New Anomalous Lieb-Robinson Bounds in Quasi-Periodic XY Chains

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|^α for some 0<α<1. In fact, we can characterize exactly the values of α for which this holds as those exceeding the upper transport exponent α_u^+ of an appropriate one-body discrete Schrödinger operator. Previous study has produced a good amount of quantitative information on α_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.