Degeneracy of Majorana bound states and fractional Josephson effect in a dirty SNS junction

TL;DR

This paper investigates the stability and degeneracy of multiple Majorana bound states in a disordered p-wave superconductor junction, revealing their resilience due to chiral symmetry and explaining the fractional Josephson effect.

Contribution

It demonstrates that Majorana bound states in a disordered junction maintain degeneracy at zero energy because of chiral symmetry, leading to a fractional Josephson effect.

Findings

Majorana bound states are degenerate at zero energy due to chiral symmetry.

Degeneracy persists even with random potential in the normal metal.

Resonant Cooper pair transmission causes fractional Josephson current at phase difference π.

Abstract

We theoretically study the stability of more than one Majorana Fermion appearing in a $p$-wave superconductor/dirty normal metal/$p$-wave superconductor junction in two-dimension by using chiral symmetry of Hamiltonian. At the phase difference across the junction $\varphi$ being $\pi$, we will show that all of the Majorana bound states in the normal metal belong to the same chirality. Due to this pure chiral feature, the Majorana bound states retain their high degree of degeneracy at the zero energy even in the presence of random potential. As a consequence, the resonant transmission of a Cooper pair via the degenerate MBSs carries the Josephson current at $\varphi=\pi-0^+$, which explains the fractional current-phase relationship discussed in a number of previous papers.