  We generalize nonlinear Luttinger liquid theory to describe the dynamics of
one-dimensional quantum critical systems at low temperatures. Analyzing
density-matrix renormalization group results for the spin autocorrelation
function in the XXZ chain we provide, in particular, direct evidence for spin
diffusion in sharp contrast to the exponential decay in time predicted by
conventional Luttinger liquid theory. Furthermore, we discuss how the
frequencies and exponents of the oscillatory contributions from the band edges
are renormalized by irrelevant interactions and obtain excellent agreement
between our finite temperature nonlinear Luttinger liquid theory and the
numerical data.
