Intraband circular photogalvanic effect in Weyl semimetals

E. L. Ivchenko; L. E. Golub

REVIEW 1 major objections 6 minor 22 references

Intraband circular photogalvanic effect in Weyl semimetals

T0 review · 1 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Two standard calculations of the circular photogalvanic effect in Weyl semimetals disagree, showing that the known semiclassical mechanisms — Berry curvature dipole, side jumps, and skew scattering — do not exhaust the intraband…

desk verdict Claims a new semiclassical/quantum discrepancy in intraband CPGE, but the quantum side is missing a diagrammatic completeness proof. read the letter →

arxiv 2507.13796 v1 pith:RU5535UQ submitted 2025-07-18 cond-mat.mes-hall quant-ph

L. E. Golub , E. L. Ivchenko This is my paper · ORCID

classification cond-mat.mes-hallquant-ph

keywords circularphotogalvaniceffectWeylsemimetalsBerrycurvaturedipolesidejumpskewscatteringintrabandopticaltransitionsdisordersemiclassicalversusquantumtheory

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 asks whether the standard semiclassical theory of the circular photogalvanic effect, which accounts for the Berry curvature dipole, side jumps, and skew scattering, gives the same intraband photocurrent as a full quantum-mechanical calculation in a Weyl semimetal. For the simplest single Weyl node with isotropic short-range disorder, the answer is no: the semiclassical constant is $\gamma = -1$ while the quantum value is $\gamma = 0$. The authors extend the comparison to Gaussian-correlated disorder of arbitrary spatial range and find that the two approaches still disagree at every correlation length. They conclude that an additional microscopic mechanism, beyond the known semiclassical corrections, is required to describe intraband CPGE in gapless systems. The discrepancy matters because the same semiclassical machinery is known to be complete in gapped gyrotropic materials, so the failure appears precisely where the spectrum is gapless.

What carries the argument

The load-bearing object is the dimensionless CPGE constant $\gamma$ defined by $\mathbf{j} = \gamma \, e^3/(h^2\omega) \, \boldsymbol{\kappa} |\mathbf{E}|^2$, which both approaches are supposed to compute. Semiclassically, $\gamma$ is the sum of the Berry curvature dipole contribution $\gamma_{\rm BCD} = -2/3$ and the side-jump contribution $\gamma_{\rm sj} = -1/3$, with skew scattering vanishing by the model's rotational symmetry. Quantum-mechanically, $\gamma$ comes from the indirect optical transition matrix element $M_{\mathbf{k}'\mathbf{k}}$ for conduction-band intermediate states plus $\delta M_{\mathbf{k}'\mathbf{k}}$ for virtual transitions through the valence band, inserted into the current formula of Eq. (20); the $|\delta M|^2$ term gives $+4/3$ and the interference term gives $-4/3$, cancelling exactly at isotropic scattering. The anisotropic-scattering extension carries the argument to arbitrary disorder range by replacing $n_i|U_0|^2$ with the disorder correlator $K(|\mathbf{k}' - \mathbf{k}|)$, producing the non-crossing curves plotted in Fig. 2.

What would settle it

Enumerate all Feynman diagrams for the second-order photon-plus-disorder response of a single Weyl node and check whether any amplitude beyond $M_{\mathbf{k}'\mathbf{k}}$ and $\delta M_{\mathbf{k}'\mathbf{k}}$ contributes to the intraband CPGE; alternatively, run a numerically exact Kubo-formula calculation of the intraband CPGE with short-range and Gaussian disorder to see whether $\gamma$ is $0$ or $-1$ and how it varies with $\alpha$. A clean experimental measurement in a material with one dominant Weyl node would also discriminate, but the decisive check is the theoretical enumeration.

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Extended reading notes

Core claim

The central claim is that, in the intraband frequency range $\hbar/\tau \ll \hbar\omega \ll \varepsilon_F$, the complete quantum-mechanical calculation of the CPGE in a single Weyl node does not coincide with the semiclassical calculation. For isotropic short-range scattering, $\gamma_{\rm SC} = \gamma_{\rm BCD} + \gamma_{\rm sj} = -2/3 - 1/3 = -1$, while the quantum-mechanical result is $\gamma_{\rm QM} = \gamma^{(1)}_{\rm QM} + \gamma^{(2)}_{\rm QM} = 4/3 - 4/3 = 0$. With a Gaussian disorder correlator the two curves $\gamma_{\rm SC}(\alpha)$ and $\gamma_{\rm QM}(\alpha)$, with $\alpha = 2(k_F d)^2$, never cross for any $\alpha > 0$: $\gamma_{\rm QM}$ vanishes only at $\alpha = 0$ and tends to $-4/3$ at large $\alpha$, while $\gamma_{\rm SC}$ has $|\gamma_{\rm SC}| = 1$ at both limits with a nonmonotonic interpolation. The paper argues that this persistent discrepancy indicates an extra contribution to intraband CPGE in gapless systems, on top of the Berry curvature dipole, side jumps, and skew scattering.

Load-bearing premise

The quantum-mechanical side is assumed complete: the paper treats the indirect-transition amplitude $M_{\mathbf{k}'\mathbf{k}}$ plus $\delta M_{\mathbf{k}'\mathbf{k}}$, together with the current formula of Eq. (20), as containing every second-order photon-plus-disorder process, without a diagrammatic proof that no other amplitude of the same order contributes.

Editorial extensions

If this is right

In a single ideal Weyl node with isotropic short-range scattering, the intraband CPGE constant is exactly zero quantum-mechanically, so the known mechanisms produce no helicity-dependent current in that limit.
For Gaussian-correlated disorder, $\gamma_{\rm QM}$ is finite for every correlation length and ranges from 0 to $-4/3$, while $\gamma_{\rm SC}$ is nonmonotonic and never matches it.
The equivalence between semiclassical and quantum approaches that holds in gapped gyrotropic materials does not extend to gapless Weyl nodes.
A missing microscopic contribution must be added to the quasiclassical description of CPGE in gapless systems, since the Berry curvature dipole, side jumps, and skew scattering together cannot reproduce the quantum result.
The side-jump term in an earlier Weyl-semimetal calculation omitted a piece; restoring it changes the isotropic semiclassical value from 0 to $-1$.

Reading between the lines

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

If the quantum result is exact, intraband helicity photocurrent in a clean single Weyl node may vanish completely at isotropic disorder, which would redirect experimental searches toward finite-range disorder or multi-node effects.
A diagrammatic enumeration of all second-order photon-plus-disorder processes would settle the discrepancy directly; if an additional amplitude exists, it would likely be a field-dependent correction to scattering beyond the overlap correction ruled out in Appendix B.
The same tension may appear in other gapless systems such as Dirac semimetals and graphene, where interband virtual transitions are kinematically available at small photon energy.
A numerically exact Kubo-formula calculation of the intraband CPGE in the single-node model, as a function of disorder strength and correlation length, would reveal which side of the discrepancy survives beyond the approximations used here.

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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

Colleague,

The headline: this paper claims to have found a genuine discrepancy between semiclassical and fully quantum-mechanical calculations of the intraband circular photogalvanic effect in a minimal Weyl node. For isotropic short-range disorder, semiclassical gives γ_SC = -1 (Berry curvature dipole -2/3 plus side-jump -1/3, skew scattering zero), while the quantum-mechanical result is γ_QM = 0, because a previously ignored interference term γ_QM^(2) = -4/3 exactly cancels the known |δM|^2 contribution +4/3. They further show the discrepancy persists for Gaussian-correlated disorder of any correlation length. If true, this says the standard semiclassical mechanisms do not exhaust intraband CPGE in gapless systems, which is a meaningful result for the photogalvanic subfield.

The paper does several things well. The algebra in Sections II and III and Appendices A and C is explicit and checkable. The correction to the earlier semiclassical value γ_SC = 0 from Ref. [13], traced to a missing term in the side-jump distribution function, is plausible and clearly explained. The Gaussian-disorder extension is a new calculation. The writing is clear and the model is minimal, so the logic is easy to follow.

The soft spot is the quantum-mechanical side. The current formula used, Eq. (20), is an injection-type expression: it multiplies the Fermi-golden-rule absorption rate by [v' τ' - v τ]. It does not by itself include the quantum counterpart of the semiclassical side-jump, which in the Boltzmann language is a field-induced correction to the scattering probability W^sj. The paper never enumerates the full set of order-E^2 diagrams. Appendix B rules out one candidate, the field-induced change of the spinor overlap |⟨u_k'|u_k⟩|^2, but it does not check the phase/shift part of the scattering amplitude (the r_{k'k} term) or diagrams with the photon vertex inserted in the impurity line. If such terms contribute, they could cancel the -4/3 interference term and bring γ_QM to -1, eliminating the discrepancy. This is not a small footnote; it is the load-bearing assumption of the paper's central claim.

There is also a minor typo: Eq. (A10) seems to contain an extra impurity density n_i on the right-hand side, inconsistent with the definitions of M and δM.

Who is this for? Specialists in photogalvanic effects, nonlinear transport, and Weyl semimetal theory. It deserves a serious referee: the claim is important enough and the calculation detailed enough that the completeness issue should be settled rather than the paper desk-rejected. My recommendation: send it to peer review with a request that the authors either provide a diagrammatic enumeration of all second-order photon-plus-disorder processes or explicitly justify why Eq. (20) captures the full current. Once that gap is closed, the result will either be a solid correction or an instructive cancellation.

On balance, I would not cite it in my own work until that completeness issue is resolved.

Referee Report

1 major / 6 minor

Summary. The paper studies the intraband circular photogalvanic effect (CPGE) in a single Weyl node described by H = Cℏv0σ·k. In the semiclassical approach, the authors evaluate the Berry curvature dipole contribution (γ_BCD = -2/3), the side-jump contribution (γ_sj = -1/3), and note the vanishing of skew scattering, obtaining γ_SC = -1 for isotropic short-range scattering. In the quantum-mechanical approach, they use the injection-current formula of Eq. (20) with indirect optical transitions mediated by disorder, decomposing the transition amplitude into a conduction-band-mediated part M and a valence-band-mediated part δM. They find that the |δM|^2 contribution (γ_QM^(1) = 4/3, from Ref. [8]) is exactly cancelled by the M*δM interference term (γ_QM^(2) = -4/3, derived in Appendix A), so γ_QM = 0 for isotropic scattering. For Gaussian-correlated disorder, they obtain γ_SC(α) and γ_QM(α) that remain different for all α = 2(k_F d)^2, with γ_QM ranging from 0 to -4/3 and γ_SC ranging only slightly around -1. The paper concludes that the established semiclassical mechanisms do not exhaust the intraband CPGE in Weyl semimetals.

Significance. If the quantum-mechanical calculation is complete, the result is significant: it would overturn the expectation, established in gapped gyrotropic systems, that Berry curvature dipole, side-jump, and skew-scattering mechanisms saturate the intraband CPGE, and it would point to a new microscopic contribution in gapless systems. The manuscript's strengths are its parameter-free analytic derivations, the explicit calculation of the interference contribution in Appendix A, and the extension to anisotropic Gaussian disorder, which yields a concrete prediction for the dependence of the CPGE constant on the dimensionless disorder range α. These are falsifiable statements that go beyond a re-derivation of known results. However, the significance is conditional: the claimed discrepancy is only meaningful if Eq. (20) truly captures the full quantum-mechanical response, and this is precisely the point that the manuscript leaves unproven.

major comments (1)

[Sec. III, Eq. (20), and Appendix B] The central claim of the paper rests on the assumption that Eq. (20) is the complete quantum-mechanical expression for the intraband CPGE current. This assumption is not demonstrated. Equation (20) is an injection-type formula: it multiplies the Fermi-golden-rule transition rate W_in by the difference [v_{k'}τ(ε_{k'}) - v_kτ(ε_k)]. It does not include the quantum counterpart of the semiclassical side-jump, i.e., the electric-field-induced correction to the scattering probability W^sj used in Sec. II.B, nor a shift/displacement contribution of the indirect optical transition. Appendix B investigates only one specific field-induced correction, the change in the spinor overlap |⟨u_{k'}|u_k⟩|^2, and shows that it vanishes; it does not examine the phase/shift part of the scattering amplitude (the r_{k'k} term in Eq. (11)) or diagrams in which the photon vertex is inserted in the impurity line. Consequently, the cancellation γ_QM^(1) + γ_QM^(2) = 0 in Eq. (25) is established only within the subset of processes contained in Eq. (20). A diagrammatic enumeration of all processes of order E^2 (one photon field and one disorder vertex) or a full quantum kinetic calculation that includes both injection and shift terms is needed to establish that no additional contribution modifies γ_QM. Without this, the discrepancy reported in Sec. V may be an artifact of an incomplete quantum calculation.

minor comments (6)

[Sec. III, after Eq. (21)] There is a typo in the sentence following Eq. (21): 'thirs' should be 'third'.
[Abstract and Sec. V] The phrase 'complete quantum-mechanical approach' overstates the scope of the calculation, which uses a specific injection-current formula; the authors should either qualify the claim or add the missing diagrammatic proof.
[Sec. II and Eqs. (24)-(25)] The factor C and the direction κ in Eq. (1) are not carried through the later formulas; the text should state explicitly that the numerical values γ_SC and γ_QM are for C = +1 and for the current component along κ.
[Sec. II.A] The remark that the signs of the Berry curvatures in Eq. (6) of Ref. [1] should be reversed would be more useful if the corrected signs were displayed explicitly.
[Appendix A, Eq. (A14)] The angular averages leading to ⟨Φ cosθ_{k'}⟩ = ⟨Φ cosθ_k⟩ = -2/9 are stated without derivation; a short indication of the integration procedure would improve reproducibility.
[Fig. 2] The upper and lower panels use different horizontal-axis limits (α up to 20 vs. α up to 10); a common range would make the comparison of γ_SC and γ_QM more direct.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: γ_SC = −1 and γ_QM = 0 are derived from explicit symbolic calculations, and the claimed discrepancy is not an input or fitted value.

full rationale

The paper's central result is a derived discrepancy between the semiclassical sum γ_SC = γ_BCD + γ_sj = −2/3 − 1/3 = −1 and the quantum-mechanical sum γ_QM = γ^(1)_QM + γ^(2)_QM = 4/3 − 4/3 = 0. Each contribution is obtained by explicit algebra or angular integration from the Weyl Hamiltonian (3) and the stated disorder model; no parameter is fitted to make the discrepancy appear. The cited prior works [8,9] supply the current formula (20) and the |δM|^2 value γ^(1)_QM = 4/3, but those are parameter-free calculations with stated assumptions that do not include the new interference contribution computed in Appendix A or the comparison itself, so under the review rules they constitute independent evidence rather than circular self-citation. The remaining concern — whether Eq. (20) exhausts all second-order photon-plus-disorder processes — is a completeness or correctness assumption, not a circular reduction of the derivation to its inputs. The paper does not define its target result into the starting formulas, does not rename a known pattern as a new mechanism, and does not invoke a self-citation chain to forbid alternative interpretations. No specific circular step can therefore be exhibited.

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

No parameters are fitted to data; v0, U0, n_i, k_F, and d are symbolic model inputs. The central result depends on the electron-hole symmetric Weyl model and on the completeness of the quantum-mechanical amplitude set, both listed as axioms.

assumptions (4)

domain assumption Single Weyl-node Hamiltonian H = C hbar v0 sigma dot k with linear, isotropic, electron-hole symmetric dispersion.
Defines the model and is used throughout, beginning at Eq. (3).
domain assumption Disorder is short-range intra-node scattering U(r) = U0 sum delta(r - R_i), generalized to a Gaussian correlator K(q) = K0 exp(-q^2 d^2) in Appendix C.
Sections II and Appendix C; no inter-node, magnetic, phonon, or Coulomb scattering is included.
domain assumption Frequency window hbar/tau much less than hbar omega much less than epsilon_F.
Eq. (2); used to expand in hbar omega / epsilon_F and to treat the collision integral in the high-frequency limit.
domain assumption The quantum-mechanical transition amplitude M + delta-M and the current formula Eq. (20), taken from Refs. [8,9], exhaust all second-order photon-plus-disorder processes.
Load-bearing for the result gamma_QM = 0; the paper asserts completeness without a diagrammatic enumeration.

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Cite this review

Pith. "Pith review of Intraband circular photogalvanic effect in Weyl semimetals." pith.science (2026). https://pith.science/paper/RU5535UQ

@misc{pith2026250713796,
  author       = {Pith},
  title        = {Pith review of: Intraband circular photogalvanic effect in Weyl semimetals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RU5535UQ}},
  note         = {Machine review of arXiv:2507.13796}
}

read the original abstract

We apply the semiclassical theory including the Berry curvature dipole, side jumps and skew scattering for a quantitative description of the circular photogalvanic effect (CPGE) in Weyl semimetals at intraband absorption. In contrast to gapped systems where they completely exhaust all contributions to the CPGE current, all previously known semiclassical mechanisms give a result different from that obtained using a complete quantum-mechanical approach. We show that this difference in the existing quasiclassical and full quantum-mechanical approaches persists at all spatial ranges of the disorder potential. Apparently, the implementation of another microscopic mechanism into the quasiclassical description of the CPGE is required.

Figures

Figures reproduced from arXiv: 2507.13796 by the authors.

**Figure 2.** The dependence of the CPGE parameter γ, Eq. (1), on the parameter of the Gaussian correlator α = 2(kFd) 2 . Upper panel: Semiclassical result, Eq. (26). Lower panel: the result of the quantum-mechanical calculation, γQM = γ (1) QM + γ (2) QM (solid curve). Dashed curves: the contributions γ (1) QM, Eq. (29) (blue), and γ (2) QM, Eq. (30) (magenta). The electric photocurrent is obtained by two approaches, semiclassic… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Semiclassical derivation The BCD contribution is independent of scattering details and γBCD = −2/3 as before. The side-jump contribution is calculated as follows. The second-order correction to the distribution function is still found from Eq. (14), where now 1 τ = X k′ W (0) k′k(1 − nk · nk′), W (0) k′k = 2π ℏ Kk′kδ(εk − εk′), Kk′k = K(|k′ − k|)|⟨uk′|uk⟩...

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(A11) and Ψ = (cos θk ∓ 1)e±iφk sc′eiϕ/2 − cs′e−iϕ/2 + (cos θk′ ± 1)e±iφk′ cs′eiϕ/2 − sc′e−iϕ/2 2

Quantum-mechanical derivation For the intraband optical matrix elements one has M ∗ k′kδMk′k = −K(|k′ − k|) ev0E ω 2 Φ 4εkℏω , |δMk′k|2 = K(|k′ − k|) ev0E ω 2 Ψ 8ε2 k , (C14) where Φ is given by Eq. (A11) and Ψ = (cos θk ∓ 1)e±iφk sc′eiϕ/2 − cs′e−iϕ/2 + (cos θk′ ± 1)e±iφk′ cs′eiϕ/2 − sc′e−iϕ/2 2 . (C15) The CPGE current is jCPGE = γQM e3 h2ω |E|2, γ QM = ...

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [13]

D. Rees, K. Manna, B. Lu, T. Morimoto, H. Borrmann, 10 C. Felser, J. E. Moore, D. H. Torchinsky, and J. Oren- stein, Helicity-dependent photocurrents in the chiral Weyl semimetal RhSi, Sci. Adv. 6, eaba0509 (2020)

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[2] [8]

L. E. Golub, E. L. Ivchenko, and B. Spivak, Electrical magnetochiral current in tellurium, Phys. Rev. B 108, 245202 (2023)

work page 2023

[3] [1]

1 − 2α exp(α cos Θ)(1 − cos Θ) sin2 Θ exp(α cos Θ) sin2 Θ # = − 1 3

Semiclassical derivation The BCD contribution is independent of scattering details and γBCD = −2/3 as before. The side-jump contribution is calculated as follows. The second-order correction to the distribution function is still found from Eq. (14), where now 1 τ = X k′ W (0) k′k(1 − nk · nk′), W (0) k′k = 2π ℏ Kk′kδ(εk − εk′), Kk′k = K(|k′ − k|)|⟨uk′|uk⟩...

work page

[4] [2]

(A11) and Ψ = (cos θk ∓ 1)e±iφk sc′eiϕ/2 − cs′e−iϕ/2 + (cos θk′ ± 1)e±iφk′ cs′eiϕ/2 − sc′e−iϕ/2 2

Quantum-mechanical derivation For the intraband optical matrix elements one has M ∗ k′kδMk′k = −K(|k′ − k|) ev0E ω 2 Φ 4εkℏω , |δMk′k|2 = K(|k′ − k|) ev0E ω 2 Ψ 8ε2 k , (C14) where Φ is given by Eq. (A11) and Ψ = (cos θk ∓ 1)e±iφk sc′eiϕ/2 − cs′e−iϕ/2 + (cos θk′ ± 1)e±iφk′ cs′eiϕ/2 − sc′e−iϕ/2 2 . (C15) The CPGE current is jCPGE = γQM e3 h2ω |E|2, γ QM = ...

work page

[5] [3]

Nagaosa, T

N. Nagaosa, T. Morimoto, and Y. Tokura, Transport, magnetic and optical properties of Weyl materials, Nat. Rev. Mater. 5, 621 (2020)

work page 2020

[6] [4]

B. Q. Lv, T. Qian, and H. Ding, Experimental perspec- tive on three-dimensional topological semimetals, Rev. Mod. Phys. 93, 025002 (2021)

2021

[7] [5]

Ideue, M

T. Ideue, M. Hirayama, H. Taiko, T. Takahashi, M. Murase, T. Miyake, S. Murakami, T. Sasagawa, and Y. Iwasa, Pressure-induced topological phase transition in noncentrosymmetric elemental tellurium, Proc. Natl. Acad. Sci. 116, 25530 (2019)

work page 2019

[8] [6]

M. M. Glazov, E. L. Ivchenko, and M. O. Nestoklon, Ef- fect of pressure on the electronic band structure and cir- cular photocurrent in tellurium, JETP 135, 575 (2022)

work page 2022