Topology Induced Oscillations in Majorana Fermions in a Quasiperiodic Superconducting Chain

Abstract

Spatial profile of the Majorana fermion wave function in a one-dimensional $p$-wave superconductors ($\cal{PWS}$) with quasi periodic disorder is shown to exhibit spatial oscillations. These oscillations damp out in the interior of the chain and are characterized by a period that has topological origin and is equal to the Chern number determining the Hall conductivity near half-filling of a two-dimensional electron gas in a crystal. This mapping unfolds in view of a correspondence between the critical point for the topological transition in $\cal{PWS}$ and the {\it strong coupling fixed point} of the Harper's equation. Oscillatory character of these modes persist in a generalized model related to an extended Harper system where the electrons also tunnel to the diagonals of a square lattice. However, beyond a bicritical point, the Majorana oscillations occur with a random period, characterized by an invariant fractal set.