Scale-free localization and PT symmetry breaking from local non-Hermiticity

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.