Topological Boundary Invariants for Floquet Systems and Quantum Walks

Abstract

A Floquet systems is a periodically driven quantum system. It can be described by a Floquet operator. If this unitary operator has a gap in the spectrum, then one can define associated topological bulk invariants which can either only depend on the bands of the Floquet operator or also on the time as a variable. It is shown how a K-theoretic result combined with the bulk-boundary correspondence leads to edge invariants for the half-space Floquet operators. These results also apply to topological quantum walks.

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Acknowledgements

The authors thank Rafeal Tiedra for discussions on quantum walks, and the Instituto de Matematicas (UNAM), Cuernavaca, for its hospitality during a visit in July 2017 when this work was written. We also thank an unknown referee for a careful reading and constructive comments. An independent contribution on similar matters by Graf and Tauber [10] appeared on the arXiv while preparing the final version of this manuscript. It also proves bulk-boundary correspondence for Floquet systems, but is restricted to two-dimensional models. It provides a more ad hoc functional analytic treatment, while here the general theory of bulk-boundary correspondence of topological insulators from [16] is combined with a K-theoretic fact (Theorem 2) not contained in [10]. Before preparing the final revision of this work, another related draft on edge states in the Chalker-Coddington model was posted by Asch, Bourget and Joye [3]. This research was partly supported by the Chilean grant FONDECYT Regular 1161651 and the DFG.

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Correspondence to Hermann Schulz-Baldes.

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Sadel, C., Schulz-Baldes, H. Topological Boundary Invariants for Floquet Systems and Quantum Walks. Math Phys Anal Geom 20, 22 (2017). https://doi.org/10.1007/s11040-017-9253-1

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Keywords

  • Floquet topological insulators
  • Quantum walks
  • Connecting maps in K-theory

Mathematics Subject Classifications (2010)

  • 46L80
  • 82C10
  • 37L05