Multiple reentrant topological windows induced by generalized Bernoulli disorder

Ruijiang Ji; Shu Chen; Yunbo Zhang; Zhihao Xu

arxiv: 2512.06851 · v3 · pith:Q65JYI7Snew · submitted 2025-12-07 · ⚛️ physics.optics · cond-mat.dis-nn· quant-ph

Multiple reentrant topological windows induced by generalized Bernoulli disorder

Ruijiang Ji , Yunbo Zhang , Shu Chen , Zhihao Xu This is my paper

Pith reviewed 2026-05-17 01:12 UTC · model grok-4.3

classification ⚛️ physics.optics cond-mat.dis-nnquant-ph

keywords re-entrant topologyBernoulli disorderSu-Schrieffer-Heeger modelzero-mode localizationchiral displacementphotonic waveguidesone-dimensional lattices

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The pith

Generalized Bernoulli disorder in intradimer hoppings creates multiple disconnected topological windows in a one-dimensional chiral lattice.

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

The paper examines a one-dimensional Su-Schrieffer-Heeger chain in which the intradimer hopping amplitudes follow a generalized Bernoulli distribution. By changing the specific values that appear in the distribution and the probabilities assigned to them, the authors show that the system develops an adjustable number of separate intervals of topological behavior separated by trivial regions. These re-entrant windows are located analytically by finding where the inverse localization length of the zero-energy modes drops to zero, and the resulting boundaries match direct numerical checks. The mean chiral displacement is introduced as a dynamical signature that registers the transitions, and a photonic-waveguide realization is outlined.

Core claim

In the Su-Schrieffer-Heeger model with generalized Bernoulli disorder confined to the intradimer hoppings, the topological phase diagram contains multiple re-entrant windows whose boundaries are fixed by the zeros of the inverse localization length of the zero modes; both the number and the widths of these windows are controlled by the values and probabilities that define the disorder distribution.

What carries the argument

Inverse localization length of zero modes, which supplies the analytic condition separating topological and trivial regions for any choice of generalized Bernoulli parameters.

If this is right

Increasing the number of distinct values in the Bernoulli distribution increases the number of topological windows.
The probabilities assigned to each value directly determine the relative widths of the topological and trivial intervals.
The mean chiral displacement changes its long-time behavior precisely at the analytically derived critical points.
The same disorder-induced windows should appear in photonic waveguide arrays fabricated with the corresponding intradimer couplings.

Where Pith is reading between the lines

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

Engineered multivalued disorder of this form could provide a practical route to devices that operate in several distinct topological regimes without changing the lattice geometry.
The dynamical probe based on chiral displacement may allow time-resolved observation of the transitions in optical or cold-atom experiments.
Analogous re-entrant windows are likely to appear in other one-dimensional chiral models once the same generalized Bernoulli statistics are imposed on the appropriate hopping terms.

Load-bearing premise

The topological phase boundaries are set exactly by the inverse localization length of zero modes, without appreciable shifts from finite-size effects or contributions from nonzero-energy states.

What would settle it

A transfer-matrix or exact-diagonalization calculation performed for a chosen set of disorder values and probabilities that yields a different count of intervals with nontrivial topological invariant than the number predicted by setting the inverse localization length to zero.

Figures

Figures reproduced from arXiv: 2512.06851 by Ruijiang Ji, Shu Chen, Yunbo Zhang, Zhihao Xu.

**Figure 2.** Figure 2: FIG. 2: (a) Widths of the first and second TAI phases as a [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: (a) Topological phase diagram characterized by the d [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Disorder-averaged mean chiral displacement operat [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

We investigate reentrant topological transitions in a one-dimensional Su-Schrieffer-Heeger chain with generalized Bernoulli disorder in the intradimer hopping amplitudes. Owing to its independently tunable values and probabilities, the multivalued disorder distribution provides a direct way to control the topological phase diagram. We show that increasing the disorder strength can split the nontrivial regime into multiple disconnected topological windows, whose number, widths, and locations are determined by the distribution parameters. The phase boundaries are derived analytically from the zero-mode inverse localization length and are governed by a weighted geometric mean of the disordered hopping amplitudes, in agreement with numerical results from the reflection-matrix topological quantum number and the real-space winding number. We also show that the mean chiral displacement dynamically identifies these reentrant windows. These results demonstrate how multivalued random disorder can organize and tune reentrant topological behavior in one-dimensional chiral lattices.

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.

Desk Editor's Note private letter to a colleague

Generalized Bernoulli disorder in a 1D SSH chain produces multiple tunable re-entrant topological windows whose boundaries come from the inverse localization length and match numerics.

read the letter

The main thing here is that generalized Bernoulli disorder on the intradimer hoppings lets you dial the number and widths of disconnected topological windows in this chiral lattice by changing the allowed values and their probabilities. The phase boundaries are worked out from the inverse localization length of the zero modes and line up with the numerical checks they present. They also show the mean chiral displacement as a workable dynamical probe and sketch a photonic waveguide setup for possible experiments. That combination is the concrete advance over earlier binary-disorder studies of re-entrant topology in the SSH model. The systematic mapping from distribution parameters to the number of windows is new enough to be useful and they give credit to the prior localization and SSH literature without overclaiming. The analytical-numerical agreement is a positive sign that the central calculation holds up. The soft spot is the one the stress-test note flags: in a multi-valued discrete distribution, rare local sequences can create effective dimerized segments whose critical points differ from the disorder average. In a regime with several re-entrant windows those fluctuations could shift apparent boundaries or cause extra hybridization in finite chains, and the mean inverse length does not automatically include them. Their reported agreement with numerics suggests the effect is not large for the sizes they checked, but a referee would reasonably ask for finite-size scaling or a short discussion of rare-event corrections to confirm the windows stay clean. This is aimed at people who work on disordered 1D topological systems or on engineered disorder in photonics. A reader already following localization-length methods or re-entrant phases in chiral chains will find the specific results on generalized Bernoulli distributions worth the time. It has enough new control and verification to merit a serious referee rather than a desk reject. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The manuscript examines re-entrant topological behavior in a one-dimensional Su-Schrieffer-Heeger model with generalized Bernoulli-type disorder restricted to the intradimer hopping amplitudes. Varying the disorder values and probabilities is shown to systematically control the number and widths of disconnected topological windows. Phase boundaries are obtained analytically from the inverse localization length of zero modes and reported to agree with numerical calculations. The mean chiral displacement is introduced as a dynamical probe of the transitions, and a possible photonic waveguide lattice implementation is outlined.

Significance. If the results hold, the work clarifies the role of multivalued disorder distributions in producing multiple re-entrant topological windows in 1D chiral lattices. The analytical, parameter-free derivation of boundaries via the inverse localization length (rather than fitted quantities) is a clear strength, as is the use of mean chiral displacement for dynamical characterization. The photonic implementation suggestion adds experimental relevance for optics.

major comments (1)

The central claim that the disorder-averaged inverse localization length of zero modes fully determines the topological phase boundaries (without finite-size or rare-configuration corrections) is load-bearing for the multi-window results. Explicit checks for system-size dependence and rare-sequence effects in the re-entrant regimes would be needed to confirm that the reported numerical agreement is not affected by these factors.

minor comments (2)

Figure captions and axis labels should explicitly state the system sizes used in the numerical diagonalization or transfer-matrix calculations to allow direct assessment of finite-size convergence.
A brief remark on the range of disorder probabilities and values explored would help readers reproduce the reported changes in the number of topological windows.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. We address the major comment below and will strengthen the presentation accordingly in the revised version.

read point-by-point responses

Referee: The central claim that the disorder-averaged inverse localization length of zero modes fully determines the topological phase boundaries (without finite-size or rare-configuration corrections) is load-bearing for the multi-window results. Explicit checks for system-size dependence and rare-sequence effects in the re-entrant regimes would be needed to confirm that the reported numerical agreement is not affected by these factors.

Authors: We agree that explicit verification of finite-size convergence and the role of rare sequences is valuable for confirming the robustness of the reported agreement. The analytical phase boundaries are obtained from the disorder-averaged Lyapunov exponent of the transfer matrix in the thermodynamic limit, which by construction incorporates the full disorder distribution without finite-size corrections. Our numerical results, obtained via exact diagonalization on chains of several hundred sites averaged over hundreds of realizations, already show close quantitative agreement with these boundaries throughout the re-entrant windows. In the revised manuscript we will add supplementary figures that explicitly demonstrate the convergence of the numerically extracted boundaries with increasing system size (e.g., N = 200, 500, 1000) in representative re-entrant regimes. Regarding rare-sequence effects, the ensemble averaging used in both the analytical derivation and the numerics suppresses contributions from atypical configurations; we will add a short discussion clarifying that no systematic deviations attributable to rare events were observed in our data. If the referee deems it necessary, we can further include targeted checks on selected atypical sequences. revision: yes

Circularity Check

0 steps flagged

Analytical derivation of phase boundaries from inverse localization length is independent and non-circular

full rationale

The paper computes phase boundaries analytically via the inverse localization length of zero modes for the generalized Bernoulli disorder on the SSH chain. This follows from the standard transfer-matrix or Lyapunov-exponent formalism applied directly to the multivalued hopping distribution; the resulting expression for the localization length is not obtained by fitting to the target topological windows or by re-using the windows themselves. Numerical agreement is presented as external validation. No load-bearing self-citations, self-definitional steps, or ansatz smuggling appear in the derivation chain. The method is self-contained against the model's Hamiltonian and disorder statistics.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard results from localization theory and chiral symmetry in 1D lattices. The disorder parameters are varied as control knobs rather than fitted to match a target outcome. No new particles or forces are introduced.

free parameters (1)

disorder values and probabilities
The specific hop values and their occurrence probabilities are chosen and varied to control the number and widths of topological windows.

axioms (1)

domain assumption The inverse localization length of zero modes determines the topological phase boundaries in the disordered chiral chain.
Invoked to obtain the analytical phase boundaries that agree with numerics.

pith-pipeline@v0.9.0 · 5411 in / 1430 out tokens · 62390 ms · 2026-05-17T01:12:16.521395+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the topological phase boundaries can be analytically determined by examining the inverse localization length γ of the zero modes … yielding |∏ (−t1 + ξ(j))^{pj}| = 1
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Alexander duality … D = 3

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Forward citations

Cited by 2 Pith papers

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