Topological Boundary Invariants for Floquet Systems and Quantum Walks
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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.
KeywordsFloquet topological insulators Quantum walks Connecting maps in K-theory
Mathematics Subject Classifications (2010)46L80 82C10 37L05
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  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  is combined with a K-theoretic fact (Theorem 2) not contained in . 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 . This research was partly supported by the Chilean grant FONDECYT Regular 1161651 and the DFG.
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