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Within the expanding field of non-Hermitian physics, non-Hermitian pumpinghas emerged as a key phenomenon, epitomized through the skin effect viaextensive boundary mode accumulation modifying the conventional Bloch picture.Beyond redefining bulk-boundary correspondences, we show that non-Hermitianpumping induces an unprecedented type of spectral topology: it admits aclassification in terms of graph topology, which is distinct from conventionaltopological classifications of the eigenstate or energy Riemann surface. Eachtopological class is characterized by a conformally invariant configuration ofspectral branching singularities, with gap-preserving transitions giving riseto emergent band geometry and Berry curvature discontinuities physicallymanifested as anomalous response kinks. By placing all Hermitian andnon-Hermitian lattice Hamiltonians on equal footing, our comprehensiveframework also enables the first analytic construction of topological phasediagrams in the presence of multiple non-reciprocal coupling scales, asprototypically demonstrated for the extended non-Hermitian Chern and Kitaevmodels. Based on general algebraic geometry properties of the energydispersion, our framework can be directly generalized to multiple bands,dimensions, and even interacting systems. Overall, it reveals the conspiracy ofband representations, spectral topology, and complex geometry as it unfolds indirectly measurable quantities.

Unraveling non-Hermitian pumping: emergent spectral singularities and anomalous responses