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REVIEW 3 major objections 5 minor 63 references

Topological finite size effect in one-dimensional chiral symmetric systems

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In finite chiral-symmetric chains, the real-space winding number fails exactly when the mid-gap edge states overlap through the bulk, and the paper proposes a bulk-conductivity criterion that tracks the anomaly and is experimentally…

desk verdict A useful finite-size diagnostic in chiral 1D chains, but the bulk-conductivity criterion is really a numerical amplitude cutoff, not a transport quantity. read the letter →

arxiv 2411.17822 v1 pith:GKIJ2R3A submitted 2024-11-26 cond-mat.mes-hall quant-ph

Guliuxin Jin , D. O. Oriekhov , Lukas Johannes Splitthoff , Eliska Greplova This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords real-spacewindingnumberSSHmodelextendedfinite-sizeeffectsbulkconductivityofedgestateschiralsymmetrydisorder
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Finite chunks of chiral-symmetric one-dimensional wires do not always show the topology that the infinite crystal predicts: the real-space winding number (RSWN) for a short SSH chain switches from trivial to non-trivial at a hopping ratio smaller than the bulk value $v/w=1$, and the extended SSH model acquires spurious $\nu=0$ patches that persist to hundreds of unit cells. The paper proposes that this finite-size anomaly is controlled by whether the two mid-gap edge states hybridize through the bulk, and it defines a bulk-conductivity criterion: look at the average wavefunction amplitude of the edge modes on the two central unit cells and call the bulk conducting if that amplitude exceeds $10^{-15}$. Numerically, the conducting/insulating boundary coincides with the anomalous RSWN transition, so a simple transport measurement can serve as an experimentally accessible indicator of the real-space topological invariant. The same criterion is then shown to survive chiral-symmetric hopping disorder and to be sharpened by chiral-symmetry-breaking chemical-potential disorder, which pushes the transition back toward the ideal $k$-space value.

What carries the argument

The mechanism is the real-space winding number formula $\nu = -\frac{1}{L}\operatorname{Tr}\{P_B Q P_A [X, P_A Q P_B]\}$, built from sublattice projectors and the flat-band Hamiltonian $Q=H/|H|$. The new ingredient is the bulk-conductivity criterion of Eq. (4): the average of the mid-gap eigenstate amplitudes on the central two unit cells must fall below $10^{-15}$ for the bulk to count as insulating. The analytic continuum model supplies the exponential splitting $E_0 \sim \sqrt{2}\,\Delta\,e^{-L\Delta/w}$ between symmetric and antisymmetric edge states, which is what lets a hybridized pair cancel one unit of RSWN through the trace of position matrix elements; when the level broadening $\Gamma$ exceeds $E_0$, the choice of localized edge-state basis becomes legal and the RSWN returns to its $k$-space value.

What would settle it

Take an SSH chain of length $N$ with $v/w$ just below the finite-size RSWN transition and compute the mid-gap amplitude at the central unit cells while sweeping the threshold from $10^{-12}$ to $10^{-18}$; if the conducting/insulating boundary moves substantially, the criterion is an artifact of the cutoff. A sharper test is to compute the actual two-terminal zero-energy transmission through the chain and check whether the transmission step lines up with the $10^{-15}$ amplitude boundary for several system sizes; if the transport step occurs at a different hopping ratio, the amplitude criterion does not track the real conductance.

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Extended reading notes

Core claim

On its own terms, the paper establishes that the covariant real-space winding number is a reliable invariant only when edge-state overlap through the bulk is negligible. In finite systems of the SSH and extended SSH families, symmetric and antisymmetric pairs of mid-gap edge states hybridize with an energy splitting $E_0 \simeq \sqrt{2}\,\Delta\,e^{-L\Delta/w}$; the RSWN then loses one unit for each hybridized pair, which shifts the apparent phase boundary below $v/w=1$ and creates the anomalous $\nu=0$ regions. The paper's new criterion classifies the mid-gap states' bulk wavefunction amplitude with a $10^{-15}$ threshold, and the resulting conducting/insulating phase diagram matches the RSWN's finite-size diagram: wherever the bulk is conducting the RSWN reads $0$, and wherever it is insulating the RSWN takes its nonzero value. The authors conclude that bulk conductance is an experimentally accessible indicator of the RSWN, and that combining it with the RSWN diagnoses topological protection in finite, disordered devices.

Load-bearing premise

The classification of a bulk as conducting or insulating rests on a fixed $10^{-15}$ threshold on the mid-gap wavefunction amplitude, which the paper takes as a stand-in for physical level broadening; if that identification is wrong, the conducting boundary is a numerical artifact rather than a measurable property.

Editorial extensions

If this is right

  • In an SSH chain of a few unit cells, the RSWN transition sits below $v/w=1$ and moves toward $1$ as the chain lengthens, and the bulk-conductivity transition moves with it.
  • In the extended SSH model, an anomalous $\nu=0$ region appears at finite size and survives to at least $N=512$; the region where the mid-gap states conduct through the bulk coincides with that $\nu=0$ region.
  • A transport experiment that measures whether the two mid-gap modes conduct through the bulk can label the finite-size topological phase even when the RSWN alone is unreliable.
  • Chiral-symmetric hopping disorder at the 5 percent level leaves the RSWN phase diagrams essentially unchanged, while chiral-symmetry-breaking chemical-potential disorder shifts the transitions closer to the $k$-space values and narrows the anomalous $\nu=0$ plateau.
  • Near each phase-transition line only the softest pair of edge states hybridizes, so in a $\nu=2$ phase one pair may be conducting while the other remains localized, making topological protection partial.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not spell out, but its exponential splitting formula implies, a design rule: because $E_0 \sim \sqrt{2}\,\Delta\,e^{-L\Delta/w}$, the anomalous RSWN regions shrink exponentially once $L\Delta/w$ is large, so the system size at which the anomaly matters can be predicted from the bulk gap alone.
  • The criterion reads only the central-cell amplitude of the zero modes, so it should transfer to any chiral-symmetric platform with local density readout, even when the full winding number cannot be measured directly.
  • A sharper version of the proposed indicator would replace the fixed $10^{-15}$ amplitude cutoff by a real two-terminal zero-energy transmission calculation; whether the transport step coincides with the amplitude boundary across system sizes is a direct test of the paper's central identification.
  • The symmetry-breaking disorder result can be inverted into a design strategy: deliberately adding weak on-site disorder to a small topological device could pull the apparent transition back to the bulk value, at the cost of partially breaking the chiral symmetry that protects the edge states.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript studies finite-size corrections to the real-space winding number (RSWN) in one-dimensional chiral-symmetric models, specifically the SSH model and the extended SSH model with third-neighbor hoppings. The authors show that the RSWN computed from Eq. (16) deviates from the momentum-space winding number for finite chains, and they propose a 'bulk conductivity' criterion, Eq. (4), that classifies the mid-gap edge states as conducting or insulating based on the average wave-function amplitude at the chain center, with a fixed threshold of 10^-15. They find numerically that the region classified as conducting coincides with the region where the RSWN is anomalously reduced (Figs. 7 and 8), and they support this with a continuum low-energy analysis in Sec. V that traces the RSWN reduction to symmetric/antisymmetric hybridization of edge states. They also study hopping and chemical-potential disorder and report that chiral-symmetry-breaking chemical-potential disorder shifts the finite-size transition toward the k-space prediction.

Significance. If the proposed criterion were physically grounded, it would offer a practical, finite-size-aware way to interpret RSWN anomalies in the experimental platforms named in the paper, such as Rydberg atom arrays, superconducting resonator chains, and semiconductor quantum dots. The manuscript has notable strengths: a pedagogical re-derivation of the covariant real-space winding number, a transparent continuum calculation in Sec. V that identifies the edge-state hybridization mechanism, a systematic numerical study across system sizes and disorder strengths, and a public code/data repository. However, the central claim that the bulk-amplitude threshold is an experimentally accessible conductance indicator is currently supported only by a numerical cutoff, not by a transport calculation or a physical broadening scale. The observed alignment between the 'conducting' region and the RSWN anomaly is a correlation between two quantities computed from the same eigenstates; it is consistent with the hybridization mechanism rather than an independent experimental benchmark. The significance is therefore conditional on the authors either upgrading Eq.

major comments (3)
  1. [Sec. II C, Eq. (4), and Sec. V] The bulk-conductivity criterion is set by the float64 machine precision 10^-15, and Sec. V explicitly identifies numerical solver accuracy with level broadening. This identification is the load-bearing step for the claim in Sec. IV B that the bulk conductance is an experimentally accessible indicator of the RSWN. For a physical system, the conductance boundary is set by comparing the hybridization energy E0 of the two mid-gap states with a physical level broadening Γ (due to leads, temperature, or environment). For the exponential edge-state tails studied here, E0/Δ ~ A^2, where A is the mid-chain amplitude; the threshold A = 10^-15 corresponds to E0/Δ ~ 10^-30, orders of magnitude below any realistic Γ/Δ (typically 10^-3 to 10^-8). The manuscript provides no Landauer or scattering calculation, no lead-coupling model, and no argument that the amplitude criterion is equivalent to a conductance measurement. As written, the overlap in Figs. 7-8 may be an artifact of the numerical cutoff rather than a physically realizable transport signature. The authors should either replace Eq. (4) with a physical threshold (e.g., E0 versus Γ) and recompute the phase diagrams, or restrict the claim to a numerical diagnostic and remove the phrase 'experimentally accessible indicator.'
  2. [Sec. IV B] The statement that 'our numerical study strongly indicates that the bulk conductance serves as an experimentally accessible indicator of the RSWN' is stronger than the evidence presented. The alignment between the bulk-amplitude criterion and the RSWN is a comparison of two quantities that are both derived from the same exact eigenstates of the same Hamiltonian; the correlation is therefore expected from the common hybridization mechanism and does not by itself constitute an independent benchmark of the criterion. To substantiate the 'indicator' claim, the manuscript should test the criterion against a genuinely independent transport quantity, such as the two-terminal conductance of a finite chain coupled to leads, or at least show that the criterion predicts the RSWN anomaly in parameter regions that were not used to define the threshold.
  3. [Sec. V, Eq. (23)] The asymptotic prefactor in Eq. (23), E0 ~ ±√2 Δ e^{-LΔ/w}, appears inconsistent with the known SSH finite-size splitting, which scales as 2(v/w)^{N+1} ≈ 2 Δ e^{-LΔ/w} near the transition. As typeset, Eq. (23) is also difficult to parse because of the nested expression under the square root. This issue does not affect the qualitative exponential-decay argument, which is the main point of the section, but the prefactor and the derivation should be corrected or carefully stated if the equation is to be used quantitatively.
minor comments (5)
  1. [Sec. II C, Eq. (4)] The criterion in Eq. (4) uses an average of wave-function amplitudes, not probabilities; this should be stated explicitly or changed to probabilities to avoid confusion with tunneling or conductance quantities.
  2. [Sec. III] The normalization of the trace in Eq. (16) should be clarified: the text uses 'trace per volume' in Eq. (7) but writes 1/L in Eq. (16), and the connection between the two conventions is not explained.
  3. [Sec. V] The sentence 'the level broadening could be replaced by the accuracy of numerical solver' should be rewritten, because numerical precision is not a physical broadening mechanism; if the authors intend this as an analogy, the limits of that analogy should be stated.
  4. [Sec. VI B] When chiral symmetry is broken by chemical-potential disorder, the RSWN formula of Eq. (16) is no longer symmetry-protected; the manuscript should state whether the plotted values remain quantized and how the invariant is defined in that case.
  5. [Throughout] There are several typographical and style issues: 'R WSN' appears instead of 'RSWN' in Sec. III, 'mig-dap' should be 'mid-gap', 'Su-Schriefer-Heeger' should be 'Su-Schrieffer-Heeger', and 'vita' in Sec. VII should be 'vital'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the RSWN and the bulk-conductivity criterion are independently defined, and their correlation is analytically explained rather than fitted.

full rationale

No circular steps were found. The real-space winding number (Eq. 16) is an independent covariant formula (Refs. 17-19) applied to the model Hamiltonians, while the bulk-conductivity criterion (Eq. 4) is a separately defined threshold on mid-chain amplitudes; no parameter is fitted to the RSWN output. The observed coincidence of the 'conducting' regions with the RSWN nu = 0 regions is supported by the analytic calculation in Sec. V, where hybridized symmetric/antisymmetric edge states are shown to reduce the winding-number trace (Eq. 26); this derivation is made from the Hamiltonian and does not invoke Eq. 4. The only self-citations (Refs. 24 and 33) are motivational or contextual and do not carry the central claim. Sec. V's statement that 'the level broadening could be replaced by the accuracy of numerical solver' is an explicit modeling assumption and a legitimate validity caveat, but it is not a circular reduction: the numerical threshold is not fitted to the predicted phase boundary, and the analytic overlap estimate (Eq. 23) is parameter-free. The paper is therefore judged not circular, with a low score reflecting only minor non-load-bearing self-citation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the RSWN formula from prior work, a linearized continuum approximation, and the ad hoc identification of numeric precision with physical level broadening. No new particles, forces, or conserved quantities are introduced.

free parameters (1)
  • bulk conductivity threshold epsilon = 10^-15
    Hand-chosen as the numerical precision of NumPy float64 (Sec. II.C). It determines where the conducting versus insulating boundary is drawn and thus the claimed phase diagram overlap.
assumptions (4)
  • domain assumption The real-space winding number formula Eq. (16) from Ref. [17] correctly computes the topological invariant for finite systems when chiral symmetry is preserved.
    The paper relies on this formula throughout; its finite-size anomaly is the starting point of the work.
  • domain assumption The low-energy continuum description near k=pi, keeping only linear terms in q, captures the edge-state physics relevant for the RSWN.
    Used in Sec. V to derive the hybridization energy Eq. (23); the derivation ignores quadratic and higher-order terms.
  • ad hoc to paper Only the pair of edge states associated with the smallest gap contributes to the RSWN reduction; other pairs remain localized and do not affect the invariant.
    Stated in Sec. V for the extended SSH model without quantitative justification; it is needed to explain the phase diagrams in Fig. 7.
  • ad hoc to paper Numerical precision (10^-15) can be identified with a physical level broadening that decides whether edge states are topologically protected.
    This identification is the bridge between the overlap criterion and the claim of experimental accessibility, appearing in Sec. V.

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Cite this review

Pith. "Pith review of Topological finite size effect in one-dimensional chiral symmetric systems." pith.science (2026). https://pith.science/paper/GKIJ2R3A

@misc{pith2026241117822,
  author       = {Pith},
  title        = {Pith review of: Topological finite size effect in one-dimensional chiral symmetric systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKIJ2R3A}},
  note         = {Machine review of arXiv:2411.17822}
}
read the original abstract

Topological phases of matter have been widely studied for their robustness against impurities and disorder. The broad applicability of topological materials relies on the reliable transition from idealized, mathematically perfect models to finite, real-world implementations. In this paper, we explore the effects of finite size and disorders on topological properties. We propose a new criterion for characterizing finite topological systems based on the bulk conductivity of topological edge modes. We analyze the behavior of bulk conductivity and real space topological invariants both analytically and numerically for the family of SSH models. We show that our approach offers practical insights for topology determination in contemporary intermediate scale experimental applications.

Figures

Figures reproduced from arXiv: 2411.17822 by the authors.

Figure 1
Figure 1. FIG. 1. Lattice geometry of SSH model and extended SSH [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Examples of edge states in Extended SSH model. [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Energy spectrum of extended SSH models. The sys [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Real space winding number(RSWN) [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Mid-gap edge states bulk conductivity phase diagram [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Winding number calculated from the Extended SSH [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Real space winding number [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. RSWN of SSH model vs [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 7
Figure 7. Figure 7: B. Disorder in chemical potential µi A fundamental question in implementing topological phases of matter in real-world experiments is: What happens when perfect symmetry is slightly broken? In the SSH model, the chiral symmetry requires equal local chemical potentials …
Figure 11
Figure 11. Figure 11: FIG. 11. RSWN of Extended SSH model vs [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

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    Right: v = 0 .25, w= 0 .25, j3 = 2 .5, j3p = 0 .0, N= 71. Both panels are in the ν = 2 phase. The inset of each panel shows a zoom in of the zero energy region. Each insets shows four nearly degenerate mid-gap modes as a manifestation of ν = 2 phase. The corresponding mid-gap ...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.