JournalsjstVol. 6, No. 3pp. 601–628

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
On anomalous Lieb–Robinson bounds for the Fibonacci XY chain cover

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Abstract

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 xvt|x|-v|t| is replaced by exponential decay in xvtα|x|-v|t|^\alpha with 0<α<10<\alpha<1. In fact, we can characterize the values of α\alpha for which such a bound holds as those exceeding αu+\alpha_u^+, 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