Chern and Majorana Modes of Quasiperiodic Systems

New 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 the quasiperiodicity and introduces an intrinsic scale, producing Chern-beats and nested regions where the fractal structure becomes smooth. Cherns also influence the zero-energy mode, that for quasiperiodic systems which exhibit exponential localization, is related to the ghost of the Majorana; the delocalized state at the onset to topological transition. The Chern and the Majorana, two distinct types of topological edge modes, exist in quasiperiodic superconducting wires.