Magnetically controlled critical non-Hermitian skin effect and Anderson localization in a superconducting nanowire

The critical non-Hermitian skin effect (cNHSE), first proposed in two coupled Hatano-Nelson chains, heralds a whole class of size-dependent transitions with rich anomalous scaling behaviors. Up to now, it remains unknown whether the cNHSE exists in a superconductor. This work uncovers the cNHSE in a superconducting nanowire with spin-dependent nonreciprocal hoppings. An arbitrarily small magnetic field, which couples different spins, plays a crucial role in the emergence of the cNHSE, marked by a discontinuous jump in eigenenergies and eigenstates. In finite-sized systems, the cNHSE is manifested by pronounced size-induced effects and remarkable scaling behaviors. Furthermore, applying a single strong magnetic perturbation at one end of the system can also induce the cNHSE. We observe a critical-conventional-critical NHSE transition, a direction reversal of NHSE, and the presence/absence of a Majorana zero mode by shifting the single magnetic perturbation from the left to right end of the chain. By introducing a class of quasiperiodic disorder, we investigate the interplay between cNHSE, magnetic field, and Anderson localization. This exploration reveals a rich array of localization phenomena, including distinctive localization transitions, mobility edges, and fascinating reentrant localization. Our study offers a significant proposal for realizing cNHSE in superconductors and enhances the understanding of its interplay with magnetic fields and Anderson localization.