  We announce and sketch the rigorous proof of a new kind of anomalous (or
sub-ballistic) Lieb-Robinson bound for an isotropic XY chain in a
quasi-periodic transversal magnetic field. By "anomalous", we mean that the
usual effective light cone defined by $|x|\leq v|t|$ is replaced by the region
$|x|\leq v|t|^\alpha$ for some $0<\alpha<1$. In fact, we can characterize
exactly the values of $\alpha$ for which this holds as those exceeding the
upper transport exponent $\alpha_u^+$ of an appropriate one-body discrete
Schr\"odinger operator. Previous study has produced a good amount of
quantitative information on $\alpha_u^+$.
  The result is obtained by mapping to free fermions, obtaining good dynamical
bounds on the one-body level by adapting techniques developed by Damanik,
Gorodetski, Tcheremchantsev, and Yessen and then "pulling back" these bounds
through the non-local Jordan-Wigner transformation, following an idea of Hamza,
Sims, and Stolz. To our knowledge, this is the first rigorous derivation of
anomalous many-body transport.
  We also explain why our method does not extend to yield anomalous LR bounds
of power-law type if one replaces the quasi-periodic field by a random dimer
field.
