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

Abstract

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 $|x|\le v|t|$ is replaced by the region $|x|\le v|t{|}^{\alpha }$ 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 Schrödinger 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.

arXiv comment5 pages, to appear in Phys. Rev. Lett

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