Abnormal Frequency Response Determined by Saddle Points in Non-Hermitian Crystal Systems

TL;DR

This paper reveals that non-Hermitian crystal systems exhibit an anomalous exponential response at saddle point frequencies, challenging the traditional belief that only modes with positive imaginary parts lead to instability.

Contribution

It introduces a new understanding of frequency response in non-Hermitian systems by identifying saddle points as key to anomalous growth and stability criteria.

Findings

Anomalous exponential response occurs at saddle point frequencies.

Stability depends on saddle point frequencies being below the real axis.

Derived formulas link excitation and response in non-Hermitian systems.

Abstract

In non-Hermitian crystal systems under open boundary condition (OBC), it is generally believed that the OBC modes with frequencies containing positive imaginary parts, when excited by external driving, will experience exponential growth in population, thereby leading to instability. However, our work challenges this conventional understanding. In such a system, we find an anomalous response that grows exponentially with the frequency aligned with those of saddle points. The frequencies of these saddle points on the complex plane are below the maximum imaginary part of OBC spectrum, but they can lie within or beyond the OBC spectrum. We derive general formulas of excitation-response relationships and find that this anomalous response can occur because the excitation of OBC modes eventually evolve toward these saddle points at long times. Only when the frequencies of all these saddle points are below the real axis do the non-Hermitian crystal systems remain stable under periodic excitation. Thus our results also provide new insights on the stability criterion of non-Hermitian crystal systems.