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We present a unified study of the effect of periodic, quasiperiodic anddisordered potentials on topological phases that are characterized by Majoranaend modes in 1D p-wave superconducting systems. We define a topologicalinvariant derived from the equations of motion for Majorana modes and, as ourfirst application, employ it to characterize the phase diagram for simpleperiodic structures. Our general result is a relation between the topologicalinvariant and the normal state localization length. This link allows us toleverage the considerable literature on localization physics and obtain thetopological phase diagrams and their salient features for quasiperiodic anddisordered systems for the entire region of parameter space.

Majorana Fermions in superconducting 1D systems having periodic, quasiperiodic, and disordered potentials