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 $\ensuremath{\infty}$, 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 $\ensuremath{\gamma}={1.375}_{\ensuremath{-}0.01}^{+0.02}$, $2\ensuremath{\nu}={1.405}_{\ensuremath{-}0.01}^{+0.02}$, and $\ensuremath{\eta}=0.043\ifmmode\pm\else\textpm\fi{}0.014$ describe all lattices and all spin. Explicit formulas are presented for the susceptibility/zero-angle scattering ${\ensuremath{\chi}}_{0}(T)$, for the inverse correlation length ${\ensuremath{\kappa}}_{1}(T)$, for the effective interaction range ${r}_{1}(T)$, and using the Fisher-Burford approximant, for the total scattering $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)$. The shape parameter ${\ensuremath{\varphi}}_{c}$ attains the "universal" value ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 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 $\frac{{\ensuremath{\kappa}}_{1}({T}_{max})}{k}\ensuremath{\simeq}0.10 (\mathrm{for} S\ensuremath{\gtrsim}2)$ to 0.15. A detailed study is made of the structure dependence of the critical-point correlations ${〈{S}_{0}^{z}{S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉}_{c}$ for various models. This leads to the revised, universal estimate ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 15 for all three cubic lattice, spin-\textonehalf{} Ising models. The results are compared d briefly with various experiments which support $\ensuremath{\eta}\ensuremath{\gtrsim}0.05$.