Supercurrent detection of topologically trivial zero-energy states in nanowire junctions

In this work we investigate phase-biased transport in SNS junctions based on nanowires, with strong Rashba spin-orbit coupling and magnetic field, strongly coupled to 2D conventional $s$-wave superconductors. We find that all the nanowire parameters are renormalized, in a manner dependent on the width of the superconductor. In particular, we show that the finite width of the superconductor induces a finite energy shift, or effective chemical potential, in the nanowire sectors coupled to the two superconductors, thus leading to a natural formation of a quantum dot at the junction. Remarkably, under these conditions, the junction supports the emergence of a pair of zero-energy states in the trivial phase, i.e. false Majorana fermion states. We demonstrate that this effect is highly tunable by the superconductor width, and suggest this as an explanation for the formation of trivial zero-energy states reported in recent experiments. Most importantly, we show that these zero-energy states produce a $\pi$-shift in the phase-biased supercurrent. This gives access to a simple tool for their unambiguous detection, thus ruling out any Majorana-like interpretation. See Phys. Rev. Lett. 123, 117001 (2019) for further details.