Bulk-boundary correspondance for Sturmian Kohmoto like models

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 onsite 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 to 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 theoretical 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.

37 pages, 7 figures

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