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Bulk–Boundary Correspondence for Sturmian Kohmoto-Like Models

  • Johannes KellendonkEmail author
  • Emil Prodan
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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. 1.CNRS UMR 5208, Institut Camille Jordan, Université de LyonUniversité Claude Bernard Lyon 1VilleurbanneFrance
  2. 2.Department of Physics and Department of Mathematical SciencesYeshiva UniversityNew YorkUSA

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