Topological Phases of Inhomogeneous Superconductivity

TL;DR

This paper explores how spatially periodic modulation of the superconducting order parameter influences the emergence of Majorana fermions in inhomogeneous superconducting systems, revealing conditions for different topological phases.

Contribution

It introduces a theoretical framework for understanding Majorana fermions in modulated superconducting systems, bridging Shiba chain and quantum wire models under various regimes.

Findings

Large magnetic exchange energy supports a single pair of Majorana fermions at system ends.

Strong superconducting modulation can induce a topological phase with two pairs of Majorana fermions.

The system transitions between trivial and topological phases depending on the relative strength of modulation and magnetic exchange.

Abstract

We theoretically consider the effect of a spatially periodic modulation of the superconducting order parameter on the formation of Majorana fermions induced by a one-dimensional system with magnetic impurities brought into close proximity to an $s$-wave superconductor. When the magnetic exchange energy is larger than the inter-impurity electron hopping we model the effective system as a chain of coupled Shiba states. While in the opposite regime, the effective system is accurately described by a quantum wire model. Upon including a spatially modulated superconducting pairing, we find, for sufficiently large magnetic exchange energy, the system is able to support a single pair of Majorana fermions with one Majorana fermion on the left end of the system and one on the right end. When the modulation of superconductivity is large compared to the magnetic exchange energy, the Shiba chain returns to a trivially gapped regime while the quantum wire enters a new topological phase capable of supporting two pairs of Majorana fermions.