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. Furthermore, when the local non-Hermitian perturbation enjoys parity-time (PT) symmetry, the PT symmetry breaking of continuous spectrum is always accompanied by the emergence of scale-free localization. This type of PT symmetry breaking is highly sensitive to boundary conditions: The continuous spectrum of a periodic system undergoes a PT symmetry breaking as long as the non-Hermitian perturbation is strong enough; however, the counterpart under open boundary condition allows PT symmetry breaking only when the band structure satisfies certain condition that we unveil here. We also obtain the precise energy window in which the PT symmetry breaking is possible. Our results uncover a generic boundary-induced non-Hermitian phenomenon, which has unexpected interplay with PT symmetry.