Spin Correlation Functions at High Temperatures

A method for the self-consistent calculation of correlation functions is described and applied to the Heisenberg model at high temperatures. The technique is based on a straightforward physical picture. It is used (i) to derive a simple analytic approximation for the autocorrelation function valid at times $t\lesssim \frac{2JS}{\hslash }$; (ii) to derive an equation given by Résibois and DeLeener which is shown to be valid at short times also; (iii) to derive a set of integrodifferential equations for the general correlation functions $〈{\mathrm{S}}_{-\mathrm{q}}·{\mathrm{S}}_{\mathrm{q}}\mathrm{}\left(t\right)\mathrm{}〉$. The latter equations are solved numerically for the case of a simple cubic lattice with nearest-neighbor interactions, and are shown to give results in excellent agreement with computer simulation calculations for the same model. A discussion is given of the physical motivation of the approximations employed and the special mathematical aspects of the problem.