Anomalous Scaling Behaviors of the Green's Function in Critical Skin Effects
We study the Green's functions in non-Hermitian systems exhibiting the critical non-Hermitian skin effect (critical NHSE) using a double-chain Hatano-Nelson model with inter-chain coupling $Δ$. For $Δ=0$, the system decouples into two independent chains, and the Green's functions follow predictable patterns based on the GBZ theory. For small $Δ$ ($Δ=1/10000$), in conventional regions with trivial OBC spectral winding numbers, inter-chain coupling induces a zigzag scaling structure in Green's functions due to competition between the two chains, explainable by first-order perturbation theory. In anomalous regions with non-trivial winding numbers, Green's functions match GBZ predictions in the bulk but diverge near boundaries, with residual contributions from the $Δ=0$ GBZ accounting for the deviations. These results reveal the unique non-perturbative features of critical NHSE and highlight the limitations of GBZ theory in capturing finite-size and boundary effects, emphasizing the need to consider both bulk and boundary dynamics in such systems.