Non-Hermitian Edge State Endocytosis

An isolated edge state observed in a finite open chain is usually expected to survive the thermodynamic limit (TDL), with a localization mechanism distinct from non-Hermitian skin accumulation, which localizes the \emph{entire} bulk continuum. We show that scale-sensitive non-Hermitian systems can generically admit a different fate: as we scale up the system size, a detached edge-localized eigenstate can remain sharply visible over a broad window until a critical scale is reached, where it forms an ephemeral bound state in the continuum (BIC) of the open-boundary bulk before being absorbed (entocytosed) at even larger system sizes. We call this phenomenon edge state endocytosis. Its mechanism is fundamentally traced to the Widom expansion of the open-chain characteristic determinant (energy dispersion equation) into contributions corresponding to admissible non-Bloch mode subsets. Each subset contribution factorizes into a boundary-projected Green's function (proj-GF) determinant, which encodes lattice truncation, and a subset-resolved bulk propagation factor, which encodes the system size dependence. We uncover the fundamental distinction: TDL edge states are zeros of the leading-subset proj-GF determinant, whereas endocytosed states are a hitherto-ignored class of hidden proj-GF zeros from subleading subsets that control the spectrum at finite sizes. Due to its fundamental mathematical origin, the endocytosis mechanism is completely platform-independent, occurring generically without fine-tuning when isolated edge states, topological or otherwise, are subject to non-Hermitian couplings that generate the requisite non-locality. Our new framework quantitatively predicts the endocytosis scale and sheds light on how its intricate competitive mechanism can be revealed through experimentally relevant Green's functions.