REVIEW 2 major objections 4 minor 34 references
Edge burst effect and scale-free localization
T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Edge bursts occur without the non-Hermitian skin effect.
desk verdict The paper gives a clean numerical example of edge burst in a coupled lossy lattice, but the central claim that NHSE is not required rests on an uncomputed winding number for the coupled system. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is a one-dimensional tight-binding Hamiltonian on two coupled sublattices A and B with nearest-neighbor couplings equal to 1, purely imaginary next-to-nearest-neighbor couplings $\pm i\gamma$, and loss rate $V$ on odd-numbered sites, connected by a weak coupling $\Delta$. Uncoupled ($\Delta=0$), each sublattice shows bipolar non-Hermitian skin effect, diagnosed by opposite winding numbers $\pm 1$ of the periodic-boundary spectrum; coupling the sublattices destroys the skin effect and produces bipolar scale-free localization, meaning open-boundary eigenstates whose localization lengths scale with the system size and which appear at both edges of each sublattice. The argument is carried by the local decay probability $P^{A,B}_j = 2V_j \int_0^\infty |\psi^{A,B}_j|^2\,dt$, whose edge-to-minimum ratios define the edge burst, and by the funneling of an initially localized wave packet toward opposite edges of the two sublattices.
What would settle it
An exact solution or very large-$N$ numerical study of the coupled-lattice open-boundary eigenstates that shows their localization lengths saturating to a constant as $N \to \infty$ would falsify the scale-free-localization identification. Alternatively, computing $P^A_N/P^A_{\min}$ and $P^B_1/P^B_{\min}$ for $N$ up to thousands at fixed $\Delta$ and $\gamma$ and finding the ratios tend to 1 would show that the edge burst disappears in the thermodynamic limit, contradicting the claim that scale-free localization supports a persistent edge burst.
Extended reading notes
Core claim
The central claim is that the edge burst effect appears in a lossy two-sublattice lattice exhibiting bipolar scale-free localization, and therefore that the non-Hermitian skin effect is not a necessary condition for edge bursts. Within each sublattice, half the scale-free localized states sit at the left edge and half at the right, with a given state localized at opposite edges in the two sublattices. Starting from a single lossless site in each sublattice, the local decay probabilities $P^A_j$ and $P^B_j$ show an asymmetric distribution with a sharp peak at the right edge of sublattice A and the left edge of sublattice B, satisfying $P^A_N/P^A_{\min} \gg 1$ and $P^B_1/P^B_{\min} \gg 1$. The peak persists as the loss rate $V$ and coupling $\Delta$ vary, grows with $\gamma$ up to a maximum near $\gamma \approx 0.45$, vanishes when the imaginary next-nearest-neighbor coupling $\gamma$ is zero, and weakens as the lattice is made larger.
Load-bearing premise
The central claim rests on identifying the coupled-lattice eigenstates as scale-free localized from finite-size numerics—open-boundary spectra expanding toward periodic-boundary loops as the size grows and nearly size-independent $\langle j\rangle/N$—rather than from an exact analytical solution; if those signatures are actually a residual finite-size skin effect, the demonstration that edge bursts occur without skin localization would not be established.
Editorial extensions
If this is right
- The edge burst effect can serve as a dynamical signature of scale-free localization, not only of skin localization.
- In this model the edge burst is not tied to the non-Hermitian skin effect; the relevant ingredient is the directed funneling produced by the imaginary next-nearest-neighbor couplings.
- Increasing system size suppresses the edge burst because the initial site lies farther from the edge and the wave packet decays in the bulk before arriving.
- The effect is tunable: edge loss grows with the loss rate $V$ up to roughly $V \approx 5$ and then declines, and vanishes if the imaginary coupling $\gamma$ is switched off.
Reading between the lines
- If the identification of scale-free localization is correct, the same two-sublattice design should be realizable in photonic or circuit lattices, and the edge burst should be observable as a boundary loss imbalance without any non-reciprocal hopping.
- The paper's comparison with a single coupling impurity suggests a sharper test: coupling-impurity scale-free localization with uniform losses should show no edge burst, so the funneling term, not scale-free localization alone, is the active ingredient.
- A natural extension is to study the time-resolved loss profile rather than the integrated local decay probability; the model predicts the wave packet moves asymmetrically toward opposite edges, which could be measured directly.
- An implication the authors leave implicit is that earlier criteria tying edge bursts to imaginary gap closing together with the skin effect are sufficient but not necessary, and scale-free systems with directed funneling form a separate class where bursts occur.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies a one-dimensional tight-binding lattice with two coupled sublattices, each of which exhibits bipolar non-Hermitian skin effect (NHSE) when uncoupled. The authors claim that a weak inter-sublattice coupling (Δ≠0) destroys the bipolar NHSE and produces bipolar scale-free localization (SFL). They then numerically demonstrate the non-Hermitian edge burst effect in this coupled system and conclude that NHSE is not a necessary condition for the edge burst.
Significance. If the central claim is correct, the paper provides a new example of the edge burst effect in a system without skin localization, specifically in a system with bipolar scale-free localization. This would strengthen the existing argument that NHSE is not required for edge bursts. The paper includes parameter scans over V, Δ, and γ, and system-size dependence, which are valuable. However, the identification of SFL for Δ≠0 rests on finite-size numerical evidence only, and no analytical proof or topological invariant calculation is provided for the coupled system.
major comments (2)
- [Section II, Eq. (1), Figs. 1 and 2] The central assertion that coupling the sublattices (Δ≠0) destroys the bipolar NHSE and produces scale-free localization is supported only by finite-size numerical evidence. The OBC spectra in Fig. 1(b,c) expanding toward the PBC loops with increasing N, and the approximately constant <j>/N in Fig. 2(b), are consistent with SFL but also with a conventional skin effect whose localization length ξ is much larger than the simulated system sizes. For N ≪ ξ, the OBC spectrum is size-dependent and <j>/N appears roughly constant. The paper does not compute the spectral winding number for the coupled Hamiltonian (Eq. (1) with Δ≠0), nor does it provide the coupled PBC spectrum. If the coupled PBC spectrum contains a loop with nonzero winding around the OBC eigenvalues, then ordinary NHSE is present at a length scale larger than N, and the edge burst signals in Fig. 3 would be finite-size skin remnants rather than an SFL phenomenon. Please provide a direct test: either compute the winding number of the coupled Bloch Hamiltonian, or extract the localization length from eigenstates over a wide range of N and show that it grows linearly with N.
- [Section II, Fig. 1] The 'corresponding PBC loops' for the coupled system are never defined. The PBC spectrum is given analytically only for the uncoupled case (Δ=0); for Δ≠0, the text refers to PBC loops without providing their expression. Since the claim that the OBC spectrum approaches the PBC loops in the N→∞ limit is central to the SFL identification, the authors should provide the analytic PBC spectrum for the coupled system or otherwise specify the limiting curves.
minor comments (4)
- [Abstract] The abstract's first sentence states that the edge burst is presented 'in a lossy lattice with bipolar non-Hermitian skin effect', but the edge burst simulation in Section III uses the coupled system (Δ=0.05) which is claimed to exhibit scale-free localization, not NHSE. Please clarify the wording to avoid confusion.
- [Section I, first paragraph] 'SFL states has recently been experimentally realized' should be 'SFL states have recently been experimentally realized'.
- [Section II, paragraph after Eq. (1)] Please use 'size-dependent localization lengths' and consider rephrasing the sentence 'Note that the analogy between NHSE and bipolar NHSE can be extended to scale-free localization and bipolar scale-free localization' for clarity.
- [Section III, Fig. 3 caption and text] The sentence 'the slight difference is due to odd number of N' should read 'odd number of sites'. Also, the phrase 'slowly decrease with it' is awkward; consider rewording.
Circularity Check
No significant circularity: the edge-burst prediction follows from direct numerical solution of the model and an external edge-burst criterion; self-citations are contextual, not load-bearing.
full rationale
A step-by-step walk of the derivation chain shows no reduction of the claimed prediction to its inputs. The edge-burst signature is defined by Eq. (2) using the external condition of Ref. [28] (P_A_N/P_A_min >> 1 and P_B_1/P_B_min >> 1), and the reported P_A_N and P_B_1 distributions in Fig. 3 are obtained by numerically solving the evolution Eq. (1) for the stated initial condition; no parameter is fitted to the edge-burst output. The Delta = 0 bipolar-NHSE starting point is established in the text by the explicit PBC dispersion E_minus(k) and the winding-number integral, so the citation to the authors' Ref. [30] is corroborative rather than load-bearing. The transition to bipolar scale-free localization for Delta != 0 is supported by independent finite-size diagnostics (OBC spectra expanding toward PBC loops in Fig. 1(b,c) and roughly size-independent <j>/N in Fig. 2(b)); these are not equivalent by construction to the edge-burst curves. Own-prior-work citations [29,30] appear as background on previously studied edge-burst settings, not as the justification for the new model's behavior. The paper also explicitly concedes a limitation: not all systems exhibiting scale-free localization display the edge burst, giving a coupling-impurity system as a counterexample; this concession confirms that the central claim is not tautological. The skeptic's concern that the coupled-system winding number is not computed is a verification/correctness gap about whether the finite-size signature is really SFL rather than large-xi skin effect; it is not a circularity, because the paper's assertions would still be genuine, if possibly wrong, numerical predictions. Overall, the central claim has independent content.
Assumptions & free parameters
assumptions (3)
- domain assumption The dynamics is governed by the coupled tight-binding equations (1) with the stated open boundary conditions.
- domain assumption The edge burst effect is defined by the conditions P_A^N/P_A^min >> 1 and P_B^1/P_B^min >> 1, following Ref. [28].
- ad hoc to paper The finite-size numerical spectra and density profiles are sufficient to conclude scale-free localization for Δ≠0.
Cite this review
Pith. "Pith review of Edge burst effect and scale-free localization." pith.science (2026). https://pith.science/paper/4PTR7L3H
@misc{pith2026250608559,
author = {Pith},
title = {Pith review of: Edge burst effect and scale-free localization},
year = {2026},
howpublished = {\url{https://pith.science/paper/4PTR7L3H}},
note = {Machine review of arXiv:2506.08559}
}
read the original abstract
The non-Hermitian edge burst is a phenomenon observed in non-Hermitian quantum dynamics, characterized by a significant accumulation of loss at the boundaries of a system. We present an example of the edge burst effect in a lossy lattice with bipolar non-Hermitian skin effect (NHSE). By introducing a weak coupling between two such non-Hermitian lattices, we demonstrate that the system exhibits bipolar scale-free localization. Through an analysis of local decay distributions and their sensitivity to system parameters, we confirm the occurrence of the edge burst effect in the system displaying scale-free localization. Our findings support the notion that the skin effect is not a necessary condition for the edge burst effect.
Figures
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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