Irreversibility in Heisenberg Spin Systems. I. General Formalism and Kinetic Equations in the High-Temperature Limit

Starting from the von Neumann equation for the spin density matrix of a Heisenberg system, we analyze the perturbation expansion of the spin autocorrelation function by the diagrammatic technique previously applied to quantum gases. We demonstrate a number of theorems which allow us to express this perturbation series in terms of renormalized graphs only; we then derive a kinetic equation for the autocorrelation function. The main feature of this equation is that the kernel, which is highly nonlinear in the autocorrelation function itself, tends to zero in the limit of long times. The results, which are exact in the high-temperature region and in the Weiss limit (number of neighbors $Z\to \infty$), allow us to consider the behavior of the autocorrelation function for times both short and long. This model is a typical example of a system with a discrete unperturbed spectrum showing an irreversible behavior.