  Within the expanding field of non-Hermitian physics, non-Hermitian pumping
has emerged as a key phenomenon, epitomized through the skin effect via
extensive boundary mode accumulation modifying the conventional Bloch picture.
Beyond redefining bulk-boundary correspondences, we show that non-Hermitian
pumping induces an unprecedented type of spectral topology: it admits a
classification in terms of graph topology, which is distinct from conventional
topological classifications of the eigenstate or energy Riemann surface. Each
topological class is characterized by a conformally invariant configuration of
spectral branching singularities, with gap-preserving transitions giving rise
to emergent band geometry and Berry curvature discontinuities physically
manifested as anomalous response kinks. By placing all Hermitian and
non-Hermitian lattice Hamiltonians on equal footing, our comprehensive
framework also enables the first analytic construction of topological phase
diagrams in the presence of multiple non-reciprocal coupling scales, as
prototypically demonstrated for the extended non-Hermitian Chern and Kitaev
models. Based on general algebraic geometry properties of the energy
dispersion, our framework can be directly generalized to multiple bands,
dimensions, and even interacting systems. Overall, it reveals the conspiracy of
band representations, spectral topology, and complex geometry as it unfolds in
directly measurable quantities.
