Non-Hermitian Edge State Endocytosis

Abstract: 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.

• The paper shows that non-Hermitian edge states persist until a critical length Lc, where they are abruptly absorbed into the continuum as a finite-size BIC.
• It employs a Widom-type expansion of boundary-projected Green's functions to differentiate competing bulk and edge contributions in finite lattices.
• The study highlights experimental diagnostics, including flux threading and biorthogonal inverse participation ratios, to distinguish pre- and post-endocytosis regimes.

Non-Hermitian Edge State Endocytosis: Scale-Dependent Absorption of Boundary Modes

Introduction

The paper "Non-Hermitian Edge State Endocytosis" (2607.07703) establishes a new universal finite-size phenomenon in non-Hermitian (NH) lattices—termed "edge state endocytosis"—that fundamentally challenges the conventional view of isolated edge modes as finite-size precursors of thermodynamic-limit (TDL) edge states. Instead, the authors demonstrate that under NH couplings, sharply localized edge eigenstates can persist up to a critical length whereupon they are abruptly absorbed into the open-boundary bulk spectrum, despite being outside of any non-Hermitian skin effect (NHSE) regime. This transition proceeds via a novel scaling-induced, ephemeral bound state in the continuum (BIC), indicating a profound competition between boundary projection and bulk propagation mechanisms unique to non-Hermitian block Toeplitz matrices.

Formal Structure and Mathematical Framework

The authors leverage the Widom-type expansion of open-chain characteristic determinants for 1D non-Hermitian systems, decomposing the finite-size determinant $D_L(E) = \det(E-H_L)$ into contributions from admissible "subsets" $S$ of non-Bloch modes. Each subset contributes two factors: a "boundary-projected Green's function" (proj-GF) determinant, sensitive to lattice truncation and boundary matching, and a bulk propagation factor, encoding the system size scaling. Notably, the subset with the leading bulk factor asymptotically controls the TDL spectrum and is responsible for standard NHSE and edge state formation.

However, the critical insight is that in systems supporting both robust localized edge states and sufficiently non-local NH couplings, subleading subsets—ignored in previous TDL analyses—can possess hidden zeros of their boundary-projected Green's function determinants. These zeros generate strongly localized, spectrally isolated states for finite $L$. As $L$ grows, the suppression of the corresponding subset bulk factor becomes dominant; the state is absorbed (endocytosed) into the open-boundary continuum once the leading term overtakes the subleading zero, as quantified by subset competition (Eq. 6 and Eq. 7).

Phenomenology of Edge State Endocytosis

The endocytosis process is fundamentally distinct from both the NHSE—which delocalizes the entire bulk—or disorder-induced broadening, with its mathematical and physical origin in the competitive hierarchy between leading and subleading subsets in finite-size determinant expansions. A remarkable numerical finding is that for a wide class of block Toeplitz NH lattices with weakly coupled "auxiliary" bands, the isolated edge eigenmode persists up to a sharply defined $L_c$, after which the state merges with the continuum via a finite-size scaling-induced BIC. The authors formalize this with a crossover formula (Eq. 7) that predicts $L_c$ based on the boundary and bulk factors, verified via explicit lattice simulations.

Empirically, the transition is tracked by the biorthogonal inverse participation ratio (bi-IPR) and analytical Green's function winding numbers. The distinguishing feature is the endpoint: the hidden proj-GF zero is rendered spectrally invisible for $L > L_c$, even though it remains nontrivial in the mathematical structure of the boundary Green's function.

Diagnostics: Green’s Function Windings and Flux Response

A crucial theoretical tool introduced is the subset-resolved Green's function winding number, which distinguishes ordinary TDL edge modes, endocytosed states, and trivial bulk states locally in the complex energy plane. The authors prove that only leading-subset proj-GF zeros yield persistent TDL edge eigenmodes, whereas hidden zeros of subleading subsets give rise to finite-$L$-only, pre-asymptotic physics, enabling clear diagnostic criteria within a boundary Green’s function framework.

Furthermore, adiabatic weak-coupling flux threading experiments (Eq. 9) are proposed as a direct means to distinguish between pre-endocytosis and post-endocytosis regimes. The flux spectral flow of the isolated eigenmode differs qualitatively from the OBC continuum, offering a sharp experimental signature for the absorption process.

Generality, Constructive Routes, and Physical Implications

The endocytosis mechanism is mathematically robust and generic: it applies irrespective of the topological character of the original edge state and does not require fine-tuning. The construction can be systematically achieved by embedding a "parent" edge mode and weakly coupling to an auxiliary band with nonreciprocal or otherwise NH hopping. The necessary condition is the presence of emergent non-locality, manifest in the finite-size determinant expansion, making the effect platform-independent. As a result, the phenomenon is expected to be observable in photonic, topolectrical, active mechanical, and acoustic systems with NH couplings, for which edge states have been probed using both transport and local spectroscopy (2607.07703).

On a theoretical level, this result has significant implications for the interpretation of bulk-boundary correspondence in non-Hermitian systems. While TDL approaches associate edge states exclusively with zeros of leading subset proj-GF determinants (i.e., boundary-matched non-Bloch modes), experimentally, finite-size systems can display robust boundary eigenvalues at scales and energies completely missed by the TDL. Thus, the work signals that subset competition and hidden zeros must be taken seriously in spectral analyses of realistic, finite NH platforms, especially in the presence of auxiliary mode mixing and weak nonlocal couplings.

Conclusion

This work identifies and analyzes a fundamentally new class of NH scaling phenomena: edge state endocytosis. By dissecting the scale-dependent interplay between the boundary Green’s function and bulk propagation factors in block Toeplitz determinant expansions, the authors demonstrate that localized edge eigenstates may generically be absorbed into the continuum with increasing system size, distinct from NHSE-driven behavior. The mechanism predicts salient, quantifiable, and broadly observable features—including scale-tunable localization, finite-size BICs, Green’s function windings, and nontrivial flux responses—that provide robust experimental smoking guns for non-TDL NH edge physics. The framework advanced here is expected to catalyze new finite-size diagnostics, theoretical developments, and applied investigations of non-Hermitian matter.

Key Result: The scale $L_c$ at which endocytosis occurs is accurately predicted by the competition of bulk and boundary factors in the subset expansion and is independent of the topological or nontopological nature of the parent edge mode. This establishes finite-size spectral separation as a robust, universal phenomenon intrinsic to non-Hermitian lattice systems with emergent non-locality.