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Machine learning prediction of the convergence criterion for a topological invariant of finite non-Hermitian chains

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Pith's one-line read Crop-length for non-Hermitian topology predicted by decay lengths

desk verdict The paper shows that the crop-length for the polar-decomposition invariant in finite non-Hermitian chains is controlled by skin-effect localization lengths, with ML predictions generalizing well across model classes. read the letter →

arxiv 2607.05900 v1 pith:5DWQKC7B submitted 2026-07-07 cond-mat.mes-hall quant-ph

Raghav Chaturvedi , Viktor Könye , Ewelina M. Hankiewicz This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords non-Hermitiantopologyskineffectpolardecompositionreal-spacetopologicalinvariantcrop-lengthlocalizationlengthrandomforestregressionHatano-Nelsonmodel
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

The paper addresses a practical problem in finite non-Hermitian chains exhibiting the non-Hermitian skin effect: a real-space topological invariant called the polar-decomposition invariant (w_PD) only converges to the correct integer winding number if one chooses a boundary cutoff parameter called the crop-length. This parameter was previously selected empirically. The authors show that the required crop-length is controlled by physical localization lengths derived from the roots of the equation H(β) − E_B = 0, where β is a spatial multiplier and E_B is a base energy in the complex plane. For chains with a single hopping range, a single decay length suffices and the crop-length is approximately a constant times that length, with the constant growing as log(1/ε) for tolerance ε. For chains with multiple hopping ranges, multiple decay channels exist; the dominant one is the root channel closest to the unit circle in the complex β-plane. A random-forest regressor trained on these root-derived features predicts the crop-length with R² above 0.99 across model classes, generalizing to unseen Hamiltonians and complex base energies, and the predictor trained on clean systems remains stable under moderate disorder.

What carries the argument

The polar decomposition H − E_B = QP yields a local topological marker d_n whose central-window average gives w_PD(ℓ). The crop-length ℓ controls the window. The roots β_i of H(β) − E_B = 0 define decay exponents κ_i = |log|β_i|| and localization lengths ξ_i = 1/(2κ_i). For mixed hopping, roots are grouped into physical channels by shared decay rate, and the channel closest to the unit circle dominates. A random forest trained on these root-derived features predicts ℓ⋆.

What would settle it

If one constructs a non-Hermitian chain where the real-space invariant w_PD systematically deviates from the Bloch winding number by an amount that does not decay exponentially with crop-length, or where the deviation depends on quantities unrelated to the roots of H(β) − E_B, the decay-length control mechanism would fail and the random-forest predictor would not generalize.

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Extended reading notes

Core claim

The convergence criterion of the polar-decomposition real-space topological invariant in finite non-Hermitian chains is governed by the decay lengths of the non-Hermitian skin effect, which are read off from the roots of H(β) − E_B = 0. For single-range hopping, one length suffices; for mixed hopping, the full signed radial root structure is needed, but the slowest-decaying channel dominates. The relationship between crop-length, tolerance, and decay length follows ℓ⋆ ≈ C_ε ξ with C_ε ≈ log(1/ε) + const, and random-forest regression on root features captures finite-size corrections beyond this simple scaling while preserving physical interpretability.

Load-bearing premise

The paper assumes that the clean momentum-space winding number computed from the Bloch Hamiltonian is the correct reference topology for the finite open chain, so that any discrepancy between the real-space invariant and this winding is entirely a finite-size boundary effect fixable by the crop-length. If the real-space invariant has systematic biases unrelated to boundary effects — for instance from the polar decomposition itself in certain parameter regimes — the croplength

Editorial extensions

If this is right

  • The crop-length can be selected automatically for finite non-Hermitian systems without empirical tuning, making real-space topological invariants practical for experimental and numerical studies.
  • The tolerance dependence ℓ⋆ ≈ ξ log(1/ε) provides a method to extract the skin-effect localization length directly from how the crop-length varies with tolerance, without knowing microscopic hopping parameters.
  • The signed full-root representation generalizes to arbitrary finite hopping range R, suggesting a universal feature set for crop-length prediction in one-dimensional non-Hermitian chains.
  • The stability of the clean-trained predictor under moderate disorder suggests that the decay-length physics is robust enough that disorder-averaged topological characterization may not require retraining.

Reading between the lines

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

  • If the exponential decay model Δ(ℓ) ∼ A e^{−ℓ/ξ} holds universally, one could derive a parameter-free crop-length formula for any non-Hermitian chain given only its root structure, eliminating the need for machine learning entirely in regimes where finite-size corrections are negligible.
  • The connection between root proximity to the unit circle and crop-length suggests that chains whose roots cluster near |β| = 1 — i.e., near a topological phase transition — may require system sizes exponentially larger than the localization length for any valid crop to exist, setting a fundamental limit on real-space topological characterization in finite systems.
  • The success of root-derived features as ML inputs raises the question of whether analogous root-based or transfer-matrix-based decay lengths control convergence of real-space invariants in higher-dimensional or interacting non-Hermitian systems.
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Editorial analysis

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

Referee Report

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Summary. This manuscript studies the crop-length parameter ℓ* that controls convergence of the polar-decomposition real-space topological invariant w_PD in finite non-Hermitian chains exhibiting the skin effect. The authors show analytically and numerically that ℓ* is governed by physical decay (localization) lengths derived from the roots of the characteristic equation H(β) − E_B = 0. For nearest-neighbor and pure m-hop Hatano–Nelson chains, ℓ* ≈ C_ε ξ with C_ε ≈ log(1/ε) + const, where ξ is the skin-effect localization length. For mixed-hopping models, multiple decay channels enter, and the dominant scale is the grouped root channel closest to the unit circle. Random-forest regression on root-derived features predicts ℓ* with R² > 0.99 across model classes, generalizing to unseen Hamiltonians and complex base energies via leave-one-JL-out validation. A secondary result tests robustness of the clean predictor under hopping disorder.

Significance. The paper addresses a practical problem—choosing the crop-length empirically—that directly limits the applicability of the real-space invariant w_PD to finite non-Hermitian systems. The central physical insight (decay lengths control convergence) is independently grounded: ξ is derived analytically from hopping parameters (Eqs. 16, 40, 58), not fitted to crop-length data. The log(1/ε) scaling of the prefactor C_ε (Eq. 32, Fig. 2d) is a falsifiable prediction that allows extraction of localization lengths from tolerance dependence alone. The leave-one-JL-out and branch-aware train-test protocols are appropriate and strengthen the generalization claims. Reproducible code and data are provided on Zenodo. The identification of when a single-length predictor fails (mixed model, |w|=1 sectors, R²=0.18) and the remedy via the full signed root vector is a honest and useful finding.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for a careful and positive assessment of our manuscript. The referee's summary accurately captures the main results: that the crop-length ℓ* governing convergence of the polar-decomposition invariant w_PD is controlled by skin-effect decay lengths derived from the roots of H(β) − E_B = 0, that the log(1/ε) scaling of the prefactor C_ε provides a falsifiable prediction, and that random-forest regression on root-derived features predicts ℓ* with high accuracy across model classes, generalizing to unseen Hamiltonians and complex base energies. The referee recommends minor revision. As the referee did not raise any major comments requiring changes to the manuscript, we have no specific revisions to report. We are grateful for the referee's recognition of the physical grounding of our approach, the appropriateness of our validation protocols, and the honest reporting of cases where the single-length predictor fails.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central physical claim is independently grounded, with only minor self-citation that is not load-bearing.

full rationale

The paper's central claim—that the crop-length ℓ⋆ is controlled by physical decay (localization) lengths derived from the roots of H(β)−E_B=0—is not circular. The localization length ξ is derived analytically from the hopping parameters (Eq. 16 for nearest-neighbor, Eq. 40 for pure m-hop, Eqs. 49-52 for mixed hopping), not fitted to the crop-length data. The proportionality constant C_ϵ is fitted, but its log(1/ϵ) dependence is explained by an independent physical argument: the exponential-decay model Δ(ℓ)∼Ae^{−ℓ/ξ} (Eq. 29), which yields ℓ⋆ > ξ log(1/ϵ) + ξ log A (Eq. 31) purely from the crop condition Δ(ℓ⋆)<ϵ. This derivation is self-contained and does not assume its conclusion. The random-forest regression on root-derived features (Eqs. 18, 42, 66) predicts ℓ⋆ with R²>0.99, but the features are constructed from the Hamiltonian parameters, not from the target variable. The paper does cite prior work by the same authors (Refs. [29, 31]) for the polar-decomposition invariant w_PD, but the invariant itself is defined from first principles in Eq. (2)-(4), and its convergence to the Bloch winding w in the thermodynamic limit is attributed to Ref. [42] (Claes and Hughes), an independent group. The disorder robustness test (Section VI) uses the clean Bloch winding as reference, which is a correctness concern (the clean winding may not be the correct topological invariant for disordered chains), but this is not circularity—it is an externally falsifiable assumption, not a definition disguised as a result. No step in the derivation chain reduces to its own inputs by construction.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities, particles, or forces. The free parameters are phenomenological fitting constants for finite-size corrections, not fundamental constants. The axioms are standard domain assumptions in non-Hermitian topology, with the exponential-decay model being a physically motivated but unrigorously derived ansatz.

free parameters (3)
  • C_ϵ = varies with ϵ; ~log(1/ϵ)+const
    Proportionality constant between crop-length and localization length, fitted to data for each tolerance. Its log(1/ϵ) dependence is explained by the exponential-decay model but the constant offset is fitted.
  • (a, b, c) in F(x)=a/(x+b)+c = (2.329, -5.194, 7.00e-3) for nearest-neighbor at N=200
    Three-parameter phenomenological finite-size correction to the crop-length scaling, fitted to numerical data in Eq. (26). No first-principles derivation provided.
  • (a_|w|, b_|w|) in Eq. 54 = (1.001, -0.096) for |w|=1; (1.001, 0.493) for |w|=2
    Sector-dependent coefficients for the tolerance scaling of the mixed model, fitted to data.
assumptions (3)
  • domain assumption The clean momentum-space winding number w from the Bloch Hamiltonian is the correct topological reference for the finite open chain.
    Used in Eq. (5) to define the convergence criterion Δ(ℓ)=|w_PD(ℓ)−w|. This is standard in the non-Hermitian topology literature but is a nontrivial assumption for finite systems where OBC and PBC spectra differ drastically.
  • domain assumption The boundary contribution to the real-space invariant decays exponentially as Ae^{−ℓ/ξ} (Eq. 29).
    Invoked in Sec. II to explain the log(1/ϵ) dependence of C_ϵ. Physically motivated by the exponential localization of skin-effect modes but not rigorously derived for the polar-decomposition invariant.
  • domain assumption Random-forest regression generalizes from training Hamiltonians to unseen Hamiltonians within the same model class.
    Validated empirically by leave-one-JL-out and held-out-Hamiltonian protocols, but assumed as a methodological premise.

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Cite this review

Pith. "Pith review of Machine learning prediction of the convergence criterion for a topological invariant of finite non-Hermitian chains." pith.science (2026). https://pith.science/paper/5DWQKC7B

@misc{pith2026260705900,
  author       = {Pith},
  title        = {Pith review of: Machine learning prediction of the convergence criterion for a topological invariant of finite non-Hermitian chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5DWQKC7B}},
  note         = {Machine review of arXiv:2607.05900}
}
read the original abstract

A topological invariant based on polar-decomposition of matrices correctly captures the topology of finite non-Hermitian chains exhibiting the non-Hermitian skin effect, provided that an appropriate crop-length parameter is chosen. This parameter, which sets the cutoff used in the calculation of the invariant, is usually chosen empirically and becomes especially important near topological phase transitions, where finite-size effects are strongest. Here we show that the required crop-length is controlled by physical decay (localization) lengths. For nearest-neighbor and pure longer-range hopping Hatano-Nelson-type chains, the crop-length is set mainly by a single localization length and is well approximated by a scalar multiple of that length. For more general longer-range hopping models, it is governed instead by a multichannel root structure of the characteristic polynomial. Random-forest regression captures finite-size and near-boundary corrections while preserving this decay-length interpretation. Trained on one set of Hamiltonians, the predictor accurately generalizes to unseen Hamiltonians and complex base energies, reproducing crop-lengths across full phase diagrams. We further show that the predictions learned from clean nearest-neighbor hopping chains remain stable under moderate hopping disorder. These results provide a practical and physically interpretable way to choose the crop-length, which in turn determines when the real-space invariant can reliably capture the topology of finite non-Hermitian chains.

Figures

Figures reproduced from arXiv: 2607.05900 by the authors.

Figure 1
Figure 1. (a) Finite-size HN chain with asymmetric nearest [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (a) Convergence of the real-space polar [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. (a) Complex spectrum of Hm(k) for m = 2. The spectrum is an ellipse in the complex-energy plane, but it winds two times around the base point EB = 0. (b-d) Real￾space polar-decomposition invariant wPD across the point-gap transition for m = 2, 3, 4, compared with the exact clean winding w. recovered from the real-space polar-decomposition invari￾ant wPD in finite open chains. This is shown in Figs. 3(b) to 3(d) for … view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: (a) One-feature logistic fits for the valid-crop [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: (a) Schematic of the open chain with first- and [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: (a) Numerical ℓ⋆ compared with the RF-predicted ℓ⋆ from a random-forest model trained on all nonzero winding sectors. (b,c) Numerical ℓ⋆ compared with the RF-predicted ℓ⋆ from sector-resolved models for w = 1 and w = 2, re￾spectively. The dashed line denotes perfect pr…
Figure 7
Figure 7. Figure 7: Random-forest prediction of the crop-length [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: (a) Exact k-space winding w(EB) over the complex energy plane for one representative sample. (b) Reconstructed real-space invariant wPD evaluated using the RF-predicted crop-length. Gray points denote base energies for which no valid crop-length exists at this N and to…
Figure 10
Figure 10. Figure 10: Numerical crop-length ℓ⋆(EB) for two mixed-hopping Hamiltonians with (a) (JR1, JL1, JR2, JL2) = (0.328, 0.238, 0.234, 1) and (b) (0.668, 1, 0.022, 0.877), respec￾tively. Here N = 256 and ϵ = 0.1. Gray regions denote the trivial w(EB) = 0 sector, while white pixels nea…
Figure 12
Figure 12. Figure 12: Fixed-crop finite-size diagnostic for pure [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

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Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.