On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

  • David Damanik

    Rice University, Houston, United States
  • Marius Lemm

    California Institute of Technology, Pasadena, USA
  • Milivoje Lukic

    Rice University, Houston, USA
  • William Yessen

    Rice University, Houston, USA


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 is replaced by exponential decay in with . In fact, we can characterize the values of for which such a bound holds as those exceeding , the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of [14], 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 [8, 9, 2, 3] to our purposes. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport.

Along the way, we prove a new result about the one-body Fibonacci Hamiltonian: the upper transport exponent agrees with the time-averaged upper transport exponent, see Corollary 2.9. We also explain why our method does not extend to yield anomalous Lieb–Robinson bounds of power-law type for the random dimer model.

Cite this article

David Damanik, Marius Lemm, Milivoje Lukic, William Yessen, On anomalous Lieb–Robinson bounds for the Fibonacci XY chain. J. Spectr. Theory 6 (2016), no. 3, pp. 601–628

DOI 10.4171/JST/133