The coalescence of eigenstates is a unique phenomenon in non-Hermitian systems. Remarkably, it has been noticed in some non-Hermitian systems under open boundary conditions that the whole set of eigenstates can coalesce to only a few eigenstates. In the parameter space, the point at which such a coalescence of macroscopic eigenstates occurs is dubbed as an infernal point. In this Letter, based on the non-Bloch band theory and amoeba formulation, we establish the criteria for the presence of infernal points in one-dimensional and higher-dimensional open boundary non-Hermitian systems. In addition, we find an explanation for the extreme localization of the wave functions and unveil the mechanism for the coalescence of enormous eigenstates at the infernal points. Our work provides a general theory for infernal points in open boundary non-Hermitian systems in arbitrary dimensions, and hence paves the way to study the intriguing infernal points systematically.

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• Edge states
• Geometric & topological phases
• Exceptional points
• Non-Hermitian systems