Anomalous scaling behavior of Green's function in critical skin effects

We study the Green's functions in non-Hermitian systems that exhibit the critical non-Hermitian skin effect (critical NHSE) using a double-chain Hatano-Nelson model with interchain coupling $\mathrm{\ensuremath{\Delta}}$. When $\mathrm{\ensuremath{\Delta}}=0$, the system decomposes into two independent chains and, in this case, the Green's functions follow predictable patterns based on the generalized Brillouin zone (GBZ) theory. For small $\mathrm{\ensuremath{\Delta}}$ $(\mathrm{\ensuremath{\Delta}}=1/10\phantom{\rule{0.16em}{0ex}}000)$, there are two cases. In conventional regions with trivial open-boundary conditions spectral winding numbers, the interchain coupling induces a zigzag scaling structure in the Green's functions, which can be explained by the first-order perturbation theory. In anomalous regions with nontrivial winding numbers, the actual Green's functions and the GBZ-based predictions match in the bulk but diverge near the boundaries and this deviation turns out to be derived from the influence of the GBZ of the $\mathrm{\ensuremath{\Delta}}=0$ case. These results reveal the unique nonperturbative features of the critical NHSE and highlight the limitations of the GBZ theory in capturing the boundary effects of finite-size systems and thus emphasize the need to consider both bulk and boundary dynamics in such systems.