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REVIEW 2 major objections 5 minor 67 references

Detecting the topological winding of superconducting nodes via Local Density of States

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Pith's one-line read The topological winding of superconducting nodes can be read from impurity-induced LDOS ripples, whose Fourier transform carries a vortex of strength equal to the winding difference.

desk verdict A useful generalization of QPI dislocation detection to nodal superconductors, with a concrete NbSe2 proposal, but the key projection step is deferred and needs verification. read the letter →

arxiv 2412.13042 v2 pith:3AIOHLVV submitted 2024-12-17 cond-mat.supr-con

Lena Engström , Pascal Simon , Andrej Mesaros This is my paper · ORCID
classification cond-mat.supr-con
keywords topologicalwindingnodalsuperconductorchiralsymmetrylocaldensityofstatesquasiparticleinterferencescanningtunnelingmicroscopyNbSe2phasevortex
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

This paper claims that the topological winding of nodal points in chiral two-dimensional systems, including nodal superconductors, can be read directly from the local density of states measured by a scanning tunneling microscope near a single impurity. A difference in winding between two nodes produces a phase vortex in the Fourier transform of the impurity-induced LDOS, with vorticity equal to the winding difference, provided the impurity and the STM tip prefer opposite chiralities of the Bogoliubov wavefunctions. The paper derives explicit conditions on the impurity and tip, expressed through their chirality-preference ratios, and shows they are met in monolayer NbSe2 under an in-plane magnetic field with a magnetic impurity and a spin-polarized tip, making all 12 nodal windings detectable. If the claim holds, a purely local STM measurement can determine lower-dimensional topological invariants that bulk transport or edge-state probes cannot reach.

What carries the argument

The central object is the real-space phase function $\theta_{K_i}(r) = \arctan[(v_\parallel/v_\perp) \tan(W_{K_i} \theta_r + \phi_i)]$, which inherits the node's winding number $W_{K_i}$ from the linearized dispersion and appears in the off-diagonal propagator $g_1$. Writing the scattering-induced LDOS as $\mathrm{Re}[\varrho(r)e^{i\varphi_r}]$ with $\varphi_r = \Delta K_{ij}\cdot r + \mathrm{Arg}[I_0 + I_\Delta e^{i\Delta\theta} + I_{-\Delta} e^{-i\Delta\theta}]$, the winding of $\mathrm{Arg}[\cdot]$ around the impurity yields the dislocation charge $W_{\delta\rho} = \pm\Delta W$. The conditions for a non-zero $W_{\delta\rho}$ reduce to two chirality-preference ratios: $\chi_M = M_A/M_B$ for the tip and $\chi_V = V_A/V_B$ for the impurity, with the requirement $|h_1(\chi_M+\chi_V)| > |h_0(\chi_M\chi_V+1)|$ and $\chi_M \neq \pm\chi_V$.

What would settle it

Compute the full Bogoliubov-de Gennes Green's function without cone-band projection for the NbSe2 model, generate the impurity LDOS, and take its Fourier transform; if the vortex vorticity at any ΔKij differs from the winding difference predicted by the projected theory, the central claim collapses.

Watch

Extended reading notes

Core claim

In a chiral Hamiltonian written in a basis of A and B eigenstates, the complex phase θn(k) of the off-diagonal Q-matrix carries the winding. After linearizing around a node and projecting to the cone band, the real-space Green's function g1(r) acquires the same phase as a function of the real-space polar angle θr, so the cone's winding WKi appears in the phase of the propagator. Scattering between two nodes i and j by an impurity then contributes LDOS terms cos(ΔKij·r ± Δθij(θr)) to the wavefront, and when the tip and impurity have different chirality preferences, the phase Δθij winds ΔW = WKi - WKj times, creating a dislocation of strength ±ΔW in real space. Crucially, in the Fourier transform of the LDOS this becomes a single vortex at ΔKij whose vorticity equals the total enclosed dislocation charge, a quantity that is robust to finite field of view. The paper's application to the twelve-node nodal superconductor proposed for monolayer NbSe2 under an in-plane field shows that a magnetic impurity combined with a spin-polarized STM tip satisfies the winding conditions, so each node's winding is in principle measurable.

Load-bearing premise

The argument assumes that after a gauge choice putting all winding into the cone band, no gapped band carries any of the node's winding, so the projected propagator measures the true node winding; the authors state that a full numerical Green's function check without projection is still needed.

Editorial extensions

If this is right

  • A single STM measurement, not quantized transport, can extract the winding numbers of superconducting nodes, which are otherwise inaccessible experimentally.
  • In a nodal superconductor such as NbSe2, the measured winding differences can discriminate pairing symmetries, since only certain pairing functions yield non-zero windings at the nodes.
  • The method extends beyond superconductors to any chiral system with linearly dispersing nodes, including graphene-like semimetals, and to nodal lines in three dimensions.
  • Momentum-space vortices remain observable even when real-space dislocations are masked by other defects, because the vorticity at ΔKij accumulates all dislocation charges inside the field of view.
  • The explicit chirality-preference conditions provide a recipe to choose an impurity and tip (or functionalized tip) for any candidate material.

Reading between the lines

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

  • Beyond the paper: a decisive numerical test would be a full, unprojected Green's function calculation of the impurity LDOS in NbSe2; if the vortex vorticity at each ΔKij still equals the total winding difference, the projection to the cone band is validated.
  • Beyond the paper: the same scattering mechanism could be used on the surfaces of three-dimensional Weyl semimetals or nodal-line superconductors, where the projected surface LDOS may carry winding signatures from the bulk nodes.
  • Beyond the paper: the requirement that tip and impurity prefer opposite chiralities suggests that functionalized or spin-polarized STM tips could act as chirality filters, potentially enabling detection of higher-winding Dirac points or multi-band nodes with |W| > 1.
  • Beyond the paper: if confirmed, the technique gives a local, real-space probe of topological invariants that does not require edge states or global Berry curvature cancellation, which could be applied to disordered or small samples.
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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

2 major / 5 minor

Summary. The paper develops a general theory for extracting the difference of chiral winding numbers of two Dirac nodes in a 2D nodal chiral system from impurity-induced LDOS modulations. Using a chiral-basis Green's function and the T-matrix/Born approximation, it derives conditions on the chirality preferences of an STM tip and an impurity under which the LDOS at a scattering vector ΔKij displays a wavefront dislocation in real space and a phase vortex in momentum space with charge equal to the winding difference ΔW = WKi - WKj. The method is applied to the proposed nodal Ising superconducting phase of monolayer NbSe2 under an in-plane magnetic field, where the authors predict that a magnetic impurity combined with a spin-polarized STM tip can reveal the winding of all 12 nodes. The paper also discusses anisotropy corrections, the relative robustness of momentum-space vortices versus real-space dislocations, and the experimental feasibility including a quantitative Born-approximation estimate.

Significance. If the central correspondence is fully substantiated, the paper provides a concrete experimental route to measure a lower-dimensional topological invariant—the winding of superconducting nodes—via STM, which is currently lacking. The work extends the graphene wavefront-dislocation technique to superconductors and other chiral nodal systems, and it gives explicit, falsifiable predictions for a specific material (NbSe2). Strengths include the detailed analytical derivation of the conditions, the generalization to anisotropic cones and complex chirality parameters, the careful treatment of experimental resolution, and the honest enumeration of approximations. The authors also provide a quantitative estimate for the Fe impurity in NbSe2, showing that the Born approximation is marginally reasonable (G0V ≈ 0.3). These features make the paper a valuable contribution if the identified gaps are addressed.

major comments (2)
  1. [III.C, Eqs. (27)-(36)] The general conditions for a non-zero dislocation charge are misstated for complex chirality preferences. With IΔ = h1 χM and I−Δ = h1 χV (Eq. (30)), the requirement |IΔ| ≠ |I−Δ| in Eq. (28) is equivalent to |χM| ≠ |χV|, not to χM ≠ ±χV as claimed in Eq. (36). Equation (34b) as written only excludes χM = χV. For instance, χM = 1, χV = i satisfies both Eq. (34b) and Eq. (36) yet has equal magnitudes, so the dislocation is absent. The correct conditions for complex χ are Eqs. (A8a)-(A8b) of Appendix A 2; the main text should either adopt them or explicitly restrict the derivation to real-valued χ.
  2. [III.B and Appendix C1, Eqs. (7), (13)-(16)] The central identification of the measured LDOS vortex charge with the physical node winding difference ΔW is not yet established, because the derivation projects the Green's function onto the single cone band n = 1 after fixing a gauge in which all winding sits in that band. As the authors state in Appendix C1, the winding number is defined only as a total over bands (Eq. (7)) and "we may reshuffle the non-trivial winding between the different bands n"; the projection to n = 1 can therefore remove winding contributions. The exact Green's function (8) includes the gapped bands, and the impurity/tip operators act in the full Hilbert space, so the weight of each band in the LDOS need not match the n = 1 projection. The paper explicitly defers the decisive check to "a full (numerical) calculation of the Green's function including the gapped bands, without projection." Since the claimed correspondence between the measured vortex and the node winding is the paper's central result, this gap must be closed, either by a rigorous argument that the gapped bands do not alter the winding of the LDOS phase at the relevant scattering vector, or by a numerical evaluation of the unprojected Green's function for a concrete model (e.g., the NbSe2 model of Eq. (39)).
minor comments (5)
  1. [I. Introduction] The statement that "their total in the Brillouin zone must vanish" is used to argue that all windings follow from the pairwise differences; a brief justification or reference for this topological constraint would be helpful.
  2. [III.A] The full T-matrix expression is written for a local impurity potential, but the text does not explicitly state that the impurity is local until later; a sentence clarifying this would avoid ambiguity.
  3. [III.B, Eqs. (11) and (15)] The same symbol θKi is used for the momentum-space phase in Eq. (11) and the real-space phase in Eq. (15); consider using a different notation (e.g., θKi^(q) and θKi^(r)) to avoid confusion.
  4. [III.E, footnote 53] Two references appear as unresolved placeholders "[?]" in the footnote about the Su-Schrieffer-Heeger chain and Theorem 1; these need to be completed.
  5. [Fig. 2 caption] The caption refers to green and purple vectors, but the figure may be rendered in grayscale; please ensure the color coding is clear for all readers or label the vectors directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the vortex-winding correspondence is derived from the model Hamiltonian via the Green's function; the projection/gauge caveat is an acknowledged assumption, not a circular reduction.

full rationale

The paper's central derivation is self-contained rather than circular. The node winding is defined from the phases of the Q-matrix eigenvalues of the chiral Hamiltonian (Eqs. 6-7), and the real-space propagator phase in Eq. (15) is obtained by Fourier-transforming the linearized cone Green's function, so the appearance of WKi in θKi(r) is a derived consequence, not an input fitted to the LDOS. The dislocation/vortex condition (Eqs. 27-29, 34-37) is obtained by an explicit analysis of the interference terms I0, IΔ, I−Δ, with no parameter tuned to force the NbSe2 result. The NbSe2 application computes the winding from the model Hamiltonian (Wn=1=±1, Wn=2=0 under the stated basis choice) and then evaluates the LDOS from the analytic expression Eq. (35); both sides come from the same model, but that is forward modeling, not circularity. The one substantive caveat is the projection onto the n=1 cone band and the gauge choice that places all winding in that band. Appendix C1 states this explicitly: 'we may reshuffle the non-trivial winding between the different bands n. The projection to n = 1 may therefore remove some winding contributions from the Green's function,' and the authors defer the decisive check: 'a full (numerical) calculation of the Green's function including the gapped bands, without projection, may be used to confirm the validity of the projection procedure.' This is an acknowledged unverified assumption about the physical LDOS, which is a correctness risk, not a circular reduction: the paper does not define the node winding in terms of the LDOS vortex, nor does it fit the winding from the LDOS. References to prior work (Dutreix et al., Ref. 22) are external experimental/theoretical support for the general mechanism, not a self-citation chain carrying the present derivation. No fitted parameter is renamed as a prediction, and no result is forced by definition. Accordingly, the circularity score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central derivation uses standard chiral-basis Green's function manipulations and Bessel-function identities. Load-bearing physical assumptions: exact or near-exact chiral symmetry protecting the windings, an isolated linearly dispersing cone band with all winding assigned to it by gauge choice, the Born approximation, and chirality-preserving impurity and tip. The NbSe2 demonstration parameters are representative rather than fitted, and the tip magnetization ratio is chosen to satisfy the derived conditions. No new physical entities are postulated: χM and χV are derived diagnostic ratios, not invented degrees of freedom.

free parameters (3)
  • NbSe2 model parameters (λ, h, Δ, μ, ω) = λ = 0.15, h = 0.06, Δ = 0.02, μ = 0.25, ω = 0.01 (units m = 1, a = 1)
    Hand-chosen inputs for the demonstration at a fixed Δ/h = 1/3 ratio, argued to transfer to the material regime (Δ ≈ 0.45 meV, H ≈ 22.5 T). They set the nodal structure and the annuli positions but are not fitted to produce the predicted vortices; the paper argues only the ratio matters for the conditions.
  • STM tip magnetization ratio Mx/My = Mx = 0.1 My (tip about 6° from the y-axis)
    Chosen by hand so that χM differs from ±χV and the winding conditions (A9) hold; the authors note the conditions are not fine-tuned to a specific tip angle.
  • Fe impurity potential strength V = V ≈ 100 meV, taken from YSR literature (Refs. 56-57)
    Input to the Born-approximation estimate G0V ≈ 0.3 in Appendix D; not fitted to the vortex prediction and the estimate is order-of-magnitude only.
assumptions (6)
  • domain assumption The bulk Hamiltonian has chiral symmetry Γ with {Γ, H} = 0, protecting the node windings.
    Section II.A, Eqs. (1)-(7). Exact by construction in the model Hamiltonian of Eq. (39); only approximate in real NbSe2, as the authors state in the Discussion.
  • domain assumption Around each node only one band n = 1 is linear and isolated: En≠1(Ki+q) ≥ ΔEn ≫ vKi·q.
    Section II.B, Eq. (9). Justified at low ω ≪ h for the NbSe2 model, where the gapped band is separated by 2h.
  • ad hoc to paper The chiral-basis gauge is fixed so that all winding is assigned to the cone band n = 1, with θn>1 trivial.
    Appendix C1: the winding is a sum over bands and can be reshuffled between them; the authors choose Wn=2 = 0 for NbSe2 and defer a full unprojected Green's function check.
  • domain assumption Born approximation: the T-matrix of the impurity equals the bare potential V.
    Section III.A and Appendix D, justified by the rough estimate πb²DOS(ω = 0.2 meV)V ≈ 0.3, which replaces G0 by its trace-imaginary part.
  • domain assumption Impurity and tip preserve chirality ([Γ, V] = [Γ, M] = 0), so A-B scattering matrix elements vanish.
    Section III.A and Appendix B. For the chosen NbSe2 combination (Vx, Mx) the A-B element is actually non-zero (Eq. C7); its contribution is argued to cancel upon summing symmetric node pairs (Appendix C2).
  • domain assumption Isotropic-cone approximation for the Hankel integrals (vKi(θq) → vKi).
    Section II.B, Eqs. (13)-(15), with corrections estimated as an order of magnitude smaller in Appendix A; general conditions with anisotropy are given in Appendix A2.

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Pith. "Pith review of Detecting the topological winding of superconducting nodes via Local Density of States." pith.science (2026). https://pith.science/paper/3AIOHLVV

@misc{pith2026241213042,
  author       = {Pith},
  title        = {Pith review of: Detecting the topological winding of superconducting nodes via Local Density of States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3AIOHLVV}},
  note         = {Machine review of arXiv:2412.13042}
}
read the original abstract

Many systems are topologically trivial in the bulk, but still have non-trivial wavefunctions locally in the Brillouin zone. For example, in a small-gap Dirac material the Berry curvature is strongly peaked, but cancels over the full Brillouin zone, while in semimetals and in nodal superconductors there may be a lower-dimensional winding topology associated to the nodes. Experimentally, it is difficult to directly observe such topology. We consider general bulk Hamiltonians with nodes and chiral symmetry, extending to them the method developed in Dutreix et al. [Nature, 574(7777):219-222 (2019)], which in particular detected the winding around Dirac cones in graphene using charge modulations around an impurity. We apply our method to nodal superconductors in 2d, in presence of a (non)magnetic impurity, measured by standard or spin-polarized STM tip. We derive general conditions on the impurity scattering and on the STM tip, expressed in terms of their preference among the two chiralities, for when the measurement near the impurity captures the winding difference between any chosen pair of (Bogoliubon) Dirac cones. We emphasize the robustness of observing vortices in momentum space, in contrast to dislocations in real space, in STM data. Testing the conditions on the topological nodal superconductor proposed for monolayer NbSe2 under an in-plane magnetic field, we find that spin-polarized STM on a magnetic impurity can detect the winding of each of the 12 nodes. We conclude that a judicious choice of impurity can be a powerful tool to determine topological quantities in 2d superconducting systems as well as any nodal chiral system.

Figures

Figures reproduced from arXiv: 2412.13042 by the authors.

Figure 1
Figure 1. FIG. 1. Example system for two isotropic Dirac cones and a combination of impurity and STM tip which has [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Nodal windings in the example of superconductor NbSe [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Two adjacent nodes and their non-trivial wind [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. General winding conditions for Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Regions where the winding conditions, Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. For NbSe [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

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Reference graph

Works this paper leans on

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