Uniform response theory of non-Hermitian systems: Non-Hermitian physics beyond the exceptional point

Non-Hermitian systems display remarkable response effects that directly reflect a variety of distinct spectral scenarios, such as exceptional points where the eigensystem becomes defective. However, present frameworks treat the different scenarios as separate cases, following the singular mathematical change between different spectral decompositions from one scenario to another. This not only complicates the coherent description near the spectral singularities where the response qualitatively changes, but also impedes the application to practical systems, as the determination of these decompositions is manifestly ill conditioned. Here we develop a general response theory of non-Hermitian systems that uniformly applies across all spectral scenarios. We unravel this response by formulating a uniform expansion of the spectral quantization condition, as well as a uniform expansion of the Green's function, where both expansions exclusively involve directly calculable data from the Hamiltonian. These data smoothly vary with external parameters and energy as spectral singularities are approached and attained, and nevertheless capture the qualitative differences of the response in these scenarios. We furthermore present two direct applications of this framework. First, in the context of the quantization condition, we determine the precise conditions for spectral degeneracies of geometric multiplicity greater than unity, as well as the perturbative behavior around these cases. Second, in the context of the Green's function, we formulate a hierarchy of spectral response strengths that varies continuously across all parameter space, and thereby also reliably determines the response strength of exceptional points. Finally, we join both themes and demonstrate, both generally and in concrete examples, that the previously inaccessible scenarios of higher geometric multiplicity result in unique variants of the super-Lorentzian response. Our approach widens the scope of non-Hermitian response theory to capture all spectral scenarios on an equal and uniform footing, identifies the exact mechanisms that lead to the qualitative changes of physical signatures, and renders non-Hermitian response theory fully applicable to numerical descriptions of practical systems.