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Chiral Floquet Systems and Quantum Walks at Half-Period

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Abstract

We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous, or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators in terms of five integer indices. On the basis of these indices, it can be decided whether the half-step operator can be obtained from a continuous Hamiltonian driving, or not. The half-step operator determines two Floquet operators, obtained by starting the driving at zero or at half-period, respectively. These are called timeframes and are chiral symmetric quantum walks. Conversely, we show under which conditions two chiral symmetric walks determine a common half-step operator. Moreover, we clarify the connection between the classification of half-step operators and the corresponding quantum walks. Within this theory, we prove bulk-edge correspondence and show that a second timeframe allows to distinguish between symmetry protected edge states at \(+\,1\) and \(-\,1\) which is not possible for a single timeframe.

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Acknowledgements

We thank the referees for helpful comments, which improved the readability of the manuscript. Moreover, we thank Janos Asbóth for inspiring discussions and for directing our interest to timeframed quantum walks in the first place. C. Cedzich acknowledges support by the projet PIA-GDN/QuantEx P163746-484124 and by DGE – Ministère de l’Industrie. T. Geib and R. F. Werner acknowledge support from the DFG through SFB 1227 DQ-mat. A. H. Werner thanks the VILLUM FONDEN for its support with a Villum Young Investigator Grant (Grant No. 25452) and its support via the QMATH Centre of Excellence (Grant No. 10059).

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Authors and Affiliations

  1. Laboratoire de Recherche en Informatique (LRI), CNRS, Centrale Supélec, Université Paris Sud, Bât 650, Rue Noetzlin, 91190, Gif-sur-Yvette, France

    C. Cedzich

  2. Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167, Hannover, Germany

    T. Geib & R. F. Werner

  3. QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    A. H. Werner

  4. NBIA, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

    A. H. Werner

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  1. C. Cedzich
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  2. T. Geib
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  3. A. H. Werner
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  4. R. F. Werner
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Correspondence to C. Cedzich.

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Communicated by Alain Joye.

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Cedzich, C., Geib, T., Werner, A.H. et al. Chiral Floquet Systems and Quantum Walks at Half-Period. Ann. Henri Poincaré 22, 375–413 (2021). https://doi.org/10.1007/s00023-020-00982-6

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