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REVIEW 3 major objections 7 minor 47 references

Anomalous Hall and Nernst effects driven by static and fluctuating spin chiralities on Kagome lattice

T0 review · 3 major / 7 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Static and fluctuating spin chirality drive opposite-signed Hall transport on Kagome lattice

desk verdict Useful framework for decomposing intrinsic vs extrinsic anomalous transport on Kagome, but the extrinsic derivation has a real 2D/3D inconsistency that undermines quantitative claims. read the letter →

arxiv 2607.06966 v1 pith:HNRK7LF2 submitted 2026-07-08 cond-mat.str-el

He-Lin Li , Zhen-Gang Zhu , Gang Su This is my paper · ORCID
classification cond-mat.str-el
keywords staticanomalousspinfluctuatingfluctuationsintrinsictransportberry
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 argues that anomalous Hall and Nernst effects on a Kagome lattice can be decomposed into two competing mechanisms with opposite signs: an intrinsic contribution from momentum-space Berry curvature generated by static noncoplanar spin textures (dominant in the skyrmion crystal phase), and an extrinsic skew-scattering contribution driven by thermal fluctuations of the scalar spin chirality (dominant near the magnetic order-disorder transition). The authors use Monte Carlo simulations of a classical spin Hamiltonian with Heisenberg and Dzyaloshinskii-Moriya interactions to extract both the thermal average of local spin configurations and the fluctuating part of the scalar spin chirality across a temperature-field phase diagram. The static averages feed into a tight-binding electronic Hamiltonian whose Berry curvature yields the intrinsic anomalous Hall and Nernst conductivities. The fluctuating chirality feeds into analytically derived skew-scattering formulas (obtained via second-Born-approximation treatment of the s-d exchange interaction under a low-energy effective-mass approximation) to yield the extrinsic conductivities. The central result is a mechanistic crossover: deep in the skyrmion crystal phase, static chirality and Berry curvature dominate; near the phase boundary where thermal fluctuations destroy static noncoplanar order, the intrinsic response collapses but dynamical chiral fluctuations sustain anomalous transport through skew scattering, with the two channels contributing opposite signs. This sign competition, the authors argue, explains experimentally observed sign reversals of the topological Hall and Nernst effects near magnetic phase transitions.

What carries the argument

Monte Carlo simulation of classical spin Hamiltonian on Kagome lattice (L=48, 3×48² sites) with Heisenberg exchange J_H, Dzyaloshinskii-Moriya interaction (D_xy=0.5, D_z=sqrt(3) in units of J_H), and external magnetic field B; tight-binding electronic Hamiltonian with s-d exchange coupling J_K=0.2t, hopping t=-1 eV, on a 12×12 magnetic supercell; intrinsic transport via Berry curvature integrals (Kubo formula); extrinsic transport via second-Born-approximation skew-scattering rate under parabolic dispersion approximation; Mott relation connecting extrinsic anomalous Nernst conductivity to energy derivative of extrinsic anomalous Hall conductivity at low temperature.

What would settle it

If the parabolic-band approximation fails at the chosen Fermi energy, the magnitude and potentially the sign of the extrinsic anomalous Hall and Nernst conductivities could differ from the computed values, undermining the claim of opposite-sign competition between intrinsic and extrinsic mechanisms.

Watch

Extended reading notes

Core claim

The paper's central object is the scalar spin chirality chi_ijk = S_i · (S_j × S_k) on triangular plaquettes of a Kagome lattice, decomposed into a static part <S_i>·(<S_j>×<S_k>) and a fluctuating part delta_chi obtained by subtracting the static background from the total thermal average. The static part generates momentum-space Berry curvature in the electronic band structure and drives intrinsic anomalous Hall and Nernst conductivities via the Kubo formula. The fluctuating part drives extrinsic anomalous conductivities through a skew-scattering mechanism derived from the third-order Kondo exchange coupling, where the antisymmetric scattering rate is proportional to both the cross productk

Load-bearing premise

The extrinsic transport formulas are derived assuming electrons have a parabolic dispersion (a simple quadratic energy-momentum relation), but the Kagome lattice band structure contains Dirac cones and flat bands that are far from parabolic, and the paper does not justify that the chosen Fermi energy (-4.0 eV) sits in a regime where this approximation is valid. The relaxation time is also treated as a free parameter rather than computed from the symmetric scattering rate.

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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

3 major / 7 minor

Summary. This manuscript investigates anomalous Hall and Nernst effects (AHE/ANE) on a 2D Kagome lattice, decomposing the total response into an intrinsic contribution from momentum-space Berry curvature (driven by static scalar spin chirality) and an extrinsic skew-scattering contribution from dynamical spin chirality (DSC) fluctuations. Monte Carlo simulations on a classical spin model generate the phase diagram and chirality data, which feed into tight-binding (intrinsic) and Boltzmann transport (extrinsic) calculations. The central claim is a mechanistic crossover: intrinsic transport dominates in the skyrmion crystal phase, while extrinsic transport dominates near the order-disorder transition, with the two mechanisms contributing opposite signs. The framework is well-motivated and the combination of MC with analytical transport is a reasonable approach. However, a dimensional inconsistency in the extrinsic scattering derivation and an unjustified parabolic-band approximation affect the quantitative reliability of the crossover claim.

Significance. The paper addresses a timely question in topological magnetism: disentangling intrinsic and extrinsic anomalous transport mechanisms driven by static versus fluctuating spin chirality. The decomposition framework and the prediction of opposite-sign competition between mechanisms near phase boundaries are potentially useful for interpreting experimental sign reversals in topological Hall/Nernst effects. The MC protocol is standard and the phase diagram construction is systematic. The analytical derivation of the extrinsic AHC/ANC from the Kondo model provides a transparent connection between DSC and skew scattering. However, the quantitative reliability of the extrinsic magnitudes—which underpin the crossover claim—is compromised by the issues detailed below.

major comments (3)
  1. Appendix A, Eq. (A7): The second-order scattering amplitude is derived using a 3D free-particle Green's function, yielding the 3D retarded Green's function ~e^{ikr_ij}/r_ij. However, the system is explicitly 2D (Section III). The 2D analogue is a Hankel function (iπ/2)H_0^{(1)}(kr_ij), which has qualitatively different spatial dependence (logarithmic in the long-wavelength limit ka≪1 used in Eq. 23). The paper converts to 2D only at the final step (Eq. 27) by substituting 2D carrier density and DOS, without rederiving the scattering rate in 2D from the outset. This affects the prefactor and potentially the k-dependence of the extrinsic AHC (Eq. 27) and ANC (Eq. 28), undermining the quantitative reliability of the crossover claim.
  2. Section V, Eqs. (27)–(28) and Fig. 5: The extrinsic AHC scales as τ², where τ is a free parameter not computed from the symmetric scattering rate. This makes the relative magnitude of extrinsic versus intrinsic contributions adjustable, and hence the crossover point in the T–B phase diagram (Fig. 5e–f) is not uniquely determined by the theory. The sign opposition (intrinsic vs. extrinsic) may be robust since it depends on J_K³ and chirality direction, but the crossover claim depends on magnitudes that are not reliably fixed. The authors should either compute τ self-consistently from the symmetric part of the scattering rate or provide a sensitivity analysis showing that the crossover persists for a reasonable range of τ.
  3. Section V and Fig. 1(b): The extrinsic transport formulas (Eqs. 27–28) are derived under a parabolic-band effective-mass approximation (ε_k = ℏ²k²/2m), but the Kagome lattice band structure (Fig. 1b) contains Dirac cones and flat bands. The Fermi energy ε_F = −4.0 eV is stated to be in a 'low energy region' but the validity of the parabolic approximation at this energy is not justified. Since the extrinsic AHC and ANC magnitudes depend on the effective mass m and the DOS at ε_F, this approximation directly affects the load-bearing crossover result. The authors should justify this approximation quantitatively (e.g., by showing the band dispersion near ε_F is approximately parabolic) or discuss its limitations explicitly.
minor comments (7)
  1. Eqs. (2) and (4): The intrinsic AHC and ANC formulas use d³k/(2π)³, appropriate for 3D, but the system is 2D. These should be d²k/(2π)². This appears to be carried over from a general 3D formalism but should be corrected for the 2D model in Section III.
  2. Eq. (17): The expression uses 3D carrier density n_e = k_F³/(6π²), but Eq. (27) correctly uses 2D n_{e,2D} = k_F²/(4π). The derivation should be consistent in 2D from the start, or the dimensional reduction should be clearly explained.
  3. Section IV: The phase boundary thresholds (e.g., χ_Q ≥ 3, δχ > 0.2, M_z > 0.1) are stated without justification. A brief discussion of how these values were chosen and their sensitivity would help reproducibility.
  4. Fig. 5: The color scales for intrinsic and extrinsic panels should use the same units and range to facilitate visual comparison of the crossover. Currently it is difficult to assess the relative magnitudes from the figures alone.
  5. The term 'fluctuating chiral (FC) phase' is introduced as a distinct phase, but the criteria (δχ > 0.2) suggest it may be a crossover regime rather than a thermodynamic phase. Clarifying whether this is a true phase or a crossover regime would strengthen the presentation.
  6. Reference [27] (Ishizuka and Nagaosa, 2018) is the primary theoretical basis for the extrinsic mechanism. The manuscript should clarify what is genuinely new in the extrinsic derivation relative to this reference, beyond the application to the Kagome lattice.
  7. Several typos: 'chi rality' (abstract), 'configu rations' (abstract), 's-d exchange' formatting inconsistencies, 'intergral' should be 'integral' where applicable. Proofreading recommended.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful reading and for identifying three substantive issues with the extrinsic transport derivation. We agree that all three points are valid concerns that require revision of the manuscript. Below we address each in turn.

read point-by-point responses

  1. Referee: Appendix A, Eq. (A7): 3D Green's function used in a 2D system. The 2D analogue is a Hankel function with different spatial dependence, affecting prefactors and k-dependence of extrinsic AHC/ANC.

    Authors: The referee is correct. The second-order scattering amplitude in Eq. (A7) was evaluated using the 3D free-particle retarded Green's function, G(r) ~ e^{ikr}/r, while the system is explicitly two-dimensional. The correct 2D Green's function is G(r) = (im/2ℏ²) H_0^{(1)}(kr), which has qualitatively different spatial dependence—particularly in the long-wavelength limit ka << 1 used in Eq. (23), where H_0^{(1)}(kr) ~ (2i/π) ln(kr). We performed the intermediate steps of the derivation in 3D and only substituted 2D carrier density and DOS at the final stage (Eqs. 27–28), which is not a consistent procedure. We will rederive the second-order T-matrix and the antisymmetric scattering rate from the outset in 2D, using the Hankel-function Green's function. We note that the key structural features of the result—the proportionality to J_K³, the scalar spin chirality S_l·(S_i×S_j), the antisymmetric (k×k') factor, and the spatial interference factor I_{ijl}—are determined by the spin algebra and geometric structure of the Kondo coupling, not by the dimensionality of the Green's function. The sign opposition between intrinsic and extrinsic contributions, which depends on these structural features, is therefore expected to be robust. However, the referee is right that the prefactor and potentially the k-dependence of Eqs. (27)–(28) will change, affecting the quantitative magnitude of the extrinsic contribution. We will revise Appendix A to present the full 2D derivation and update Eqs. (27)–(28) and Fig. 5(c–f) accordingly. revision: yes

  2. Referee: Extrinsic AHC scales as τ², where τ is a free parameter not computed from the symmetric scattering rate. The crossover point in Fig. 5(e–f) is not uniquely determined. Authors should compute τ self-consistently or provide sensitivity analysis.

    Authors: This is a fair criticism. The relaxation time τ enters the extrinsic AHC as τ² and is treated as a free parameter, which means the absolute magnitude of the extrinsic contribution—and hence the precise location of the intrinsic-extrinsic crossover in the T–B phase diagram—is not uniquely fixed by the theory as currently presented. We agree that the sign opposition is robust, since it depends on J_K³ and the chirality direction, not on τ. However, the crossover claim as a quantitative statement about where in the phase diagram extrinsic transport overtakes intrinsic transport does depend on the ratio of magnitudes, and thus on τ. We will address this in two ways. First, we will compute the symmetric scattering rate W^S (arising from |F^{(1)}|²) self-consistently from the same Kondo coupling and MC spin configurations, yielding τ = 1/(2π W^S/ℏ) as a function of temperature and field. This provides a first-principles estimate of τ within the same framework. Second, we will provide a sensitivity analysis: we will show the crossover boundary in Fig. 5(e–f) for a range of τ values spanning at least one order of magnitude around the self-consistent estimate, demonstrating that the qualitative picture—extrinsic dominance in the FC phase near the order-disorder transition and intrinsic dominance in the SkX phase—persists across a reasonable parameter range. We acknowledge that without this analysis, the quantitative crossover claim is underdetermined. revision: yes

  3. Referee: Extrinsic transport formulas derived under parabolic-band approximation, but Kagome lattice has Dirac cones and flat bands. Validity of parabolic approximation at ε_F = −4.0 eV is not justified. This affects extrinsic AHC/ANC magnitudes and the crossover result.

    Authors: The referee raises a legitimate point. The extrinsic AHC and ANC formulas (Eqs. 27–28) are derived assuming a parabolic dispersion ε_k = ℏ²k²/2m, which enters through the effective mass m, the DOS ρ(ε_F), and the carrier density n_e. The Kagome lattice band structure (Fig. 1b) contains Dirac cones and flat bands, and the parabolic approximation is not obviously valid at ε_F = −4.0 eV. We will address this by: (1) explicitly examining the band dispersion near ε_F = −4.0 eV in Fig. 1(b) to assess whether it is locally approximately parabolic, and presenting this analysis in the revised manuscript; (2) if the parabolic approximation is found to be inadequate at this energy, we will either adjust ε_F to a regime where the approximation is better justified, or discuss the limitations explicitly and note that the extrinsic magnitudes should be regarded as order-of-magnitude estimates rather than quantitative predictions. We note that the intrinsic AHC/ANC calculations (Eqs. 2–4) use the full Kagome band structure and are not affected by this approximation; the issue is specific to the extrinsic analytical formulas. The sign opposition between intrinsic and extrinsic mechanisms does not depend on the band structure approximation, but the referee is correct that the crossover magnitudes do. We will make the status of the parabolic approximation transparent in the revised manuscript. revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the extrinsic AHC is linear in the MC-computed chirality by construction, but this is a straightforward computation of a derived quantity, not a fit renamed as a prediction.

full rationale

The paper's central claim is a mechanistic decomposition of anomalous transport into intrinsic (Berry curvature) and extrinsic (skew scattering) channels. The extrinsic AHC (Eq. 27) is derived analytically from the Kondo Hamiltonian via second Born approximation and Boltzmann transport, yielding a formula proportional to ⟨S_l · (S_i × S_j)⟩. The DSC δχ (Eq. 30) is defined as total chirality minus static chirality, and substituting MC-computed δχ into Eq. 27 yields the extrinsic transport. This is a computation of a derived quantity from its microscopic inputs, not a fit renamed as a prediction. The linearity in the input chirality is a consequence of the perturbative (J_K^3) derivation, not a circular definition. The intrinsic contribution (Eqs. 2-4) is computed independently via Kubo formula from the static spin configuration. The two mechanisms producing opposite signs is a non-trivial result of the J_K^3 prefactor and the chirality direction, not forced by construction. The paper cites Ishizuka & Nagaosa [27] for the theoretical framework, but this is an external citation to a different author group, not a self-citation. The relaxation time τ being a free parameter affects quantitative magnitudes but does not make the derivation circular. The 3D/2D formalism mixing flagged by the reader is a correctness concern, not a circularity issue. No step in the derivation chain reduces to its own inputs by definition or by self-citation. The paper is self-contained against external benchmarks (experimental sign reversals in Refs. [20, 41]). Score 2 reflects the minor concern that the extrinsic result is a direct linear map of MC inputs rather than an independently falsifiable prediction, but this is standard for ab initio-style transport calculations and does not constitute circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

Legacy field mirrored from axiom_ledger.axioms.

free parameters (5)
  • τ (relaxation time) = not stated numerically
    Appears in Eqs. 17, 27, 28 as a free parameter; not computed from the symmetric scattering rate. Directly scales all extrinsic transport coefficients.
  • ε_F (Fermi energy) = -4.0 eV
    Chosen to evaluate transport in 'low energy region'; affects which band features contribute.
  • J_K (Kondo coupling) = 0.2t = 0.2 eV
    Set following Ref. 44; controls both intrinsic and extrinsic magnitudes.
  • D_xy, D_z (DM interaction) = 0.5, √3 (in units of J_H)
    Chosen to stabilize skyrmion and helical phases; determines the phase diagram topology.
  • Phase boundary thresholds = χ_Q≥3, δχ>0.2, M_z>0.1, M_total>0.2, etc.
    Post-hoc numerical criteria for phase identification; not derived from theory.
assumptions (4)
  • domain assumption Quasistatic approximation: spin fluctuations are slow compared to electron dynamics, making scattering elastic
    Invoked in Sec. II.B to justify elastic scattering (Eq. 6). Validity depends on the ratio of spin dynamics timescale to electronic hopping timescale, not quantitatively checked.
  • domain assumption Low-energy effective mass approximation: parabolic dispersion ε_k = ℏ²k²/2m
    Used in Sec. II.B and Appendix A to evaluate the q-summation analytically. The Kagome band structure (Fig. 1b) is non-parabolic; the regime of validity is not established.
  • domain assumption Skew scattering rate takes the form w⁻ ∝ Ṽ·(k×k')/k² (Eq. 13)
    Assumed compact form for the antisymmetric scattering rate to solve the Boltzmann equation. Justified by the long-wavelength expansion on Kagome triangles but is an ansatz for the general case.
  • domain assumption Intrinsic and extrinsic contributions are additive with no cross-terms
    The total transport (Figs. 5e,f) is computed as σ^int + σ^ext. Potential interference or cross-terms between static and fluctuating chirality are not discussed.
invented entities (1)
  • Fluctuating chiral (FC) phase independent evidence
    purpose: Phase regime where dynamical spin chirality fluctuations dominate extrinsic transport
    Defined by δχ > 0.2 extracted from MC; produces falsifiable predictions of extrinsic AHC/ANC enhancement near phase boundaries. However, the phase is identified by the same quantity (δχ) that drives the extrinsic transport, creating a degree of definitional circularity in the phase-transport correlation.

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Cite this review

Pith. "Pith review of Anomalous Hall and Nernst effects driven by static and fluctuating spin chiralities on Kagome lattice." pith.science (2026). https://pith.science/paper/HNRK7LF2

@misc{pith2026260706966,
  author       = {Pith},
  title        = {Pith review of: Anomalous Hall and Nernst effects driven by static and fluctuating spin chiralities on Kagome lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HNRK7LF2}},
  note         = {Machine review of arXiv:2607.06966}
}
read the original abstract

We theoretically investigate the anomalous Hall and Nernst effects (AHE and ANE) in a two dimensional Kagome lattice to uncover the distinct roles of static and fluctuating scalar spin chi ralities. Employing Monte Carlo simulations incorporating with a tight binding Hamiltonian via the s-d exchange interaction, we explicitly evaluate the anomalous transport coefficients. A key finding is the systematic disentanglement of the macroscopic responses into an intrinsic contribu tion, governed by momentum space Berry curvature induced by static chirality, and an extrinsic skew scattering contribution driven by real space dynamical spin fluctuations. We demonstrate a pronounced mechanistic crossover: deep in magnetically ordered phases like the skyrmion crystal, the intrinsic Berry curvature dictates the transport behavior. However, approaching the magnetic order-disorder critical regime, strong thermal fluctuations disrupt static noncoplanar spin configu rations, drastically suppressing intrinsic responses. Here, dynamical chiral fluctuations emerge as the dominant driving force. By delineating the phase regimes governed by static versus fluctuating chiralities, this work elucidates the distinct microscopic mechanisms dictating anomalous transport in frustrated magnetic systems.

Figures

Figures reproduced from arXiv: 2607.06966 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic illustration of the Kagome lattice. The [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Intrinsic anomalous Hall conductivity [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Snapshots of the real space spin configurations [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Total scalar spin chirality averaged over a sin [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Distribution of the anomalous transport coefficients [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Phase diagrams of thermodynamic and topological quantities calculated via Monte Carlo simulations on a 60 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Thermodynamic phase diagrams as a function of temperature [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Real space spin configurations obtained from Monte Carlo simulations on a finite lattice ( [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

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