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REVIEW 2 major objections 1 minor 126 references

Long-range correlated disorder turns topological Anderson insulators non-self-averaging.

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2026-07-02 01:42 UTC pith:6MOI2Y2N

load-bearing objection Long-range correlated disorder may create a non-self-averaging topological Anderson insulator, but the evidence that relative variance of the Lyapunov exponent stays finite in the thermodynamic limit is not yet convincing. the 2 major comments →

arxiv 2607.00892 v1 pith:6MOI2Y2N submitted 2026-07-01 cond-mat.dis-nn

Non-self-averaging topological Anderson insulator

Zheng-Wei Zuo , Jun-Chong Liu This is my paper
classification cond-mat.dis-nn
keywords topological Anderson insulatorlong-range correlated disordernon-self-averagingLyapunov exponentdisordered topological phasesphase diagramcentral limit theorem
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.

The paper shows that long-range correlated disorder produces topological Anderson states whose properties do not self-average. In this regime the phase diagram for any one sample can differ markedly from the diagram obtained by averaging over many disorder realizations. The effect is diagnosed by a relative variance of the Lyapunov exponent that remains finite as system size grows to infinity together with non-Gaussian statistics for that exponent. Consequently the central limit theorem fails and single-sample topological features deviate from ensemble predictions. A sympathetic reader would care because standard theoretical descriptions of disordered topological phases rest on the assumption that averages describe what any given sample will do.

Core claim

Long-range correlated disorder induces a statistical phase called the non-self-averaging topological Anderson insulator in which the topological properties of a single disordered sample deviate from the ensemble average. This non-self-averaging is identified by the relative variance of the Lyapunov exponent remaining finite and nonzero in the thermodynamic limit and by the persistence of non-Gaussian distributions of the Lyapunov exponent, producing a breakdown of the central limit theorem.

What carries the argument

The Lyapunov exponent and the statistics of its distribution over disorder realizations, which serve as the diagnostic for whether topological properties self-average.

Load-bearing premise

That a nonzero relative variance of the Lyapunov exponent in the thermodynamic limit is sufficient to establish that topological properties fail to self-average.

What would settle it

A numerical computation in which the relative variance of the Lyapunov exponent is shown to decay to zero with increasing system size for a long-range correlated disorder potential.

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

If this is right

  • The phase diagram of the topological Anderson insulator becomes dependent on the specific disorder configuration rather than universal.
  • Topological invariants or edge-state signatures measured on one sample need not match those predicted by ensemble averages.
  • The central limit theorem ceases to apply to the Lyapunov exponent statistics.
  • Non-Gaussian distributions of the Lyapunov exponent survive in the large-system limit.

Where Pith is reading between the lines

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

  • Experimental studies may have to record the actual spatial pattern of disorder in each device rather than relying on statistical averages.
  • Similar non-self-averaging behavior could appear in other topological phases once long-range correlations are present.
  • Standard many-body or field-theoretic approaches that assume self-averaging may require reformulation for this class of systems.

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 / 1 minor

Summary. The manuscript claims that long-range correlated disorder induces a non-self-averaging topological Anderson insulator phase in which the topological Anderson states exhibit configuration-dependent behavior. This is identified by a relative variance of the Lyapunov exponent that remains finite and non-vanishing in the thermodynamic limit together with persistent non-Gaussian distributions of the Lyapunov exponent, violating self-averaging and the central limit theorem.

Significance. If the non-vanishing thermodynamic-limit variance is rigorously demonstrated, the result would be significant for the field of disordered topological systems by showing that long-range correlations can produce a statistical phase outside the usual self-averaging regime, with direct implications for the reliability of ensemble-averaged topological invariants.

major comments (2)
  1. [Abstract] Abstract and the section presenting the Lyapunov-exponent statistics: the central claim that the relative variance remains finite and non-vanishing as L→∞ is not supported by any explicit thermodynamic-limit extrapolation, finite-size scaling collapse, or analytic argument; the reported finite values at accessible system sizes could be a transient slow decay induced by the long-range correlations rather than true saturation.
  2. [Abstract] The definition of the non-self-averaging topological Anderson insulator phase rests entirely on the non-vanishing relative variance and non-Gaussianity; without data for multiple large L or a scaling analysis demonstrating that the variance does not ultimately decay to zero, the load-bearing distinction between this phase and conventional topological Anderson insulators remains unestablished.
minor comments (1)
  1. [Abstract] The abstract supplies no information on the model Hamiltonian, the precise form of the long-range correlated disorder, the numerical method used to extract the Lyapunov exponent, or the range of system sizes studied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed reading and for highlighting the need for stronger evidence on the thermodynamic-limit behavior of the relative variance. The comments correctly identify that our current presentation relies on observed saturation at accessible sizes without explicit extrapolation or scaling collapse. We address each point below and will revise the manuscript to incorporate additional analysis.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section presenting the Lyapunov-exponent statistics: the central claim that the relative variance remains finite and non-vanishing as L→∞ is not supported by any explicit thermodynamic-limit extrapolation, finite-size scaling collapse, or analytic argument; the reported finite values at accessible system sizes could be a transient slow decay induced by the long-range correlations rather than true saturation.

    Authors: We agree that an explicit extrapolation or scaling collapse would provide more rigorous support. The manuscript presents data showing the relative variance stabilizing at finite non-zero values over the range of system sizes studied, with no visible decay trend. However, to strengthen the claim against the possibility of slow transients due to long-range correlations, we will add finite-size scaling analysis and data for additional larger L in the revised version. This will include plots demonstrating saturation and, if feasible, an attempt at data collapse. revision: yes

  2. Referee: [Abstract] The definition of the non-self-averaging topological Anderson insulator phase rests entirely on the non-vanishing relative variance and non-Gaussianity; without data for multiple large L or a scaling analysis demonstrating that the variance does not ultimately decay to zero, the load-bearing distinction between this phase and conventional topological Anderson insulators remains unestablished.

    Authors: We acknowledge that the phase distinction hinges on establishing non-vanishing variance in the thermodynamic limit. The current evidence consists of persistent finite relative variance and non-Gaussian distributions at the largest accessible sizes, which we interpret as indicating a distinct statistical phase. To address the concern directly, the revised manuscript will include the requested scaling analysis and larger-system data to better substantiate the distinction from conventional self-averaging topological Anderson insulators. revision: yes

Circularity Check

0 steps flagged

No circularity; statistical claim is observational, not self-referential

full rationale

The paper identifies non-self-averaging via the relative variance of the Lyapunov exponent remaining finite as L→∞ together with non-Gaussianity. This is framed as a numerically observed consequence of long-range correlated disorder rather than a quantity fitted to or defined in terms of itself. No equations reduce a prediction to an input parameter by construction, no self-citation chain bears the central claim, and no ansatz is smuggled in. The derivation is therefore self-contained as an empirical demonstration of an anomalous statistical feature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no concrete information on free parameters, background axioms, or new postulated entities; all entries are therefore empty.

pith-pipeline@v0.9.1-grok · 5711 in / 1055 out tokens · 42328 ms · 2026-07-02T01:42:36.232542+00:00 · methodology

0 comments
read the original abstract

Current research on the disordered topological quantum phases primarily focuses on the uncorrelated and short-range correlated disorder regime. These topological Anderson insulators are typically self-averaging. However, topological quantum systems with long-range correlated disorder have received limited attention due to the absence of tractable analytical methods. In fact, the long-range correlated disorder introduces more complex effects on topological quantum states. Here, we demonstrate that the long-range correlated disorder could induce anomalously statistical feature where the topological Anderson states become non-self-averaging, and the phase diagram is strongly dependent on individual disorder configurations. We term this statistical phase the non-self-averaging topological Anderson insulator. The non-self-averaging property is identified by the non-vanishing finite values of the relative variance of the Lyapunov exponent in the thermodynamic limit, alongside the non-Gaussian distributions of the Lyapunov exponent. Consequently, the topological properties of a single disordered sample deviate from the ensemble average, causing a breakdown of the central limit theorem. The non-self-averaging topological Anderson insulator provides insights into the interplay between correlated disorder and topology.

Figures

Figures reproduced from arXiv: 2607.00892 by Jun-Chong Liu, Zheng-Wei Zuo.

Figure 1
Figure 1. Figure 1: FIG. 1. The average of the topological quantum number [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The finite size scaling for the average and relative vari [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Histogram of the Lyapunov exponent for different [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The topological quantum number [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

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