Journal of Spectral Theory

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Volume 6, Issue 3, 2016, pp. 601–628
DOI: 10.4171/JST/133

Published online: 2016-09-15

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

David Damanik[1], Marius Lemm[2], Milivoje Lukic[3] and William Yessen[4]

(1) Rice University, Houston, United States
(2) California Institute of Technology, Pasadena, USA
(3) Rice University, Houston, USA
(4) 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 $|x|-v|t|$ is replaced by exponential decay in $|x|-v|t|^\alpha$ with $0<\alpha<1$. In fact, we can characterize the values of $\alpha$ for which such a bound holds as those exceeding $\alpha_u^+$, 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.

Keywords: Quantum dynamics, anomalous many-body transport, Jordan–Wigner transformation

Damanik David, Lemm Marius, Lukic Milivoje, Yessen William: On anomalous Lieb–Robinson bounds for the Fibonacci XY chain. J. Spectr. Theory 6 (2016), 601-628. doi: 10.4171/JST/133