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

TL;DR

This paper proves a new type of anomalous Lieb-Robinson bounds in quasi-periodic XY chains, showing sub-ballistic transport characterized by a fractional light cone, and connects these bounds to the spectral properties of an associated Schrödinger operator.

Contribution

It provides the first rigorous derivation of anomalous many-body transport bounds in quasi-periodic XY chains using a novel mapping to free fermions and spectral analysis.

Findings

Established sub-ballistic light cone with fractional exponent

Linked anomalous transport to upper transport exponent of Schrödinger operator

Demonstrated method does not extend to random dimer fields

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 $|x|\leq v|t|$ 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.