Bulk-boundary correspondance for Sturmian Kohmoto like models

TL;DR

This paper establishes a bulk-boundary correspondence for Sturmian Kohmoto-like models, linking topological gap labels to edge state winding numbers, with implications for understanding edge states in quasicrystals and related systems.

Contribution

It provides a theoretical framework connecting gap labels to winding numbers of edge states in Sturmian models, including the effects of phason flips and experimental relevance.

Findings

Bulk-boundary correspondence for Sturmian models proven

Winding number linked to edge state properties

Experimental observations with polaritonic waveguides explained theoretically

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.