  The non-Hermitian skin effect, anomalous localization of an extensive number
of eigenstates induced by nonreciprocal dissipation, plays a pivotal role in
non-Hermitian topology and significantly influences the open quantum dynamics.
However, its genuinely quantum characterization in many-body systems has yet to
be developed. Here, we elucidate that the skin effect manifests itself as
multifractality in the many-body Hilbert space. This multifractality does not
accompany the single-particle skin effect and hence is intrinsic to the
many-body skin effect. Furthermore, we demonstrate that the many-body skin
effect can coexist with spectral statistics of random matrices, in contrast to
multifractality associated with the many-body localization, which necessitates
the absence of ergodicity. We also illustrate multifractality caused by the
Liouvillian skin effect in Markovian open quantum systems. Our work establishes
a defining characterization of the non-Hermitian skin effect and uncovers a
fundamental relationship between multifractality and ergodicity in open quantum
many-body systems.
