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On Anomalous Lieb-Robinson Bounds for the Fibonacci XY Chain

Damanik, David and Lemm, Marius and Lukic, Milivoje and Yessen, William (2016) On Anomalous Lieb-Robinson Bounds for the Fibonacci XY Chain. Journal of Spectral Theory, 6 (3). pp. 601-628. ISSN 1664-039X. https://resolver.caltech.edu/CaltechAUTHORS:20141209-084133546

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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 |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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.4171/JST/133DOIArticle
http://www.ems-ph.org/journals/show_abstract.php?issn=1664-039X&vol=6&iss=3&rank=5PublisherArticle
http://arxiv.org/abs/1407.4924arXivDiscussion Paper
ORCID:
AuthorORCID
Damanik, David0000-0001-5924-3849
Additional Information:© 2016 European Mathematical Society. Received September 3, 2014; revised February 22, 2015. David Damanik was supported in part by NSF grants DMS–1067988 and DMS–1361625. Milivoje Lukic was supported in part by NSF grant DMS–1301582. William Yessen was supported by NSF grant DMS–1304287.
Funders:
Funding AgencyGrant Number
NSFDMS-1067988
NSFDMS-1361625
NSFDMS-1301582
NSFDMS-1304287
Subject Keywords:Quantum dynamics, anomalous many-body transport, Jordan–Wigner transformation
Issue or Number:3
Classification Code:2010 MSC: Primary 81Q10; Secondary 82C10, 47B36
Record Number:CaltechAUTHORS:20141209-084133546
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20141209-084133546
Official Citation: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
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:52488
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:09 Dec 2014 18:53
Last Modified:09 Mar 2020 13:18

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