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arxiv: 2606.07274 · v2 · pith:PQAVC4CWnew · submitted 2026-06-05 · ❄️ cond-mat.dis-nn

Topological Anderson insulators and reentrant topological transitions in a quasiperiodic long-range Su-Schrieffer-Heeger model

Fang-Ming Meng , Qi-Bo Zeng This is my paper

Pith reviewed 2026-06-27 20:18 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords topological Anderson insulatorquasiperiodic disorderSu-Schrieffer-Heeger modelwinding numberreentrant topological transitionlong-range hoppingone-dimensional topological phase
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The pith

Quasiperiodic disorder in a long-range Su-Schrieffer-Heeger chain produces topological Anderson insulators whose winding numbers change in discrete steps even when the spectral gap nearly closes.

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 chain with long-range hoppings up to third neighbors plus a quasiperiodic on-site potential. It finds that this disorder creates new topological Anderson insulating phases labeled by winding numbers -1, 0, 1 and 2. These phases survive into the strong-disorder regime where the energy gap is almost closed. The real-space winding number changes in a staircase pattern with rising disorder strength, and reentrant transitions appear when either disorder or hopping amplitudes are varied.

Core claim

In the clean limit the model hosts phases with winding numbers W = -1, 0, 1 and 2. The introduction of quasiperiodic disorder modifies the phase diagram, inducing topological Anderson insulating phases that persist even when the spectral gap becomes nearly closed. The real-space winding number evolves through successive quantized steps with increasing disorder strength, accompanied by multiple reentrant topological transitions when disorder strength or hopping amplitudes are varied.

What carries the argument

The real-space winding number, a topological invariant computed directly from the real-space wave functions that assigns an integer to each phase under quasiperiodic disorder.

If this is right

  • TAI phases with distinct winding numbers appear because of the competition between topological dimerization and localization.
  • The real-space winding number changes in successive quantized steps as disorder strength is increased, producing staircase-like transitions.
  • Reentrant topological transitions occur both when the quasiperiodic disorder strength is varied and when the hopping amplitudes are varied.

Where Pith is reading between the lines

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

  • The same real-space invariant might reveal staircase behavior in other one-dimensional chains that combine long-range hopping with incommensurate potentials.
  • Cold-atom or photonic-lattice experiments could test whether the nearly gapless TAI phases remain stable when weak additional random disorder is added.

Load-bearing premise

The real-space winding number remains a faithful topological invariant that correctly classifies phases under quasiperiodic disorder, with finite-size numerical computations free of significant artifacts when the gap is nearly closed.

What would settle it

Direct computation of the real-space winding number on system sizes several times larger than those used in the paper, at strong disorder where the gap is closed, checking whether the quantized staircase steps remain stable or disappear.

Figures

Figures reproduced from arXiv: 2606.07274 by Fang-Ming Meng, Qi-Bo Zeng.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color online) Phase diagram of the clean long-range [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) Schematic illustration of the dimeriza [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) Topological phase diagrams in the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Topological phases of the long-range [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) Topological phases of the long-range [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

We study a one-dimensional long-range Su-Schrieffer-Heeger model with third-nearest-neighbor hopping and subject to quasiperiodic disorder. In the clean limit, the model hosts phases characterized by winding numbers $W=-1,0,1$ and $2$. The introduction of quasiperiodic disorder profoundly modifies the phase diagram and induces a series of topological phase transitions. Owing to the competition between topological dimerization and localization, topological Anderson insulating (TAI) phases with different winding numbers emerge and can persist even when the spectral gap becomes nearly closed in the strong-disorder regime. In addition, we uncover multiple reentrant topological phase transitions induced by varying either the quasiperiodic disorder strength or the hopping amplitudes. Remarkably, the system exhibits staircase-like topological Anderson transitions, where the real-space winding number evolves through successive quantized steps with increasing disorder strength. Our results demonstrate that the interplay between long-range hopping and quasiperiodic disorder generates a rich landscape of disorder-induced topological phases and reentrant topological transition phenomena.

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 examines a one-dimensional long-range Su-Schrieffer-Heeger chain with third-nearest-neighbor hopping subject to quasiperiodic disorder. In the clean limit the model realizes phases with winding numbers W = −1, 0, 1 and 2. Introduction of quasiperiodic disorder is shown to generate topological Anderson insulator phases with the same winding numbers, reentrant topological transitions, and staircase-like evolution of the real-space winding number with increasing disorder strength; these phases are reported to survive even when the bulk gap becomes nearly closed.

Significance. If the numerical invariants remain faithful in the near-gapless regime, the work would establish that long-range hopping combined with quasiperiodic disorder produces an unusually rich set of disorder-induced topological phases, including multiple reentrant transitions and quantized staircase behavior. The explicit use of the real-space winding number as a diagnostic provides a concrete, falsifiable classification of the phases.

major comments (2)
  1. [strong-disorder regime] § on strong-disorder regime (where gap closure is reported): the central claim that TAI phases with quantized W persist “even when the spectral gap becomes nearly closed” rests on finite-size computations of the real-space winding number. When the gap falls below the finite-size level spacing, hybridization of edge modes or localization length exceeding system size can produce spurious non-quantized values or artificial jumps in W. The manuscript must supply explicit checks (e.g., gap versus system size, localization length versus chain length, or convergence of W with increasing system size) in the strong-disorder windows to confirm that the reported staircase steps remain faithful topological invariants rather than numerical artifacts.
  2. [Methods / winding-number definition] Definition and validation of real-space winding number: the paper adopts the real-space winding number as the primary order parameter for all phases, including the quasiperiodic case. A short derivation or reference establishing its topological protection under quasiperiodic (rather than purely random) potentials, together with a demonstration that it remains integer-valued when the gap is small but finite, is required to support the classification of the TAI phases with W = −1, 0, 1, 2.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the system sizes used for the winding-number calculations and whether open or periodic boundary conditions were employed.
  2. [Abstract / Introduction] The abstract lists winding numbers as W = −1, 0, 1 and 2; the main text should confirm that the same set is recovered in the disordered case and that no additional values appear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and checks.

read point-by-point responses

  1. Referee: [strong-disorder regime] § on strong-disorder regime (where gap closure is reported): the central claim that TAI phases with quantized W persist “even when the spectral gap becomes nearly closed” rests on finite-size computations of the real-space winding number. When the gap falls below the finite-size level spacing, hybridization of edge modes or localization length exceeding system size can produce spurious non-quantized values or artificial jumps in W. The manuscript must supply explicit checks (e.g., gap versus system size, localization length versus chain length, or convergence of W with increasing system size) in the strong-disorder windows to confirm that the reported staircase steps remain faithful topological invariants rather than numerical artifacts.

    Authors: We agree that explicit finite-size scaling is necessary to rule out artifacts when the gap is small. In the revised manuscript we will add supplementary figures showing (i) the spectral gap versus system size in the strong-disorder windows and (ii) the convergence of the real-space winding number with increasing chain length. These data will confirm that the quantized staircase steps remain stable and are not due to level-spacing or localization-length effects. revision: yes

  2. Referee: [Methods / winding-number definition] Definition and validation of real-space winding number: the paper adopts the real-space winding number as the primary order parameter for all phases, including the quasiperiodic case. A short derivation or reference establishing its topological protection under quasiperiodic (rather than purely random) potentials, together with a demonstration that it remains integer-valued when the gap is small but finite, is required to support the classification of the TAI phases with W = −1, 0, 1, 2.

    Authors: The real-space winding number is known to be topologically protected for gapped one-dimensional systems with quasiperiodic potentials, which can be treated as a deterministic form of disorder. We will insert a concise derivation in the Methods section, citing the relevant literature on its application to quasiperiodic models, and include a numerical panel demonstrating that the invariant remains strictly integer-valued for small but finite gaps in our parameter regimes. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on direct numerical evaluation of real-space winding number

full rationale

The paper defines the model Hamiltonian explicitly, introduces quasiperiodic disorder as an external parameter, and computes the real-space winding number W from the resulting eigenstates or transfer matrix on finite chains. No step equates a claimed transition or phase label to a fitted quantity by construction, nor does any load-bearing premise reduce to a self-citation whose content is itself unverified within the paper. The reported staircase transitions and persistence of quantized W are outputs of that computation rather than inputs renamed as predictions. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions of topological condensed-matter physics; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption The real-space winding number is a valid topological invariant for quasiperiodically disordered 1D chains.
    Invoked to label phases with W = -1,0,1,2 and to detect the staircase transitions.

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