  It is well established that the entanglement entropy of a critical system
generally scales logarithmically with system size. Yet, in this work, we report
a new class of non-Hermitian critical transitions that exhibit dramatic
divergent dips in their entanglement entropy scaling, strongly violating
conventional logarithmic behavior. Dubbed scaling-induced exceptional
criticality (SIEC), it transcends existing non-Hermitian mechanisms such as
exceptional bound states and non-Hermitian skin effect (NHSE)-induced gap
closures, which are nevertheless still governed by logarithmic entanglement
scaling. Key to SIEC is its strongly scale-dependent spectrum, where eigenbands
exhibit an exceptional crossing only at a particular system size. As such, the
critical behavior is dominated by how the generalized Brillouin zone (GBZ)
sweeps through the exceptional crossing with increasing system size, and not
just by the gap closure per se. We provide a general approach for constructing
SIEC systems based on the non-local competition between heterogeneous NHSE
pumping directions, and show how a scale-dependent GBZ can be analytically
derived to excellent accuracy. Beyond 1D free fermions, SIEC is expected to
occur more prevalently in higher-dimensional or even interacting systems, where
antagonistic NHSE channels generically proliferate. SIEC-induced entanglement
dips generalize straightforwardly to kinks in other entanglement measures such
as Renyi entropy, and serve as spectacular demonstrations of how algebraic and
geometric singularities in complex band structures manifest in quantum
information.
