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REVIEW 5 major objections 4 minor 7 references

Direct and inverse spin Hall effect: Lorentz force and Zeeman energy

T0 review · 5 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Lorentz and Zeeman forces, seen from each spin's rest frame, account for both the spin Hall effect and the inverse spin Hall effect, without invoking quantum spin-orbit scattering.

desk verdict Elegant classical picture of SHE/ISHE, but the frame argument only works for full polarization, α=1 is unrealistic, and Eq. (7) has a sign problem. read the letter →

arxiv 1908.07515 v1 pith:OEWLM2VT submitted 2019-08-19 cond-mat.mes-hall

Antonio Hernando , Fernando Gálvez , Francisco Guinea This is my paper · ORCID
classification cond-mat.mes-hall PACS 72.25.-b72.25.Ba
keywords spinHalleffectinverseZeemanforceLorentzpurecurrentaccumulationanglemagneticmoment
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 spin Hall effect and the inverse spin Hall effect are consequences of classical magnetic forces rather than of quantum spin-orbit scattering. The idea is to view each spin sub-band in the reference frame where its carriers are at rest: the neutral lattice then moves as a background current, and its inhomogeneous magnetic field exerts a Zeeman force $\mp\mu\nabla B$ on the carrier spins. When a battery drives an unpolarized current, these opposite forces on the two sub-bands produce the transverse spin accumulation of the spin Hall effect. When no net current flows but the two spin species drift in opposite directions, the forces add up instead of cancelling, producing a net transverse force on the conduction band and hence the charge imbalance and Hall voltage of the inverse spin Hall effect. If the derivation is right, both effects follow from electrodynamics plus the spin magnetic moment, with strengths of the order reported in experiment.

What carries the argument

The machinery is the Zeeman force $\mp\mu\nabla B$ evaluated in the inertial rest frames of the two spin sub-bands, together with the Lorentz (Hall) force between sub-bands and between each sub-band and the moving positive background; the inhomogeneous background field is $B(x) = \alpha \mu_0 M_s$, with $\alpha$ a geometric coefficient taken as 1 in the main text. The frame equivalence, namely that the electrical current in the lab frame equals the relative lattice current in the carrier rest frame, converts a steady charge current into a magnetic field acting on the spins, the same mechanism as atomic spin-orbit coupling with the orbital current replaced by a linear current. Equations (4), (6), (5), and (7) carry the argument: (4) gives the force each sub-band feels from the background, (6) the equal-and-opposite force between sub-bands, and (5) and (7) combine them into the total transverse force identified with the inverse spin Hall effect and the spin-separating force identified with the direct spin Hall effect.

What would settle it

Re-derive the transverse force using the full current present in the spin-up rest frame, namely the moving background plus the oppositely drifting spin-down carriers, rather than the single-species current $j_\uparrow$, and check whether the coefficient in $F_{\rm tot} = -\mu_0 M_s (j_\uparrow - j_\downarrow)$ survives unchanged; a shift would break the quantitative claim. Experimentally, sweep the spin polarization in a partially polarized film and test whether the inverse spin Hall voltage scales strictly with $M_s$ and $(j_\uparrow - j_\downarrow)$ as Eq. (5) demands, or follows the polarization dependence of a two-fluid correction.

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Extended reading notes

Core claim

On the paper's own terms, the claim is that magnetic forces account for both effects: the Lorentz force exerted on electric currents, and the force $\mp\mu\nabla B$ exerted on electron spins at rest. The load-bearing step is a frame equivalence: for an electrically neutral solid the current seen in the laboratory frame, due to carrier motion, is the same as the current seen in a frame in which the carriers are at rest, due to the opposite motion of the lattice. Working in the rest frames of the two spin sub-bands, the paper computes the force each sub-band feels from the moving background current and the mutual force between sub-bands, and combines them into two results: the net transverse force on the whole conduction band, $F_{\rm tot} = -\mu_0 M_s (j_\uparrow - j_\downarrow)$, which is nonzero for a pure spin current despite zero net electric current and therefore drives the inverse spin Hall charge imbalance; and the force difference between sub-bands, $F_{\rm spin} = -\mu_0 M_s (j_\uparrow + j_\downarrow) + \mu_0 M (j_\uparrow - j_\downarrow)$, which separates the spins for an unpolarized battery current. Four scenarios are worked through: unpolarized band with a charge current (spin Hall effect), unpolarized band with a spin current (inverse spin Hall effect), fully polarized band (the two effects coincide), and partially polarized band (both imbalances appear, with weights fixed by the ratio of spin currents).

Load-bearing premise

The argument assumes that the current seen in the laboratory frame (carrier motion) is the same as the current seen in the carrier rest frame (lattice motion in the opposite direction); that equality is exact only when one spin species carries the whole current, and in the mixed two-spin case the background current in a carrier's rest frame is set by the total electron density, not by that species alone, so the coefficients in Eqs. (4)-(5) can shift.

Editorial extensions

If this is right

  • A pure spin current, although it carries zero net electric charge, exerts a net transverse force $-\mu_0 M_s (j_\uparrow - j_\downarrow)$ on the conduction band, so spin injection into an unpolarized film should produce a transverse Hall voltage, the inverse spin Hall effect, whose magnitude is set by the saturation magnetization.
  • An unpolarized charge current produces opposite transverse forces on the two spin sub-bands ($F_{\rm spin} = \mu_0 M_s j$ in the depolarized case), so spin accumulates at the film edges with a Hall angle of the order seen in spin Hall experiments.
  • In a fully spin-polarized band the distinction between the two effects collapses: a spin current and a battery-driven current generate the same transverse force, so the direct and inverse effects coincide quantitatively.
  • In a partially polarized band both imbalances appear at once, with the charge-imbalance force and the spin-separating force governed by the ratio of spin-up to spin-down current (Eqs. 12-13), giving a polarization dependence that can be tested.
  • When the lattice itself is magnetized, the force formulas acquire additional terms proportional to the lattice magnetization times $(j_\uparrow + j_\downarrow)$, which the paper connects to the anomalous Hall effect (supplementary, Eq. 2').

Reading between the lines

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

  • A clean experimental discriminator follows: in one host metal with fixed spin-orbit coupling, tune the magnetization by temperature or dilute magnetic doping; the classical mechanism predicts the inverse spin Hall voltage tracks $M_s(T)$, whereas scattering-based conversion tracks the resistivity.
  • Because the field $B(x) = \alpha \mu_0 M_s$ carries a geometric factor, the derived Hall angle should depend on the film's width-to-length ratio; spin-orbit scattering models have no such shape dependence, so aspect-ratio sweeps of the same material could separate the two channels.
  • If the mechanism is operative, the inverse spin Hall effect should appear in light metals with negligible atomic spin-orbit coupling once a spin current is injected, a prediction the quantum scattering models would not make and one that is testable with existing spin-injection geometries.
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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

5 major / 4 minor

Summary. The paper claims that the transverse spin imbalance in the spin Hall effect and the transverse charge imbalance in the inverse spin Hall effect can both be explained by classical magnetic forces: the Lorentz force on electric currents and the Zeeman force ∓μ∇B on electron spins. The authors consider a thin metallic film with spin-up and spin-down currents along y, introduce rest frames for each spin species, and relate the forces exerted by the positive background and by the opposite-spin sub-band to the currents. They derive a total transverse force F_total = -μ0 M_s (j↑ - j↓) (Eq. 5) and a spin-separating force F_diff (Eq. 7), then apply these to four scenarios: spin Hall effect in a depolarized film, inverse spin Hall effect in a pure spin current, fully polarized band, and partially polarized band. The paper concludes that both effects are natural consequences of classical electrodynamics plus the spin magnetic moment.

Significance. The manuscript puts forward a clear, falsifiable proposal: that the Zeeman force ∓μ∇B on spin magnetic moments, together with the Lorentz force on the background current, quantitatively explains the spin Hall and inverse spin Hall effects in metallic films. The strength of the paper is its ambition to reduce both effects to classical electrodynamics, and its explicit formulas (Eqs. 5, 8, 9) are testable predictions. The paper does not fit data, and it has essentially one free parameter (the geometry coefficient α), which is a point in favor of falsifiability. However, as detailed below, the reference-frame argument underlying the derivation is not valid for a two-fluid conductor, the geometric factor α is not established and appears to be orders of magnitude smaller than assumed, and an algebraic error affects Eq. (7). If these issues can be resolved, the mechanism would be significant; in its present form the quantitative claim is not supported.

major comments (5)
  1. [Main text, second paragraph; Supplementary a; Eq. (4)] The frame-equivalence assertion used to derive Eq. (4) is not valid for the two-fluid conductor described in the paper. The paper states that in a neutral solid the current seen in the laboratory frame due to carrier motion equals the current seen in the frame where the carriers are at rest due to relative lattice motion. For a single carrier species this is true, but here there are two species with different drift velocities. In S↑, the lattice (positive background) has density n=n↑+n↓ and moves with velocity -v↑, so the lattice current is -e n v↑, whereas the lab spin-up current is j↑=-e n↑ v↑. These coincide only for a fully polarized band (n↓=0). In the depolarized case used for Eqs. (5), (8), and (9), n↑=n↓=n/2, so the lattice current in S↑ is 2j↑, not j↑. Consequently the magnetic field B=α μ0 x j_lattice used in the Zeeman force has the wrong magnitude and the derived forces are incorrect by a factor of order unity in the depolarized case; the derivation of Eq. (4) is therefore not established.
  2. [Supplementary a and Eqs. (5), (8), (9)] The assignment α=1 is not a harmless simplification. For a thin-film strip of width W and thickness t carrying a uniform current density along y, the transverse magnetic field is of order μ0 j t, so the gradient along x is at most μ0 j t / W. Thus the geometric coefficient α in B(x)=α μ0 x j_s is of order t/W, which is 10^-3 to 10^-4 for typical spin Hall geometries (W/t ~ 10^3–10^4), not 1. The main text sets α=1 for clarity, and the paper uses this value to claim order-of-magnitude agreement with experiment. With a realistic α, the forces in Eqs. (5) and (8) and the inverse spin Hall field in Eq. (9) would be several orders of magnitude smaller than claimed. Since no value or estimate of α is provided, the central quantitative claim is unsupported.
  3. [Eq. (7) and the partially polarized case] Equation (7) does not follow from Eqs. (3) and (6). Substituting n↑ f↑↓ = -μ0 M (j↑-j↓) from Eq. (6) into Eq. (3) gives the internal-force contribution 2 n↑ f↑↓ = -2 μ0 M (j↑-j↓), whereas Eq. (7) contains + μ0 M (j↑-j↓). The sign and coefficient error propagates to the partially polarized expressions in Eqs. (12) and (13), so the derived spin-separation force is not correctly stated.
  4. [Paragraph after Eq. (1) and Ref. [1]] The central quantitative premise of the paper—that the force ∓μ∇B produces a transverse spin imbalance of the same order as the observed spin Hall effect—is not derived in this manuscript but is attributed to the authors' own unpublished preprint [1]. The present paper therefore does not independently establish the magnitude of the effect; the order-of-magnitude agreement with experiments rests on an unreviewed citation. This missing derivation is load-bearing because the subsequent calculation in Eqs. (4)–(9) assumes the Zeeman-force mechanism without proving that it is large enough.
  5. [Second paragraph; Eqs. (4)–(9)] The derivation keeps only magnetic forces and neglects electric fields produced by the Lorentz transformation between the lab frame and S↑/S↓. In a frame where the lattice moves, the positive background is no longer neutral as seen by the carriers, and the resulting electric field exerts a force qE' on the charge carriers that is of the same order as the magnetic forces considered. These electric forces must be included in the force balance that leads to Eqs. (5) and (7); without them, the net transverse force on each spin sub-band is incomplete.
minor comments (4)
  1. [Throughout the manuscript] The typesetting is severely corrupted: Greek letters, subscripts, and mathematical symbols appear as placeholder strings (e.g., '∓/g2020∇B', '/g1∗62↑', '/g8041') throughout the abstract, main text, and supplementary information. The equations are very hard to read and must be reset.
  2. [Supplementary a] Supplementary a defines α as a 'geometrical coefficient' but does not give its expression or an order-of-magnitude estimate; since the main text sets α=1 for clarity, the reader cannot assess whether the numerical comparisons are meaningful. A value or bound for α should be stated in the main text.
  3. [Figures 1–3] Figures 1–3 contain no dimension labels, coordinate axes, or markers for the spin/charge accumulation regions, making it difficult to connect the schematics to the geometry assumed in the derivation.
  4. [Eq. (9)] The effective Hall field expression in Eq. (9) is stated without derivation; the factor 2 in the first form is not explained and appears inconsistent with the force density in Eq. (5).

Circularity Check

2 steps flagged · score 4.0 of 10

SHE premise and field profile B(x)=αμ₀xj_s are imported from the authors' own arXiv [1]; ISHE derivation inherits this unverified self-citation, though the ISHE algebra is newly developed.

  1. self citation load bearing [Main text, second paragraph (after abstract)]
    "Under this assumption it has been shown [1] that the force ∓μ∇B induces a transverse spin imbalance similar to that known as spin Hall effect [2] and with strength of the order of magnitude of that reported for spin Hall experiments with polarized conduction band [1,2,7]. In particular spin accumulation and Hall angle were shown [1] to be of the order of magnitude of that predicted by assuming intrinsic or extrinsic impurities and atomic spin-orbit scattering [3-7]."

    The paper's central premise — that the Zeeman force ∓μ∇B produces a transverse spin imbalance of the observed magnitude — is not re-derived here. It is taken as established from the authors' own prior arXiv preprint [1], which overlaps with the present authorship. All subsequent results, including the inverse spin Hall force (Eq. 5) and effective field (Eq. 9), are built on this same premise. Since [1] is not independently verified in the paper, the load-bearing argument reduces to a self-citation rather than to a self-contained derivation.

  2. ansatz smuggled in via citation [Supplementary information a)]
    "the inhomogeneous magnetic field produced by the lattice current in S↑BF is given by B(x) = α μ0 x j_s [1]. Where α is the geometrical coefficient defined as ... In the text of the article, for the sake of clarity we have considered α=1 and have omitted the directions of forces..."

    The quantitative field profile B(x)=αμ₀xj_s is the seed of Eq. (4) and hence of every derived force, including the inverse spin Hall field (Eq. 9). It is adopted from the authors' own prior work [1] without derivation in this paper, and the main text then sets α=1 'for clarity'. Thus the magnitude of the predicted spin/charge imbalance is inherited from a prior same-author relation. If [1] originally proposed this B(x) as an assumption rather than deriving it from independent electrodynamics, the present predictions are not independent of their own input.

full rationale

No fitted-parameter circularity is present: the paper does not fit any parameter to a dataset and then predict a closely related quantity. The derivation is algebraic, starting from Lorentz and Zeeman forces. The main circularity concern is the reliance on the authors' own previous work [1]. The paper's claim to 'show' that the ∓μ∇B force accounts for the spin Hall effect is delegated to [1], and the field profile B(x)=αμ₀xj_s used in the inverse spin Hall derivation is likewise cited to [1]. Both are load-bearing and unverified here, so the central mechanism is not self-contained. However, the inverse spin Hall derivation itself (Eqs. 4–9) is newly constructed in this paper and does not simply restate [1]; it extends the cited mechanism to a different observable. Therefore the circularity is partial. A separate algebraic inconsistency in Eq. (7) with respect to Eqs. (3) and (6) is a correctness risk rather than a circularity and is not counted in the score.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

All quantitative content rests on the Zeeman-force premise, the inertial-frame equivalence, the same-current assumption, and the neglect of electric fields in moving frames. No new particles or interactions are proposed.

free parameters (1)
  • Geometry coefficient α = 1 (assumed in main text)
    Defined in supplementary a via B(x)=α μ0 x j_s; controls the magnitude of every force and Hall field, but the main text sets α=1 without a geometric calculation or experimental comparison.
assumptions (6)
  • domain assumption The Zeeman force ∓μ∇B acts on electron spins at rest in their comoving frame and produces spin separation.
    Standard Stern-Gerlach force; the paper cites its own prior work [1] for the claim that it produces spin imbalance of the observed order, and uses it throughout without re-deriving it.
  • domain assumption For stationary currents, the laboratory frame and the frames in which spin-up and spin-down carriers are at rest are inertial frames, so Newton's third law can be applied between sub-bands and the background via Lorentz forces.
    Used in the derivation of eqs. (4)-(7); ignores scattering and the finite velocity distribution of electrons in a metal.
  • ad hoc to paper The lattice current seen in each spin rest frame equals that spin sub-band's lab current.
    Appears in paragraph 2 and in supplementary a; exact only for a one-species conductor and not justified for the depolarized two-fluid cases treated for SHE and ISHE.
  • ad hoc to paper Electric fields produced by Lorentz transformation between frames can be neglected; only magnetic forces on spins and Lorentz forces on currents matter.
    The paper computes transverse forces without considering the electric field seen in the carrier rest frame, which is load-bearing for the force balance.
  • domain assumption The conduction band can be modeled as two uniform fluids with definite drift velocities j_up/(n_up e) and j_down/(n_down e).
    Needed to define spin currents and reference frames; a strong simplification of the actual Fermi surface and disorder scattering.
  • standard math Maxwell's equations and the Lorentz force law apply inside the metallic film.
    Foundation of all force calculations; standard background assumed without proof.

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Cite this review

Pith. "Pith review of Direct and inverse spin Hall effect: Lorentz force and Zeeman energy." pith.science (2026). https://pith.science/paper/OEWLM2VT

@misc{pith2026190807515,
  author       = {Pith},
  title        = {Pith review of: Direct and inverse spin Hall effect: Lorentz force and Zeeman energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OEWLM2VT}},
  note         = {Machine review of arXiv:1908.07515}
}
read the original abstract

It is shown that magnetic forces as the Lorentz force, exerted on electric currents, and the force {\mu}Div(B), exerted on electron spins at rest, account for both the transverse spin imbalance typical of spin Hall effect and the transverse charge imbalance associated with pure spin currents (inverse spin Hall effect). Considering that for stationary currents the laboratory reference frame and those for which the spin up and spin down carriers are at rest are inertial systems, one can easily find the forces exerted by the lattice on both spin sub-bands, as well as the force between sub-bands.

Figures

Figures reproduced from arXiv: 1908.07515 by the authors.

Figure 1
Figure 1. Thin metallic film with two currents of e [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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    Hernando A, Guinea F, Garcia MA 2019 Spin imbal ance of charge carriers induced by an electric current arXiv:1906.04851 [cond-mat.mes-hall]

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    Valenzuela SO, Tinkham M 2006 Direct electronic measurement of the spin Hall effect Nature 442 , 176

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    Kato Y, Myers RC, Gossard AC, Awschalom DD 2004 Observation of the Spin Hall effect in Semiconductors Science 306 1910 Figure 1 Figure 2 Figure 3 Supplementary information a) Consider the case of a full spin polarized conduct ion band, thus, /g1∗62/g8052= /g1∗62/g8052↑. Since the carriers current seen in /g1∗45/g8008/g1∗82 and the lattice current seen is ...

    work page 2004

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