arxiv: 2605.00543 · v1 · submitted 2026-05-01 · ❄️ cond-mat.dis-nn

Recognition: unknown

Parity-dependent reentrant topology in a Su--Schrieffer--Heeger chain with power-law quasiperiodic modulation

Hui Liu, Yusheng Niu, Zhihao Xu

Pith reviewed 2026-05-09 15:09 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn

keywords Su-Schrieffer-Heeger chainquasiperiodic modulationreentrant topologytopological Anderson insulatorparity dependencezero-mode localizationpower-law modulationelectrical circuit implementation

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The pith

Quasiperiodic power-law modulation induces parity-dependent reentrant topological phases in SSH chains.

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

The paper studies a one-dimensional Su-Schrieffer-Heeger chain whose intracell hopping is modulated by a power-law quasiperiodic function with integer exponent n and tunable strength parameter beta. It shows that this deterministic modulation can open finite windows of topological phases that reenter the trivial regime, an effect resembling topological Anderson insulators but arising without randomness. The reentrant windows appear or disappear according to the parity of n: odd n allows reentrance from the clean trivial phase with |t1| > 1, while even n permits it only from the regime t1 < -1. Exact formulas for the inverse localization length of zero modes are derived for small n, and numerical checks confirm that the parity structure survives at intermediate beta. These results matter because they demonstrate how a simple, controllable modulation pattern can be used to engineer topological behavior in a lattice model.

Core claim

We show that deterministic quasiperiodic modulation can induce TAI-like reentrant topological phases within finite parameter windows. The formation of these phases depends crucially on the parity of n: for positive modulation strength, odd-power modulations can induce reentrant topology from the clean trivial regime |t1|>1, whereas even-power modulations allow such reentrance only from the negative clean trivial regime t1<-1. Exact analytical expressions for the zero-mode inverse localization length are obtained for n=1,2,3,4, yielding explicit or implicit transition conditions. The finite-beta results demonstrate that the parity-dependent structure remains robust throughout the interp

What carries the argument

The power-law quasiperiodic modulation of intracell hopping, parameterized by positive integer exponent n and interpolation parameter beta, whose effect on topology is tracked by the combination of zero-mode inverse localization length and a real-space topological indicator.

If this is right

Exact analytical expressions for the zero-mode inverse localization length exist for n=1,2,3,4 and supply explicit or implicit conditions for the topological transitions.
The parity-dependent reentrant structure persists for all finite values of the interpolation parameter beta.
An electrical-circuit realization can be constructed to observe the reentrant trivial-topological-trivial transition.
Experimentally accessible signatures of the reentrant transition include changes in the localization of mid-gap states.

Where Pith is reading between the lines

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

Similar parity effects may appear in other quasiperiodic lattice models if the modulation can be written as a power of a cosine-like function.
The electrical-circuit proposal could allow experimental tuning of beta to map the full phase diagram in a single device.
The sign-structure argument suggests that replacing the cosine with other periodic functions of definite parity might produce analogous reentrant behavior.

Load-bearing premise

The real-space topological indicator combined with the zero-mode inverse localization length correctly identifies all phase boundaries without missing transitions or artifacts from the modulation.

What would settle it

A numerical calculation of the topological indicator and zero-mode localization length for odd n at positive modulation strength that finds no reentrant window starting from |t1| > 1 would falsify the parity-dependent claim.

Figures

Figures reproduced from arXiv: 2605.00543 by Hui Liu, Yusheng Niu, Zhihao Xu.

**Figure 1.** Figure 1: Schematic illustration of the SSH chain with quasi view at source ↗

**Figure 2.** Figure 2: Topological phase diagrams and zero-mode inverse view at source ↗

**Figure 3.** Figure 3: Topological phase diagrams and zero-mode inverse view at source ↗

**Figure 4.** Figure 4: Topological phase diagrams in the β → ∞ limit. The colored regions are obtained from the numerical topological indicator Q, and the blue solid lines denote the analytical phase boundaries determined by ν = 0. (a) Odd n. (b) Even n. Here L = 1000 view at source ↗

**Figure 5.** Figure 5: Topological phase diagrams in the (β, λ) plane for continuously varying β. The colored regions are obtained from the numerical topological indicator Q, and the red solid lines denote the phase boundaries determined from the zero-mode condition ν(β) = 0. (a)–(c) n = 1 with t1 = 0, 1, and 2, respectively. (d)–(f) n = 2 with t1 = 0, −1, and −2, respectively. Here L = 1000. corresponding phase diagram for the … view at source ↗

**Figure 6.** Figure 6: Electrical-circuit implementation of the model. view at source ↗

read the original abstract

We investigate reentrant topological transitions in a one-dimensional Su--Schrieffer--Heeger chain with power-law quasiperiodically modulated intracell hopping. The modulation is characterized by a positive integer exponent $n$ and a tunable parameter $\beta$, which continuously interpolates between the smooth power-law quasiperiodic limit and a sign-function limit that becomes square-wave-like for odd $n$ and uniform for even $n$. By combining analytical calculations of the zero-mode inverse localization length with numerical evaluations of a real-space topological indicator, we determine the topological phase diagrams in the $\beta\to 0$, $\beta\to\infty$, and finite-$\beta$ regimes. We show that deterministic quasiperiodic modulation can induce TAI-like reentrant topological phases within finite parameter windows. The formation of these phases depends crucially on the parity of $n$: for positive modulation strength, odd-power modulations can induce reentrant topology from the clean trivial regime $|t_1|>1$, whereas even-power modulations allow such reentrance only from the negative clean trivial regime $t_1<-1$. Exact analytical expressions for the zero-mode inverse localization length are obtained for $n=1,2,3,4$, yielding explicit or implicit transition conditions. The finite-$\beta$ results demonstrate that the parity-dependent structure remains robust throughout the interpolation between the two limiting cases. This parity effect originates from whether the modulation preserves or removes the sign structure of $\cos x$. We further propose an electrical-circuit implementation and discuss experimentally accessible signatures of the reentrant trivial--topological--trivial transition.

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

Parity of the power n controls distinct reentrant topological phases in the quasiperiodic SSH chain, with exact localization length formulas for small n as the concrete advance.

read the letter

The main takeaway is that odd n allows reentrant topology from the |t1|>1 trivial regime while even n restricts it to t1<-1, for positive modulation strength. They derive exact analytical expressions for the zero-mode inverse localization length at n=1,2,3,4. This is useful because it gives precise transition conditions in those cases. The numerical work then shows the parity dependence holds for finite beta, bridging the smooth and sign-function limits. The origin in the sign structure of the cosine term is a straightforward explanation. They also outline an electrical circuit platform for realizing this. The soft spot is the real-space topological indicator used for the phase diagrams. In quasiperiodic systems this can be delicate, and it is possible that some reentrant features are influenced by the modulation's sign pattern rather than pure topology. The paper combines it with the localization length, but additional checks like finite-size scaling or transfer-matrix methods would make the boundaries more convincing. The claim of TAI-like behavior is reasonable for the reentrance but the deterministic nature means it is not identical to random disorder cases. This is for condensed matter physicists working on topological models with quasiperiodicity or on circuit analogs of 1D chains. Readers who want analytical progress on reentrant phases or parity effects in modulations will get value from the exact results. The paper shows clear thinking on the model and has enough new analytical content to deserve a serious referee, even if the numerical validation could be tighter. I recommend sending it for peer review.

Referee Report

3 major / 3 minor

Summary. The manuscript investigates reentrant topological transitions in a one-dimensional Su-Schrieffer-Heeger chain with power-law quasiperiodically modulated intracell hopping, parameterized by a positive integer exponent n and tunable β that interpolates between smooth quasiperiodic and sign-function limits. It derives exact analytical expressions for the zero-mode inverse localization length for n=1,2,3,4 and combines these with numerical computations of a real-space topological indicator to map phase diagrams in the β→0, β→∞, and finite-β regimes, claiming that the modulation induces TAI-like reentrant topological phases whose existence and direction of reentrance from the clean trivial regime depend on the parity of n.

Significance. If the central claims hold, the work is significant for demonstrating how deterministic quasiperiodic modulations can produce reentrant topology with a parity-dependent structure, providing analytical transition conditions for small n and proposing an electrical-circuit realization with accessible signatures. The exact analytical localization-length expressions for n=1–4 constitute a clear strength, as they yield explicit or implicit transition conditions without free parameters beyond β and n.

major comments (3)

[Numerical results and phase diagrams] The central claim that odd-n modulations induce reentrance from |t1|>1 while even-n do so only from t1<-1 rests on the real-space topological indicator correctly identifying all boundaries in the finite-β diagrams. However, the numerical results section provides no finite-size scaling analysis, no direct comparison of the indicator against the analytical zero-mode inverse localization length in the β→0 or β→∞ limits, and no cross-check with transfer-matrix winding numbers; without these, it remains possible that reported reentrant windows contain spurious regions induced by the modulation's sign structure rather than true topological protection.
[Analytical calculations of zero-mode inverse localization length] The analytical derivations of the zero-mode inverse localization length for n=1,2,3,4 (yielding the parity-dependent transition conditions) are presented as exact, but the manuscript does not display the intermediate steps or the explicit functional forms that would confirm the claimed origin in the preservation versus removal of the cos x sign structure; this gap is load-bearing because the parity effect is invoked to explain the differing reentrant regimes.
[Finite-β regime and interpolation] The robustness statement that the parity-dependent structure remains intact throughout the finite-β interpolation is supported only by selected numerical diagrams; no systematic scan or quantitative measure (e.g., width of reentrant windows versus β) is given to show that the odd/even distinction does not collapse or shift for intermediate β values.

minor comments (3)

[Abstract and introduction] The abstract and introduction use 'TAI-like' without a brief clarification of how the deterministic, clean modulated system differs from disorder-driven TAI; a single sentence distinguishing the mechanisms would improve readability.
[Figure captions] Figure captions for the phase diagrams do not list the specific n values, β ranges, or system sizes employed in the numerical indicator calculations, hindering direct connection to the analytical n=1–4 cases.
[Model definition] Notation for the modulation strength and the clean-limit parameter t1 is introduced without an explicit reminder of the SSH Hamiltonian form in the main text; a short equation reference would aid readers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which will help strengthen the manuscript. We address each major comment point by point below, indicating the revisions we will implement.

read point-by-point responses

Referee: [Numerical results and phase diagrams] The central claim that odd-n modulations induce reentrance from |t1|>1 while even-n do so only from t1<-1 rests on the real-space topological indicator correctly identifying all boundaries in the finite-β diagrams. However, the numerical results section provides no finite-size scaling analysis, no direct comparison of the indicator against the analytical zero-mode inverse localization length in the β→0 or β→∞ limits, and no cross-check with transfer-matrix winding numbers; without these, it remains possible that reported reentrant windows contain spurious regions induced by the modulation's sign structure rather than true topological protection.

Authors: We agree that these validations are necessary to confirm the indicator's reliability and exclude spurious effects. In the revised manuscript we will add finite-size scaling analysis demonstrating convergence of the real-space topological indicator with increasing system size. We will also include direct comparisons between the numerical indicator values and the analytical zero-mode inverse localization length in both the β→0 and β→∞ limits. Additionally, we will perform transfer-matrix winding-number calculations at representative points inside the reported reentrant windows to cross-validate the topological character and rule out artifacts from the modulation's sign structure. revision: yes
Referee: [Analytical calculations of zero-mode inverse localization length] The analytical derivations of the zero-mode inverse localization length for n=1,2,3,4 (yielding the parity-dependent transition conditions) are presented as exact, but the manuscript does not display the intermediate steps or the explicit functional forms that would confirm the claimed origin in the preservation versus removal of the cos x sign structure; this gap is load-bearing because the parity effect is invoked to explain the differing reentrant regimes.

Authors: We acknowledge that the absence of intermediate steps obscures the origin of the parity dependence. In the revision we will insert the complete derivations for the zero-mode inverse localization length at n=1,2,3,4, together with the explicit functional forms. These steps will explicitly demonstrate how the parity of n determines whether the cos x sign structure is preserved or removed, thereby grounding the distinct reentrant regimes for odd and even n. revision: yes
Referee: [Finite-β regime and interpolation] The robustness statement that the parity-dependent structure remains intact throughout the finite-β interpolation is supported only by selected numerical diagrams; no systematic scan or quantitative measure (e.g., width of reentrant windows versus β) is given to show that the odd/even distinction does not collapse or shift for intermediate β values.

Authors: We agree that a more quantitative demonstration is warranted. The revised manuscript will contain a systematic scan over a dense grid of β values for representative odd and even n. We will add plots of the reentrant-window widths versus β, together with a quantitative measure of the separation between odd-n and even-n boundaries, confirming that the parity-dependent structure persists without collapse or significant shift throughout the interpolation. revision: yes

Circularity Check

0 steps flagged

No circularity: phase boundaries obtained from direct solution of the modulated Hamiltonian

full rationale

The derivation computes the zero-mode inverse localization length analytically for n=1–4 and evaluates a real-space topological indicator numerically on the SSH chain with explicit power-law quasiperiodic term; neither step reduces to a fitted parameter renamed as a prediction nor to a self-citation whose content is presupposed. The parity-dependent reentrance follows from the algebraic sign structure of cos(x) under the modulation, which is an input definition rather than an output. The reported phase diagrams are therefore independent of the target claims and do not collapse by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard SSH Hamiltonian with added modulation, the assumption that the topological indicator tracks phases correctly, and standard math for localization length. No new entities are postulated.

free parameters (2)

beta
Tunable parameter interpolating between smooth power-law quasiperiodic and sign-function limits
n
Positive integer exponent setting modulation power and parity properties

axioms (2)

domain assumption The real-space topological indicator correctly identifies the topological phase for the modulated chain
Used to map phase diagrams with the analytical localization length
standard math The zero-mode inverse localization length is analytically obtainable for n=1,2,3,4 and marks transitions
Basis for exact transition conditions in the abstract

pith-pipeline@v0.9.0 · 10780 in / 1440 out tokens · 69737 ms · 2026-05-09T15:09:28.250030+00:00 · methodology

discussion (0)

Reference graph

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