$\mathcal{P}\mathcal{T}$ phase transition and non-Hermitian skin effect in vacancy-induced subbands of the Haldane model with gain and loss

Defects can profoundly alter the properties of quantum materials, inducing emergent phenomena beyond the scope of their parent counterparts. Here, we construct effective models to investigate the properties of vacancy-induced subbands in the Haldane model with gain and loss, revealing that their behaviors are critically dependent on the spatial configuration of the vacancy array. Specifically, a sequence of parity–time ($\mathcal{P}\mathcal{T}$) phase transitions occurs for the one-dimensional vacancy array across different sublattices, whereas the non-Hermitian skin effect (NHSE) — with its nontrivial spectral winding — manifests in a chain confined to a single sublattice. Notably, the NHSE arises in this subband even though the parent system lacks it. Our results demonstrate that vacancy defect engineering serves as a powerful approach to generate subsystems whose properties are decoupled from the parent Hamiltonian.