Long-time magnetization relaxation of spins diffusing in a random field

We study the free-induction decay and the spin-echo attenuation at long times for spins diffusing in a random magnetic field. We show that the decay of transverse magnetization in a Gaussian random longitudinal field is asymptotically M(t)∼${\mathit{t}}^{\gamma \mathrm{-}1}$${\mathit{e}}^{\mathrm{-}\mathit{t}/\mathrm{\tau }}$, where γ is the exponent of a self-avoiding walk. The usual result given by a cumulant expansion fails due to correlations arising from multiple self-intersections in d≤4. Experimental relevance of our analysis is discussed along with the question of universality and the existence of large fluctuations in finite-size systems.