  We show that a local non-Hermitian perturbation in a Hermitian lattice system
generically induces scale-free localization for the continuous-spectrum
eigenstates. When the perturbation lies at a finite distance to the boundary,
the scale-free eigenstates are promoted to exponentially localized modes, whose
number is proportional to the distance. Furthermore, when the local
non-Hermitian perturbation respects parity-time (PT) symmetry, the PT symmetry
breaking is always accompanied by the emergence of scale-free or exponential
localization. Intriguingly, we find a concise band-structure condition, which
tells not only when the continuous-spectrum PT breaking of scale-free modes can
occur, but also the precise PT-breaking energy window. Our results uncover a
series of unexpected generic phenomena induced by a local non-Hermitian
perturbation, which has interesting interplay with PT symmetry.
