Chiral Anomaly Induced Transverse Planar Transport Phenomena in Three Dimensional Spin-Orbit Coupled Metals

Binayyak B. Roy; Gargee Sharma; Rishi G. Gopalakrishnan; Sumanta Tewari

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Chiral Anomaly Induced Transverse Planar Transport Phenomena in Three Dimensional Spin-Orbit Coupled Metals

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Pith's one-line read This paper aims to establish that the chiral anomaly, not the Lorentz force or the usual anomalous velocity, generates measurable transverse planar Hall, Nernst, and mixed electrothermal responses in three-dimensional spin-orbit coupled…

desk verdict The paper has a new and testable chiral-anomaly planar transport calculation, but it drops a same-order term from its own first-order solution, so the central coefficients are incomplete until that is fixed. read the letter →

arxiv 2505.21498 v3 pith:HFPIP5Y7 submitted 2025-05-27 cond-mat.mes-hall

Rishi G. Gopalakrishnan , Binayyak B. Roy , Gargee Sharma , Sumanta Tewari This is my paper

classification cond-mat.mes-hall

keywords chiralanomalyplanarHalleffectNernstspin-orbitcoupledmetalBerrycurvaturenonlineartransportMottrelationsemiclassicalBoltzmann

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 aims to establish that the chiral anomaly, the non-conservation of chiral charge when an electric field or thermal gradient is parallel to a magnetic field, produces measurable transverse planar transport in a three-dimensional spin-orbit coupled metal with a single band-degeneracy point and two Fermi surfaces of opposite Berry curvature flux. Using semiclassical Boltzmann transport in the chiral limit, the authors derive exact angular and field dependencies for the linear and quadratic planar Hall, planar Nernst, and mixed electrothermal coefficients. The linear coefficients scale as $B^2\sin\theta\cos\theta$ and peak at $\theta=\pi/4$, while the second-order coefficients scale as $B^3\cos^2\theta\sin\theta$ and peak at the smaller angle $\theta=\arctan(1/\sqrt{2})$. Because the underlying Fermi-surface moments contain exponential temperature factors, both the usual and nonlinear Mott relations fail at relatively low temperatures, giving a distinctive thermoelectric signature. If correct, these predictions provide a practical way to detect chiral-anomaly physics in ordinary spin-orbit coupled metals rather than only in Weyl semimetals.

What carries the argument

The argument is carried by the Berry-flux moments $C_s^\nu=\int [dk](-\partial f_{\rm eq}/\partial\epsilon_s)(e/\hbar)(\mathbf v_s\cdot\boldsymbol\Omega_s)(\epsilon_s-\mu)^\nu$ and their second-derivative counterparts $\Sigma_s^\nu$, together with the anomaly-induced imbalances $\delta\mu_s$ and $\delta T_s$ between the two chiral centers. These moments convert the component of $\mathbf E$ or $\nabla T$ parallel to $\mathbf B$ into a transverse current through the chiral magnetic velocity term $(\mathbf v_s\cdot\boldsymbol\Omega_s)\mathbf B$. Their low-temperature limits are dominated by the constants $F_0\to 1$, $F_2\to\pi^2/3$, and $G_1\to\beta$, while all other pieces decay exponentially, and that exponential structure is what generates the $B^2$ and $B^3$ field scalings, the $\sin\theta\cos\theta$ and $\cos^2\theta\sin\theta$ angular laws, and the early breakdown of the Mott relations. The chiral limit $\tau_\nu\gg\tau_0\simeq\tau_*$ ensures that only the anomaly-pumping terms contribute at leading order.

What would settle it

Measure the angular and field dependence of the transverse planar conductivity and thermopower in a clean 3D spin-orbit coupled metal at fixed temperature; the anomaly-only scenario predicts a peak at $\theta=\pi/4$ with $B^2$ scaling for the linear coefficients and a peak at $\theta=\arctan(1/\sqrt{2})$ with $B^3$ scaling for the quadratic coefficients, and a Mott number $\alpha_{yx}/(T\,\partial_\mu\sigma_{yx})$ that departs from unity by about 10% near 20 K. Observing a different peak angle, a different field exponent, or Mott-relation validity to high temperature would falsify the chiral-limit truncation.

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Extended reading notes

Core claim

The central claim is that in a 3D spin-orbit coupled metal described by $H=\sum_{\mathbf k} c^\dagger_{\mathbf k}(\hbar^2 k^2/2m\,\sigma_0+\alpha\,\boldsymbol\sigma\cdot\mathbf k)c_{\mathbf k}$, the chiral anomaly, rather than the Lorentz force or anomalous velocity, dominates transverse planar transport. Each band has a Fermi surface carrying a nonvanishing Berry flux, opposite for the two chiralities, and with $\mathbf E$ or $nabla T$ parallel to an in-plane $\mathbf B$, the anomaly pumps charge and energy between these chiral centers. The resulting first-order transverse coefficients obey $\sigma_{yx}\propto B^2\sin\theta\cos\theta$ and $\alpha_{yx}\propto B^2\sin\theta\cos\theta$, while the second-order coefficients obey $\sigma_{yxx},\alpha_{yxx},\beta_{yxx}\propto B^3\cos^2\theta\sin\theta$. The paper also gives closed low-temperature forms, for example $\sigma^{\rm LT}_{yx}\propto (2+\tilde\mu)\sqrt{1+\tilde\mu}\,\tilde\mu^{-2}\,B^2\cos\theta\sin\theta$. Because the fundamental moments contain exponentially decaying temperature pieces, the linear and nonlinear Mott relations hold only at very low temperatures and are violated at relatively modest temperatures, which the authors identify as the key observable signature.

Load-bearing premise

Everything depends on the assumption that electrons scatter much more slowly between the two chirality centers than within a center, so that the only important correction is the anomaly-induced pumping of charge and heat between the centers.

Editorial extensions

If this is right

At linear order, the planar Hall and planar Nernst coefficients vanish when $\mathbf B\parallel\mathbf E$ and when $\mathbf B\perp\mathbf E$, peak at $\theta=\pi/4$, and grow as $B^2$, so measuring those dependencies tests whether the anomaly channel dominates.
At second order, the coefficients peak at $\theta=\arctan(1/\sqrt{2})$, are $2\pi$-periodic in $\theta$, and grow as $B^3$; the paper notes that an $n$th-order coefficient would vary as $\cos^n\theta\sin\theta$ with its maximum at $\arctan(1/\sqrt{n})$.
The mixed coefficient $\beta_{yxx}$ produces a transverse current only when an electric field and a thermal gradient act together, offering an independent probe of anomaly-driven response.
Both the linear and nonlinear Mott relations fail at relatively low temperatures, with the paper's numerical estimates showing roughly 10% deviations by about 20 K for the linear response and 40 K for the nonlinear response.
At very low temperature the nonlinear Mott relation is recovered in the form $\alpha_{yxx}=(\pi^2 k_B^2/3e^2)\sigma_{yxx}$.

Reading between the lines

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

The predicted angular shift from $\pi/4$ to $\arctan(1/\sqrt{2})$ is a sharp fingerprint that future experiments could check against conventional Fermi-liquid planar Hall effects, which would not share the same field-scaling exponents.
The early Mott-relation breakdown could serve as a discriminator: a candidate material that satisfies the Mott relation up to high temperature would be evidence against anomaly-dominated transverse transport, while the predicted departure near 20-40 K would support it.
Measuring $\beta_{yxx}$ in a simultaneous electric-field and thermal-gradient experiment may offer a route to extract the inter-chirality relaxation time $\tau_\nu$, since the coefficient is proportional to $\tau_*\tau_\nu$.
If the spin-orbit coupling is anisotropic, the extremal angles will likely shift, but the general $n$th-order $\cos^n\theta\sin\theta$ form and the $B^{n+1}$ scaling should survive as long as the chiral-limit mechanism remains dominant.

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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

Quick take: this is a legitimate new calculation, but the main quantitative results are built on an f^(1) that is missing a term from the paper's own Appendix A, and that term contributes at the same order. I would send it to referees, but only with a strong request to redo the central coefficients.

What is actually new: the application of chiral kinetic theory to the 3D Rashba-like SOC metal, where the two chiral sectors are Fermi surfaces around the same degeneracy point, and the explicit low-T analytic forms for the linear and nonlinear planar Hall/Nernst coefficients, including the mixed beta. The B^2/B^3 scalings, the sin(theta)cos(theta) versus cos^2(theta)sin(theta) angular forms, and the exponential-T moments behind the Mott-relation breakdown are clean and worth taking seriously. The moment integrals in Eqs. (33)-(40) are consistent with the corrected Berry curvature.

The soft spot is not minor. Eq. (11) in the main text defines f^(1) using only the inter-chirality collision term. But the first-order solution derived in Appendix A, Eq. (A11), contains an additional explicit driving term: -tau* D [v + (e/hbar)(v·Omega)B]·[eE + ((epsilon-mu)/T)grad T](-df/depsilon). For this model v·Omega is isotropic, so when this term is multiplied by the chiral magnetic velocity in the current, it gives a transverse current of the form -tau*(e^3/hbar^2) B^2 sin(theta)cos(theta) E_x sum_s integral (v·Omega)^2(-df/depsilon). This has exactly the same sin(theta)cos(theta) and B^2 structure as the retained contribution and is not suppressed by the chiral limit; tau*/tau_nu times the tau_nu in delta_mu leaves both terms of order tau*. So Eqs. (42)-(46), and the low-temperature Mott analysis, are missing a comparable contribution. The qualitative signatures may survive, but the numerical coefficients do not stand as written.

Other weaknesses are proportionate. The authors do not quantify the conventional Lorentz-force and anomalous-velocity planar Hall background, which is essential for the claimed experimental tests. The chiral limit tau_nu >> tau_0 is assumed without material-specific estimates. Both are fixable. Eq. (32) also needs a typographical cleanup: write it as -s k/(2k^3) = -s k-hat/(2k^2).

Who this is for: condensed matter theorists working on chiral anomaly transport and experimental groups looking for new material classes to test planar Hall/Nernst predictions. The idea is good enough that a serious referee should engage; the missing term is not a desk-reject level problem if the authors are willing to redo the calculation. Recommend: send to peer review, and make sure the referee sees the Appendix-A discrepancy.

Referee Report

2 major / 3 minor

Summary. This paper studies linear and second-order transverse planar transport in three-dimensional spin-orbit coupled metals using semiclassical Boltzmann transport with chiral kinetic theory. For the model H = ℏ²k²/2m + α σ·k, the authors derive anomaly-induced planar Hall and Nernst coefficients at linear order (σ_yx, α_yx) and second order (σ_yxx, α_yxx, β_yxx), reporting sinθcosθ B² and cos²θ sinθ B³ dependencies, extrema at θ = π/4 and θ = arctan(1/√2), and explicit low-temperature forms in Eqs. (42)-(46). They further show that the energy moments have exponential temperature dependence, causing violation of the (non)linear Mott relation at relatively low temperatures. The derivation is analytic and is supplemented by an appendix with the perturbative solution of the Boltzmann equation.

Significance. If the results hold, the paper extends chiral-anomaly transport beyond Weyl semimetals to systems with paired Fermi surfaces of opposite Berry flux, and the predicted angular/field scalings and the mixed electrothermal coefficient β_yxx are experimentally testable. The manuscript is strong in providing analytic closed forms, explicit low-T asymptotics, and numerical maps of the validity of the low-T expansion. However, the first-order current calculation omits a driving term of the same order as the retained anomaly contribution, so the reported linear coefficients and the Mott-relation analysis built on them need revision before the quantitative predictions can be accepted.

major comments (2)

[III.A, Eq. (16); Appendix A, Eq. (A11)] The linear current is computed from the inter-chirality part of f^(1) only. However, the complete first-order solution derived in Appendix A, Eq. (A11), contains the additional explicit term -τ* D [v_s + (e/ℏ)(v_s·Ω_s)B]·[eE + ((ϵ−µ)/T)∇T](-∂f/∂ϵ). For the Hamiltonian in Eq. (29), v_s and Ω_s are both radial, so after angular integration the v_s part of this term does not contribute to j_y, but the chiral-magnetic-velocity part contributes δj_y^(1) = -τ*(e³/ℏ²) B² cosθ sinθ E_x Σ_s ∫[dk] (v_s·Ω_s)² (-∂f/∂ϵ), with an analogous contribution from ∇T. This has exactly the same sinθcosθ and B² structure as the retained term in Eq. (17) and is of the same order in τ*; the chiral limit τν≫τ0≃τ* does not suppress it, because the retained anomaly term is also of order τ* after δµ∼τν. Thus Eqs. (42)-(43) and the Mott-relation analysis built on them are incomplete unless the authors explicitly restrict their claim to the inter-chirality (imbalance) contribution and show that the omitted term is negligible or include it in the final coefficients.
[Section II, after Eq. (4)] The decision to omit the E×Ω_s and v_s×B terms from the equations of motion is justified for isolating the chiral-anomaly mechanism, but the paper does not estimate their contributions to the planar transverse response for the model in Eq. (29). Since the abstract presents σ_yx, α_yx and the nonlinear coefficients as the planar transport coefficients of the system, a statement of the relative magnitude, or a symmetry argument that they vanish identically for this isotropic model, is needed to make the predicted angular and field scalings falsifiable in experiments.

minor comments (3)

[Eq. (32)] The Berry curvature Ω_s = -s/(2k³) k is equivalent to -s/(2k²) k̂; please write it in the latter form to avoid ambiguity about the vector in the numerator and to make the dimension clearer.
[Section IV.B, Figs. 4, 5, 7, 8] The numerical heatmaps and Mott-number plots would be more useful for experimental comparison if the authors stated representative values of the model parameters (e.g., ϵα, µ, τ0, τν, carrier density) used to generate the temperature and chemical-potential scales.
[Section V] The sentence beginning "A new feature of our calculations in the nonlinear order is the emergence of a mixed transport coefficient" appears nearly verbatim in both the main text and the conclusion; consider keeping only one occurrence and editing the repeated phrasing.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: coefficients follow from a model Hamiltonian and Boltzmann equation with no fitted inputs; self-citations provide background but are not load-bearing.

full rationale

The derivation is self-contained. The linear coefficients (17)-(18) and nonlinear coefficients (26)-(28) are obtained by inserting the model Hamiltonian (29) and its Berry curvature (32) into the Boltzmann-equation moments (8), (9), (21), and (23); no parameter is fitted to the predicted quantities and no target angular or field dependence is assumed. The sinθcosθ and cos²θsinθ forms and the B²/B³ scalings follow algebraically from (B·E) and (v_s·Ω_s)B contractions, and the low-temperature expressions (42)-(46) are controlled limits of the exact Fermi integrals (35)-(40). The Mott-relation violation is a numerical consequence of those integrals, not an input. Self-citations to earlier work (refs [85], [89]-[92], [100], [101]) supply the Boltzmann collision form and prior comparisons, but the cited results are published derivations, partly by non-overlapping groups ([85], [89]-[91]), and are not used as an unverified premise; even the overlapping [92] is corroborated by [89]-[91]. The skeptical concern about the explicit driving term in Eq. (A11) being dropped from Eq. (11)/(16) is an internal-completeness issue, not circularity: inclusion would alter the predicted coefficients rather than reproduce them by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard semiclassical Boltzmann formalism, the chiral-limit scattering hierarchy, and the specific isotropic model Hamiltonian. No new physical entities or fitted parameters are introduced; the free parameters are the relaxation times τ0 and τν, whose ratio is assumed large but not material-specified.

free parameters (2)

τ0 (intra-chirality relaxation time)
Introduced in the Boltzmann collision term (Eq. 3); the final transport coefficients are proportional to τ* ≈ τ0 in the chiral limit. Its value is not fixed by the paper.
τν (inter-chirality relaxation time)
Introduced in the Boltzmann collision term; the chiral limit τν≫τ0 is assumed. The dimensionless ratio τν/τ0 determines the validity of the central predictions.

assumptions (4)

domain assumption Semiclassical Boltzmann equations with Berry curvature corrections (Eqs. 1-4) accurately describe transport in the 3D SOC metal.
The paper adopts the standard chiral kinetic theory from refs [85,91,102] without derivation.
domain assumption The two Fermi surfaces s=±1 act as independent chiral centers with well-defined chemical potentials and temperatures, and the chiral limit τν≫τ0 applies.
Section II, Eq. (4) and text: 'We work in the chiral limit...'. If τν is not much larger than τ0, the expressions fail.
ad hoc to paper Cross-product terms E×Ω_s and v_s×B in the equations of motion can be omitted because they do not contribute to chiral anomaly.
Section II, paragraph after Eq. (4). This is a modeling choice that restricts the calculation to the anomaly-induced part; conventional planar Hall contributions are neglected.
domain assumption The 3D SOC metal is described by the isotropic Hamiltonian H=ℏ²k²/(2m)σ0+ασ·k with μ≥0.
Section IV.A, Eq. (29). The paper notes anisotropic spin-orbit terms do not change the arguments, but the explicit coefficients are computed only for this isotropic form.

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Pith. "Pith review of Chiral Anomaly Induced Transverse Planar Transport Phenomena in Three Dimensional Spin-Orbit Coupled Metals." pith.science (2026). https://pith.science/paper/HFPIP5Y7

@misc{pith2026250521498,
  author       = {Pith},
  title        = {Pith review of: Chiral Anomaly Induced Transverse Planar Transport Phenomena in Three Dimensional Spin-Orbit Coupled Metals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HFPIP5Y7}},
  note         = {Machine review of arXiv:2505.21498}
}

read the original abstract

We investigate linear and nonlinear transverse planar transport phenomena (viz. linear and nonlinear Hall and Nernst coefficients) induced by chiral anomaly in three-dimensional spin-orbit coupled metallic systems. Unlike Weyl semimetals, these systems do not possess multiple Weyl nodes located at isolated points in the momentum space but instead host a pair of Fermi surfaces characterized by opposite Berry curvature fluxes enclosing the same band-degeneracy point. Using semiclassical Boltzmann transport formalism within the relaxation time approximation, we derive first- and second-order transverse planar transport coefficients induced by electrical and thermal gradients in the presence of an in-plane magnetic field. Our analysis reveals distinctive angular dependencies of the transport coefficients, along with characteristic scaling behavior with the magnetic field strength. Furthermore, we demonstrate that the anomaly-induced transport coefficients exhibit an exponential temperature dependence. This unconventional behavior leads to the violations of the Mott relation at comparatively low temperatures, highlighting unique thermoelectric signatures that can be probed experimentally in 3D spin-orbit coupled metallic systems.

Figures

Figures reproduced from arXiv: 2505.21498 by the authors.

**Figure 2.** Band structure (left) and corresponding Fermi sur [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Normalized planar Hall conductivity σyx/σ0 yx (with σ 0 yx = τ ∗e 4 8 √ 2π2ℏm3/2ϵ 1/2 α ) as a function of (top) the angle θ between the magnetic field and current for field strengths 0.5 ≤ B ≤ 1.5 (color scale), and (bottom) the field magnitude B for angles 0 ≤ θ ≤ π/4 (color scale). We can notice that the maximum value for the field strength occurs when θ = π/4. α LT yx =   τ ∗ e 3kB 24√ 2βℏm 3 2 ϵ 3 2 α   B … view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: Variation of the Mott number as a function of tem [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: Variation of the non-linear Mott number as a func [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 6.** Figure 6: Non-linear normalized planar Hall conductivity [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 8.** Figure 8: Heatmap of the low-temperature expansion validity [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Boltzmann transport calculation finds sign-reversing planar Hall conductance and anisotropic angular responses in tilted pseudospin-1 fermions arising from chiral anomaly effects.

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