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Bulk–Boundary Correspondence for Sturmian Kohmoto-Like Models

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Abstract

We consider one-dimensional tight binding models on \(\ell ^2({{\mathbb {Z}}})\) whose spatial structure is encoded by a Sturmian sequence \((\xi _n)_n\in \{a,b\}^{{\mathbb {Z}}}\). An example is the Kohmoto Hamiltonian, which is given by the discrete Laplacian plus an on-site potential \(v_n\) taking value 0 or 1 according to whether \(\xi _n\) is a or b. The only non-trivial topological invariants of such a model are its gap labels. The bulk–boundary correspondence we establish here states that there is a correspondence between the gap label and a winding number associated with the edge states, which arises if the system is augmented and compressed onto half-space \(\ell ^2({{\mathbb {N}}})\). This has been experimentally observed with polaritonic waveguides. A correct theoretic explanation requires, however, first a smoothing out of the atomic motion via phason flips. With such an interpretation at hand, the winding number corresponds to the mechanical work through a cycle which the atomic motion exhibits on the edge states.

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Authors and Affiliations

  1. CNRS UMR 5208, Institut Camille Jordan, Université de Lyon, Université Claude Bernard Lyon 1, 69622, Villeurbanne, France

    Johannes Kellendonk

  2. Department of Physics and Department of Mathematical Sciences, Yeshiva University, New York, USA

    Emil Prodan

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  1. Johannes Kellendonk
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Correspondence to Johannes Kellendonk.

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Communicated by Jean Bellissard.

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Kellendonk, J., Prodan, E. Bulk–Boundary Correspondence for Sturmian Kohmoto-Like Models. Ann. Henri Poincaré 20, 2039–2070 (2019). https://doi.org/10.1007/s00023-019-00792-5

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