Chern and Majorana modes of quasiperiodic systems

Different types of self-similar states are found in quasiperiodic systems characterized by topological invariants---the Chern numbers. We show that the topology introduces a competing length in the self-similar band edge states transforming peaks into doublets of size equal to the Chern number. This length intertwines with quasiperiodicity and introduces an intrinsic scale, producing Chern beats and nested regions where the fractal structure becomes smooth. Chern numbers also influence the zero-energy mode that, for quasiperiodic systems, which exhibit exponential localization, is related to the ghost of the Majorana: the remnant of the edge localized topological state that delocalizes at the onset to a topological transition. In superconducting wires, the exponentially decaying profile of the edge localized Majorana modes also encode fingerprints of the Chern states that reside in close proximity to zero energy.