi }_c\simeq 0$. 15 for all three cubic lattice, spin-½ Ising models. The results are compared d briefly with various experiments which support $\eta \gtrsim 0.05$.