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arxiv: 2510.20804 · v2 · submitted 2025-10-23 · ❄️ cond-mat.mes-hall · cond-mat.str-el· cond-mat.supr-con

Anomalous Hall effect in rhombohedral graphene

Vera Mikheeva , Daniele Guerci , Daniel Kaplan , Elio J. K\"onig This is my paper

Pith reviewed 2026-05-18 04:15 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-elcond-mat.supr-con
keywords anomalous Hall effectrhombohedral grapheneimpurity scatteringKubo-Streda formulaskew scatteringwarping effectsquarter metal
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0 comments X

The pith

The anomalous Hall conductivity in rhombohedral graphene arises from intrinsic, side-jump, and multiple skew-scattering processes computed via diagrammatic expansion for different impurity regimes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper calculates the anomalous Hall conductivity for rhombohedral stacked multilayer graphene in a spontaneous spin-valley polarized quarter metal state. It uses the Kubo-Streda diagrammatic approach to treat two impurity types: weak dense impurities via Gaussian disorder and strong sparse impurities. Non-crossing diagrams capture intrinsic, side-jump, and Gaussian skew-scattering contributions, while additional crossing diagrams X and Ψ account for diffractive skew-scattering, and the Mercedes star diagram handles non-Gaussian skew-scattering. Analytical solutions are derived for an isotropic model, with warping effects included only perturbatively because they shape the low-energy bands.

Core claim

The anomalous Hall conductivity σ_xy is obtained by summing all non-crossing diagrams for intrinsic, side-jump and Gaussian skew-scattering contributions together with diagrams X and Ψ that represent diffractive skew-scattering processes in the Gaussian disorder model, plus the Mercedes star diagram for non-Gaussian skew scattering when impurities are strong and sparse, after first solving the isotropic case analytically and then adding warping perturbatively.

What carries the argument

Kubo-Streda diagrammatic expansion that systematically includes non-crossing impurity diagrams plus selected crossing diagrams for skew-scattering contributions to the Hall conductivity.

If this is right

  • Different impurity regimes produce qualitatively different balances among intrinsic, side-jump, and skew-scattering contributions to the Hall conductivity.
  • Warping, although treated perturbatively, modifies the low-energy Hall response in a measurable way.
  • The diagrammatic results supply a microscopic basis for the anomalous Hall effect reported in recent experiments on rhombohedral multilayer graphene.

Where Pith is reading between the lines

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

  • The same diagrammatic bookkeeping could be applied to other spontaneously valley-polarized 2D systems to predict how impurity engineering alters the Hall response.
  • Varying the ratio of weak to strong impurities might allow experimental control over the relative weight of diffractive versus non-Gaussian skew scattering.
  • The calculated conductivity expressions could be inserted into transport simulations to test consistency with finite-temperature or finite-size data.

Load-bearing premise

The main analytical solutions assume an isotropic dispersion relation and treat band warping only as a small perturbation.

What would settle it

A direct measurement of the anomalous Hall conductivity versus impurity density or scattering strength that cannot be reproduced by any combination of the calculated intrinsic, side-jump, Gaussian skew, diffractive skew, and non-Gaussian skew terms would falsify the result.

Figures

Figures reproduced from arXiv: 2510.20804 by Daniele Guerci, Daniel Kaplan, Elio J. K\"onig, Vera Mikheeva.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) Schematic band structure of rhombohedral [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Exemplary diagrams for the anomalous Hall [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Example diagram of the side-jump contribution [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Intrinsic Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Full anomalous Hall conductivity (one val [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

Motivated by recent experiments on rhombohedral stacked multilayer graphene and the observation of the anomalous Hall effect in a spontaneous spin-valley polarized quarter metal state, we calculate the anomalous Hall conductivity for this system in the presence of two types of impurities: weak and dense as well as sparse and strong. Our calculation of $\sigma_{xy}$ is based on the Kubo-Streda diagrammatic approach. In a model with Gaussian disorder applicable to weak dense impurities, this involves all non-crossing diagrams (intrinsic, side-jump and Gaussian skew-scattering contributions) and additionally diagrams with two intersecting impurities, X and $\Psi$, representing diffractive skew-scattering processes. A "Mercedes star" diagram (non-Gaussian skew scattering) is furthermore included to treat in the case of strong, sparse impurities. We supplement our asymptotically exact analytical solutions for an isotropic model without warping effects by semi-numerical calculations accounting perturbatively for warping, which plays a crucial role in the low-energy band structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript calculates the anomalous Hall conductivity σ_xy in rhombohedral stacked multilayer graphene in the spontaneous spin-valley polarized quarter-metal state. It uses the Kubo-Středa diagrammatic approach, incorporating intrinsic, side-jump, Gaussian skew-scattering, diffractive skew-scattering (X and Ψ diagrams with intersecting impurities), and non-Gaussian skew-scattering (Mercedes-star diagram) contributions for weak-dense and strong-sparse impurity regimes. Asymptotically exact analytical solutions are derived for an isotropic model without warping, supplemented by semi-numerical calculations that include warping perturbatively.

Significance. If the central results hold, this provides a systematic microscopic theory for the anomalous Hall effect in rhombohedral graphene quarter metals, addressing recent experimental observations through a comprehensive treatment of disorder scattering channels. The inclusion of higher-order diagrams beyond standard non-crossing approximations and the attempt at asymptotically exact analytics for the isotropic case are strengths. The work could help interpret the role of impurities in stabilizing the observed AHE.

major comments (1)
  1. Abstract: The abstract states that warping 'plays a crucial role in the low-energy band structure,' yet the asymptotically exact analytical solutions for σ_xy are derived exclusively for an isotropic model without warping effects, with warping added only perturbatively in later semi-numerical calculations. If warping is essential, this raises a load-bearing concern that the leading analytical expressions may miss non-analytic or higher-order contributions dominant in the quarter-metal regime, and the validity of the perturbative correction is not demonstrated.
minor comments (1)
  1. The abstract describes the diagrammatic program in detail but does not include explicit final expressions for the conductivity contributions or direct numerical outputs; providing these in the main text or supplementary material would aid verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below and propose revisions to improve clarity.

read point-by-point responses

  1. Referee: Abstract: The abstract states that warping 'plays a crucial role in the low-energy band structure,' yet the asymptotically exact analytical solutions for σ_xy are derived exclusively for an isotropic model without warping effects, with warping added only perturbatively in later semi-numerical calculations. If warping is essential, this raises a load-bearing concern that the leading analytical expressions may miss non-analytic or higher-order contributions dominant in the quarter-metal regime, and the validity of the perturbative correction is not demonstrated.

    Authors: We appreciate the referee highlighting this potential inconsistency. The abstract emphasizes warping's role in the low-energy band structure of rhombohedral graphene to motivate the overall study and the semi-numerical results. The asymptotically exact analytical expressions are obtained for the isotropic model (without warping) because this limit permits a complete summation of all diagrams (intrinsic, side-jump, Gaussian skew-scattering, X/Ψ diffractive, and Mercedes-star non-Gaussian) within the Kubo-Středa formalism for both weak-dense and strong-sparse impurity regimes. Warping is then added perturbatively in the semi-numerical calculations, as it represents a small correction relative to the dominant energy scales in the quarter-metal regime. We maintain that non-analytic or higher-order contributions arising from warping enter only at sub-leading orders in the perturbative parameter and do not alter the leading analytical results. To resolve the concern, we will revise the abstract to explicitly distinguish the isotropic analytical results from the perturbative warping treatment and add a short paragraph in the main text justifying the perturbative validity. revision: partial

Circularity Check

0 steps flagged

No circularity: conductivity derived from microscopic model via standard diagrams

full rationale

The paper derives the anomalous Hall conductivity σ_xy from the Kubo-Streda diagrammatic expansion applied directly to a microscopic Hamiltonian with Gaussian or non-Gaussian disorder. All listed contributions (intrinsic, side-jump, Gaussian skew, diffractive X/Ψ, Mercedes-star) are computed from the model parameters and impurity scattering without any parameter being fitted to the target conductivity or renamed as a prediction. The isotropic analytical solutions without warping are supplemented by perturbative corrections; this is an explicit approximation choice rather than a self-definitional or self-citation reduction. No load-bearing step collapses to a prior self-citation or to the result itself by construction, so the derivation chain remains independent of its outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on standard transport theory for disordered electrons together with the assumption that selected diagram classes capture the leading impurity effects in each regime.

axioms (2)
  • standard math The Kubo-Streda formula correctly gives the Hall conductivity when all relevant impurity diagrams are summed.
    Invoked as the basis for the entire calculation.
  • domain assumption Warping can be added perturbatively after solving the isotropic model.
    Stated explicitly when moving from analytical to semi-numerical results.

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    Intrinsic contribution for isotropic model In this appendix, we show details for the intrinsic con- tribution, Eq. (5), which in the Kubo-Streda approach is given by σint xy = e2 h ∫ dp2 (2π) 2 Tr⎛ ⎝jx(p)G R p jy(p)G A p ⎞ ⎠.(B5) After taking the trace we can get σint xy = e2 h ∫ ∞ 0 pdp 2π ∫ 2π 0 dθ 2π υ2n2(4ϵmi(i0)p 2n−2+iυ 2p4n−2(e−2iθ−e2iθ)) (υ2p2n 0 ...

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  61. [61]

    (7) of the main text

    Vertex correction In this appendix, we calculate a single-impurity vertex correction, which is given by Eq. (7) of the main text. Let us consider the first order correction: δjx/y=α ϵ nρn(ϵ) ∫ d2p (2π) 2 GA p jx/yGR p ∝ ∫ d2p (2π) 2(a(∣⃗p∣2)+b(∣⃗p∣2)P n+c(∣⃗p∣2) ¯P n)P n−1+h.c.=0. (B7) The matrix-valued isotropic functions a(∣⃗p∣2), b(∣⃗p∣2), c(∣⃗p∣2)may ...

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  62. [62]

    (13a), (13b) of the main text

    Diffractive skew-scattering In this appendix we provide calculations about diffrac- tive skew-scattering contribution from X and Ψ diagram Eq. (13a), (13b) of the main text. σX xy = e2 h ⎛ ⎝ ϵ nρn(ϵ) α⎞ ⎠ 2 ∫{pi} Tr[GA p1 jx(p1)GR p1 GR p3 GR p2 jy(p2)GA p2 GA p4](2π)2δ(p 1+p 2−p3−p4).(B10) We rewrite the average Green’s function as GR p =N(p)G R p , N(p)...

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  63. [63]

    Calculation of integral over Bessel functions In this section, we present calculations of the Bessel- function integrals appearing in (13a)–(13b) and obtain asymptotic expressions in the limitn→∞. We need the following integrals: In10n = ∫ ∞ 0 dxJnJ1J0 ˜Yn,(B11a) Inn10 = ∫ ∞ 0 dxJ2 nJ1 ˜Y0,(B11b) Inn01 = ∫ ∞ 0 dxJ2 nJ0 ˜Y1,(B11c) In±00= ∫ ∞ 0 dxJn(Jn−1−Jn...

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  64. [64]

    (2b), of the disorder potential Eq

    Case of sparse and strong impurities In this appendix we deduce the contribution of the higher-order moment, Eq. (2b), of the disorder potential Eq. (16). The contribution from ”Mercedes star” diagram: σMC xy =σ Y xy+σ Y xy = e2 h (βυ 3p3n−4 0 )⋅ ∫{pi} ⎛ ⎝Tr[GA p1 jx(p1)GR p1 GR p3 GR p2 jy(p2)GA p2]+ Tr[GA p1 jx(p1)GR p1 GR p2 jy(p2)GA p2 GA p3]⎞ ⎠. This...

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  65. [65]

    Firstly, let us define the Fermi momentum

    Necessary designations In this section, we introduce all the necessary defini- tions which we use in this paper for the warped model, see Eqs.(17)-(23). Firstly, let us define the Fermi momentum. Using the notationϵ 2−m2 =v 2p2n 0 we find ξ2 =m 2+v 2p2n+2vwp 2n−3cos(3ϕ)+w 2p2n−6.(C1) Solvingξ=ϵusing ¯p=p/p 0,˜w=w/vp 3 0 leads to the con- dition 1=¯p2n+˜w¯...

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  66. [66]

    (20a), (25), (27), (29) from the main text

    Trilayer graphene in warped model In this section, we provide calculations for trilayer graphene, i.e.n=3 in warped case, for Eqs. (20a), (25), (27), (29) from the main text. We start from the effective HamiltonianH (w) 3 : H (w) 3 = ⃗d(p)⋅⃗σ=( m d − d+ −m),(C5) withd −=d x−iy(p)=vP 3+w;d z =m. a. Intrinsic contribution Firstly, we consider intrinsic cont...

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  67. [67]

    Tetralayer graphene in warped model In this section we provide calculations for tetralayer graphene, i.e.n=4 in the warped case, i.e. Eqs. (20b), (25), (28), (29) from the main text. We do the similar steps as for the trilayer case. We start from the effective HamiltonianH (w) 4 : H (w) 4 = ⃗d(p)⋅⃗σ=( m d − d+ −m),(C20) withd −=d † + =d x−iy(p)=vP 4+wP;d ...

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