Duality breaking, mobility edges, and the connection between topological Aubry-Andr\'e and quantum Hall insulators in atomic wires with fermions

It is well known that the Aubry-Andr{\'e} model lacks mobility edges due to its energy-independent self-duality but may exhibit edge states. When duality is broken, we show that mobility regions arise and non-trivial topological phases emerge. By varying the degree of duality breaking, we identify mobility regions and establish a connection between Aubry-Andr{\'e} atomic wires with fermions and quantum Hall systems for a family of Hamiltonians that depends on the relative phase of laser fields, viewed as a synthetic dimension. Depending on the filling factor and the degree of duality breaking, we find three classes of non-trivial phases: conventional topological insulator, conventional topological Aubry-Andr{\'e} insulator, and unconventional (hybrid) topological Aubry-Andr{\'e} insulator. Finally, we discuss appropriate Chern numbers that illustrate the classification of topological phases of localized fermions in atomic wires.