Anomalous Scaling Behaviors of the Green's Function in Critical Skin Effects

TL;DR

This paper investigates the Green's functions in non-Hermitian systems with critical skin effects, revealing anomalous scaling behaviors and boundary effects that challenge existing theoretical predictions.

Contribution

It uncovers non-perturbative features of critical NHSE and demonstrates limitations of GBZ theory near boundaries and finite sizes.

Findings

Green's functions follow GBZ predictions in bulk regions.

Inter-chain coupling induces zigzag scaling in conventional regions.

Deviations near boundaries explained by residual GBZ contributions.

Abstract

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 $\Delta$. For $\Delta=0$, the system decouples into two independent chains, and the Green's functions follow predictable patterns based on the GBZ theory. For small $\Delta$ ($\Delta=1/10000$), 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 $\Delta=0$ 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.