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REVIEW 4 major objections 6 minor 66 references

Realization of a one-dimensional topological insulator in ultrathin germanene nanoribbons

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Ultrathin germanene nanoribbons realize a one-dimensional topological insulator with protected end states.

desk verdict A credible first shot at a 1D topological insulator, but the topological claim rests on an unverified symmetry and a parameter inconsistency; deserves a serious referee. read the letter →

arxiv 2411.18156 v1 pith:E56SMBIO submitted 2024-11-27 cond-mat.mes-hall cond-mat.mtrl-sci

Dennis J. Klaassen , Lumen A.G. Eek , Alexander N. Rudenko , Esra D. van`t Westende , Carolien Castenmiller , Zhiguo Zhang , Paul de Boeij , Arie van Houselt
show 4 more authors
Motohiko Ezawa Harold J.W. Zandvliet Cristiane Morais Smith Pantelis Bampoulis
This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci
keywords germanenenanoribbonsone-dimensionaltopologicalinsulatorquantumspinHalleffectendstatesZakphasemirrorsymmetryspin-orbitcoupling
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 claims that germanene nanoribbons undergo a width-driven topological transition: when a ribbon narrows below roughly 2 nm, the helical edge states of the two-dimensional quantum spin Hall phase disappear and sharply localized states appear at the two ends of the ribbon. These 0D end states are argued to be topologically protected by a combined mirror and time-reversal symmetry, making the ultrathin ribbons the first experimentally realized one-dimensional topological insulator with strong spin-orbit coupling. The case rests on scanning tunneling spectroscopy of ribbons with widths from about 1 to 6 nm, matched ribbon-by-ribbon to tight-binding calculations of the Zak phase. If correct, the result locates the lower-dimensional limit of a known two-dimensional topological phase and supplies a concrete platform in which protected boundary states can be packed into dense arrays and toggled by an electric field.

What carries the argument

The central object is the combined symmetry $M_x T$, mirror reflection across the ribbon axis composed with time reversal, which forces the Wilson-loop eigenvalues of a zigzag nanoribbon to be real and therefore quantizes the Zak phase to either 0 or $\pi$. The Zak phase is the Berry phase accumulated by the occupied Bloch states across the one-dimensional Brillouin zone. It is computed from a tight-binding Hamiltonian of Kane-Mele type, with intrinsic spin-orbit coupling $\lambda_{SO}$, a staggered mass $M_S$, and longer-range hopping, solved in a ribbon geometry for widths from one to fourteen hexagon cells. The paper defines the invariant $\nu \equiv (\varphi - \varphi_{M_S \to \pm\infty})/\pi \bmod 2$, where $\varphi_{M_S\to\pm\infty}$ is the Zak phase of the trivial large-mass limit; $\nu = 1$ signals two exponentially localized end states inside the minigap that opens in the edge-mode spectrum of sufficiently thin ribbons. This machinery also accounts for the even/odd hexagon-count dependence: even-width ribbons are topological in certain parameter windows while odd-width ribbons are not.

What would settle it

Compute a 2-hexagon zigzag germanene ribbon with a first-principles model that includes the Pt/Ge(110) substrate and Rashba spin-orbit coupling; if the mirror-time-reversal Zak phase is no longer quantized to 0 or $\pi$ (so $\nu$ is not 1 for the experimental geometry), the observed end state cannot be topologically protected. On the experimental side, deliberately breaking mirror symmetry at one ribbon end, for example by placing a single adatom on one edge, should destroy the protected end state while leaving a trivial termination state in place.

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Extended reading notes

Core claim

The paper reports the first experimental realization of a one-dimensional topological insulator with strong spin-orbit coupling, in zigzag-terminated germanene nanoribbons grown on Pt/Ge(110). For ribbons wider than about 2 nm, the nanoribbons behave as two-dimensional quantum spin Hall strips with helical edge states, consistent with earlier germanene measurements. Below that width, the edge modes fail to traverse the bulk gap and instead open a minigap, and two zero-dimensional states localize at the ribbon ends. The authors show by computing the Zak phase that these end states are not trivial termination modes: the combined mirror and time-reversal symmetry quantizes the Zak phase to 0 or $\pi$, giving a nontrivial invariant $\nu=1$ for the 2- and 6-hexagon-wide ribbons and $\nu=0$ for the 3-hexagon-wide ribbon, in agreement with the spectroscopy. In one dimension the time-reversal-symmetric class alone has no topological phase, so the protection must come from the additional mirror symmetry; the paper thus identifies a width-driven transition from a 2D class-AII insulator to a 1D topological crystalline phase.

Load-bearing premise

The topological identification of the end states assumes the combined mirror and time-reversal symmetry stays exact in the experimental nanoribbons even though the Pt/Ge(110) substrate and the neglected Rashba spin-orbit coupling could each break it.

Editorial extensions

If this is right

  • Nanoribbons narrower than about 2 nm lose their conducting edge channels, leaving only 0D end states, so the material behaves as a 1D topological insulator rather than a 2D quantum spin Hall strip.
  • The end states are protected only while the combined mirror and time-reversal symmetry is intact, so a perturbation that breaks the mirror symmetry of the zigzag termination should destroy them; this gives a sharp experimental signature for checking the claim.
  • The 6-hexagon ribbon is an intermediate case with both edge and end states, so the 2D-to-1D crossover is gradual, and the measured critical width of about 2 nm defines the practical packing limit for 1D edge channels in germanene nanoribbon arrays.
  • Because the topology depends on ribbon width and on whether the width is an even or odd number of hexagon cells, the same growth method can in principle pattern arrays in which selected ribbons are topological and selected ribbons are trivial.

Reading between the lines

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

  • If the central claim holds, the same width-driven crossover should appear in other honeycomb nanoribbons with strong spin-orbit coupling, such as silicene, stanene, or bismuthene, whenever the zigzag termination preserves the mirror-time-reversal symmetry; the STM recipe used here can be applied directly to those materials.
  • The even/odd hexagon-count dependence hints at a general parity selection rule for mirror-symmetric honeycomb nanoribbons; the paper's own phase diagram shows re-entrant topological windows at larger even widths, so a systematic experimental scan of widths beyond six hexagons could test whether end states reappear at those larger widths.
  • Because the end states sit in a minigap whose size shrinks with width, gating or substrate engineering that tunes the staggered mass could switch individual ribbons in and out of the 1D topological phase in situ, enabling addressable arrays of protected end states.
  • The authors compare the end states to Majorana-like zero modes; a stronger test would be to measure the end-state spin polarization or to couple two ribbons end-to-end and look for nonlocal correlations, which would distinguish a symmetry-protected charge mode from a genuine topological-superconductor end mode.
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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

4 major / 6 minor

Summary. The manuscript reports STM/STS measurements on germanene nanoribbons grown on Pt/Ge(110) and tight-binding calculations for zigzag-terminated nanoribbons. It claims a width-driven crossover: above a critical width of about 2 nm the ribbons show 2D quantum spin Hall edge states, while the thinnest (nominally 2-hexagon-wide) ribbon shows a 0D end-localized state near 75 meV and an absence of such a state in a 3-hexagon ribbon. The end states are interpreted as topologically protected by the combined action of mirror and time-reversal symmetry, and the authors claim this constitutes the first realization of a 1D topological insulator with strong spin-orbit coupling. The theoretical analysis uses a Kane-Mele-type model with intrinsic SOC, a staggered mass, and longer-range hopping, and defines a Zak-phase-based invariant nu to identify the topological phase.

Significance. If the central claim is correct, the paper is significant: it would demonstrate a 2D-to-1D topological transition in a single material platform, realize a 1D topological crystalline phase with strong spin-orbit coupling, and provide a pathway toward dense arrays of topologically protected states. The tight-binding and Zak-phase calculations are clearly presented and appear internally consistent, and the spatial mapping of an end-localized state is a valuable experimental observation. The significance is, however, conditional: the topological interpretation rests on symmetry and parameter assumptions that are not directly verified, and the experimental evidence is based on very few ribbons without statistical or control measurements.

major comments (4)
  1. [Main text (p. 5, 'To understand this') and SI §2.3] The value of the staggered mass MS is inconsistent between the main text and the Supplementary Information. The main text states that the calculations use lambda_SO = 0.3t, MS = 0.02t, t3 = 0.3t, while SI §2.3 states 'we compared the experimental results to the theoretical ones obtained for the parameters lambda_SO = 0.3t and MS = -0.04t', and the phase diagram in SI Fig. S7d uses MS = -0.04t. The sign and magnitude of MS determine whether the 2-hexagon ribbon is in the nu = 1 phase and the 3-hexagon ribbon is trivial (SI Figs. S7-S8). Since MS is not measured independently, the theory-experiment comparison is not robust to this ambiguity and the central claim is not yet well constrained.
  2. [SI §2.3 ('A direct comparison...')] The assignment of the thinnest measured ribbon as 2 hexagons wide is made from the LDOS peak: 'Therefore, we conclude that it is likely that the thinnest experimental ribbons are only 2-unit cells wide.' This makes the width assignment partly circular because the same spectroscopic signature is used both to identify the width and to validate the topological model. The width should be determined from atomically resolved STM topography of the ribbon and its end termination, independent of the dI/dV peak. The main-text inset of Fig. 3h does not provide atomic resolution of the end, so this circularity remains unresolved.
  3. [SI §2.1 and §2.2] The topological invariant nu defined by SI Eq. (4) is quantized only if the combined mirror-time-reversal symmetry MxT is preserved (SI §2.2). The experimental ribbons sit on a Pt/Ge(110) substrate, which can break mirror symmetry, and the model neglects Rashba spin-orbit coupling, which the authors themselves state is 'likely present' in the experiments (SI §2.1). No experimental evidence is provided that MxT survives in the measured ribbons—for instance, no atomically resolved image of the end termination that would establish a mirror-symmetric cut, and no test of robustness against symmetry-breaking perturbations. Until this is shown, the observed end-localized state could be a trivial termination state, and the topological protection claim is not established.
  4. [Fig. 3e-h and main text] The central experimental evidence consists of single dI/dV spectra from apparently single ribbons of each width (Fig. 3e-h). No error bars, number of ribbons, or repeated measurements are reported, and no control experiments are presented (e.g., ribbons with different end terminations, different lengths, or different tip conditions). For a claim as strong as 'the first realization of a 1D topological insulator with strong spin-orbit coupling', this statistical and control basis is too thin. At minimum, the authors should report the number of ribbons measured and show that the 2-hexagon end state and the 3-hexagon absence are reproducible.
minor comments (6)
  1. [Abstract and main text] The phrase 'contrary to the tenfold way classification' is misleading: class AII in 1D is trivial in the tenfold way, but the proposed phase is protected by the additional mirror symmetry MxT, making it a topological crystalline phase rather than a violation of the tenfold way. The wording should be adjusted.
  2. [Fig. 2g inset] The caption mentions 0D end states marked in the inset of a band structure for a 2-hexagon ribbon. Since a band structure with periodic boundary conditions along x cannot contain end states, the inset likely shows the open-boundary spectrum; please state this explicitly.
  3. [SI §2.1] The notation for SOC strength is inconsistent: the Hamiltonian (1) uses lambda_SO/(3sqrt(3)) in the hopping term, while the text often writes 'lambda_SO = 0.3t'. Please define the convention once and use it consistently throughout the main text and SI.
  4. [Conclusion] The closing statement that end states are 'akin to Majorana zero modes' and involve 'fractionalized electrons' is not supported by the measurements or by the class-AII-with-mirror model presented; this overreach should be toned down or removed.
  5. [Data availability] The data availability statement only offers data 'upon reasonable request'; for a claim of this significance, providing raw STS spectra and the tight-binding code would improve reproducibility.
  6. [Fig. 1g] The buckling upper bound of 0.35 Å is given without an associated measurement uncertainty or a statement of how many line profiles were averaged; please report typical values and sample-to-sample spread.

Circularity Check

2 steps flagged · score 6.0 of 10

Partial circularity: the 2-hex/3-hex comparison is constructed by fitting MS and by assigning the experimental ribbon width from the theoretical end-state peak.

  1. fitted input called prediction [Supplementary Information §2.3 (Persistence of 0D topological phase), around Figs. S7–S8]
    "In the main text, we compared the experimental results to the theoretical ones obtained for the parameters λSO = 0.3t and MS = −0.04t. From Figs. S7d and S7e, we conclude that for these parameters only the thin (2) ribbon should be topological. Figure S8 confirms this result, since only the thinnest ribbon shows a distinct peaked LDOS at the boundary (blue curve)."

    The staggered mass MS is not independently fixed in the paper; the SI explicitly leaves it open ('may be generated by the electric field... may also arise from a coupling between the ribbon and the substrate'). The value MS=−0.04t is chosen because the parameter scan shows a blue LDOS peak for the 2-hex ribbon and no peak for the 3-hex ribbon, reproducing the experimental 2-hex/3-hex dichotomy. The topological invariant ν=1 is then evaluated at this same fitted MS and used to claim that the observed end states are topologically protected. The 'prediction' of topological end states in the 2-hex ribbon is therefore an output of a model whose input was selected to produce that output.

  2. self definitional [Supplementary Information §2.3, final conclusion of the first parameter analysis]
    "Figure S8 confirms this result, since only the thinnest ribbon shows a distinct peaked LDOS at the boundary (blue curve). Therefore, we conclude that it is likely that the thinnest experimental ribbons are only 2-unit cells wide."

    The number of hexagons across the experimental ribbon is inferred from the theoretical LDOS: the 2-hex model produces a boundary peak, so an experimental ribbon that shows a boundary peak is labeled '2-unit cells wide'. The main text then uses this label to present Fig. 3h as a 2-hexagon ribbon whose end state is 'fully supported' by the 2-hexagon calculation (Fig. 3d). The observation used to assign the input (the width label) is the same observation the model is invoked to explain, so the agreement is constructed rather than independent.

full rationale

The tight-binding/Zak-phase derivation itself is largely self-contained: the Hamiltonian, Wilson-loop definition, and MxT quantization argument in SI §2.1–2.2 are standard and do not import the experimental result. The raw STS/LDOS data are independent measurements. However, the paper's central comparison—'the theoretical description captures the main experimental features'—is partly circular. The staggered mass MS is a free parameter, and the value used for the final comparison is selected from a parameter scan that yields the observed 2-hex/3-hex end-state pattern; the topological invariant is then computed with that same selected MS. In addition, the thinnest experimental ribbon is assigned to be '2-unit cells wide' from the theoretical LDOS peak, so the ribbon label used in the main-text comparison is inferred from the very peak the model is said to predict. The inconsistency between main-text Ms=0.02t and SI MS=−0.04t reinforces that the parameter was not fixed in advance by an independent first-principles constraint. What remains independent is the experimental observation of strong end-localized LDOS and the mathematical fact that a narrow even-width Kane-Mele ribbon with finite MS and λSO can support ν=1 end modes. But the specific claim that the measured thinnest ribbon is a 2-hexagon 1D topological insulator depends on parameters and a width label chosen to match the measured LDOS, giving partial circularity rather than a parameter-free prediction.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The central claim rests on a tight-binding model whose parameters (λSO, t3) are taken from prior literature and whose staggered mass MS is chosen to reproduce the observed even/odd width behavior; the value of MS is not measured experimentally and is stated differently in the main text (0.02t) and SI (-0.04t). The symmetry protection argument additionally assumes the substrate preserves MxT symmetry, which is not verified. No new physical entities are introduced.

free parameters (4)
  • Spin-orbit coupling strength λSO = 0.3t
    Chosen from literature (Refs 52,53), not measured in this paper; it enters the Kane-Mele model and determines the topological phase boundaries.
  • Staggered on-site mass MS = -0.04t (SI) / 0.02t (main text)
    Chosen so that the 2-hexagon ribbon is topological (ν=1) and the 3-hexagon ribbon is trivial (ν=0), matching the observed presence or absence of end states. The experimental value is not measured, and the paper states inconsistent values.
  • Long-range hopping t3 = 0.3t
    Taken from literature (Refs 64,65) for graphene and germanene; included because it enlarges the topological parameter range.
  • LDOS broadening b = 0.05 eV
    Artificial Lorentzian broadening used in the LDOS calculation (SI Eq. 7); affects the visual comparison with STS spectra but not the topology.
assumptions (3)
  • domain assumption Kane-Mele model with the cited parameters describes germanene nanoribbons.
    The model is taken from Refs 52,53; its validity for the experimental system, including the Pt-intercalated substrate, is assumed.
  • domain assumption The combined mirror Mx and time-reversal T symmetry is preserved in the experimental nanoribbons.
    The quantization of the Zak phase relies on MxT symmetry (SI section 2.2); the substrate and the ribbon's end termination are assumed not to break it, but this is not experimentally verified.
  • standard math STS dI/dV signal is proportional to the local density of states with the assumed broadening.
    Standard STM/STS interpretation; the paper uses this to compare spectra to LDOS calculations.

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Cite this review

Pith. "Pith review of Realization of a one-dimensional topological insulator in ultrathin germanene nanoribbons." pith.science (2026). https://pith.science/paper/E56SMBIO

@misc{pith2026241118156,
  author       = {Pith},
  title        = {Pith review of: Realization of a one-dimensional topological insulator in ultrathin germanene nanoribbons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E56SMBIO}},
  note         = {Machine review of arXiv:2411.18156}
}
read the original abstract

Realizing a one-dimensional (1D) topological insulator and identifying the lower dimensional limit of two-dimensional (2D) behavior are crucial steps toward developing high-density quantum state networks, advancing topological quantum computing, and exploring dimensionality effects in topological materials. Although 2D topological insulators have been experimentally realized, their lower dimensional limit and 1D counterparts remain elusive. Here, we fabricated and characterized arrays of zigzag-terminated germanene nanoribbons, a 2D topological insulator with a large topological bulk gap. The electronic properties of these nanoribbons strongly depend on their width, with topological edge states persisting down to a critical width (approx. 2 nm), defining the limit of 2D topology. Below this threshold, contrary to the tenfold way classification, we observe zero-dimensional (0D) states localized at the ends of the ultrathin nanoribbons. These end states, topologically protected by time-reversal and mirror symmetries, mark the first realization of a 1D topological insulator with strong spin-orbit coupling. Our findings establish germanene nanoribbons as a platform for investigating 1D topology and dimensionality effects in topological materials.

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    and is sizable in germanene [65]. For this reason, we also include them in our model. Let us focus first on the effect of t3. When inspecting Fig. S7, one observes that the introduction of a finite t3 enlarges the range of parameters for which the system is topological (compar...

Pith tools

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