On correlations at long times in Heisenberg paramagnets

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Abstract
The low-frequency, long-wavelength properties of a paramagnet in d dimensions are studied within the framework of a non-linear Langevin equation. The frequency and wave-vector dependent transport coefficient Gamma (k, omega ) has a singular part which varies as omega d/2 at k=0 so its time Fourier transform decays as t-(1+d2)/, in contrast to the assumptions of linearised hydrodynamics according to which Gamma (k,t) decays rapidly in a microscopic time. Both reversible and irreversible processes give contributions to this effect but with opposite signs. The latter contribution is non vanishing as k to 0 only because we include a drift term in the Langevin equation which depends explicitly on the energy density. The consequences of this generalisation of the usual Langevin equation are investigated in detail in a perturbation expansion.