Non-Hermitian Multipole Skin Effects Challenge Localization

We study the effect of quenched disorder on the non-Hermitian skin effect in systems that conserve a U(1) charge and its associated multipole moments. In particular, we generalize the Hatano-Nelson argument for an Anderson transition in a disordered system with non-reciprocal hopping to the interacting case. When only U(1) charge is conserved, we show that there is a transition between a skin effect phase, in which charges are localized at a boundary, and a many-body localized phase, in which the charges are localized at random positions. In periodic boundary conditions, the skin effect gives way to a delocalized phase with a unidirectional current. If additional multipole moments like dipoles are conserved, we show that the non-Hermitian skin effect remains stable for any amount of disorder. As a result, under periodic boundary conditions, the system is always delocalized regardless of disorder strength.