pith. sign in

arxiv: 2603.20330 · v3 · pith:7MCNZCQRnew · submitted 2026-03-20 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· cond-mat.stat-mech· quant-ph

Competing skin effect and quasiperiodic localization in the non-Hermitian Su-Schrieffer-Heeger chain: Reentrant delocalization, spectral topology destruction, and entanglement suppression

Souvik Ghosh This is my paper

Pith reviewed 2026-05-21 10:07 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-scicond-mat.stat-mechquant-ph
keywords non-Hermitian skin effectAubry-André-Harper disorderSu-Schrieffer-Heeger chainreentrant delocalizationpoint-gap topologyentanglement entropyquasiperiodic localization
0
0 comments X

The pith

In non-Hermitian SSH chains, intermediate quasiperiodic disorder first disrupts skin-effect accumulation then restores localization, producing a reentrant delocalization window.

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

The paper maps the full phase diagram of a non-Hermitian Su-Schrieffer-Heeger chain that combines directional skin accumulation with Aubry-André-Harper quasiperiodic modulation. It identifies five regimes, one of which is a previously unreported competition regime in which moderate disorder strengths undo the directional piling of states before stronger disorder localizes everything. An analytical similarity transformation yields a modified localization boundary that depends on an effective intra-cell hopping reduced by the dimerization parameter. The same disorder also unwinds complex spectral loops, eliminating point-gap topology at a strength different from the band-topological transition, and it modulates entanglement entropy by first suppressing it through the skin effect and then partially restoring it. The dimerized lattice structure is shown to be essential for generating this five-phase landscape.

Core claim

The central claim is that the interplay between non-Hermitian skin effect and AAH quasiperiodic disorder produces a distinct competition regime (V) with reentrant partial delocalization: intermediate modulation strengths disrupt directional skin accumulation before ultimately Anderson-localizing all states. This regime is characterized by a non-monotonic inverse participation ratio that sharpens with system size. The modified localization boundary is given by λ_c(δ) = 2√(v_eff w) with v_eff = √(v² - δ²), derived via similarity transformation and confirmed by Lyapunov exponent calculations. Quasiperiodic disorder additionally destroys point-gap topology and alters entanglement entropy in a SK

What carries the argument

The modified localization boundary λ_c(δ)=2√(v_eff w) with v_eff=√(v²-δ²), obtained from a similarity transformation that absorbs the nonreciprocal and dimerized hoppings into an effective Hermitian problem.

If this is right

  • The reentrant regime persists under finite-size scaling and phase averaging, confirming its robustness in the thermodynamic limit.
  • Quasiperiodic disorder unwinds complex spectral loops and destroys point-gap topology at a critical strength distinct from the band-topological transition.
  • The skin effect drives entanglement entropy to near zero, while sufficiently strong AAH disorder partially restores finite entanglement.
  • The SSH sublattice structure is required to produce the five-regime landscape; removing dimerization collapses the diagram to the simpler non-Hermitian AAH case.

Where Pith is reading between the lines

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

  • Similar reentrant competition windows could appear in higher-dimensional or interacting non-Hermitian quasiperiodic models once an analogous similarity mapping is constructed.
  • Photonic or mechanical metamaterial platforms with tunable nonreciprocity and quasiperiodic potentials offer a direct route to observe the predicted non-monotonic participation ratio.
  • The destruction of point-gap topology at a separate critical disorder strength suggests that transport signatures of skin localization may vanish before full Anderson localization occurs.

Load-bearing premise

The similarity transformation that produces the analytical localization boundary remains accurate for all modulation strengths and does not miss higher-order corrections from the interplay of dimerization and nonreciprocity.

What would settle it

A direct numerical computation of the Lyapunov exponent versus AAH strength that deviates systematically from the predicted curve λ_c(δ)=2√(v_eff w) at moderate δ would falsify the boundary.

Figures

Figures reproduced from arXiv: 2603.20330 by Souvik Ghosh.

Figure 1
Figure 1. Figure 1: FIG. 1. Energy spectrum Re( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase diagram at [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Superimposed probability density (blue) and near [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Line-cut diagnostics at [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the analytical prediction (5) alongside numerically extracted D2 = 0.5 crossings from phase￾averaged data. The analytical curve λc(δ) = 2√ veffw FIG. 6. Entanglement entropy S(ℓ) for four regimes. (c) Near-total suppression by the skin effect; (d) partial re￾covery upon adding AAH disorder. v/w = 0.5, N = 50. FIG. 7. Localization boundary in the (λ/w, δ/w) plane. Solid blue: analytical prediction λc … view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Phase-averaged diagnostics ( [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of SSH ( [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: presents targeted finite-size analysis at five parameter points representative of the identified phases, with all quantities averaged over ϕ. Panel (a) shows the mean IPR versus system size L = 2N on a log-log scale. The delocalized Hermitian case (blue circles) follows IPR ∝ L −1 , confirming extended￾state behavior. All localized and skin-dominated cases show size-independent IPR, confirming true locali… view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Schematic phase diagram at [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
read the original abstract

We investigate the interplay between the non-Hermitian skin effect and Aubry-Andr\'e-Harper (AAH) quasiperiodic disorder in a one-dimensional Su-Schrieffer-Heeger (SSH) chain with nonreciprocal hopping. By exact diagonalization, transfer-matrix analysis, and an analytical similarity-transformation argument, we map the full ( , $\delta$) phase diagram, where A is the AAH modulation strength and the nonreciprocity parameter. We identify five distinct regimes: ( ) topological with extended bulk, (II) AAH-localized, (III) skin-localized, (IV) fully localized, and a previously unreported (V) competition regime exhibiting reentrant partial delocalization, in which intermediate quasiperiodic disorder disrupts the directional skin accumulation before ultimately Anderson-localizing all states. Using phase-averaged diagnostics and finite-size scaling, we confirm that the reentrant regime is robust, characterized by a non-monotonic inverse participation ratio that sharpens with increasing system size. We derive an analytical expression for the modified localization boundary $\lambda_{c}(\delta)=2\sqrt{v_{eff}w}$ with $v_{vff}=\sqrt{v^{2}-\delta^{2}}$, which agrees with numerical Lyapunov exponent calculations. We further show that quasiperiodic disorder progressively unwinds the complex spectral loops, destroying the point-gap topology at a critical strength distinct from the band-topological transition ; that the skin effect suppresses entanglement entropy to near-zero values while sufficiently strong AAH disorder partially restores it ; and that the SSH sublattice structure absent in the widely studied non-Hermitian AAH chain is essential for producing the five-phase landscape, as demonstrated by direct comparison with the non-dimerized limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript investigates the interplay between the non-Hermitian skin effect and Aubry-André-Harper quasiperiodic disorder in a one-dimensional nonreciprocal Su-Schrieffer-Heeger chain. Using exact diagonalization, transfer-matrix analysis, and a similarity-transformation argument, the authors map the (A, δ) phase diagram and identify five regimes, including a novel competition regime (V) with reentrant partial delocalization. They derive an analytical localization boundary λ_c(δ)=2√(v_eff w) with v_eff=√(v²-δ²) that agrees with Lyapunov-exponent numerics, and report progressive unwinding of complex spectral loops together with suppression and partial restoration of entanglement entropy. The SSH dimerization is shown to be essential by comparison with the non-dimerized limit.

Significance. If the central claims hold, the work would be significant for non-Hermitian topology and localization physics. The identification of a reentrant partial-delocalization window arising from competition between skin accumulation and quasiperiodic disorder, together with an analytical boundary and the demonstration that dimerization is required for the five-regime structure, would provide concrete, testable predictions for photonic or cold-atom realizations and would clarify the role of sublattice structure absent in simpler non-Hermitian AAH models.

major comments (2)
  1. [Analytical derivation of localization boundary] Analytical similarity-transformation argument (paragraph deriving λ_c(δ)): the mapping that produces v_eff=√(v²-δ²) and the boundary λ_c(δ)=2√(v_eff w) assumes the transformation absorbs the AAH potential without generating additional δ-dependent corrections. It is not shown that this remains exact for arbitrary modulation strength A; a systematic bias could shift or eliminate the reentrant window in regime (V).
  2. [Phase diagram and regime (V)] Definition and diagnostics of regime (V) (section presenting the five-regime phase diagram): the reentrant partial delocalization is diagnosed via non-monotonic inverse-participation-ratio behavior and finite-size scaling. The boundaries separating regimes (IV) and (V) appear to be determined post hoc from the same Lyapunov and IPR data used to validate the analytical expression, raising the possibility of circularity in the reported agreement.
minor comments (3)
  1. [Abstract] Abstract: the symbol is written as v_{vff} instead of v_eff.
  2. [Model Hamiltonian] Notation: the nonreciprocity parameter is introduced as δ but later appears interchangeably with other symbols; a single consistent symbol should be used throughout.
  3. [Numerical diagnostics] Figures showing IPR scaling: the sharpening with system size should be quantified with explicit scaling exponents or collapse plots rather than qualitative statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below. We have revised the manuscript to improve the presentation of the analytical derivation and the definition of the regimes, while maintaining that the central results remain robust.

read point-by-point responses

  1. Referee: Analytical similarity-transformation argument (paragraph deriving λ_c(δ)): the mapping that produces v_eff=√(v²-δ²) and the boundary λ_c(δ)=2√(v_eff w) assumes the transformation absorbs the AAH potential without generating additional δ-dependent corrections. It is not shown that this remains exact for arbitrary modulation strength A; a systematic bias could shift or eliminate the reentrant window in regime (V).

    Authors: The similarity transformation rescales the wave-function amplitudes on the two sublattices to absorb the non-reciprocal parameter δ, yielding the effective intra-cell hopping v_eff = √(v² - δ²). Because the AAH potential is strictly on-site and diagonal in the original basis, the transformation does not generate additional δ-dependent corrections to the leading Lyapunov exponent; the quasiperiodic term remains unchanged in form after the rescaling. This is confirmed by the quantitative agreement between the analytical boundary and the numerically computed Lyapunov exponents over the full range of A examined. To address the concern explicitly, we have added a supplementary derivation showing that higher-order corrections in A do not modify the localization criterion at the level relevant for the reentrant window. revision: yes

  2. Referee: Definition and diagnostics of regime (V) (section presenting the five-regime phase diagram): the reentrant partial delocalization is diagnosed via non-monotonic inverse-participation-ratio behavior and finite-size scaling. The boundaries separating regimes (IV) and (V) appear to be determined post hoc from the same Lyapunov and IPR data used to validate the analytical expression, raising the possibility of circularity in the reported agreement.

    Authors: The five regimes are identified from qualitative changes in the A-dependence of the IPR and Lyapunov exponent. Regime (V) is defined by the distinctive non-monotonic IPR behavior (initial decrease followed by increase), which is observed directly in the data and is independent of the analytical formula. The expression λ_c(δ) = 2√(v_eff w) is obtained from the similarity transformation without reference to the numerical boundaries and is subsequently compared to the Lyapunov data for validation. In the revised manuscript we have restructured the relevant section to state the diagnostic criteria for each regime before presenting any numerical results, thereby separating the analytical derivation from the numerical confirmation. revision: yes

Circularity Check

0 steps flagged

Analytical similarity transformation for localization boundary is independent and numerically verified

full rationale

The paper derives λ_c(δ)=2√(v_eff w) with v_eff=√(v²-δ²) via an explicit similarity-transformation argument that maps the non-Hermitian SSH Hamiltonian onto an effective problem; this is a standard mathematical reduction, not a self-definition or renaming of inputs. The resulting boundary is cross-checked against independent Lyapunov-exponent numerics and finite-size scaling of the inverse participation ratio, providing external falsifiability. No load-bearing self-citation, fitted-parameter-as-prediction, or ansatz-smuggling steps appear in the derivation chain. The five-regime phase diagram and reentrant delocalization claims rest on direct diagonalization and transfer-matrix methods rather than reducing to the effective-parameter construction alone.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; model parameters v, w, δ and AAH modulation A are introduced as standard but no explicit free parameters or invented entities are detailed beyond the effective v_eff constructed in the localization formula.

axioms (1)
  • domain assumption The system is described by a one-dimensional non-Hermitian SSH Hamiltonian with nonreciprocal hopping and superimposed AAH quasiperiodic modulation.
    This is the starting model whose phase diagram is mapped.

pith-pipeline@v0.9.0 · 5874 in / 1488 out tokens · 51829 ms · 2026-05-21T10:07:50.525215+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    The (λ/w, δ/w) phase diagram containsfivedis- tinct regimes, including a previously unreported competition regime(region V) exhibiting reentrant partial delocalization, confirmed by phase-averaged diagnostics and finite-size scaling

    work page

  2. [2]

    We derive an analytical expressionλ c(δ) = 2√veffw for the modified localization boundary via an imag- inary gauge transformation, wherev eff = √ v2 −δ 2 is the effective Hermitian hopping

    work page

  3. [3]

    The point-gap topological transition and the band- topological transition occur atdistinctparameter values, demonstrating decoupled topological tran- sitions

    work page

  4. [4]

    The NHSE suppresses entanglement entropy to ex- ponentially small values; sufficiently strong AAH disorder partiallyrestoresentanglement by disrupt- ing the directional skin accumulation

    work page

  5. [5]

    The paper is organized as follows

    Direct comparison with the non-dimerized limit (v=w) confirms that the SSH sublattice structure is essential for the five-phase landscape. The paper is organized as follows. Section II de- fines the model Hamiltonian. Section III describes the computational methods and the analytical similarity- transformation argument. Section IV presents the nu- merical...

    work page

  6. [6]

    W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett.42, 1698 (1979)

    work page 1979

  7. [7]

    Aubry and G

    S. Aubry and G. Andr´ e, Ann. Israel Phys. Soc.3, 133 (1980)

    work page 1980

  8. [8]

    P. G. Harper, Proc. Phys. Soc. Sect. A68, 874 (1955)

    work page 1955

  9. [9]

    Yao and Z

    S. Yao and Z. Wang, Phys. Rev. Lett.121, 086803 (2018)

    work page 2018

  10. [10]

    F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Phys. Rev. Lett.121, 026808 (2018)

    work page 2018

  11. [11]

    Okuma, K

    N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Phys. Rev. Lett.124, 086801 (2020)

    work page 2020

  12. [12]

    C. H. Lee and R. Thomale, Phys. Rev. B99, 201103(R) (2019); C. H. Lee, L. Li, R. Thomale, and R.-J. Slager, Phys. Rev. Lett.124, 056802 (2020)

    work page 2019

  13. [13]

    Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi- gashikawa, and M. Ueda, Phys. Rev. X8, 031079 (2018)

    work page 2018

  14. [14]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys. Rev. X9, 041015 (2019)

    work page 2019

  15. [15]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda, Adv. Phys.69, 249 (2020)

    work page 2020

  16. [16]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys.93, 015005 (2021)

    work page 2021

  17. [17]

    Jiang, L.-J

    H. Jiang, L.-J. Lang, C. Yang, S.-L. Zhu, and S. Chen, Phys. Rev. B100, 054301 (2019)

    work page 2019

  18. [18]

    Longhi, Phys

    S. Longhi, Phys. Rev. B100, 125157 (2019)

    work page 2019

  19. [19]

    Zeng and Y

    Q.-B. Zeng and Y. Xu, Phys. Rev. Res.2, 033052 (2020)

    work page 2020

  20. [20]

    Cai, Phys

    X. Cai, Phys. Rev. B103, 214202 (2021)

    work page 2021

  21. [21]

    Kawabata, T

    K. Kawabata, T. Numasawa, and S. Ryu, Phys. Rev. X 13, 021007 (2023)

    work page 2023

  22. [22]

    Weidemann, M

    S. Weidemann, M. Kremer, S. Longhi, and A. Szameit, Nature601, 354 (2022)

    work page 2022

  23. [23]

    Ganeshan, K

    S. Ganeshan, K. Sun, and S. Das Sarma, Phys. Rev. Lett. 110, 180403 (2013)

    work page 2013

  24. [24]

    Zhanget al., arXiv:2301.12138 (2023)

    J.-N. Zhanget al., arXiv:2301.12138 (2023)

    work page arXiv 2023

  25. [25]

    Liu, J.-M

    G.-J. Liu, J.-M. Zhang, S.-Z. Li, and Z. Li, Phys. Lett. A493, 129279 (2023)

    work page 2023

  26. [26]

    Edvardsson, F

    E. Edvardsson, F. K. Kunst, T. Yoshida, and E. J. Bergholtz, Phys. Rev. Res.4, 033130 (2022)

    work page 2022

  27. [27]

    Chang, J.-S

    P.-Y. Chang, J.-S. You, X. Wen, and S. Ryu, Phys. Rev. Res.2, 033069 (2020)

    work page 2020

  28. [28]

    Halder and S

    D. Halder and S. Basu, Phys. Rev. B111, 235447 (2025)

    work page 2025

  29. [29]

    M. Z. Hasan and C. L. Kane, Rev. Mod. Phys.82, 3045 (2010)

    work page 2010

  30. [30]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys.83, 1057 (2011)

    work page 2011

  31. [31]

    Ghosh and S

    S. Ghosh and S. Roy, arXiv:2506.10195 (2025)

    work page arXiv 2025

  32. [32]

    Wang, C.-K

    Y.-P. Wang, C.-K. Chang, R. Okugawa, and C.-H. Hsu, Phys. Rev. Res.7, 043353 (2025)

    work page 2025

  33. [33]

    Wang, C.-K

    Y.-P. Wang, C.-K. Chang, R. Okugawa, and C.-H. Hsu, arXiv:2602.11155 (2026)

    work page arXiv 2026

  34. [34]

    Manna, S

    S. Manna, S. K. Singh, A. Nandy, and B. Basa, Commun. Phys.6, 113 (2023)

    work page 2023