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REVIEW 4 major objections 5 minor 64 references

This paper argues that helicity, circular dichroism, and ellipticity dependence of high harmonics are reliable all-optical diagnostics of topological phases in lower-order topological insulators, while bulk-only harmonics from breathing Kag

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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Simulated HHG in three tight-binding models shows observable differences between topological phases, but controlled comparisons are missing and the HOTI claim is a null result.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Interesting new Kagome HOTI calculation, but the Haldane topology attribution fails on controlled comparison and the abstract overclaims boundary-state inclusion. the 4 major comments →

arxiv 2508.15631 v2 pith:6V3FOSGK submitted 2025-08-21 cond-mat.mtrl-sci

High Harmonic Spectroscopy from Lower-Order to Higher-Order Topological Insulators

Bryan Lorenzo , Carlos Batista , Milad Jangjan , Dasol Kim , Jean Menotti , Feng Liu , Wenlong Gao , Shambhu Ghimire
show 2 more authors
Camilo Granados Alexis Chacon
This is my paper
classification cond-mat.mtrl-sci PACS 42.65.Ky
keywords high-harmonic spectroscopyChern insulatorHaldane modelKane-Mele modelbreathing Kagome latticehigher-order topological insulatorcircular dichroismellipticity dependence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 tries to settle an open controversy: can high-harmonic spectroscopy act as a universal all-optical readout of electronic topology, or do harmonic features mostly follow the band structure rather than the topological invariant? It answers with numerical experiments on three paradigmatic 2D models. For Chern insulators (Haldane model), it reports that helicity, circular dichroism, and ellipticity dependence all change character between trivial and topological phases, with circular dichroism reaching exactly -1, the value of the Chern number. For 2D topological insulators (Kane-Mele model), helicity and circular dichroism stay negligible but the ellipticity dependence is anomalous, matching experiments on Bi2Se3. For the breathing Kagome higher-order insulator, bulk-only harmonics show no topological signatures; the topological semimetal phase, however, emits harmonics two to three orders of magnitude brighter. A careful reader would care because the paper delineates which materials HHS can diagnose and identifies corner states as the likely missing ingredient for higher-order insulators.

Core claim

The authors' central claim is that the three standard high-harmonic observables—helicity, circular dichroism, and ellipticity dependence—carry topological information for lower-order topological insulators but not, from bulk states alone, for the higher-order Kagome insulator studied here. In the Haldane Chern insulator, the helicity of low-order harmonics flips sign between trivial and topological phases, the circular dichroism changes from near zero to exactly −1.0, matching the Chern number C = −1, and the ellipticity dependence becomes asymmetric. In the Kane-Mele topological insulator, helicity and circular dichroism are negligible, but the harmonic yield versus driving-laser ellipticit

What carries the argument

The argument runs through numerical solution of the time-dependent density matrix (TDDM) for each tight-binding model, using the Wannier gauge to handle the dipole matrix and Berry-phase terms smoothly, then Fourier-transforming the computed current to obtain harmonic spectra. The three observables—helicity (the RCP/LCP asymmetry of harmonics from a linear driver), circular dichroism (the RCP versus LCP intensity difference from circular drivers), and ellipticity dependence (harmonic yield versus driver ellipticity)—are read off from the same spectra. Three model Hamiltonians carry the comparison: the Haldane model with Chern number C, the Kane-Mele model with Z2 index, and the breathing Kag

Load-bearing premise

The load-bearing premise is that the trivial and topological Haldane calculations are a controlled comparison: the two phases are meant to have comparable bandgaps, but their hopping parameters and bandwidths differ, so the spectral changes could in principle come from those parameter shifts rather than from the topological invariant.

What would settle it

Repeat the Haldane-model HHG calculation along a parameter path that interpolates between the trivial and topological phases while holding the K-point gap and bandwidth fixed. If the helicity sign flip and CD = −1 track the Chern number and appear only after the gap closes, the paper's interpretation is supported; if they vary continuously with the gap or hopping parameters instead, the claim that the observables encode topology fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the Chern-insulator result holds, circular dichroism values can be read as a direct all-optical proxy for the Chern number (CD = −1 matching C = −1), and helicity sign flips between phases could serve as a quick phase diagnostic.
  • For time-reversal-invariant topological insulators, the anomalous ellipticity dependence—maximum emission at circular polarization, minimum at linear—becomes the observable of choice, consistent with existing Bi2Se3 experiments.
  • Bulk-only harmonics from breathing Kagome higher-order insulators will not reveal corner-state topology; all-optical probing of HOTIs requires calculations or experiments that include corner states, or a different observable.
  • The predicted two-to-three-order-of-magnitude harmonic enhancement in the Kagome topological semimetal gives a concrete experimental target: compare harmonic yields from a topological semimetal Kagome sample with those from a trivial semimetal such as graphene.
  • The paper implies that high-harmonic spectroscopy is not a universal topology probe but is material- and observable-specific: each topological class needs its own diagnostic.

Where Pith is reading between the lines

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

  • If the parameter-comparison worry is set aside, the exact equality CD = C hints that circular dichroism may be quantitatively tied to the Chern number rather than merely correlated; extending the calculation to other Chern numbers would test whether the mapping is one-to-one.
  • The Kagome semimetal enhancement is computed for bulk bands only; the paper's own corner-state argument suggests that including corner states could push the HOTI response even higher, so the absence of signatures for HOTIs should not be read as a no-go for HHS in all higher-order systems.
  • A natural next test is to vary driving wavelength and field strength across the topological semimetal phase: if the enhancement is tied to the vanishing gap rather than the P3 index, it would separate trivial-semimetal effects from topological ones.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper reports time-dependent density-matrix simulations of high-harmonic generation in three model systems: the Haldane model (Chern insulator), the Kane-Mele model (2D topological insulator), and a breathing Kagome model (higher-order topological insulator). The authors compute helicity, circular dichroism, and ellipticity dependence for trivial and topological phases, and conclude that helicity and circular dichroism can reveal topological character in Chern insulators, that the Kane-Mele model shows an anomalous ellipticity dependence, and that the bulk response of the Kagome HOTI shows no clear topological signatures. The abstract additionally claims that edge and corner states were explicitly incorporated and that channel-resolved intensity yields reveal distinct spectral signatures. The body text, however, states repeatedly that edge and corner states were not included, and channel-resolved yields are neither defined nor presented.

Significance. If the central claim were established, the paper would strengthen the case that all-optical high-harmonic spectroscopy can serve as a tabletop probe of topological invariants in Chern insulators and 2D topological insulators, while also delineating a limitation for higher-order topological insulators. The work also identifies a potentially large harmonic-yield enhancement in a Kagome semimetal phase, which would be of experimental interest. The numerical method is standard and the authors have implemented it in a parallelized code that reproduces known trends. However, the central attribution of observed HHG differences to topology is not yet supported because the trivial and topological Haldane phases are not compared at fixed bandgap, bandwidth, and field-to-bandwidth ratio. The abstract/body contradictions about edge/corner contributions further weaken the paper in its present form.

major comments (4)
  1. [LOTI section, Fig. 2 caption] The controlled-comparison premise is not satisfied. The caption states trivial Haldane parameters t1=0.0043, t2=0.00132, M/t2=16.77 and topological parameters t1=0.0152, t2=0.0049, M/t2=0.0520. These give K-point gaps of roughly 0.83 eV and 1.37 eV respectively, and E0/t1 differs by a factor of ~3.5 at fixed E0. The text says 'when a comparable bandgap is fixed for both phases,' but the listed parameters contradict this. The observed helicity sign change and CD ≈ -1 cannot therefore be attributed to the Chern number rather than to the simultaneously changed gap, bandwidth, and field-to-bandwidth ratio. This is the load-bearing comparison for the paper's central claim, and it must be redone with matched gaps and bandwidths, or with a controlled scan over parameters.
  2. [Abstract and HOTI section] The abstract claims the work 'explicitly incorporat[es] contributions from bulk, edge, and corner states' and that 'distinct spectral signatures associated with edge and corner contributions [are] revealed through channel-resolved intensity yields.' The body does not support this. The HOTI section states 'Those conducting corner states are not included in the analysis or model here used to compute the high harmonic emission,' and the Conclusions state 'we emphasize that corner and edge states were not considered in our simulations.' No definition or calculation of 'channel-resolved intensity yields' appears in the manuscript. The abstract must be corrected to match the actual scope, or the calculations must be extended to include edge/corner states.
  3. [Table I and LOTI section] The interpretation of CD = -1.0 as 'matching the topological Chern number C = -1' is an overinterpretation. Circular dichroism is a normalized intensity ratio, not an integer topological invariant; its numerical value coinciding with -1 does not constitute a measurement of the Chern number. Moreover, the Fig. 2 caption labels the topological Haldane phase as C = +1 while the text invokes C = -1. The inconsistency must be resolved and the language toned down to refer to a qualitative correlation or asymmetry, not an equality with the invariant.
  4. [HOTI section, Fig. 4 caption, Conclusions] The reported semimetal enhancement is quantitatively inconsistent. The text says 'a relative enhancement about one or two orders of magnitude' for LP harmonics, the Fig. 4 caption says 'about three orders of magnitude,' and the Conclusions says 'more than three orders of magnitude.' The baseline also needs definition (trivial vs HOTI phase, and which harmonic order). This claim is presented as a unique result, so the reported enhancement factor must be stated consistently and with a clear comparison protocol.
minor comments (5)
  1. [Introduction] Typo: 'The resent realization' should be 'The recent realization.'
  2. [HOTI section] The statement 'we believe the correct prediction must describe the system while including the topological corner states' is reasonable, but it undercuts the earlier framing that HHS does not encode HOTI topology. The conclusion would be more accurate if reframed as 'not observable in the bulk-only response considered here.'
  3. [Appendix B] The dephasing time T2 appears in Eq. (B1), but its value is never given in the main text or appendix. The k-grid density, time step, and convergence criteria are also not reported; these are needed to assess numerical robustness of the claimed effects.
  4. [Fig. 3 and Table I] Only the 4th, 7th, and 10th harmonics are discussed. The rationale for selecting these orders should be stated, and the analysis should show whether the reported helicity/CD trends are robust across the plateau or specific to these orders.
  5. [Kane-Mele section] The statement that results are 'in direct agreement with experimental observations reported by Baykusheva et al.' is stronger than what a 2D Kane-Mele model can establish against a 3D Bi2Se3 experiment. Rephrase as qualitative agreement with the observed anomalous ellipticity dependence.

Circularity Check

0 steps flagged

No significant circularity: the HHG observables are computed from fixed model Hamiltonians, not fitted to the claimed topological invariants.

full rationale

The paper's derivations are numerical solutions of the time-dependent density matrix for fixed tight-binding models. The observables (helicity, circular dichroism, ellipticity dependence) are defined by Eqs. (1)–(2) from harmonic intensities and are not defined in terms of the topological invariants C, Z2, or P3; no parameter is fitted to the harmonic output to produce the claimed signatures. The topological phases are identified by independent invariants (Appendix A), and the HHG spectra are computed afterward. The comparison between trivial and topological Haldane phases changes multiple parameters (t1, t2, M, gap, and E0/t1), which undermines causal attribution to topology, but this is a control/validity issue rather than a circular reduction. The HOTI null result is explicitly scoped to bulk states, with corner states omitted, so the conclusion is not forced by definition. Self-citations [17,27,61] provide the numerical method and are anchored by independent experimental reproduction [18], so they are not load-bearing circularity. The statement that CD = -1 'matches' C = -1 is an interpretive overreach (and internally inconsistent with the C=+1 caption), but it is a post-hoc numerical coincidence, not a derivation of the observable from the invariant. No equation in the paper reduces a predicted quantity to an input by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 1 invented entities

The central claim rests on model parameters chosen to realize topological phases (free parameters), on the TDDM approximation from self-cited work, on the exclusion of edge/corner states, on an assumed matched-bandgap comparison, on the interpretation of the semimetal enhancement as topological, and on an ad hoc reading of CD as a Chern number. These assumptions, not the HHG observables themselves, carry most of the paper's weight.

free parameters (4)
  • Haldane model hopping parameters (t1, t2) and mass-to-NNN ratio M/t2 = trivial: t1=0.0043, t2=0.00132 a.u., M/t2=16.77; topological: t1=0.0152, t2=0.0049 a.u., M/t2=0.0520
    Chosen to realize trivial vs Chern phases (Fig. 2 caption). The two phases differ in all parameters simultaneously, so HHG differences cannot be uniquely attributed to topology.
  • Kagome hopping ratio ta/tb = HOTI: ta/tb ~ 0.142; trivial: ta/tb ~ 7.02; SM-TM: ta=tb
    Chosen to realize HOTI, trivial, and semimetal phases (Fig. 4 caption). No intermediate sweep across the phase transition is shown.
  • Laser field parameters = E0 = 0.0007 a.u., hbar omega = 0.013 a.u., 7 optical cycles FWHM
    Driving field. The ratio E0/t1 differs by about 3.5x between trivial and topological Haldane phases, confounding intensity comparisons.
  • Dephasing time T2 = not stated
    The TDDM in Eq. (B1) includes a dephasing term with time T2, but no value is given in the text. T2 strongly affects harmonic linewidths and yields.
axioms (5)
  • domain assumption The TDDM in the Wannier gauge with dipole terms neglected (approx <0m|r|Rn> ~ delta_mn Delta_n) accurately captures HHG in these tight-binding models.
    Appendix B, Eqs. (B3)-(B4). This approximation eliminates transition dipoles and Berry connection contributions; it is drawn from self-cited Refs [27, 61] and is not validated against full dipolar simulations here.
  • domain assumption Bulk-only periodic-boundary-condition simulations are the relevant object for comparing HHS topological signatures; edge and corner states can be omitted without loss.
    The authors explicitly exclude edge/corner states ('not included in the analysis', HOTI section) yet the abstract claims those contributions are incorporated. For HOTIs the corner states are the defining feature (P3 invariant), so this assumption makes the null result expected rather than informative.
  • ad hoc to paper Trivial and topological phases are matched in bandgap and bandwidth so that differences in HHG are attributable to topology.
    Statement in LOTI section ('comparable bandgap is fixed') contradicted by the parameter sets in Fig. 2: computed K-point gaps are about 0.83 eV (trivial) versus 1.37 eV (topological) and E0/t1 differs by about 3.5x. This assumption is necessary for the central claim to hold.
  • domain assumption The gapless Kagome semimetal (ta=tb) with Z2 = 1 is a 'topological semimetal' whose harmonic enhancement is a topological signature rather than a generic gap-closing effect.
    HOTI section and Fig. 4. No trivial semimetal (e.g., graphene) baseline is computed to separate gap-closing from topology.
  • ad hoc to paper Interpretation of CD = -1.0 as 'matching the topological Chern number C = -1'.
    LOTI section, Table I. CD is an intensity ratio in Eq. (2), not a topological invariant; the Haldane phase in Fig. 2 is labeled C = +1, so the claim is internally inconsistent.
invented entities (1)
  • channel-resolved intensity yields no independent evidence
    purpose: Advertised as the observable that reveals edge and corner contributions to HHG.
    Abstract and Conclusions rely on this observable, but it is never defined or computed in the paper; no channel decomposition of the current is shown.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of High Harmonic Spectroscopy from Lower-Order to Higher-Order Topological Insulators." pith.science (2026). https://pith.science/paper/6V3FOSGK

@misc{pith2026250815631,
  author       = {Pith},
  title        = {Pith review of: High Harmonic Spectroscopy from Lower-Order to Higher-Order Topological Insulators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6V3FOSGK}},
  note         = {Machine review of arXiv:2508.15631}
}
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read the original abstract

Over the past decades, high-harmonic spectroscopy (HHS) has emerged as a powerful tool for all-optical probing of topological properties of solids. There are outstanding questions regarding universal nature of the spectral features of harmonics in their relationship to the non-trivial topological properties. Here, we present a systematic theoretical study of HHS in topological materials, including lower-order and higher-order topological insulators (LOTIs and HOTIs), focusing on observables such as helicity, circular dichroism, ellipticity dependence, and channel-resolved intensity yields. Using the Haldane, Kane-Mele, and breathing Kagome lattice models, we theoretically extend all-optical approaches from the LOTI to the HOTI regime by explicitly incorporating contributions from bulk, edge, and it corner states. Depending on the crystalline system, our calculations suggest that these observables can encode topological information through distinct modifications of the HHG spectra in topological phases. In particular, we identify significant enhancements of the harmonic intensity yields, reaching up to two orders of magnitude relative to trivial phases, together with distinct spectral signatures associated with edge and corner contributions revealed through channel-resolved intensity yields. These results show that channel-resolved HHS provides a promising route for probing topological states in both LOTIs and HOTIs.

Figures

Figures reproduced from arXiv: 2508.15631 by Alexis Chacon, Bryan Lorenzo, Camilo Granados, Carlos Batista, Dasol Kim, Feng Liu, Jean Menotti, Milad Jangjan, Shambhu Ghimire, Wenlong Gao.

Figure 1
Figure 1. Figure 1: FIG. 1. High-harmonic generation in two-dimensional topological materials. A strong and ultrashort mid infrared (IR) driving [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Band dispersions and high-harmonic spectra for lower-order topological insulators: Chern insulators and topological [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. High harmonic spectroscopy in lower-order topological insulators (LOTIs): the ellipticity dependence. a) (topologically [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. High-harmonic generation from higher-order topological insulators: Panels a) and b) show the band structure of our [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. High-harmonic spectroscopy in higher-order topological insulator (HOTI): ellipticity dependence. Panels a), b), and c) [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

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Reference graph

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.