pith. sign in

arxiv: 2606.26067 · v1 · pith:7ZLHDN32new · submitted 2026-06-24 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· cond-mat.str-el

Strong photogalvanic effect in Weyl materials due to magnetic resonances

Vahid Hassanzade , Vladyslav Kozii This is my paper

Pith reviewed 2026-06-25 19:07 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-el
keywords photogalvanic effectshift currentWeyl semimetalsmagnetic resonancesKubo formalismBoltzmann equationLandau levels
0
0 comments X

The pith

A magnetic field produces resonances that enhance the shift current in Weyl semimetals.

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

The paper derives a general analytic expression for the photogalvanic shift current in Weyl semimetals under a magnetic field. The expression comes from the Kubo formalism applied to an ideal clean Weyl node at zero temperature and remains valid for any light frequency, Fermi energy, and field strength. It displays a series of magnetic resonances. A semiclassical Boltzmann calculation solves the transport equation exactly for the magnetic field rather than perturbatively and recovers the low-frequency resonances while showing how finite scattering affects them.

Core claim

Using the Kubo formalism we obtain an analytic expression for the photogalvanic shift current in an ideal Weyl node that remains valid for arbitrary frequency, Fermi level, and magnetic field. The expression exhibits a series of resonances that arise from the magnetic-field-induced Landau level structure. The semiclassical Boltzmann solution, treated non-perturbatively in the field, recovers the same low-frequency resonances and incorporates the effects of finite scattering.

What carries the argument

The non-perturbative analytic expression for the shift current obtained from the Kubo formalism on a Weyl node in a magnetic field, which encodes a ladder of magnetic resonances.

If this is right

  • The resonances remain accessible to experiment across a wide range of frequencies and field strengths.
  • Finite scattering broadens the resonances but leaves their low-frequency features intact.
  • The exact treatment of the magnetic field in the Boltzmann equation captures effects missed by perturbative approaches.
  • The result applies for arbitrary Fermi energy without additional approximations.

Where Pith is reading between the lines

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

  • Magnetic fields could provide a practical knob for controlling the strength of photocurrents in topological semimetals.
  • The same resonance mechanism may govern other nonlinear optical responses beyond the shift current.
  • Experiments could map the current versus magnetic field at fixed light frequency to isolate the resonance ladder.
  • The framework invites extension to cases with weak disorder or elevated temperature while retaining the exact magnetic-field solution.

Load-bearing premise

The Weyl node is treated as ideal and perfectly clean at zero temperature in the Kubo calculation while the Boltzmann treatment assumes semiclassical theory remains valid.

What would settle it

Measurement of the frequency-dependent photogalvanic current in a magnetic field that shows no series of resonances at the predicted positions as field strength or frequency is varied.

Figures

Figures reproduced from arXiv: 2606.26067 by Vahid Hassanzade, Vladyslav Kozii.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic picture of the setup presented in the paper. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The second order dc shift photocurrent [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Density plot of log [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Density plot of log [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

We study the photogalvanic effect in Weyl semimetals under a magnetic field, focusing on the shift current. Using the Kubo formalism for an ideal, clean Weyl node at zero temperature, we derive a general analytic expression valid for arbitrary light frequency, Fermi energy, and magnetic field strength. We identify a series of resonances that can be probed experimentally. To complement the microscopic analysis, we employ the semiclassical Boltzmann approach, which allows us to incorporate finite scattering phenomenologically. Unlike most previous studies using this method, we do not treat the magnetic field perturbatively; instead, we solve the Boltzmann equation for a Weyl node exactly within the limits of validity of the semiclassical theory. Our solution reproduces the low-frequency resonances and elucidates the role of finite scattering.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 1 minor

Summary. The paper claims to derive a general analytic expression for the shift-current photogalvanic response of an ideal clean Weyl node at T=0 using the Kubo formalism, valid for arbitrary light frequency ω, Fermi energy E_F, and magnetic field B; it identifies a series of magnetic resonances. This is complemented by a semiclassical Boltzmann treatment that incorporates phenomenological scattering and solves the equation exactly (non-perturbatively) in B, reproducing the low-frequency resonances.

Significance. If the analytic Kubo expression and the claimed reproduction of resonances by the exact-B Boltzmann solution hold, the work would provide parameter-free predictions for photogalvanic resonances in Weyl materials that could be tested experimentally, strengthening the link between microscopic quantum response and semiclassical transport under strong B.

major comments (1)
  1. [Abstract and Boltzmann section] The central claim that the semiclassical Boltzmann solution reproduces the low-frequency resonances from the Kubo analysis is load-bearing for the complementarity of the two approaches. The Kubo formalism yields discrete inter-Landau-level transitions with energies scaling as √(nB), while the semiclassical Boltzmann equation (with anomalous velocity and Lorentz force) does not quantize levels and is formally valid only when level spacing is negligible. The manuscript must explicitly demonstrate (e.g., via plots or analytic limits as γ→0) that peaks appear at matching frequencies; the abstract alone does not resolve this.
minor comments (1)
  1. [Abstract] The abstract states that derivations exist but provides no equations, error analysis, or verification steps; the full manuscript should include at least the key Kubo formula and the exact Boltzmann solution to allow checking.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the major comment below and will revise the manuscript to provide the requested explicit demonstration.

read point-by-point responses

  1. Referee: [Abstract and Boltzmann section] The central claim that the semiclassical Boltzmann solution reproduces the low-frequency resonances from the Kubo analysis is load-bearing for the complementarity of the two approaches. The Kubo formalism yields discrete inter-Landau-level transitions with energies scaling as √(nB), while the semiclassical Boltzmann equation (with anomalous velocity and Lorentz force) does not quantize levels and is formally valid only when level spacing is negligible. The manuscript must explicitly demonstrate (e.g., via plots or analytic limits as γ→0) that peaks appear at matching frequencies; the abstract alone does not resolve this.

    Authors: We agree that an explicit demonstration is necessary to substantiate the claim of complementarity. Although the semiclassical Boltzmann treatment is formally valid only when Landau level spacing is negligible compared to other energy scales, our exact (non-perturbative in B) solution within the semiclassical regime produces resonant features at frequencies that match the low-frequency inter-Landau-level transitions identified in the Kubo calculation. In the revised manuscript we will add direct comparison plots of the photogalvanic response versus frequency from both methods, together with an analytic analysis in the γ→0 limit, to make the matching of peak positions fully transparent. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; ledger populated from stated assumptions in abstract. No free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Ideal, clean Weyl node at zero temperature
    Explicitly stated for the Kubo formalism derivation.
  • domain assumption Validity of semiclassical theory for exact Boltzmann solution
    Invoked to justify solving the Boltzmann equation exactly without perturbative magnetic-field treatment.

pith-pipeline@v0.9.1-grok · 5669 in / 1193 out tokens · 27560 ms · 2026-06-25T19:07:10.489852+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

53 extracted references · 1 linked inside Pith

  1. [1]

    + (E2 n+1 −E 2 n)(−E2 n+1E2 n +E 2 0 E2 n+1 + 2E2 0 E2 n) (ℏω)4 −2(ℏω) 2(E2n +E 2 n+1) + (E2n −E 2 n+1)2 13 + Θ(εn+1) E2 0 En+1E2n d dζ ζ2(E2 n+1 + 3E2 n)−(E 2 n −E 2 n+1)2 ζ4 −2ζ 2(E2n +E 2 n+1) + (E2n −E 2 n+1)2 ζ→ℏω − ℏω E2nE3 n+1 (ℏω)2E2 n(E2 n+1 −E 2

  2. [2]

    (16), which is also visualized in Fig

    + (E2 n −E 2 n+1)(−E2 n+1E2 n +E 2 0 E2 n + 2E2 0 E2 n+1) (ℏω)4 −2(ℏω) 2(E2n +E 2 n+1) + (E2n −E 2 n+1)2 .(C9) Performing integration overk z and simplifying the result, we end up with Eq. (16), which is also visualized in Fig. 5. The resonant frequencies ofσ xyz(ω) are given by the same Eq. (13) and can be clearly seen in Fig. 5 as sharp red lines. In th...

  3. [3]

    Expanding the exact solution in Eq

    Low-field limitω B →0 First, we calculate the components of the second-order current in the limit of the vanishing magnetic fieldω B →0. Expanding the exact solution in Eq. (D10) in powers ofω 2 B, we find for the photocurrentj dc 2 : jdc 2 ≈ iηe3ωτ 2 6π2ℏ2(1 +ω 2τ 2)Eω ×E ∗ ω 16 + ηe3ω2 Bτ 2 24π2ℏµ(1 +ω 2τ 2)2     2(ExE∗ z +E ∗ xEz) 2(EyE∗ z +E ∗ y E...

  4. [4]

    Clean low-field limit,τ→ ∞followed byω B →0 Next, we consider the clean limitτ→ ∞followed by the low-field limitω B →0. Expanding the exact solution to the orderO(1/τ 0) and subsequently to the leading order inω B, we find for the photocurrent: jdc 2 ≈ iηe3 4π2ωℏ2   EyE∗ z −E zE∗ y EzE∗ x −E xE∗ z2 3(ExE∗ y −E yE∗ x)   − iηe3ω4 B 20µ2 +ℏ 2ω2 480π2µ4ω3...

  5. [5]

    Clean low-frequency limit,τ→ ∞followed byω→0 Analogously, we find the asymptotic expressions in the clean limitτ→ ∞followed by the low-frequency limit ω→0. Expanding again the exact solution to the orderO(1/τ 0) and subsequently to the leading order inω, we obtain for the photocurrent: jdc 2 ≈ 3ηe3ω6 Bℏ3 160π2µ5ω2   0 0 |Ez|2   + iηe3 4π2ωℏ2   EyE∗ ...

  6. [6]

    Asymptotic behavior near resonances To derive asymptotic expressions for the second-order conductivity components near low-frequency resonances ω=ℏω 2 B/(2µ) andω=ℏω 2 B/(4µ), we introduce new dimensionless variables λ≡ℏω B/µ, γ≡ µω ℏω2 B , δ≡ωτ.(D25) We then expand the exact expressions for the current components (D10) atλ→0, keeping the leading nonvanis...

  7. [7]

    M. L. Brongersma, N. J. Halas, and P. Nordlander, Plasmon-induced hot carrier science and technology, Na- ture Nanotechnology10, 25 (2015)

    2015

  8. [8]

    Nagaosa and T

    N. Nagaosa and T. Morimoto, Concept of quantum ge- ometry in optoelectronic processes in solids: Application to solar cells, Advanced Materials29, 1603345 (2017)

    2017

  9. [9]

    J. Liu, F. Xia, D. Xiao, F. J. Garc´ ıa de Abajo, and D. Sun, Semimetals for high-performance photodetec- tion, Nature Materials19, 830 (2020)

    2020

  10. [10]

    A. M. Cook, B. M. Fregoso, F. de Juan, S. Coh, and J. E. Moore, Design principles for shift current photovoltaics, Nature Communications8, 14176 (2017)

    2017

  11. [11]

    M.-M. Yang, D. J. Kim, and M. Alexe, Flexo- photovoltaic effect, Science360, 904 (2018)

    2018

  12. [12]

    L. Z. Tan, F. Zheng, S. M. Young, F. Wang, S. Liu, and A. M. Rappe, Shift current bulk photovoltaic effect in po- lar materials—hybrid and oxide perovskites and beyond, npj Computational Materials2, 16026 (2016)

    2016

  13. [13]

    Morimoto and N

    T. Morimoto and N. Nagaosa, Topological nature of nonlinear optical effects in solids, Science Advances2, e1501524 (2016)

    2016

  14. [14]

    M. M. Glazov and S. D. Ganichev, High frequency elec- tric field induced nonlinear effects in graphene, Physics Reports535, 101 (2014)

    2014

  15. [15]

    K. N. Okada, N. Ogawa, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Enhanced pho- togalvanic current in topological insulators via Fermi en- ergy tuning, Physical Review B93, 081403 (2016)

    2016

  16. [16]

    J. E. Sipe and A. I. Shkrebtii, Second-order optical re- sponse in semiconductors, Physical Review B61, 5337 (2000)

    2000

  17. [17]

    S. M. Young and A. M. Rappe, First principles calcu- lation of the shift current photovoltaic effect in ferro- electrics, Physical Review Letters109, 116601 (2012)

    2012

  18. [18]

    Shockley and H

    W. Shockley and H. J. Queisser, Detailed balance limit of efficiency ofp−njunction solar cells, Journal of Applied Physics32, 510 (1961)

    1961

  19. [19]

    Ahn, G.-Y

    J. Ahn, G.-Y. Guo, and N. Nagaosa, Low-frequency di- vergence and quantum geometry of the bulk photovoltaic effect in topological semimetals, Phys. Rev. X10, 041041 (2020)

    2020

  20. [20]

    de Juan, A

    F. de Juan, A. G. Grushin, T. Morimoto, and J. E. Moore, Quantized circular photogalvanic effect in Weyl semimetals, Nature Communications8, 15995 (2017)

    2017

  21. [21]

    Flicker, F

    F. Flicker, F. de Juan, B. Bradlyn, T. Morimoto, M. G. Vergniory, and A. G. Grushin, Chiral optical response of multifold fermions, Phys. Rev. B98, 155145 (2018)

    2018

  22. [22]

    D. Rees, K. Manna, B. Lu, T. Morimoto, H. Bor- rmann, C. Felser, J. E. Moore, D. H. Torchinsky, and J. Orenstein, Helicity-dependent photocurrents in the chiral Weyl semimetal RhSi, Science Advances6(2020)

    2020

  23. [23]

    Rostami and M

    H. Rostami and M. Polini, Nonlinear anomalous pho- tocurrents in Weyl semimetals, Phys. Rev. B97, 195151 (2018)

    2018

  24. [24]

    Z. Ji, G. Liu, Z. Addison, W. Liu, P. Yu, H. Gao, Z. Liu, A. M. Rappe, C. L. Kane, E. J. Mele, and R. Agar- wal, Spatially dispersive circular photogalvanic effect in a Weyl semimetal, Nature Materials18, 955 (2019)

    2019

  25. [25]

    Ma, S.-Y

    Q. Ma, S.-Y. Xu, C.-K. Chan, C.-L. Zhang, G. Chang, Y. Lin, W. Xie, T. Palacios, H. Lin, S. Jia, P. A. Lee, P. Jarillo-Herrero, and N. Gedik, Direct optical detec- tion of Weyl fermion chirality in a topological semimetal, Nature Physics13, 842 (2017)

    2017

  26. [26]

    G. B. Osterhoudt, L. K. Diebel, M. J. Gray, X. Yang, J. Stanco, X. Huang, B. Shen, N. Ni, P. J. W. Moll, Y. Ran, and K. S. Burch, Colossal mid-infrared bulk photovoltaic effect in a type-I Weyl semimetal, Nature Materials18, 471 (2019)

    2019

  27. [27]

    L. Wu, S. Patankar, T. Morimoto, N. L. Nair, E. Thewalt, A. Little, J. G. Analytis, J. E. Moore, and J. Orenstein, Giant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetals, Nature Physics 13, 350 (2017)

    2017

  28. [28]

    Li, Y.-Q

    Z. Li, Y.-Q. Jin, T. Tohyama, T. Iitaka, J.-X. Zhang, and H. Su, Second harmonic generation in the Weyl semimetal TaAs from a quantum kinetic equation, Phys. Rev. B97, 085201 (2018)

    2018

  29. [29]

    A. A. Zyuzin and A. Y. Zyuzin, Chiral anomaly and second-harmonic generation in Weyl semimetals, Phys. Rev. B95, 085127 (2017)

    2017

  30. [30]

    L. E. Golub, E. L. Ivchenko, and B. Z. Spivak, Photocur- rent in gyrotropic Weyl semimetals, JETP Letters105, 782 (2017)

    2017

  31. [31]

    Zhang, H

    Y. Zhang, H. Ishizuka, J. van den Brink, C. Felser, B. Yan, and N. Nagaosa, Photogalvanic effect in Weyl semimetals from first principles, Physical Review B97, 241118 (2018)

    2018

  32. [32]

    C.-K. Chan, N. H. Lindner, G. Refael, and P. A. Lee, Photocurrents in Weyl semimetals, Physical Review B 95, 041104 (2017)

    2017

  33. [33]

    L. E. Golub and E. L. Ivchenko, Circular and magnetoin- duced photocurrents in Weyl semimetals, Phys. Rev. B 98, 075305 (2018)

    2018

  34. [34]

    Yao and A

    X. Yao and A. Belyanin, Nonlinear optics of graphene in a strong magnetic field, Journal of Physics: Condensed Matter25, 054203 (2013)

    2013

  35. [35]

    Yao and A

    X. Yao and A. Belyanin, Giant optical nonlinearity of graphene in a strong magnetic field, Phys. Rev. Lett.108, 255503 (2012)

    2012

  36. [36]

    Tamashevich, L

    Y. Tamashevich, L. D. M. Villari, and M. Ornigotti, Two- dimensional Weyl materials in the presence of constant magnetic fields, Phys. Rev. B107, 245425 (2023)

    2023

  37. [37]

    Bednik and V

    G. Bednik and V. Kozii, Magnetic field induces giant non- linear optical response in Weyl semimetals, Phys. Rev. B 109, 045106 (2024)

    2024

  38. [38]

    Dzero, M

    M. Dzero, M. Khodas, A. Levchenko, and V. Kozii, Non- linear Hall effect in topological Dirac semimetals in par- allel magnetic field, arXiv:2508.20159 (2025)

    Pith/arXiv arXiv 2025

  39. [39]

    Dzero, J

    M. Dzero, J. Hasan, and A. Levchenko, Resonant sec- ond harmonic generation in a two-dimensional electron system, Phys. Rev. B112, 045416 (2025)

    2025

  40. [40]

    Avdoshkin, V

    A. Avdoshkin, V. Kozii, and J. E. Moore, Interactions re- move the quantization of the chiral photocurrent at Weyl points, Phys. Rev. Lett.124, 196603 (2020)

    2020

  41. [41]

    The analytic continuation symmetric with respect to Ω, iω1 →ω+ Ω/2 +i0 andiω 2 → −ω+ Ω/2 +i0, signifi- cantly simplifies the calculation and automatically elim- inates unphysical contributions to the photocurrent

  42. [42]

    E. J. K¨ onig, H.-Y. Xie, D. A. Pesin, and A. Levchenko, Photogalvanic effect in Weyl semimetals, Phys. Rev. B 96, 075123 (2017)

    2017

  43. [43]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on 20 electronic properties, Rev. Mod. Phys.82, 1959 (2010)

    1959

  44. [44]

    Morimoto, S

    T. Morimoto, S. Zhong, J. Orenstein, and J. E. Moore, Semiclassical theory of nonlinear magneto-optical re- sponses with applications to topological Dirac/Weyl semimetals, Phys. Rev. B94, 245121 (2016)

    2016

  45. [45]

    D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron density of states in solids, Phys. Rev. Lett.95, 137204 (2005)

    2005

  46. [46]

    Sodemann and L

    I. Sodemann and L. Fu, Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal in- variant materials, Phys. Rev. Lett.115, 216806 (2015)

    2015

  47. [47]

    Golub and E

    L. Golub and E. Ivchenko, Intraband circular photogal- vanic effect in Weyl semimetals, Annals of Physics488, 170415 (2026)

    2026

  48. [48]

    Guo, X.-Y

    Z. Guo, X.-Y. Liu, H. Wang, L.-k. Shi, and K. Chang, Dissipation-shaped quantum geometry in nonlinear transport, Phys. Rev. Lett.136, 206303 (2026)

    2026

  49. [49]

    For componentsσ yzx 2ω and other related by symmetry, we only consider the analytic part of the microscopic answer, which is proportional toω 4 B

  50. [50]

    Belinicher, E

    V. Belinicher, E. Ivchenko, and B. Sturman, Kinetic the- ory of the displacement photovoltaic effect in piezoelec- tric, Sov. Phys. JETP56, 359 (1982)

    1982

  51. [51]

    Zhu and A

    P. Zhu and A. Alexandradinata, Anomalous shift and optical vorticity in the steady photovoltaic current, Phys. Rev. B110, 115108 (2024)

    2024

  52. [52]

    Z. Rao, H. Li, T. Zhang, S. Tian, C. Li, B. Fu, C. Tang, L. Wang, Z. Li, W. Fan, J. Li, Y. Huang, Z. Liu, Y. Long, C. Fang, H. Weng, Y. Shi, H. Lei, Y. Sun, T. Qian, and H. Ding, Observation of unconventional chiral fermions with long Fermi arcs in CoSi, Nature567, 496 (2019)

    2019

  53. [53]

    Z. Ni, K. Wang, Y. Zhang, O. Pozo, B. Xu, X. Han, K. Manna, J. Paglione, C. Felser, A. G. Grushin, F. de Juan, E. J. Mele, and L. Wu, Giant topological longitudinal circular photo-galvanic effect in the chiral multifold semimetal CoSi, Nature Communications12, 154 (2021)

    2021