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Universal Spreading Dynamics in Quasiperiodic Non-Hermitian Systems

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Pith's one-line read In the non-Hermitian Aubry-André model, wave packets spread as t^{1/3} in the localized phase and t^{1/2} in the delocalized phase, instead of halting or moving ballistically.

desk verdict Solid, clean numerics and a useful LE-based method; the central scaling derivation leans on an iid assumption that does not literally apply to the quasiperiodic model, but the gap looks repairable and the results are probably right. read the letter →

arxiv 2412.01301 v1 pith:TJLYB2NT submitted 2024-12-02 cond-mat.dis-nn cond-mat.mes-hallcond-mat.quant-gasphysics.opticsquant-ph

Ze-Yu Xing , Shu Chen , Haiping Hu This is my paper · ORCID
classification cond-mat.dis-nncond-mat.mes-hallcond-mat.quant-gasphysics.opticsquant-ph MSC 82B4481Q12
keywords non-HermitianAubry-Andrémodelquasiperiodicdisorderwave-packetspreadingdynamicaldelocalizationimaginarydensityofstatesVanHovesingularityLyapunovexponentsubdiffusion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Non-Hermitian systems can keep a wave packet spreading even when every eigenstate is localized, because eigenstates with larger imaginary energies dominate the time evolution. This paper studies the non-Hermitian Aubry-André model, a one-dimensional lattice with a complex quasiperiodic potential, and claims that normalized wave packets always spread algebraically: with exponent $\delta=1/3$ in the localized regime and $\delta=1/2$ in the delocalized regime, in place of the Hermitian behaviors of halted and ballistic motion. The authors connect both exponents to a single spectral feature, a Van Hove singularity with exponent $\beta=-1/2$ at the tail of the imaginary density of states, through scaling relations $\delta=(\beta+1)/(d+\beta+1)$ and $\delta=(\beta+1)/d$. They also give a Lyapunov-exponent method for extracting the spreading exponent without diagonalizing large non-Hermitian matrices. If the claim holds, slow algebraic spreading is a generic dynamical signature of non-Hermitian disorder, not a special property of one model.

What carries the argument

The load-bearing object is the imaginary density of states, $\rho_I(s)$, the normalized distribution of $\operatorname{Im} E$ over the spectrum. The engine of the argument is an extreme-value identity for that distribution: if the imaginary parts of the eigenvalues in a local region of size $N=X^d$ are drawn independently from $\rho_I(s)$, then the expected tail integral beyond the largest one, $\mathbb{E}[\int_{\lambda_{\max}}^{\infty}\rho_I(s)\,ds]$, equals $1/(N+1)$ and therefore decays as $X^{-d}$. Plugging that into the steepest-descent conditions $\partial\lambda/\partial X\sim t^{-1}$ and $X\partial\lambda/\partial X\sim t^{-1}$ converts a spectral tail exponent $\beta$ into the spreading exponents $\delta=(\beta+1)/(d+\beta+1)$ and $\delta=(\beta+1)/d$. The paper's practical tool is the generalized Thouless relation $\rho(E)=\frac{1}{2\pi}\nabla^2\gamma(E)$, which gives the density of states from the Laplacian of the Lyapunov exponent in the complex plane and lets the authors read off $\rho_I$ without exact diagonalization.

What would settle it

For the non-Hermitian Aubry-André model at $|V|=5$, compute the largest imaginary eigenvalue among blocks of length $X$ and check whether the ensemble-averaged tail integral $\mathbb{E}[\int_{\lambda_{\max}}^{\infty}\rho_I(s)\,ds]$ decays as $1/X$; if it does not, Eq. (12) fails and the claimed $\delta=1/3$ need not follow. One can also simulate the wave-packet spreading for much longer times and larger systems and check whether the fitted exponent stays at $1/3$ in the localized regime and $1/2$ in the delocalized regime.

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Extended reading notes

Core claim

The paper's central claim is that in the non-Hermitian Aubry-André model with complex on-site potential $V=|V|e^{i\phi_V}$, the ensemble-averaged spreading distance obeys $X(t)\sim t^{1/3}$ for $|V|>2t$ (localized regime) and $X(t)\sim t^{1/2}$ for $|V|<2t$ (delocalized regime), with an intermediate exponent near $0.57$ at the transition. To explain this, the authors propose a general random-variable mechanism: a wave packet that has spread over a volume $X^d$ is controlled by the eigenstate with the largest imaginary energy $\lambda_{\max}$ in that region, and the tail integral of the imaginary density of states satisfies $\int_{\lambda_{\max}}^{\infty}\rho_I(s)\,ds\sim X^{-d}$. Combining this with the maximization conditions $\partial\lambda/\partial X\sim t^{-1}$ (localized) and $X\partial\lambda/\partial X\sim t^{-1}$ (delocalized) yields the universal scaling relations $\delta=(\beta+1)/(d+\beta+1)$ and $\delta=(\beta+1)/d$ when the tail is $\rho_I(s)\sim(s_0-s)^{\beta}$. For this model the tail is a Van Hove singularity with $\beta=-1/2$, which produces $\delta=1/3$ and $\delta=1/2$; the authors show perturbatively in the deep localized regime that this singularity is stable.

Load-bearing premise

The scaling law assumes that the imaginary parts of eigenvalues inside a local region behave like independent random samples from the thermodynamic imaginary density of states, whereas the quasiperiodic model's spectrum is deterministic and correlated.

Editorial extensions

If this is right

  • In any $d$-dimensional non-Hermitian disordered system whose imaginary density of states has an algebraic tail with exponent $\beta$, normalized wave-packet spreading should follow these scaling laws; the quasiperiodic model is the $\beta=-1/2$ case of that general statement.
  • The Hermitian dichotomy of halted versus ballistic transport is replaced, whenever such a tail singularity exists, by a non-Hermitian dichotomy of subdiffusion versus diffusion governed by the imaginary spectrum.
  • Spreading exponents can be computed from Lyapunov exponents in the complex plane through $\rho(E)=\frac{1}{2\pi}\nabla^2\gamma(E)$, a route that avoids large-scale exact diagonalization of non-Hermitian matrices.
  • In the deep localized regime, second-order perturbation theory shows the $\beta=-1/2$ Van Hove singularity is stable, so the $1/3$ exponent is not a transition artifact but persists throughout the localized phase.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The most fragile step is the independent-samples treatment of eigenvalue imaginary parts: in a quasiperiodic system these values are deterministic and strongly correlated, so a finite-size scaling study of $\lambda_{\max}$ within blocks of length $X$ would directly test whether the $1/X$ tail decay really holds in the thermodynamic limit.
  • If the scaling relations are right, a two-dimensional quasiperiodic non-Hermitian lattice with the same $\beta=-1/2$ tail should spread with $\delta=1/5$ (localized) and $\delta=1/4$ (delocalized); those exponents are distinct enough to be checked in photonic or ultracold-atom platforms.
  • The same random-variable argument would predict different spreading behavior for disorder whose imaginary-spectrum tail is not algebraic, for example Gaussian, since the extreme-value statistics would change; this makes the quasiperiodic model's deterministic algebraic tail the special feature that produces clean power laws.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This manuscript studies wave-packet spreading in the one-dimensional non-Hermitian Aubry-André model with complex on-site potential V cos(2παj + ϕ). The authors report three numerical facts: in the delocalized phase |V| < 2 the normalized second-moment width grows as X(t) ∼ t^{1/2}, in the localized phase |V| > 2 it grows as t^{1/3}, and at the transition the fitted exponent is approximately 0.57. They propose a scaling theory in which the largest imaginary part among O(X^d) local eigenvalues controls the propagation, leading to Eqs. (9), (11), and (12) and to the general relations δ = (β+1)/(d+β+1) (localized) and δ = (β+1)/d (delocalized), where β is the Van Hove tail exponent of the imaginary density of states. Using the generalized Thouless relation, they extract ρ_I(λ) from the Lyapunov exponent and find β = −1/2 in both regimes, reproducing δ = 1/3 and δ = 1/2. A supplemental model with a mobility edge yields a fitted β ≈ −0.435 and a predicted δ ≈ 0.36, confirmed by direct dynamics.

Significance. If the derivation were made rigorous, the result would establish a useful universality: non-Hermitian dynamics converts the Hermitian halted/ballistic dichotomy into subdiffusion/diffusion controlled by the tail of the imaginary density of states. The main numerical observation (δ ≈ 1/3 and δ ≈ 1/2 across the entire phase diagram) is clean and internally consistent, and the Lyapunov-exponent method is a practical alternative to exact diagonalization. The mobility-edge test in SM Section IV is a genuine dynamical confirmation of the scaling relation once β is known. The main reservation is that the key relation Eq. (12) is derived in SM Section II under an iid assumption that the quasiperiodic model does not satisfy; the universality claim therefore currently rests on an unproven empirical-tail statement.

major comments (3)
  1. [SM Section II and Eq. (12)] The derivation of Eq. (12) is load-bearing and is not justified for the model studied. SM Section II explicitly says "By treating λ_j as random variables drawn from the distribution ρ_I(s)," and the calculation E[∫_{λ_max}^∞ ρ_I(s) ds] = 1/(N+1) uses the product structure P(λ_max ≤ λ) = [F(λ)]^N, which is exact only for independent draws. For the quasiperiodic potential in model (1), the imaginary parts of eigenvalues are deterministic functions of the site index in the deep localized limit (SM Section III gives λ(k) = |V| sin ϕ_V cos k [1 + O(t^2)]), so the extremes are order statistics of a low-discrepancy deterministic sequence, not of independent samples. The authors need to replace the iid step with a proof or explicit statement that the upper empirical tail of the first N sites has measure ∼ N^{−1}; if the empirical tail exponent were α ≠ 1, Eq. (13) would become δ = (β+1)/(α d + β + 1) and δ = (β+1)/(α d). The numerical agreement in Fig. 3 and the mobility-edge test in SM Section IV corroborate the phenomenology but do not validate this step; SM Section III only checks the stability of β = −1/2, not the order-statistics relation.
  2. [Eqs. (8), (9), and (11)] The saddle-point/maximization conditions (9) and (11) treat λ(X) as a smooth function of the localization center X and replace the discrete set of eigenstates by a continuum. In the quasiperiodic model, λ(X) is a deterministic pseudo-random sequence in X, and the derivative ∂λ/∂X is not defined in the usual sense; the propagator ansatz (8) also assumes a purely exponential spatial profile with a single localization length ξ. These are reasonable heuristic scaling arguments, but the paper does not state the conditions under which the saddle point is valid. Since Eqs. (9) and (11) are used together with Eq. (12) to obtain the exponents, the authors should either justify the continuum approximation for this model or clearly label it as an assumption on the same footing as Eq. (12).
  3. [Conclusion, final paragraph] The claim that the framework applies to "generic disordered non-Hermitian systems, whether the disorder is correlated or uncorrelated" is broader than the derivation supports. Equation (12) requires knowledge of the empirical tail of local eigenvalues; for uncorrelated random disorder the iid argument is plausible, but for correlated or quasiperiodic disorder it is not automatic. The authors should either prove a deterministic analogue of Eq. (12), state it as an explicit assumption, or restrict the universality claim to the cases where the empirical-tail relation can be verified.
minor comments (5)
  1. [Fig. 2 caption and text] In the description of Fig. 2(b), the text says the phase boundary separates "the delocalized regime (|V| < 2) from the localized regime (|V| < 2)"; the second inequality should be |V| > 2.
  2. [Supplemental Material headings] The supplement has two sections labeled "(III)": "Perturbative analysis of the spectral structure" and "Dynamical spreading in the presence of mobility edge." The latter should be numbered (IV).
  3. [Main text, Eq. (12)] Equation (12) is attributed to reference [44] (the supplemental material), but the derivation is in SM Section II; the citation should be made explicit at first use so that readers know where the proof is located.
  4. [Fig. 3] The caption for Fig. 3 refers to colored dots and dashed fitting lines but does not specify the color coding or the time window used for the steady-evolution fit; a legend or colorbar would help the reader assess how robust the extracted slopes are.
  5. [Text after Eq. (7)] The sentence "δ > 1/2, δ = 1/2, and δ < 1/2 corresponds to superdiffusive, diffusive, and subdiffusive transport" has a subject-verb agreement error; "corresponds" should be "correspond."

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spreading exponents are derived from the model's spectral iDOS and independently verified against direct wave-packet dynamics.

full rationale

The derivation chain is self-contained: the dynamical exponents δ = 1/3 and δ = 1/2 are obtained by combining the maximization conditions (9)/(11) with the cumulative-tail relation (12) and the iDOS tail exponent β = −1/2. The tail exponent is extracted from the Lyapunov-exponent/DOS analysis (Eqs. (14)-(15)) and from the perturbative spectral calculation in SM Section (III), not from the spreading data X(t). Likewise, the mobility-edge example in SM Section (IV) fits β from the iDOS and then predicts δ, which is then checked against the independent time-evolution simulation; this is a prediction, not a post-diction of the same fitted quantity. The only mildly questionable step is the iid order-statistics assumption explicitly stated in SM Section (II) ('By treating λ_j as random variables drawn from the distribution ρ_I(s)'), which may not be automatically valid for a deterministic quasiperiodic sequence. However, that is an assumption about the applicability of the extreme-value calculation, not a circular reduction: Eq. (12) is derived from the stated distributional premise, and the final exponent is not an input to that derivation. Self-citations in the paper point to Avila's global theory (an external mathematical theorem) or to the paper's own Supplemental derivations, and they do not carry the central claim by themselves. Overall, no equation in the paper is equivalent to its claimed result by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The framework rests on four modeling assumptions: Avila's theorem for the LE, the Thouless/Laplacian relation for the DOS, an extreme-value (iid) treatment of the imaginary spectrum, and a saddle-point ansatz for the propagator. The only fitted spectral input is the tail exponent β, which is not tuned to the dynamics data. No new physical entities are introduced.

free parameters (1)
  • Van Hove tail exponent β = -1/2 in main model; -0.4345 in mobility-edge model
    The scaling relations (13) depend on β. For the main model β=-1/2 comes from a perturbative analysis plus a numerical fit; for the mobility-edge model it is fitted from the iDOS and then used to predict δ.
assumptions (5)
  • standard math Avila's global theory for one-frequency Schrödinger operators
    Used to establish γ(E)=max(ln|V/2t|,0) inside the spectrum and the Anderson transition at |V|=2t (Eq. 4).
  • domain assumption Generalized Thouless relation ρ(E)=(1/2π)∇²γ(E)
    Assumed to hold for the quasiperiodic non-Hermitian model to extract DOS and iDOS from LE (Eq. 14), citing Refs. [49,50].
  • ad hoc to paper Independent and identically distributed imaginary eigenvalues in Eq. (12)
    SM Section (II) treats λ_j as iid draws from ρ_I(s) to derive ∫_λ^∞ ρ ~ X^{-d}; no justification is given for quasiperiodic disorder, where eigenvalues are correlated.
  • domain assumption Propagator ansatz and maximization condition
    Eqs. (8)-(11) assume the dominant contribution to ⟨X|e^{-iHt}|x0⟩ comes from eigenstates maximizing λ(X)t - X/ξ, with ∂λ/∂X ~ t^{-1}; this is a saddle-point heuristic.
  • domain assumption Fourier representation for delocalized states
    Eq. (10) assumes extended states can be represented as plane-wave-like with localization length ξ̃ in reciprocal space and ⟨X|K⟩ ~ X^{-d-1}.

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Pith. "Pith review of Universal Spreading Dynamics in Quasiperiodic Non-Hermitian Systems." pith.science (2026). https://pith.science/paper/TJLYB2NT

@misc{pith2026241201301,
  author       = {Pith},
  title        = {Pith review of: Universal Spreading Dynamics in Quasiperiodic Non-Hermitian Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJLYB2NT}},
  note         = {Machine review of arXiv:2412.01301}
}
abstract

Non-Hermitian systems exhibit a distinctive type of wave propagation, due to the intricate interplay of non-Hermiticity and disorder. Here, we investigate the spreading dynamics in the archetypal non-Hermitian Aubry-Andr\'e model with quasiperiodic disorder. We uncover counter-intuitive transport behaviors: subdiffusion with a spreading exponent $\delta=1/3$ in the localized regime and diffusion with $\delta=1/2$ in the delocalized regime, in stark contrast to their Hermitian counterparts (halted vs. ballistic). We then establish a unified framework from random-variable perspective to determine the universal scaling relations in both regimes for generic disordered non-Hermitian systems. An efficient method is presented to extract the spreading exponents from Lyapunov exponents. The observed subdiffusive or diffusive transport in our model stems from Van Hove singularities at the tail of imaginary density of states, as corroborated by Lyapunov-exponent analysis.

Figures

Figures reproduced from arXiv: 2412.01301 by the authors.

Figure 1
Figure 1. FIG. 1. Comparison of wave spreading in (a) Hermitian and [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Anderson localization transition in the quasiperiodic [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spreading dynamics of model ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Lyapunov exponent (LE) analysis of the spreading [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

51 extracted references · 32 canonical work pages · cited by 2 Pith papers

  1. [33]

    B. Li, C. Chen, and Z. Wang, Universal non-Hermitian transport in disordered systems, arXiv: 2411.19905

    arXiv

  2. [1]

    P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492–1505 (1958)

    work page 1958

  3. [2]

    Wiersma, D

    Lagendijk, A., van Tiggelen, B. Wiersma, D. S. Fifty years of Anderson localization, Phys. Today 62, 24–29 (2009)

    work page 2009

  4. [3]

    C. M. Bender, Making sense of non-Hermitian Hamilto- nians, Rep. Prog. Phys. 70, 947 (2007)

    2007

  5. [4]

    El-Ganainy, K

    R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non- Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018)

    2018

  6. [5]

    Moiseyev, Non-Hermitian Quantum Mechanics(Cam- bridge University Press, New York, 2011)

    N. Moiseyev, Non-Hermitian Quantum Mechanics(Cam- bridge University Press, New York, 2011)

    2011

  7. [6]

    Ashida, Z

    Y. Ashida, Z. Gong, and Masahito Ueda, Non-Hermitian Physics, Advances in Physics 69, 3 (2020)

    work page 2020

  8. [7]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021)

    2021

Show all 51 references
  1. [8]

    K. Ding, C. Fang, and G. Ma, Non-Hermitian Topol- ogy and Exceptional-Point Geometries, Nature Reviews Physics 4, 745 (2022)

    2022

  2. [9]

    Hatano and D

    N. Hatano and D. R. Nelson, Localization Transitions in Non-Hermitian Quantum Mechanics, Phys. Rev. Lett. 77, 570, (1996)

    1996

  3. [10]

    X. L. Luo, T. Ohtsuki, and R. Shindou, Universal- ity classes of the Anderson Transitions Driven by non- Hermitian Disorder, Phys. Rev. Lett.126, 090402 (2021)

    2021

  4. [11]

    Zhang, H

    Z.-Q. Zhang, H. Liu, H. Liu, H. Jiang and X.C. Xie, Bulk- boundary correspondence in disordered non-Hermitian systems, Sci. Bull. 68 (2), 157–164 (2023)

    2023

  5. [12]

    X. Luo, Z. Xiao, K. Kawabata, T. Ohtsuki and R. Shin- dou, Unifying the Anderson transitions in Hermitian and non-Hermitian systems, Phys. Rev. Research 4, L022035 (2022)

    2022

  6. [13]

    Y. Liu, X. Jiang, J. Cao, and S. Chen, Non-Hermitian mobility edges in one-dimensional quasicrystals with parity-time symmetry, Phys. Rev. B 101, 174205 (2020)

    2020

  7. [14]

    Y. Liu, Y. Wang, Z. Zheng, and S. Chen, Exact non- Hermitian mobility edges in one-dimensional quasicrystal lattice with exponentially decaying hopping and its dual lattice, Phys. Rev. B 103, 134208 (2021)

    2021

  8. [15]

    Y. Liu, Q. Zhou, and S. Chen, Localization transi- tion, spectrum structure, and winding numbers for one- dimensional non-Hermitian quasicrystals,Phys. Rev. B 104, 024201 (2021)

    2021

  9. [16]

    Y. Liu, Y. Zeng, L. Li, and S. Chen, Exact solution of the single impurity problem in nonreciprocal lattices: Impurity-induced size-dependent non-Hermitian skin ef- fect, Phys. Rev. B 104, 085401 (2021)

    2021

  10. [17]

    T. Liu, H. Guo, Y. Pu, and S. Longhi, Generalized Aubry-Andre self-duality and mobility edges in nonHer- mitian quasiperiodic lattices, Phys. Rev. B 102, 024205 (2020)

    2020

  11. [18]

    Zeng and Y

    Q.-B. Zeng and Y. Xu, Winding Numbers and Gener- alized Mobility Edges in Non-Hermitian Systems, Phys. Rev. Res. 2, 033052 (2020)

    2020

  12. [19]

    Longhi, Spectral deformations in non-Hermitian lat- tices with disorder and skin effect: A solvable model, Phys

    S. Longhi, Spectral deformations in non-Hermitian lat- tices with disorder and skin effect: A solvable model, Phys. Rev. B 103, 144202 (2021)

    2021

  13. [20]

    L. Wang, Z. Wang and S. Chen, Non-Hermitian butterfly spectra in a family of quasiperiodic lattices, Phys. Rev. B 110, L060201 (2024)

    2024

  14. [21]

    D. W. Zhang, L. Z. Tang, L. J. Lang, H. Yan, and S. L. Zhu, Non-Hermitian Topological Anderson Insulators, Sci. China-Phys. Mech. Astron. 63, 267062 (2020)

    2020

  15. [22]

    L. Z. Tang, L. F. Zhang, G. Q. Zhang, and D. W. Zhang, Topological Anderson insulators in two- dimensional non-Hermitian disordered systems, Phys. Rev. B 110, L060201

  16. [23]

    Liu, J.-K

    H. Liu, J.-K. Zhou, B. L. Wu, Z.-Q. Zhang, and H. Jiang, Real space topological invariant and higher-order topological Anderson insulator in two-dimensional non- Hermitian systems, Phys. Rev. B 103, 224203 (2021)

    2021

  17. [24]

    H. F. Liu, Z. X. Su, Z-Q. Zhang, and H. Jiang, Topologi- cal Anderson insulator in two-dimensional non-Hermitian systems, Chin. Phys. B 29, 050502 (2020)

    2020

  18. [25]

    Q. Lin, T. Li, L. Xiao, K. Wang, W. Yi, P. Xue, Obser- vation of non-Hermitian topological Anderson insulator in quantum dynamics, Nat. Commun. 13, 3229 (2022)

    2022

  19. [26]

    Li, C.-H

    L. Li, C.-H. Lee and J. Gong, Impurity induced scale-free localization, Nat. Commun. 13, 3229 (2022)

    2022

  20. [27]

    Guo, C.-H

    C.-X. Guo, C.-H. Liu, X.-M. Zhao, Y. Liu and S. Chen, Exact solution of non-Hermitian systems with general- ized boundary conditions: Size-dependent boundary ef- fect and fragility of the skin effect. Phys. Rev. Lett. 127, 116801 (2021)

    2021

  21. [28]

    C.-X. Guo, X. Wang, H. Hu and S. Chen, Accumula- tion of scale-free localized states induced by local non- Hermiticity. Phys. Rev. B 107, 134121 (2023)

    2023

  22. [29]

    Li, H.-R

    B. Li, H.-R. Wang, F. Song and Z. Wang, Scale-free lo- calization and PT symmetry breaking from local non- Hermiticity. Phys. Rev. B 108, L161409 (2023)

    2023

  23. [30]

    Bergholtz, E

    Molignini, P., Arandes, O. Bergholtz, E. J. Anoma- lous skin effects in disordered systems with a single non-Hermitian impurity. Phys. Rev. Research 5, 033058 (2023)

    2023

  24. [31]

    C.-X. Guo, L. Su, Y. Wang, L., J. Wang, X. Ruan, Y. Du, D. Zheng, S. Chen and H. Hu, Scale-tailored localization and its observation in non-Hermitian electrical circuits, Nat Commun 15, 9120 (2024)

    2024

  25. [32]

    Weidemann, M

    S. Weidemann, M. Kremer, S. Longhi and A. Szameit, Coexistence of dynamical delocalization and spectral lo- calization through stochastic dissipation, Nat. Photon. 15, 576–581 (2021)

    2021

  26. [34]

    Avila, Global theory of one-frequency Schr¨ odinger op- erators, Acta Math

    A. Avila, Global theory of one-frequency Schr¨ odinger op- erators, Acta Math. 215 (1) 1 - 54, (2015)

    2015

  27. [35]

    Aubry and G

    S. Aubry and G. Andr´ e, Analyticity breaking and An- 6 derson localization in incommensurate lattices, Ann. Isr. Phys. Soc. 3, 133 (1980)

    1980

  28. [36]

    Yao and Z

    S. Yao and Z. Wang, Edge States and Topological Invari- ants of Non-Hermitian Systems, Phys. Rev. Lett. 121, 086803 (2018)

    2018

  29. [37]

    V. M. Alvarez, J. B. Vargas, and L. F. Torres, Non- Hermitian robust edge states in one dimension: Anoma- lous localization and eigenspace condensation at excep- tional points, Phys. Rev. B 97, 121401 (2018)

    2018

  30. [38]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Correspondence between Winding Numbers and Skin Modes in Non- Hermitian Systems, Phys. Rev. Lett. 125, 126402 (2020)

    2020

  31. [39]

    Okuma, K

    N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological Origin of Non-Hermitian Skin Effects, Phys. Rev. Lett. 124, 086801 (2020)

    2020

  32. [40]

    Xiong and H

    Y. Xiong and H. Hu, Graph morphology of non- Hermitian bands, Phys. Rev. B 109, L100301

  33. [41]

    Hu, Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra, Sci

    H. Hu, Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra, Sci. Bull. (2024)

    2024

  34. [42]

    C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103 (2019)

    2019

  35. [43]

    Xiong, Z.-Y

    Y. Xiong, Z.-Y. Xing and H. Hu, Non-Hermitian skin effect in arbitrary dimensions: non-Bloch band theory and classification, arXiv:2407.01296

    arXiv

  36. [44]

    (12) in the main text; (III) Perturbative analysis of the spectral structure; (IV) Dynamical spreading in the presence of mobility edge

    See SM for details on (I) Lyapunov exponent inside the energy spectra; (II) Derivation of Eq. (12) in the main text; (III) Perturbative analysis of the spectral structure; (IV) Dynamical spreading in the presence of mobility edge

  37. [45]

    L Xiao, T Deng, K Wang, G Zhu, Z Wang, W Yi, P Xue, Non-Hermitian bulk–boundary correspondence in quantum dynamics, Nature Physics 16 (7), 761-766

  38. [46]

    L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders and P. Xue, Observa- tion of topological edge states in parity–time-symmetric quantum walks, Nature Phys 13, 1117 (2017)

    2017

  39. [47]

    X. Zhan, L. Xiao, Z. Bian, K. Wang, X. Qiu, B. C. Sanders, W. Yi, and P. Xue, Detecting Topological In- variants in Nonunitary Discrete-Time Quantum Walks, Phys. Rev. Lett. 119, 130501 (2017)

    2017

  40. [48]

    Z. Yang, K. Zhang, C. Fang, and J. Hu, Non-Hermitian Bulk-Boundary Correspondence and Auxiliary General- ized Brillouin Zone Theory, Phys. Rev. Lett.125, 226402 (2020)

    2020

  41. [49]

    D. J. Thouless, J. Phys. C 5 77 (1972)

    1972

  42. [50]

    Derrida, J.L

    B. Derrida, J.L. Jacobsen and R. Zeitak, Lyapunov Ex- ponent and Density of States of a One-Dimensional Non- Hermitian Schr¨ odinger Equation, Journal of Statistical Physics 98, 31–55 (2000)

    2000

  43. [51]

    Yang and Y

    C. Yang and Y. Wang, Level-spacing distribution of lo- calized phases induced by quasiperiodic potentials, Phys. Rev. B. 109, 214210 (2024). 7 Supplemental Material This supplemental material provides additional details on (I) Lyapunov exponent inside the energy spectra; (II) ...

    2024

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