On the angular dependence of anomalous Hall current

Junwen Sun; Ke Xia; Lei Wang; Lulu Li; X. R. Wang

REVIEW 4 major objections 5 minor 89 references

On the angular dependence of anomalous Hall current

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read At metal/ferromagnet interfaces, the anomalous Hall current is generally not perpendicular to the magnetization, so the standard product rule j_H = Θ m × j_c fails except at high-symmetry crystal orientations.

desk verdict A solid symmetry-based argument that interface crystal symmetry bends the AHE product rule, but the experimental re-evaluation claim outruns the layer-resolved clean-interface calculations. read the letter →

arxiv 2412.06630 v1 pith:ZDMEUAMT submitted 2024-12-09 cond-mat.mes-hall

Lulu Li , Junwen Sun , Lei Wang , X. R. Wang , Ke Xia This is my paper · ORCID

classification cond-mat.mes-hall

keywords anomalousHalleffectproductrulebreakdowninterfacechiralitychiralcrystalsymmetryspintransportfirst-principlesCu|Co

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

For spintronics devices, the anomalous Hall effect is normally summarized by the product rule $j_H = \Theta m \times j_c$, which says the transverse Hall current is always perpendicular to the magnetization and reverses with it. Using first-principles transport calculations for Cu|Co interfaces, the paper argues that this rule fails at realistic interfaces: the Hall current in the first ferromagnetic layer deviates from the perpendicular direction by up to 20 degrees and does not simply switch sign when the magnetization is reversed. The deviation is controlled by the interface's discrete rotational symmetry, and the paper provides a symmetry-adapted expansion in which the leading term is the conventional Hall term and higher harmonics encode the crystal's angular texture. For threefold-symmetric ($C_{3v}$) interfaces this produces a chiral anomalous Hall effect, and in superlattices the higher-order terms can dominate the conventional one. If the argument is right, Hall measurements on devices containing such interfaces should be re-evaluated.

What carries the argument

The central object is the symmetry-constrained tensor expansion of the anomalous Hall conductivity. The response tensor $\rho_{ij}(m)$ is expanded in powers of the magnetization direction, $\rho_{ij} = \rho^0_{ij} + \rho_{ijk} m_k + \rho_{ijkl} m_k m_l + \cdots$, and the interface point group decides which angular harmonics survive; for the fcc (111) interface this yields $j_H/j_z = \sum_{i=3n-2} p_i (\sin i\alpha, -\cos i\alpha) + \sum_{j=3n-1} p_j (\sin j\alpha, \cos j\alpha)$, with the $i=1$ term being the usual $j_H = \Theta m \times j_c$ and terms with $i \neq 1$ describing the deviation and the chiral magnitude asymmetry. The calculation pairs this expansion with fully relativistic first-principles scattering-wave transport through the interface, and the resulting layer-resolved current in the first Co layer is fitted to the expansion to extract the coefficients $p_i$.

What would settle it

Calculate the total Hall conductance summed over all atomic layers of the Cu|Co(111) junction, or measure the Hall voltage on a clean epitaxial sample as the in-plane magnetization angle is rotated; if the summed or measured current always lies along $m \times j_c$ and reverses sign with $m$ at every angle, the paper's central claim is refuted because its conclusions rest on the first-layer quantity.

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Extended reading notes

Core claim

The paper's central claim is that at a nonmagnetic metal|ferromagnet interface the anomalous Hall current is set by the discrete point-group symmetry of the interface, not by the continuum effective-mass picture, and therefore for generic magnetization directions it is neither perpendicular to $m$ nor antisymmetric under $m \rightarrow -m$. For a Cu|Co interface along the fcc (111) direction with charge current along $z$ and magnetization $m=(\cos\alpha,\sin\alpha,0)$ in the interface plane, the calculated layer-resolved current in the first Co layer follows the $C_{3v}$ symmetry of the lattice translation vectors and deviates from $m \times j_c$ by up to about 20 degrees, with perpendicularity restored only when $m$ aligns with one of those translation vectors. The symmetry-constrained expansion $j_H/j_z = \sum_{i=3n-2} p_i (\sin i\alpha, -\cos i\alpha) + \sum_{j=3n-1} p_j (\sin j\alpha, \cos j\alpha)$ reproduces the calculation with $n=3$; the $i=1$ term is exactly the conventional product rule, and the higher harmonics quantify how the discrete lattice breaks the isotropy of the continuum model. At $C_{3v}$ interfaces the same expansion gives $j_H(m) \neq -j_H(-m)$, a chiral anomalous Hall effect tied to interface chirality $Z = j_c \cdot (m \times c_1)$, and this chirality is visible only in transverse transport because the $k_\parallel$-resolved transmission asymmetries cancel when integrated for the longitudinal conductance. At the higher-symmetry fcc (001) interface ($C_{4v}$) the antisymmetry under magnetization reversal is restored, but the directional deviation from $m \times j_c$ remains.

Load-bearing premise

The load-bearing premise is that the Hall current computed in the first Co layer next to a perfectly clean interface represents the total, experimentally measured Hall signal; the paper itself notes that random disorder equalizes the crystal axes and restores the conventional product rule, so a disorder-averaged or full-thickness calculation could erase the effect.

Editorial extensions

If this is right

In spin-orbit torque and Hall-sensor devices built on low-symmetry interfaces, the transverse voltage becomes a function of higher angular harmonics of the magnetization angle, so simple cosine fits will miss the response.
At $C_{3v}$ interfaces, reversing the magnetization changes the magnitude of the Hall current, giving a controllable chiral signal that is absent from longitudinal conductance.
Superlattices such as [Cu2|Co2]_n can make the $p_2$ term dominate the conventional $p_1$ term, so the Hall current direction no longer reverses with $m$, an engineering route to nonreciprocal transverse responses.
Because the mirror symmetry of the $m$-$j_c$ plane is broken by the lattice, the chiral effect appears only for magnetization directions that are not invariant under the interface mirrors; $m$ along $\pm y$ at $C_{3v}$ is expected to be nonchiral.
Existing AHE-based magnetic characterization on devices with interfaces should be rechecked for higher-harmonic contributions before attributing angular anomalies to other physics.

Reading between the lines

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

The same point-group expansion applies to other current-induced transverse responses at the same interface, such as the spin Hall current or planar Hall effects; the paper does not compute those, but the symmetry argument is generic.
A direct testable extension is to grow Cu|Co(111) junctions with controlled interfacial disorder: the size of the deviation angle $\beta$ should correlate with interface quality and vanish as disorder increases.
The dominance of the $p_2$ harmonic in superlattices suggests that a Fourier analysis of Hall-angle-versus-$\alpha$ data could be used to extract the multipolar Berry-curvature moments of the interface, making the effect a spectroscopic probe.
If a full-thickness or disorder-averaged calculation restores the product rule, the practical impact is limited to the clean-interface, first-layer regime; that calculation would be the decisive check of the paper's broader conclusion.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First, the thing to know: this is a serious first-principles study with a clean symmetry argument, but the headline conclusion outruns the evidence. The paper shows for Cu|Co(111) that the anomalous Hall current in the first Co layer is not generally perpendicular to m, and that under m reversal the magnitude changes (j_H(m) ≠ -j_H(-m)), which they call a chiral anomalous Hall effect. The symmetry basis for this is sound: the C3v plane has three inequivalent mirror axes, and when m is along the high-symmetry directions the transverse current still deviates because the crystal axes break the mirror symmetry of the m–j_c plane. The tensor expansion in Eq. (3) gives the allowed harmonics (sin iα, -cos iα) for i=3n-2 and (sin jα, cos jα) for j=3n-1, which is useful and standard. The k-resolved transmission maps in Fig. 2 clearly show chiral hot spots that cancel after BZ integration, making the point about hidden chirality concrete. The superlattice idea to amplify the p2 term is clever. I credit the FR-EMTO calculation as an appropriate tool, and the paper is generally honest about its scope, though the conclusion is not.

The soft spots are structural. The whole quantitative case rests on the current in the first Co layer next to the interface. A measured Hall voltage is an integral over the FM layer thickness and over disorder configurations, and the paper never provides that integral. The authors' own statement that introducing random disorders equalizes the axes and restores j_H = Θ m × j_c means the effect may be a clean-interface boundary artifact. That is not a minor footnote; it undermines the claim that all AHE measurements should be re-evaluated. Second, the expansion coefficients p_i are fitted to the same j_H data they claim to explain, so Eq. (3) is a parametrization, not an independent derivation. Third, the superlattice period n for [Cu2|Co2]_n is never specified, so the dominant p2 result cannot be checked. There are also no error bars or convergence tests.

Who is this for? People working on interface AHE, spin-orbit torque, and spin Hall magnetoresistance would benefit from the symmetry framework. It deserves a serious referee, because the CAHE prediction is concrete and testable, and the symmetry analysis will likely be reused. But the authors should be required to show thickness-integrated conductances, address disorder, specify the superlattice, and revise the sweeping conclusion. A desk rejection would be wrong; as is, it needs major revision.

Referee Report

4 major / 5 minor

Summary. The paper studies the anomalous Hall current (j_H) at Cu|Co interfaces using first-principles FR-EMTO transport calculations. It reports that, for a charge current injected along the fcc (111) direction and magnetization rotated in the interface plane, the direction of j_H is generally not perpendicular to the magnetization, i.e., the product rule j_H = Θ m × j_c breaks down. The authors fit the angular dependence to a symmetry-based tensor expansion (Eq. 3 for C3v, Eq. 4 for C4v) and interpret the higher-order harmonics as arising from discrete crystal symmetry and interface chirality. They further identify a chiral anomalous Hall effect (CAHE) at C3v interfaces, where j_H(m) ≠ -j_H(-m), and show that in a [Cu2|Co2]_n superlattice the second harmonic can dominate the first. They also analyze k-resolved transmission chirality and argue that the total longitudinal conductance remains symmetric while the transverse AHE can reveal the interface chirality.

Significance. If the central claim holds, the conventional assumption that AHE signals are always antisymmetric under magnetization reversal would need to be revisited for interface-dominated devices, and the symmetry-based angular expansion would provide a useful framework for describing such effects. The paper identifies a concrete physical system (Cu|Co interfaces) and a specific symmetry condition (C3v) for the chiral AHE, and it presents k-resolved transmission data that illustrate the chirality. However, the evidence currently rests on layer-resolved currents in ideal clean interfaces, with no thickness-integrated or disorder-averaged Hall signal, so the broader experimental significance is not yet established. The symmetry arguments and the fitting procedure are credible, but the distinction between symmetry-derived predictions and fitted parameters should be made clearer.

major comments (4)

[Results, Fig. 1 and Fig. 3] The central quantitative results (angular dependence of j_H, deviation angle β, and the chiral AHE Θ(m||x) ≠ -Θ(m||-x)) are all computed for the first Co layer near the interface ('ji_H'), as stated in the Results section. A measurable Hall voltage, however, is determined by the transverse current integrated over the full ferromagnetic layer and averaged over disorder configurations. The manuscript does not provide a thickness-integrated total transverse conductance or any disorder-averaged Hall current, and therefore does not demonstrate that the reported first-layer effect survives these integrations.
[Results, paragraph on random disorders] The text states that 'introducing random disorders equalizes the differences between these primary axes, resulting in the direction of the anomalous Hall current eventually adhering to jH = Θm×jc'. This is an explicit admission that the predicted deviation vanishes in disordered systems, yet experimental interfaces generally contain disorder. The manuscript does not quantify the disorder strength or the crossover between the clean and disordered regimes, which makes the abstract and conclusion claim that 'all experimental measurements related to the AHE should be re-evaluated' unsupported.
[Fig. 4 and Table I, [Cu2|Co2]_n] The superlattice [Cu2|Co2]_n is used to demonstrate that higher-order harmonics can dominate the conventional AHE, and Table I lists fitting parameters for this system. However, the value of n (the number of repeating units) used in Fig. 4 is never specified, and no convergence with respect to n is shown. Without this parameter, the result is not reproducible and the claim that higher-order terms dominate cannot be independently checked.
[Eq. (3) and fitting description] The paper says that fitting the calculated j_H with Eq. (3) using n = 3 'provides a sufficiently accurate representation'. Because the coefficients p_i and p_j are fitted to the same data that the formula is supposed to describe, the agreement is a fit rather than a predictive test. A stronger validation would be to derive the coefficients from the tensor expansion coefficients ρ_ijk etc. for a separately computed response tensor, or to compare the fitted angular form with a symmetrically constrained but coefficient-free prediction. As presented, the claim that Eq. (3) 'can describe this effect well' is circular.

minor comments (5)

[Fig. 1(c)] The deviation angle β is reported as ranging in [-20, 20] without specifying the units; please state explicitly that the values are in degrees.
[Eqs. (3) and (4)] The symbol n is used both as the summation index for the harmonic order and as the number of repeating units in [Cu2|Co2]_n, which is confusing. Consider using a different index (e.g., k) for the harmonic order.
[Fig. 4 caption] Several α labels in the caption are corrupted (e.g., 'α=???°', 'α=???°', 'α=2??°'), making it difficult to map the data points to the corresponding magnetization directions.
[Supplementary information] Many essential details, including the derivation of Eqs. (3) and (4) and the disorder calculations, are relegated to the supplementary information and cited only as [70]. A brief outline of the tensor derivation in the main text would improve the paper's self-containedness.
[Abstract and Conclusion] The phrase 'all experimental measurements related to the AHE should be re-evaluated' overstates the scope of the findings, which concern layer-resolved clean-interface currents. The claim should be qualified to reflect the disorder dependence and the interface-specific nature of the effect.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central breakdown claim is an ab initio result; the symmetry expansion is a labeled fit.

full rationale

The paper's main claim is the first-principles result that at clean Cu|Co interfaces j_H is not generally parallel to m×j_c and, for C3v symmetry, j_H(m) ≠ −j_H(−m). This is determined by the FR-EMTO scattering calculation displayed in Figs. 1–3, independent of any later fitting. Equation (3) is a symmetry-constrained expansion of the conductivity tensor under C3v and My, derived from standard tensor theory and the supplementary material; the coefficients p_i are explicitly fitted to the same calculated j_H ('By fitting our calculated j_H in Fig. 1(b) using Eq. (3)'). Because the paper labels this as fitting rather than as an independent prediction, the agreement of the dashed lines is a compact parametrization rather than a circular derivation. The perpendicular configurations α_p = lπ/3 and α_p = lπ/4 follow from the symmetry-allowed harmonic content (all terms in j_H·m involve sin(3kα) or sin(4kα)), not from fitted coefficient values. The superlattice statement that p_2 dominates over p_1 is likewise read off from the fitted coefficients of the [Cu2|Co2]n calculation, not an independent forecast. No load-bearing uniqueness theorem or external self-citation is invoked; reference [70] is the paper's own supplementary derivation. Two non-circular limitations weaken the experimental generalization: the calculation reports only the first-Co-layer current (ji_H), and the authors themselves state that random disorder restores j_H = Θm×j_c; thus the 'all experimental measurements must be re-evaluated' conclusion is an extrapolation, but it is not an input–output identity and does not constitute circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The central quantitative claims are carried by the fitted coefficients p_i, which are adjusted to reproduce the same first-principles data they are used to explain. The symmetry expansion itself is a descriptive parametrization. The only independent input is the FR-EMTO calculation, which is not independently verifiable from the paper. The invented entity (interface chirality) is a definition rather than a new physical degree of freedom.

free parameters (6)

p1 for Cu|Co(111) = -0.021
Leading AHE coefficient in Eq. (3); fitted to the calculated j_H angular dependence.
p2 for Cu|Co(111) = 0.0065
First higher-order harmonic in Eq. (3); this coefficient is the main source of the deviation from the product rule.
p5 for Cu|Co(111) = -0.0017
Higher-order harmonic in Eq. (3); contributes to the angular deviation and chirality.
p1 for [Cu2|Co2]_n superlattice = -0.0077
Leading AHE coefficient for the superlattice; fitted to calculated data.
p2 for [Cu2|Co2]_n superlattice = 0.061
Dominant higher-order coefficient in the superlattice; larger than p1, leading to a Hall current that barely changes on magnetization reversal.
p1 for Cu|Co(001) = -0.019
Leading AHE coefficient for the (001) interface; no higher-order terms listed for this case.

assumptions (5)

domain assumption Linear response expansion of the conductivity tensor in powers of the magnetization m (Eq. 2) converges and is physically meaningful.
The paper assumes that the anomalous Hall current can be expanded as rho_ij(m) = rho0_ij + rho_ijk m_k + ... and that truncating at low order is valid.
domain assumption The in-plane point group of the Cu|Co(111) interface is C3v and of Cu|Co(001) is C4v, with the stated mirror symmetries.
These symmetry groups determine which angular harmonics appear in Eqs. (3) and (4). The assignment is based on the crystal structure and is standard, but it is an assumption about the ideal interface geometry.
domain assumption The FR-EMTO first-principles scattering method accurately describes the interface transport.
The authors rely on their 'homemade' code, referenced in refs. [39, 40, 65], to produce the numerical j_H values. No independent code or data is supplied for verification.
domain assumption Random disorder restores the conventional product rule j_H = Theta m times j_c.
The text states this result (with details relegated to supplementary [70]), and it is used to reconcile the new effect with existing experiments that apparently follow the product rule.
domain assumption The layer-resolved anomalous Hall current in the first Co layer is representative of the experimentally measurable AHE.
All main-text results are for ji_H in the first Co layer, and the conclusion extrapolates to experimental measurements without a total-current or disorder-averaged calculation.

invented entities (1)

Interface chirality (IC) with chiral index Z = jc dot (m cross c1)
purpose: To classify right- and left-handed configurations of the magnetization, charge current, and crystal axis at the interface, and to explain the chiral Hall effect.
This is a bookkeeping index, not a new physical object. The associated 'hidden chirality' is a symmetry statement about transmission probabilities and does not constitute a falsifiable entity outside the paper's definitions.

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Cite this review

Pith. "Pith review of On the angular dependence of anomalous Hall current." pith.science (2026). https://pith.science/paper/ZDMEUAMT

@misc{pith2026241206630,
  author       = {Pith},
  title        = {Pith review of: On the angular dependence of anomalous Hall current},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZDMEUAMT}},
  note         = {Machine review of arXiv:2412.06630}
}

read the original abstract

The transverse current (j_H) due to anomalous Hall effect (AHE) is usually assumed to be perpendicular to the magnetization (m) in ferromagnetic materials, which governs the experiments in spintronics. Generally, this assumption is derived from a continuum model, where the crystal's discrete symmetry is effectively represented by the concept of an effective mass from the band structure. In this paper, we calculate the spin transport through the nonmagnetic metal (NM) | ferromagnetic metal (FM) interfaces and find that the corresponding Hall current is generally not perpendicular to m with only a few exceptions at high symmetry crystal orientations. The calculation illustrates the breakdown of j_H={\theta}m{\times}j_c, where {\theta} denotes the anomalous Hall angle and j_c represents the injecting charge current. An analytical formula based on the discrete symmetry of the solid can describe this effect well. In this framework, the leading order corresponds to the conventional AHE, while higher-order terms account for deviations in the Hall current. Additionally, we identify the presence of a chiral anomalous Hall effect (CAHE) at interface with odd rotational symmetry (e.g., C_{3v}) and the higher-order terms can even dominate the AHE by constructing superlattices. The general existence of hidden chirality in spin transport is also revealed, with a specific focus on interface chirality (IC). Our results highlight the significance of discrete atomic positions in solids for spin transport, which extends beyond the conventional continuum model. Moreover, considering the important application of the AHE in spintronics and the wide existence of the interfaces in the devices, the breakdown of j_H={\theta}m{\times}j_c suggests that all experimental measurements related to the AHE should be re-evaluated.

Figures

Figures reproduced from arXiv: 2412.06630 by the authors.

**Figure 2.** FIG. 2. (a) The fcc structure features a Cu [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) The layer-resolved anomalous Hall angles (Θ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. The angle ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5. (a) Top view of the Cu [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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