Bulk-boundary correspondance for Sturmian Kohmoto like models

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.