  This work uncovers a new class of criticality where eigenenergies and
eigenstates of non-Hermitian lattice systems jump discontinuously across a
critical point in the thermodynamic limit, unlike established Hermitian and
non-Hermitian critical scenarios where spectrum remains continuous across a
transition. Such critical behavior, dubbed the "critical skin effect", is
rather generic, occuring whenever subsystems with dissimilar non-Hermitian skin
localization lengths are coupled, however weakly. Due to the existence of this
criticality, the thermodynamic limit and the zero-coupling limit cannot be
exchanged, thus challenging the celebrated generalized Brillouin zone (GBZ)
approach when applied to finite-size systems. As manifestations of the critical
skin effect in finite-size systems, we present stimulating examples with
anomalous scaling behavior regarding spectrum, correlation functions,
entanglement entropy, and scale-free wavefunctions that decay exponentially
rather than power-law. More spectacularly, topological in-gap modes can even be
induced by changing the system size.
