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 a localization transition in disordered, nonreciprocal systems 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 cluster at a boundary, and a many-body localized phase, in which charges localize at random positions. In the dynamics of entanglement, this coincides with an area-to-volume-law transition. For systems without boundaries, the skin effect becomes a delocalized phase with a unidirectional current. If dipoles or higher multipoles are conserved, we show that the non-Hermitian skin effect remains stable to arbitrary disorder. Counterintuitively, the system is therefore always delocalized under periodic boundary conditions, regardless of disorder strength.

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• Many-body localization
• Open quantum systems & decoherence
• Skin effect
• Non-Hermitian systems