High-Temperature Expansions for the Classical Heisenberg Model. II. Zero-Field Susceptibility

Abstract

The high-temperature series expansion of the zero-field magnetic susceptibility, $\frac{\chi }{{\chi }_{\mathrm{Curie}}}=1+\Sigma {l=1\mathrm{}}^{\infty }{a}_l{\left(\frac{J}{\mathrm{kT}}\right)}^l$, is related to the diagrammatic representation of the corresponding high-temperature expansion of the zero-field static spin correlation function ${〈{\mathbf{S}}_f\mathbf{·}{\mathbf{S}}_g〉}_{\beta }$ presented elsewhere. The first nine terms $a_l$ for loose-packed lattices and the first seven terms for close-packed lattices in the susceptibility series are explicitly obtained in terms of Domb's "general lattice constants" $p_{\mathrm{lx}}$. The general lattice expressions are then used to evaluate these $a_l$ numerically for three two-dimensional lattices and for three cubic lattices. Finally, the $a_l$ are employed to discuss two questions of current interest: (1) Does the critical exponent $\gamma$—in the assumed form of the divergence of $\chi$, $\chi \sim {(T-T_c)}^{-\gamma }$ as $T\to {T_c}^{+}$—have the value $\frac{4}{3}$ for the fcc, bcc, and sc lattices? (2) Do high-temperature expansions suggest a phase transition ($T_c\ne 0$) for some two-dimensional lattices with nearest-neighbor ferromagnetic interactions? It is argued that extrapolation suggests $\gamma$ is definitely greater than $\frac{4}{3}$ for the fcc, bcc, and sc lattices, and that $T_c$ is appreciably different from zero for the plane square and triangular lattices.