The memory function in the mode-mode coupling theory for the Heisenberg paramagnet

Abstract

The equation for the relaxation shape function given by the mode-mode theory is studied in the critical region. The emphasis is put on the long-time behaviour of the memory function. A selfconsistent solution of the approximate equation for the generalized wave-vector-dependent diffusion constant is found in the limits q<<K and K<<q. This solution is used to obtain the explicit time dependence of the memory function. In the hydrodynamic region, q<<K, it has the asymptotic form t-5/2 exp(-Dq2t/2) with an inverse fractional power dependence on t. This leads to the conclusion that the equation of motion in the mode-mode theory does not have a characteristic spin diffusion solution with pure exponential decay for asymptotically large times.