Spin Fluctuations in Heisenberg Paramagnets. I. Diagrammatic Expansion for the Moments of the Spectral Density at Finite Temperature

A diagrammatic method for calculating the moments of the spectral density for the paramagnet is presented. The method yields exact results at infinite temperatures and values for the moments that are correct to lowest order in $\frac{1}{c}$ at finite temperatures, where $c$ is the effective number of spins in the range of the exchange interaction. Explicit formulas for the second and fourth moments for the simple-cubic and body-centered-cubic lattices are presented. It is shown that to lowest order in $\frac{1}{c}$, the moments ${〈{\omega }^{2n}〉}_q$ are proportional to ${\left[\frac{1}{3}S\right(S+1\mathrm{}\left)\right]}^n$, with the constants of proportionality independent of spin. As a consequence, there is a simple "law of corresponding states" relating the spectral densities for different values of the spin. The resummation of diagrams to obtain self-consistent equations for the spectral density is discussed.