  The problem of characterizing low-temperature spin dynamics in
antiferromagnetic spin chains has so far remained elusive. We reinvestigate it
by focusing on isotropic antiferromagnetic chains whose low-energy effective
field theory is governed by the quantum non-linear sigma model. We outline an
exact non-perturbative theoretical approach to analyse the low-temperature
behaviour in the vicinity of non-magnetized states, and obtain explicit
expressions for the spin diffusion constant and the NMR relaxation rate, which
we compare with previous theoretical results in the literature. Surprisingly,
in SU(2)-invariant spin chains in the vicinity of half-filling we find a
crossover from the semi-classical regime to a strongly interacting quantum
regime characterized by zero spin Drude weight and diverging spin conductivity,
indicating super-diffusive spin dynamics. The dynamical exponent of spin
fluctuations is argued to belong to the Kardar-Parisi-Zhang universality class.
Furthermore, by employing numerical tDMRG simulations, we find robust evidence
that the anomalous spin transport persists also at high temperatures,
irrespectively of the spectral gap and integrability of the model.
