Green's Functions of Non-Hermitian Systems at Exceptional Deficiency

Exceptional deficiency (ED) is a newly discovered broadband non-Hermitian condition, at which the system's spectrum is entirely composed of exceptional points and all eigenvectors pairwise coalesce. Here, we analyze the steady-state and time-domain responses of non-Hermitian lattices at ED by considering their frequency- and time-domain Green's functions (GFs). We show that the frequency-domain GFs are characterized by the emergence of second-order poles across the entire spectrum, producing broadband second-order super-Lorentzian line shapes, enhanced magnitude, and sequential 2π phase shifts across each resonance. These second-order poles also underpin unconventional responses to global dissipation, by which we uncover a loss-revival of defective skin effect (LRDSE): the skin modes that are "missing" due to the system's defectiveness at ED can re-emerge in the responses. In the time-domain responses, a linear-in-time amplification factor is identified in the evolution kernel, which can enhance skin-effect dynamics that would otherwise be suppressed by global loss. The amplification strength scales with the degree of defectiveness of the system. Our work establishes a theoretical framework for analyzing steady-state and dynamic signatures of ED, and opens new routes for dissipation-controlled broadband exceptional responses.