Fragile non-Bloch spectrum and unconventional Green's function

TL;DR

This paper reveals that non-Hermitian systems with the non-Hermitian skin effect can have spectra that are extremely sensitive to local boundary perturbations, characterized by unconventional Green's function behaviors and a real-to-complex spectral transition.

Contribution

It uncovers the fragility of non-Hermitian spectra under local boundary perturbations and links this to unique Green's function asymptotics, advancing understanding of non-Hermitian spectral sensitivity.

Findings

Spectra can be significantly altered by exponentially small boundary perturbations.

Green's functions exhibit unconventional V-shape asymptotic behaviors.

A real-to-complex transition of the bulk spectrum occurs due to boundary perturbations.

Abstract

In non-Hermitian systems, it is a counterintuitive feature of the non-Hermitian skin effect (NHSE) that the energy spectrum and eigenstates can be totally different under open or periodic boundary conditions, suggesting that non-Hermitian spectra can be extremely sensitive to non-local perturbations. Here, we show that a wide range of non-Hermitian models with NHSE can even be highly sensitive to local perturbation under open boundary conditions. The spectrum of these models is so fragile that it can be significantly modified by adding only exponentially small perturbations on boundaries. Intriguingly, we show that such fragile spectra are quantified by the Green's function exhibiting unconventional V-shape asymptotic behaviors. Accordingly, bi-directional exponential amplification can be observed. As an interesting consequence, we find a real-to-complex transition of the bulk spectrum induced by exponentially small boundary perturbations. Finally, we reveal a hierarchy of the asymptotic behaviors of non-Hermitian Green's functions, which restricts the frequency range for the presence of unconventional Green's functions.