On Anomalous Lieb-Robinson Bounds for the Fibonacci XY Chain

We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in |x| − v|t| is replaced by exponential decay in |x| − v|t|^α with 0 < α < 1. In fact, we can characterize the values of α for which such a bound holds as those exceeding α^+_u, the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of [12], we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan-Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in [6, 7, 2, 3] to our purposes. We also discuss the extension to the more general class of Sturmian potentials and we explain why our method does not extend to yield anomalous Lieb-Robinson bounds of power-law type for the random dimer model.