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arxiv: 2509.13502 · v3 · submitted 2025-09-16 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech· quant-ph

Effective delocalization in the one-dimensional Anderson model with stealthy disorder

Carlo Vanoni , Jonas Karcher , Mikael C. Rechtsman , Boris L. Altshuler , Paul J. Steinhardt , Salvatore Torquato This is my paper

Pith reviewed 2026-05-18 16:28 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mechquant-ph
keywords Anderson localizationstealthy disorderone-dimensional modellocalization lengthcorrelated disorderself-energy expansionhyperuniform systems
0
0 comments X

The pith

Stealthy disorder allows localization length to exceed system size in finite 1D Anderson chains at fixed small disorder strength.

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

The paper examines the one-dimensional Anderson model where disorder is stealthy, meaning its power spectrum vanishes over a continuous band of wave numbers. For fixed energy and small but finite disorder strength W, the authors show there is a range of the stealthiness parameter chi such that the localization length exceeds the finite system size. This effective delocalization arises because the correlated nature of stealthy disorder suppresses scattering contributions in the perturbative expansion of the self-energy, causing leading-order terms to vanish and pushing the scaling of localization length to higher inverse powers of W. The mechanism depends only on the spectral properties of the disorder and extends to other wave systems.

Core claim

In the one-dimensional Anderson model with stealthy disorder, for fixed energy and small but finite disorder strength W, there exists for any finite length system a range of stealthiness chi for which the localization length exceeds the system size. The leading terms in the perturbation expansion of the localization length vanish identically for a progressively larger number of terms as chi increases, so that the localization length scales as W to the power of minus 2n with arbitrarily large n, unlike the W to the minus 2 scaling for uncorrelated disorder.

What carries the argument

Stealthy disorder, defined as disorder whose power spectrum vanishes over a continuous band of wave numbers; it suppresses low-wave-number scattering contributions in the self-energy expansion.

If this is right

  • The scaling of the localization length with disorder strength changes from W^{-2} to W^{-2n} for arbitrarily large n as stealthiness increases.
  • The same spectral mechanism for effective delocalization applies directly to photonic and phononic wave systems.
  • Numerical simulations of the model confirm the analytical predictions for the range of chi where localization length exceeds system size.

Where Pith is reading between the lines

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

  • Materials engineered with stealthy spectral gaps might sustain extended states over longer distances than expected from uncorrelated disorder alone.
  • The approach could be tested in optical or acoustic layered media by measuring transmission lengths as a function of the stealthiness parameter.
  • Similar spectral engineering might alter localization properties in higher-dimensional tight-binding models or in systems with other forms of correlated disorder.

Load-bearing premise

The perturbative expansion of the self-energy remains valid and captures the leading scaling for small but finite disorder strength W and finite system sizes, with the power spectrum exactly vanishing over the defined continuous band.

What would settle it

Compute the localization length numerically for fixed small W, fixed energy, and increasing system size L while holding chi in the predicted range; the length should continue to exceed L rather than saturate or fall below it.

Figures

Figures reproduced from arXiv: 2509.13502 by Boris L. Altshuler, Carlo Vanoni, Jonas Karcher, Mikael C. Rechtsman, Paul J. Steinhardt, Salvatore Torquato.

Figure 1
Figure 1. Figure 1: FIG. 1. Graphical representation of Dyson equation, see [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase diagram showing the dependence of the lo [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Localization length as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Localization length as a function of log [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Different types of stealthy disordered correlations. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Level splitting of two degenerate energy levels in the zero disorder limit as a function of [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Level splitting of two degenerate energy levels in the zero disorder limit for fixed [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
read the original abstract

We study analytically and numerically the Anderson model in one dimension with "stealthy" disorder, defined as having a power spectrum that vanishes in a continuous band of wave numbers. Motivated by recent studies on the optical transparency properties of stealthy hyperuniform layered media, we compute the localization length using a perturbative expansion of the self-energy. We find that, for fixed energy and small but finite disorder strength $W$, there exists for any finite length system a range of stealthiness $\chi$ for which the localization length exceeds the system size. This kind of "effective delocalization" is the result of the novel kind of correlated disorder that spans a continuous range of length scales, a defining characteristic of stealthy systems. Unlike uncorrelated disorder, for which the localization length $\xi$ scales as $W^{-2}$ to leading order for small W, the leading order terms in the perturbation expansion of $\xi$ for stealthy disordered systems vanish identically for a progressively large number of terms as $\chi$ increases such that $\xi$ scales as $W^{-2n}$ with arbitrarily large $n$. Moreover, we support our analytical results with numerical simulations. Our results introduce stealthy disorder into quantum tight-binding models and show that enforcing a low-$k$ spectral gap markedly alters the scattering landscape, enabling localization lengths that exceed the system size at fixed disorder strength. Since this mechanism relies only on the spectral properties of the disorder, it carries over directly to photonic and phononic wave systems.

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 / 2 minor

Summary. The manuscript studies the 1D Anderson model with stealthy disorder whose power spectrum vanishes over a continuous band of wave numbers. Using a perturbative expansion of the self-energy, the authors show that the leading low-order contributions to the inverse localization length vanish identically as the stealthiness parameter χ increases; consequently the localization length scales as W^{-2n} with n growing with χ. This implies that, for fixed energy and small but finite disorder strength W, there exists a range of χ such that the localization length exceeds any finite system size L, producing effective delocalization. Numerical simulations are presented in support of the analytic scaling.

Significance. If the perturbative analysis remains controlled in the regime of interest, the result identifies a concrete spectral mechanism—vanishing of a continuous band in the disorder power spectrum—that systematically suppresses localization to arbitrarily high orders in W. This mechanism is independent of microscopic details and therefore transfers directly to photonic and phononic wave equations. The work thereby adds a new, tunable route to engineering long localization lengths at fixed disorder strength.

major comments (2)
  1. [§3] §3 (perturbative self-energy): when the first n−1 diagrams vanish identically because S(k)=0 for |k|<k_c(χ), the radius of convergence of the remaining series in W is not estimated. For fixed finite W the suppression of leading terms can shrink the domain of validity; without an explicit bound on the remainder or a comparison of the retained term against the neglected higher orders, it is unclear whether the W^{-2n} scaling still dominates the inverse localization length.
  2. [§4] §4 (numerics): the finite-L realizations possess only discrete Fourier modes that approximate the continuous-band vanishing. The manuscript should demonstrate that the observed ξ>L is quantitatively consistent with the perturbative prediction rather than arising from finite-size corrections or from the discrete approximation to S(k)=0; a direct comparison of the numerically extracted scaling exponent versus the analytic n(χ) is needed.
minor comments (2)
  1. [Introduction] The definition of the stealthiness parameter χ and the precise relation between the gap width k_c and χ should be stated explicitly in the introduction before the perturbative calculation begins.
  2. [Figure 2] Figure captions should specify the number of disorder realizations and the precise range of system sizes used to extract the localization length.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript to incorporate additional discussion and comparisons where appropriate.

read point-by-point responses

  1. Referee: [§3] §3 (perturbative self-energy): when the first n−1 diagrams vanish identically because S(k)=0 for |k|<k_c(χ), the radius of convergence of the remaining series in W is not estimated. For fixed finite W the suppression of leading terms can shrink the domain of validity; without an explicit bound on the remainder or a comparison of the retained term against the neglected higher orders, it is unclear whether the W^{-2n} scaling still dominates the inverse localization length.

    Authors: We agree that an explicit estimate of the radius of convergence would provide additional rigor. The perturbative expansion follows the standard weak-disorder approach, with the exact vanishing of low-order diagrams enforced by the continuous spectral gap in S(k). For sufficiently small W the first non-vanishing term of order W^{2n} dominates higher-order contributions (O(W^{2(n+1)}) and beyond) whenever W^2 ≪ 1. We have added a paragraph in the revised §3 that discusses this regime of validity and supplies a heuristic bound showing that the leading term controls the inverse localization length for the parameter range of interest. A fully rigorous non-perturbative bound on the remainder lies beyond the scope of the present work. revision: yes

  2. Referee: [§4] §4 (numerics): the finite-L realizations possess only discrete Fourier modes that approximate the continuous-band vanishing. The manuscript should demonstrate that the observed ξ>L is quantitatively consistent with the perturbative prediction rather than arising from finite-size corrections or from the discrete approximation to S(k)=0; a direct comparison of the numerically extracted scaling exponent versus the analytic n(χ) is needed.

    Authors: We appreciate this suggestion for strengthening the numerical validation. In the revised manuscript we have added a direct comparison in §4: for several values of χ we extract an effective scaling exponent from the numerical localization lengths and compare it to the analytic prediction n(χ). The extracted exponents agree with the expected n(χ) within statistical errors for the system sizes employed, and we include a brief discussion showing that finite-size corrections and the discrete-mode approximation become negligible for the L values used. These additions confirm that the observed ξ > L is consistent with the perturbative scaling. revision: yes

Circularity Check

0 steps flagged

No circularity: scaling follows directly from perturbative expansion on the spectral definition of stealthy disorder

full rationale

The paper defines stealthy disorder by the property that its power spectrum vanishes identically over a continuous band of wave numbers. It then applies the standard perturbative expansion of the self-energy (standard in 1D Anderson localization) and observes that this definition forces successive low-order diagrams to vanish, yielding the higher-order scaling ξ ~ W^{-2n} with n increasing in χ. This is a direct algebraic consequence of the input spectral condition rather than a fitted parameter, self-referential definition, or load-bearing self-citation. The central claim that ξ can exceed finite L for small but finite W and sufficient χ is therefore derived from the given disorder class plus perturbation theory; numerical checks are invoked as independent support. No step reduces by the paper's own equations to a tautology or presupposed result.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The analysis rests on the standard perturbative treatment of the Anderson model self-energy together with the external definition of stealthy disorder via its power spectrum; no new free parameters are fitted to data beyond the conventional disorder strength W and the stealthiness parameter χ that defines the spectral gap.

free parameters (2)
  • disorder strength W
    Amplitude scaling the random potential; treated as a small but finite control parameter in the expansion.
  • stealthiness χ
    Parameter setting the width of the continuous k-gap in the disorder power spectrum.
axioms (2)
  • domain assumption The disorder potential is drawn from a distribution whose power spectrum vanishes identically over a continuous interval of wave numbers.
    This is the defining property of stealthy disorder invoked throughout the perturbative calculation.
  • standard math Standard diagrammatic or self-energy perturbation theory for the Anderson model applies directly when the disorder is correlated.
    The expansion used to obtain the localization length assumes the usual weak-disorder perturbative framework.

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