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

TL;DR

This paper investigates how periodic, quasiperiodic, and disordered potentials influence topological phases with Majorana end modes in 1D p-wave superconductors, establishing a relation between topological invariants and localization length.

Contribution

It introduces a unified framework to analyze topological phases in 1D superconductors with various potentials, linking topological invariants to localization properties.

Findings

Topological phase diagrams characterized for different potentials.

A relation between topological invariant and localization length.

Extension of phase diagram analysis to disordered systems.

Abstract

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