Anderson Localization for Electric Quantum Walks and Skew-Shift CMV Matrices
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- Anderson Localization for Electric Quantum Walks and Skew-Shift CMV Matrices
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We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the Lyapunov exponent of this model. Making use of a connection between quantum walks and CMV matrices our result implies Anderson localization for CMV matrices with a particular choice of skew-shift Verblunsky coefficients as well as for quasi-periodic unitary band matrices.
Originalsprog | Engelsk |
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Tidsskrift | Communications in Mathematical Physics |
Vol/bind | 387 |
Udgave nummer | 3 |
Sider (fra-til) | 1257-1279 |
ISSN | 0010-3616 |
DOI | |
Status | Udgivet - 2021 |
Bibliografisk note
Funding Information:
The authors thank Jake Fillman for clarifications of the literature on localization proofs for quasi-periodic Schrödinger operators and critical reading of this manuscript, and Luis Velázquez for illuminating discussions about the connection between squares of quantum walks and CMV matrices. C. Cedzich acknowledges support by the projet PIA-GDN/QuantEx P163746-484124 and by DGE – Ministère de l’Industrie. A. H. Werner thanks the Villum Fonden for its support via a Villum Young Investigator Grant.
Publisher Copyright:
© 2021, The Author(s).
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