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REVIEW 2 major objections 5 minor 97 references

Parameter windows that host quasiperiodic critical states under periodic boundaries can still show the non-Hermitian skin effect once open boundaries are imposed.

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2026-07-10 09:52 UTC pith:4ORC5BCY

load-bearing objection Clean analytic LE for unequal quasiperiodic hoppings, solid numerics linking PBC-critical windows to OBC skin; the Hermitian LE=0 step is the only real soft spot and is not fatal. the 2 major comments →

arxiv 2607.08294 v1 pith:4ORC5BCY submitted 2026-07-09 cond-mat.mes-hall quant-ph

Interplay of Quasiperiodic Criticality and the Non-Hermitian Skin Effect

Zhangyuan Chen , Xianqi Tong , Xiaosen Yang This is my paper
classification cond-mat.mes-hall quant-ph
keywords non-Hermitian skin effectquasiperiodic criticalityHatano–Nelson modelLyapunov exponentnon-unitary gauge transformationinverse participation ratioRice–Mele lattice
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies a Hatano–Nelson chain whose right and left hoppings are both quasiperiodically modulated, but with unequal modulation strengths, so the hoppings are nonreciprocal. A non-unitary gauge transformation converts the open-boundary problem into an ordinary Hermitian quasiperiodic chain; the Lyapunov exponent of the original wave functions then collapses to a simple integral over the log-ratio of the two hoppings, which can be evaluated exactly. The zero of that Lyapunov exponent fixes the left-skin versus right-skin phase boundary under open boundaries. Under periodic boundaries the same parameter windows are shown, by inverse-participation-ratio scaling, to support critical (multifractal) rather than extended or localized eigenstates. The comparison therefore establishes that critical quasiperiodic states are not immune to the non-Hermitian skin effect: when the boundaries are opened they pile up at one edge. The same pattern survives when longer-range hoppings or a two-band Rice–Mele structure are added.

Core claim

In an off-diagonal Hatano–Nelson model with unequal quasiperiodic modulation strengths, the parameter regimes that host critical eigenstates under periodic boundary conditions exhibit a non-Hermitian skin effect under open boundary conditions; the left/right skin boundary is given exactly by the vanishing of the thermodynamic-limit Lyapunov exponent obtained after a non-unitary gauge map.

What carries the argument

Non-unitary gauge transformation ψ_n = ϕ_n G_n that removes the nonreciprocity, after which the Lyapunov exponent reduces to the spatial average of (1/2) ln|J^R/J^L| and can be evaluated in closed form (Eqs. 8–10).

Load-bearing premise

After the gauge map, the resulting Hermitian quasiperiodic chain is assumed not to be exponentially localized, so its own Lyapunov exponent vanishes and does not shift the analytic skin boundary.

What would settle it

Compute the Lyapunov exponent of the gauged Hermitian quasiperiodic chain inside the claimed critical windows; if it remains finite and nonzero in the thermodynamic limit, the analytic left/right skin boundary of Eqs. 9–10 is incorrect.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The skin direction of critical quasiperiodic states is completely fixed by a single integral of the log hopping ratio and needs no further diagonalization.
  • Long-range and multiband generalizations inherit the same skin-versus-critical correspondence once the same gauge map is applied.
  • Boundary condition (open versus periodic) can convert a critical spectrum into a skin-localized spectrum without changing any bulk parameters.
  • Experimental platforms that already realize quasiperiodic hopping can test the predicted skin reversal simply by tuning the relative modulation strengths.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same gauge-plus-Lyapunov construction should apply to any off-diagonal nonreciprocal model whose hoppings factor into a common quasiperiodic envelope times left/right amplitudes.
  • If the Hermitian partner develops a mobility edge, the skin boundary will itself become energy-dependent, producing a hybrid skin-critical spectrum.
  • Realizing the model in a photonic or cold-atom lattice would allow direct imaging of the predicted left-to-right skin reversal across the analytic line μ_R = t + √(t²−1).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript studies a quasiperiodically modulated off-diagonal Hatano–Nelson chain in which nonreciprocity arises from unequal modulation strengths of the right and left hoppings. Under open boundary conditions a non-unitary gauge transformation maps the model onto a Hermitian quasiperiodic chain; the Lyapunov exponent is then reduced to an explicit integral over the logarithmic hopping imbalance (Eqs. 6–9), yielding an analytic left/right skin boundary (Eq. 10). Numerical dMIPR diagrams under OBC agree with this boundary. Under periodic boundary conditions, MIPR and finite-size fractal-dimension analysis identify extended and critical regimes; the comparison of the two boundary conditions shows that parameter windows hosting quasiperiodic critical states under PBC can exhibit the non-Hermitian skin effect under OBC. The same qualitative interplay is demonstrated for a long-range hopping extension and a two-band non-Hermitian Rice–Mele model.

Significance. The work supplies a concrete, analytically controlled link between quasiperiodic criticality and the non-Hermitian skin effect. The closed-form thermodynamic-limit Lyapunov exponent (Eqs. 9–10) is a genuine strength: it is parameter-free once the Hermitian contribution is discarded, and it matches the numerical OBC phase diagrams to high accuracy. The observation that PBC-critical regimes can skin under OBC is of clear interest to the non-Hermitian and quasiperiodic communities and is shown to survive long-range and multiband generalizations. If the vanishing of the Hermitian Lyapunov exponent can be placed on firmer footing, the paper would constitute a clean, citable reference for boundary-sensitive critical localization in modulated non-Hermitian systems.

major comments (2)
  1. Section III, Eqs. (6)–(9): the analytic skin boundary rests on setting the first term of Eq. (6) (the Lyapunov exponent of the gauge-transformed Hermitian quasiperiodic chain) identically to zero, on the grounds that “the corresponding eigenstates are not exponentially localized.” This is asserted rather than proved for the critical windows later identified under PBC (0 < β < 1). If those Hermitian states are exponentially localized for some energies or parameters, λ acquires an extra nonzero contribution and the claimed exact phase boundary (Eq. 10) shifts. The excellent numerical agreement of dMIPR with the λ = 0 curve (Fig. 1) is consistent but does not replace a bound or a spectral argument that the Hermitian LE vanishes throughout the critical regime. A short analytic or numerical demonstration that the Hermitian LE remains zero (or is o(1) in the thermodynamic limit) inside the win
  2. Section III and Fig. 2: the identification of “quasiperiodic critical regimes” under PBC relies on finite-size MIPR scaling for two representative points and a single fractal dimension β ≈ 0.529. While standard, this is insufficient to delineate the white transition line drawn in Fig. 2(a) across the whole (t, μ_R) plane. Additional scaling data (or an independent diagnostic such as the multifractal spectrum or level-spacing statistics) at several points along that line would strengthen the claim that entire parameter regions, rather than isolated points, host critical states that subsequently skin under OBC.
minor comments (5)
  1. Eq. (2) and surrounding text: the global phase θ is stated not to affect localization, yet all finite-size averages are performed over 10^3 realizations of θ. A brief remark clarifying that the thermodynamic-limit λ is θ-independent while finite-N IPR fluctuates would avoid confusion.
  2. Fig. 1(a) and Eq. (10): the four branches of the analytic boundary are written for μ_L = 1; it would help the reader if the general expression (Eq. 9) were also plotted or tabulated for a second value of μ_L to illustrate robustness.
  3. Section IV, Eq. (14): the product definition of long-range amplitudes J_{n,p}^{R/L} is natural but not unique; a short sentence explaining why this particular factorization preserves the gauge map would improve clarity.
  4. Typographical: “HA T ANO-NELSON” and “ST A TES” in section headings contain spurious spaces; “GENERALIZA TION” and “MUL TIBAND” likewise. Standardize to “Hatano–Nelson”, “States”, etc.
  5. References: several recent works on non-Hermitian quasiperiodic criticality and skin effect (e.g., the mobility-edge and multifractal literature) are cited, but a brief comparison with the Hermitian off-diagonal Aubry–André critical line would situate the present phase boundary more clearly.

Circularity Check

0 steps flagged

No circularity: the thermodynamic-limit Lyapunov exponent is obtained by an exact ergodic integral after a standard non-unitary gauge map; PBC MIPR/scaling diagnostics are independent numerical probes, not fitted inputs that force the OBC skin boundary.

full rationale

The central analytic object is the Lyapunov exponent after the non-unitary gauge transformation ψ_n = φ_n G_n. The paper writes λ = lim (1/n) ln|φ_n/φ_1| + lim (1/n) ln|G_n| and discards the first term because the equivalent Hermitian quasiperiodic chain is asserted not to be exponentially localized in the regimes of interest; the second term then reduces by the ergodic theorem to the explicit integral (Eq. 8) whose closed form is Eq. (9). Setting λ = 0 yields the phase boundary (Eq. 10). This chain is self-contained: no free parameter is fitted to data, no uniqueness theorem is imported from the authors’ prior work, and the integral does not reduce by construction to any quantity later “predicted.” The PBC analysis (MIPR and finite-size fractal dimension β) is performed separately and used only for comparison; the numerical dMIPR under OBC is an independent diagnostic that happens to track the analytic curve. The long-range and Rice–Mele extensions reuse the same gauge map without introducing new fitted quantities. The soft spot is the unproved vanishing of the Hermitian LE inside the critical windows (already flagged by the reader as a correctness assumption, not a circular reduction). Because no step equates a claimed prediction to its own input by definition or by self-citation, the circularity score is zero.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The paper rests on standard non-Hermitian and quasiperiodic machinery plus one modeling choice (vanishing Hermitian LE) and a handful of conventional numerical parameters. No new particles or forces are postulated; free parameters are ordinary model knobs, not fits to external data.

free parameters (3)
  • modulation frequency α
    Fixed to the inverse golden mean (and its Fibonacci approximants); conventional but still a modeling choice that selects the quasiperiodic class.
  • system sizes N (Fibonacci numbers)
    N=233, 89 etc. chosen for numerical convenience; finite-size scaling is performed but the precise sequence is conventional.
  • global phase θ averaging
    10^3 random realizations used for MIPR averages; the number is arbitrary though standard.
axioms (3)
  • domain assumption After the non-unitary gauge transformation the Lyapunov exponent of the equivalent Hermitian quasiperiodic chain vanishes in the thermodynamic limit for the regimes considered.
    Stated after Eq. 6; required to reduce λ to the pure gauge contribution.
  • domain assumption The irrational frequency α can be replaced by rational Fibonacci approximants without changing the localization classification in the large-N limit.
    Standard quasiperiodic practice, used throughout the numerics.
  • standard math Inverse-participation-ratio scaling with fractal dimension 0<β<1 diagnoses critical (multifractal) states.
    Invoked in Sec. III and Fig. 2; conventional diagnostic in the localization literature.

pith-pipeline@v1.1.0-grok45 · 17574 in / 2428 out tokens · 34271 ms · 2026-07-10T09:52:32.057455+00:00 · methodology

0 comments
read the original abstract

Quasiperiodic lattices can host critical eigenstates, whereas nonreciprocal hopping in non-Hermitian lattices can induce non-Hermitian skin effect. In this work, we investigate localization phenomena in a Hatano--Nelson model with quasiperiodically modulated hopping amplitudes, where nonreciprocity arises from unequal modulation strengths of the right and left hoppings. Using a non-unitary gauge transformation, we map the non-Hermitian system into a Hermitian quasiperiodic system and obtain an exact analytical expression for the Lyapunov exponent in the thermodynamic limit. Under periodic boundary conditions, inverse participation ratios and finite-size scaling analysis are used to identify the quasiperiodic critical regimes. The comparison shows that parameter regimes hosting quasiperiodic critical states under periodic boundary conditions can exhibit the non-Hermitian skin effect under open boundary conditions. Furthermore, the non-Hermitian skin effect associated with quasiperiodic critical regimes is also observed in representative long-range hopping models and multiband extensions. Our results provide an analytically controlled perspective on how quasiperiodicity, modulated nonreciprocity, and boundary conditions jointly shape the non-Hermitian skin effect in critical regimes.

Figures

Figures reproduced from arXiv: 2607.08294 by Xianqi Tong, Xiaosen Yang, Zhangyuan Chen.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The dMIPR under OBC as a function of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The MIPR under PBC as a function of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Schematic illustration of the HN model with long-range hoppings. (b) The dMIPR under OBC as a function of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Schematic of the Rice-Mele model with nonrecip [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

discussion (0)

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Reference graph

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