Majorana Fermions in superconducting wires: effects of long-range hopping, broken time-reversal symmetry and potential landscapes

TL;DR

This paper investigates how long-range hopping, broken time-reversal symmetry, and potential landscapes affect Majorana fermions in 1D superconducting wires, revealing new topological phases and their relation to localization phenomena.

Contribution

It extends the Kitaev model by analyzing long-range interactions and inhomogeneous potentials, introducing topological invariants, and connecting localization properties with topological phases.

Findings

Multiple Majorana modes emerge with long-range hopping.

Extended gapless regimes occur with broken time-reversal symmetry.

Localization properties relate to topological phases in disordered systems.

Abstract

We present a comprehensive study of two of the most experimentally relevant extensions of Kitaev's spinless model of a 1D p-wave superconductor: those involving (i) longer range hopping and superconductivity and (ii) inhomogeneous potentials. We commence with a pedagogical review of the spinless model and, as a means of characterizing topological phases exhibited by the systems studied here, we introduce bulk topological invariants as well as those derived from an explicit consideration of boundary modes. In time-reversal invariant systems, we find that the longer range hopping leads to topological phases characterized by multiple Majorana modes. In particular, we investigate a spin model, which respects a duality and maps to a fermionic model with multiple Majorana modes; we highlight the connection between these topological phases and the broken symmetry phases in the original spin model. In the presence of time-reversal symmetry breaking terms, we show that the topological phase diagram is characterized by an extended gapless regime. For the case of inhomogeneous potentials, we explore phase diagrams of periodic, quasiperiodic, and disordered systems. We present a detailed mapping between normal state localization properties of such systems and the topological phases of the corresponding superconducting systems. This powerful tool allows us to leverage the analyses of Hofstadter's butterfly and the vast literature on Anderson localization to the question of Majorana modes in superconducting quasiperiodic and disordered systems, respectively. We briefly touch upon the synergistic effects that can be expected in cases where long-range hopping and disorder are both present.