Abnormal frequency response determined by saddle points in non-Hermitian crystals

In non-Hermitian crystal systems under open boundary conditions (OBCs), 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 frequencies aligned with those of saddle points. The frequencies of these saddle points on the complex plane are below the maximum imaginary part of the 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 evolves 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 insights on the stability criterion of non-Hermitian crystal systems.