Complex Frequency Detection in a Subsystem

In this study, we systematically explore the non-Hermitian skin effect (NHSE) and its associated complex-frequency detection in the context of a frequency-dependent non-Hermitian Hamiltonian. This Hamiltonian arises from the self-energy correction of the subsystem and can be calculated exactly within our theoretical model, without the need for any approximations. Additionally, complex frequency detection, which encompasses complex frequency excitation, synthesis, and fingerprint, enables us to detect the physical responses induced by the complex driving frequencies. Our calculations reveal that both complex frequency excitation and synthesis are incompatible with the non-Hermitian approximation and are unable to characterize the presence or abscence of the NHSE. In contrast, the complex-frequency fingerprint successfully detects the novel responses induced by the NHSE through the introduction of a double-frequency Green's function. Our work provides a platform for studying non-Hermitian physics and their novel response in quantum systems rigorously without relying on any approximations.