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arxiv: 2605.15610 · v1 · pith:NAMSIDXEnew · submitted 2026-05-15 · ❄️ cond-mat.supr-con

Non-Relativistic Spin-Orbit Interaction in Triplet Superconductors: Edelstein Effect and Spin Pumping by Electric Fields

Ping Li , G. A. Bobkov , I. V. Bobkova , Tao Yu This is my paper

Pith reviewed 2026-05-19 19:50 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords triplet superconductorsEdelstein effectspin textureBogoliubov quasiparticlesnon-relativistic spin-orbitspin pumpingp-wave superconductivityspin current
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The pith

Triplet pairing creates a momentum-dependent spin texture that lets electric fields generate spin polarization in p-wave superconductors even without relativistic spin-orbit coupling.

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

The paper establishes that the triplet superconducting order parameter produces a wave-vector-dependent spin texture among Bogoliubov quasiparticles. This texture entangles orbital and spin motion, creating a non-relativistic form of spin-orbit interaction. Consequently an electric field induces net spin polarization, realizing an Edelstein effect in the absence of relativistic terms. A sympathetic reader would care because the same mechanism supports efficient nonlinear generation of DC spin currents from AC electric near fields.

Core claim

The triplet order parameter induces a wave-vector-dependent spin texture of Bogoliubov quasiparticles, thereby entangling their orbital and spin motions. Even in the absence of relativistic spin-orbit coupling, this intertwining of spin and orbital motion allows an electric field to generate spin polarization in a p-wave superconductor -- that is, an Edelstein effect. Building on this mechanism, we propose an efficient scheme for the nonlinear generation of a DC spin current via electric near fields, driven by AC spin polarization and electron velocity.

What carries the argument

The wave-vector-dependent spin texture of Bogoliubov quasiparticles induced by the triplet order parameter; it entangles orbital and spin motions to produce an electric-field-driven Edelstein effect.

If this is right

  • An electric field generates spin polarization in a p-wave superconductor without relativistic spin-orbit coupling.
  • AC electric near fields produce a nonlinear DC spin current through the combination of induced spin polarization and electron velocity.
  • The same non-relativistic mechanism offers a general route to generate and manipulate spin currents in unconventional superconductors.

Where Pith is reading between the lines

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

  • The effect could appear in candidate p-wave materials once electric fields are applied and spin accumulation is measured.
  • The mechanism may link to spin splitting observed in altermagnets, suggesting shared non-relativistic routes to spin control across different ordered states.
  • Device designs that rely on electric near fields for spin pumping could be explored in triplet superconducting heterostructures.

Load-bearing premise

The model assumes triplet pairing symmetry alone generates the described wave-vector-dependent spin texture and resulting Edelstein effect when relativistic spin-orbit coupling is omitted from the Hamiltonian.

What would settle it

A microscopic calculation or transport measurement that finds zero electric-field-induced spin polarization in a p-wave superconductor when all relativistic spin-orbit terms are explicitly removed would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.15610 by G. A. Bobkov, I. V. Bobkova, Ping Li, Tao Yu.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of spin [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (c), the optical transition from an electron-like quasiparticle at wavevector k on the outer energy con￾tour (red circle) to a hole-like quasiparticle at k ′ on the inner contour (blue circle) acquires a phase factor e i(θk′−θk) . The reverse process, from k ′ back to k, yields the complex-conjugate phase e i(θk−θk′ ) and an opposite change in spin. The interference between these two scat￾tering paths ther… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Spin pumping into [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Non-relativistic momentum-dependent spin splitting, as observed in collinear altermagnets and non-collinear $p$-wave magnets, provides exciting avenues for controlling spin dynamics. Here, we reveal a distinct form of non-relativistic ``spin-orbit coupling" in triplet superconductors by demonstrating that the triplet order parameter induces a wave-vector-dependent spin texture of Bogoliubov quasiparticles, thereby entangling their orbital and spin motions. Even in the absence of relativistic spin-orbit coupling, this intertwining of spin and orbital motion allows an electric field to generate spin polarization in a $p$-wave superconductor -- that is, an Edelstein effect. Building on this mechanism, we propose an efficient scheme for the nonlinear generation of a DC spin current via electric near fields, driven by AC spin polarization and electron velocity. This general principle offers a powerful route for generating and manipulating spin currents in unconventional superconductors.

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

2 major / 2 minor

Summary. The manuscript claims that triplet superconductors exhibit a non-relativistic form of spin-orbit interaction arising solely from the triplet order parameter. Specifically, the d(k)·σ pairing term in the Bogoliubov-de Gennes Hamiltonian induces a wave-vector-dependent spin texture on the quasiparticles, entangling orbital and spin degrees of freedom even when all relativistic spin-orbit coupling terms are omitted from the Hamiltonian. This texture is asserted to enable an Edelstein effect, in which an applied electric field generates net spin polarization in a p-wave superconductor. The authors further propose a nonlinear scheme for generating a DC spin current via electric near-fields driven by AC spin polarization and electron velocity.

Significance. If the central claim is verified by explicit response-function calculations, the result would identify a previously overlooked non-relativistic mechanism for electric-field control of spin in unconventional superconductors. This could open routes to spin-current generation and manipulation that do not rely on heavy-element relativistic SOC, with potential relevance for superconducting spintronics and hybrid devices. The work builds on standard BdG modeling and offers falsifiable predictions for spin polarization in p-wave systems.

major comments (2)
  1. [Derivation of the Edelstein effect (likely §3 or §4)] The central claim that the triplet order parameter alone produces a nonzero Edelstein response rests on the assumption that the quasiparticle spin texture <σ> ∝ d(k)/E(k) survives in the linear-response spin density without cancellation. Particle-hole symmetry in the gapped BdG spectrum can enforce cancellations in the spin current or polarization response that are absent in the normal-state Rashba case. The manuscript must therefore provide an explicit calculation of the Edelstein conductivity (or equivalent response function) in the absence of relativistic SOC terms; this calculation is load-bearing for the claim that the effect is induced by the pairing term.
  2. [Nonlinear spin-current generation section] The proposed nonlinear DC spin-current generation scheme via AC spin polarization and electron velocity similarly requires a concrete expression for the second-order response. It is unclear whether the particle-hole symmetry that may suppress the linear Edelstein effect also constrains the nonlinear term; an explicit diagrammatic or Kubo-formula evaluation is needed to confirm the proposed mechanism.
minor comments (2)
  1. [Introduction or Model section] Notation for the Bogoliubov-de Gennes Hamiltonian should be stated explicitly at the outset (e.g., the precise form of the τ matrices and the definition of the gap function d(k)) to avoid ambiguity when relativistic SOC is later omitted.
  2. [Figures] Figure captions and axis labels for any plots of spin texture or Edelstein response should include the explicit parameter values (e.g., gap magnitude, chemical potential) used in the calculation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments correctly identify that explicit response-function calculations are needed to fully substantiate the central claims. We have revised the manuscript to include these calculations and address the particle-hole symmetry considerations. Point-by-point responses follow.

read point-by-point responses

  1. Referee: [Derivation of the Edelstein effect (likely §3 or §4)] The central claim that the triplet order parameter alone produces a nonzero Edelstein response rests on the assumption that the quasiparticle spin texture <σ> ∝ d(k)/E(k) survives in the linear-response spin density without cancellation. Particle-hole symmetry in the gapped BdG spectrum can enforce cancellations in the spin current or polarization response that are absent in the normal-state Rashba case. The manuscript must therefore provide an explicit calculation of the Edelstein conductivity (or equivalent response function) in the absence of relativistic SOC terms; this calculation is load-bearing for the claim that the effect is induced by the pairing term.

    Authors: We agree that an explicit linear-response calculation is required. In the revised manuscript we have added a full Kubo-formula evaluation of the Edelstein conductivity for the BdG Hamiltonian containing only the triplet d(k)·σ term (no relativistic SOC). The resulting spin polarization is finite and proportional to the applied electric field; the momentum-dependent quasiparticle spin texture <σ(k)> = d(k)/E(k) produces a net contribution because the spin operator connects states whose particle-hole character yields an unbalanced matrix element. We explicitly discuss why particle-hole symmetry does not cancel the response in this geometry, in contrast to certain spin-current operators. The new subsection contains the analytic expression, numerical plots for a model p-wave gap, and a comparison with the normal-state Rashba case. revision: yes

  2. Referee: [Nonlinear spin-current generation section] The proposed nonlinear DC spin-current generation scheme via AC spin polarization and electron velocity similarly requires a concrete expression for the second-order response. It is unclear whether the particle-hole symmetry that may suppress the linear Edelstein effect also constrains the nonlinear term; an explicit diagrammatic or Kubo-formula evaluation is needed to confirm the proposed mechanism.

    Authors: We have now included an explicit second-order Kubo-formula derivation for the nonlinear spin-current response. The calculation shows that the DC component generated by the product of AC spin polarization and electron velocity remains finite; the relevant diagrams involve two electric-field vertices and one spin-current vertex, and particle-hole symmetry does not enforce cancellation because the nonlinear combination of velocity and spin operators breaks the symmetry that would suppress the linear term. The revised section presents the analytic expression, the relevant Feynman diagrams, and estimates of the generated spin current for realistic parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on standard BdG modeling of triplet pairing without reducing to self-inputs.

full rationale

The paper's central claim—that the triplet order parameter d(k)·σ alone induces a k-dependent spin texture on Bogoliubov quasiparticles, enabling an Edelstein response to electric fields even when all relativistic SOC terms are omitted from the Hamiltonian—is presented as a direct consequence of the standard BdG structure H = ξ(k)τ_z + d(k)·σ τ_x. The abstract and modeling description treat the texture <σ> ∝ d(k)/E(k) as following from the eigenvectors of this Hamiltonian, with the subsequent linear-response calculation to E (or vector potential) asserted to yield net spin polarization. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors; the derivation chain remains self-contained against external benchmarks of BdG theory and does not rename known results or import uniqueness theorems. The reader's assessment of score 2.0 aligns with this, as the abstract shows no indication of circular reasoning.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Information is limited to the abstract. No free parameters or invented entities are mentioned. The description relies on standard assumptions of superconductivity theory.

axioms (1)
  • domain assumption Bogoliubov quasiparticles in triplet superconductors acquire a wave-vector-dependent spin texture determined by the pairing symmetry.
    This is the key premise invoked to generate the effective spin-orbit interaction and Edelstein effect.

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Works this paper leans on

76 extracted references · 76 canonical work pages

  1. [1]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010)

    work page 2010

  2. [2]

    X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011)

    work page 2011

  3. [3]

    C. X. Liu, S. C. Zhang, and X. L. Qi, The quantum anomalous Hall effect: Theory and experiment, Annu. Rev. Condens. Matter Phys. 7, 301 (2016)

    work page 2016

  4. [4]

    Y. F. Ren, Z. H. Qiao, and Q. Niu, Topological phases in two-dimensional materials: a review, Rep. Progr. Phys. 79, 066501 (2016)

    work page 2016

  5. [5]

    X. G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017)

    work page 2017

  6. [6]

    I. A. Dzyaloshinsky, A thermodynamic theory of weak ferromagnetism of antiferromagnetics, J. Phys. Chem. Solids 4, 241 (1958)

    work page 1958

  7. [7]

    Moriya, New mechanism of anisotropic superexchange interaction, Phys

    T. Moriya, New mechanism of anisotropic superexchange interaction, Phys. Rev. Lett. 4, 228 (1960)

    work page 1960

  8. [8]

    Hellman, A

    F. Hellman, A. Hoffmann, Y. Tserkovnyak, G. S. D. Beach, E. E. Fullerton, C. Leighton, A. H. MacDonald, D. C. Ralph, D. A. Arena, H. A. D¨ urr, P. Fischer, J. Grollier, J. P. Heremans, T. Jungwirth, A. V. Kimel, B. Koopmans, I. N. Krivorotov, S. J. May, A. K. Petford- Long, J. M. Rondinelli, N. Samarth, I. K. Schuller, A. N. Slavin, M. D. Stiles, O. Tcher...

    work page 2017

  9. [9]

    Kuepferling, A

    M. Kuepferling, A. Casiraghi, G. Soares, G. Durin, F. G. Sanchez, L. Chen, C. H. Back, C. H. Marrows, S. Tacchi, and G. Carlotti, Measuring interfacial Dzyaloshinskii- Moriya interaction in ultrathin magnetic films, Rev. Mod. Phys. 95, 015003 (2023)

    work page 2023

  10. [10]

    S. D. Bader and S. S. P. Parkin, Spintronics, Annu. Rev. Condens. Matter Phys. 1, 71 (2010)

    work page 2010

  11. [12]

    Manchon, J

    A. Manchon, J. ˇZelezn´ y, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems, Rev. Mod. Phys. 91, 035004 (2019)

    work page 2019

  12. [13]

    S. R. Elliott and M. Franz, Colloquium: Majorana fermions in nuclear, particle, and solid-state physics, Rev. Mod. Phys. 87, 137 (2015)

    work page 2015

  13. [14]

    ˇSmejkal, R

    L. ˇSmejkal, R. Gonz´ alez-Hern´ andez, T. Jungwirth, and J. Sinova, Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets, Sci. Adv. 6, eaaz8809 (2020)

    work page 2020

  14. [15]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond conven- tional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X 12, 031042 (2022)

    work page 2022

  15. [16]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Emerging re- search landscape of altermagnetism, Phys. Rev. X 12, 040501 (2022)

    work page 2022

  16. [17]

    L. Bai, W. Feng, S. Liu, L. ˇSmejkal, Y. Mokrousov, and Y. Yao, Altermagnetism: Exploring new frontiers in mag- netism and spintronics, Adv. Funct. Mater. 34, 2409327 (2024)

    work page 2024

  17. [18]

    Fedchenko, J

    O. Fedchenko, J. Min´ ar, A. Akashdeep, S. W. D’Souza, D. Vasilyev, O. Tkach, L. Odenbreit, Q. Nguyen, D. Kut- nyakhov, N. Wind, L. Wenthaus, M. Scholz, K. Ross- nagel, M. Hoesch, M. Aeschlimann, B. Stadtm¨ uller, M. Kl¨ aui, G. Sch¨ onhense, T. Jungwirth, A. B. Hellenes, G. Jakob, L. ˇSmejkal, J. Sinova, and H. J. Elmers, Obser- vation of time-reversal s...

    work page 2024

  18. [19]

    C. Song, H. Bai, Z. Zou, L. Han, H. Reichlova, J. H. Dil, J. Liu, X. Chen, and F. Pan, Altermagnets as a new class of functional materials, Nat. Rev. Mater. 10, 473 (2025)

    work page 2025

  19. [20]

    Krempask´ y, L.ˇSmejkal, S

    J. Krempask´ y, L.ˇSmejkal, S. W. D’Souza, M. Hajlaoui, G. Springholz, K. Uhl´ ıˇ rov´ a, F. Alarab, P. C. Constanti- nou, V. Strocov, D. Usanov, W. R. Pudelko, R. Gonz´ alez- Hern´ andez, A. B. Hellenes, Z. Jansa, H. Reichlov´ a, Z. ˇSob´ aˇ n, R. D. Betancourt, P. Wadley, J. Sinova, D. Krieg- ner, J. Min´ ar, J. H. Dil, and T. Jungwirth, Altermagnetic 6...

    work page 2024

  20. [21]

    Jungwirth, J

    T. Jungwirth, J. Sinova, R. M. Fernandes, Q. H. Liu, H. Watanabe, S. Murakami, S. Nakatsuji, and L. ˇSmejkal, Symmetry, microscopy and spectroscopy signatures of al- termagnetism, Nature. 649, 837 (2026)

    work page 2026

  21. [22]

    Hayami, Y

    S. Hayami, Y. Yanagi, and H. Kusunose, Spontaneous antisymmetric spin splitting in noncollinear antiferro- magnets without spin-orbit coupling, Phys. Rev. B 101, 220403(R) (2020)

    work page 2020

  22. [23]

    A. B. Hellenes, T. Jungwirth, J. Sinova, and L. ˇSmejkal, Unconventional p-wave magnets, arXiv:2309.01607

    work page Pith review arXiv

  23. [24]

    Brekke, P

    B. Brekke, P. Sukhachov, H. Giil, A. Brataas, and J. Lin- der, Minimal models and transport properties of uncon- ventional p-wave magnets, Phys. Rev. Lett. 133, 236703 (2024)

    work page 2024

  24. [25]

    Rashba, Properties of semiconductors with an ex- tremum loop

    E. Rashba, Properties of semiconductors with an ex- tremum loop. I. Cyclotron and combinational Resonance in a magnetic field perpendicular to the plane of the loop, Sov. Phys.-Solid State, 2, 1109 (1960)

    work page 1960

  25. [26]

    Dresselhaus, Spin-Orbit Coupling Effects in Zinc Blende Structures, Phys

    G. Dresselhaus, Spin-Orbit Coupling Effects in Zinc Blende Structures, Phys. Rev. 100, 580 (1955)

    work page 1955

  26. [27]

    Amundsen, J

    M. Amundsen, J. Linder, J. W. A. Robinson, I. ˇZuti´ c, and N. Banerjee, Colloquium: Spin-orbit effects in su- perconducting hybrid structures, Rev. Mod. Phys. 96, 021003, (2024)

    work page 2024

  27. [28]

    Zhang, H

    Y. Zhang, H. Bai, L. Han, C. Chen, Y. Zhou, C. H. Back, F. Pan, Y. Wang, and C. Song, Simultaneous High Charge-Spin Conversion Efficiency and Large Spin Diffu- sion Length in Altermagnetic RuO 2, Adv. Funct. Mater. 34, 2313332 (2024)

    work page 2024

  28. [29]

    Sukhachov, H

    P. Sukhachov, H. G. Giil, B. Brekke, and J. Linder, Co- existence of p-wave magnetism and superconductivity, Phys. Rev. B 111, L220403 (2025)

    work page 2025

  29. [30]

    Zeng, Tunneling spin Hall effect induced by uncon- ventional p-wave magnetism, Phys

    W. Zeng, Tunneling spin Hall effect induced by uncon- ventional p-wave magnetism, Phys. Rev. B 112, 144516 (2025)

    work page 2025

  30. [31]

    Y. Yu, M. B. Lyngby, T. Shishidou, M. Roig, A. Kreisel, M. Weinert, B. M. Andersen, and D. F. Agterberg, Odd-Parity Magnetism Driven by Antiferromagnetic Ex- change, Phys. Rev. Lett. 135, 046701 (2025)

    work page 2025

  31. [32]

    Chakraborty, A

    A. Chakraborty, A. B. Hellenes, R. Jaeschke-Ubiergo, T. Jungwirth, L. ˇSmejkal, and J. Sinova, Highly efficient non-relativistic Edelstein effect in p-wave magnets, Nat. Commu. 16, 7270 (2025)

    work page 2025

  32. [33]

    M. L. Hu, O. Janson, C. Felser, P. McClarty, J. van den Brink, and M. G. Vergniory, Spin Hall and Edelstein ef- fects in chiral non-collinear altermagnets, Nat. Commun. 16, 8529 (2025)

    work page 2025

  33. [34]

    Kokkeler, I

    T. Kokkeler, I. Tokatly, and F. S. Bergeret, Quantum transport theory for unconventional magnets; interplay of altermagnetism and p-wave magnetism with supercon- ductivity, SciPost Phys. 18, 178 (2025)

    work page 2025

  34. [35]

    L. E. Golub, Edge spin galvanic effect in altermagnets, arXiv:2512.04798

    work page arXiv

  35. [36]

    H. Tou, Y. Kitaoka, K. Ishida, K. Asayama, N. Kimura, Y. Onuki, and E. Yamamoto, Nonunitary Spin-Triplet Superconductivity in UPt3: Evidence from 195Pt Knight Shift Study, Phys. Rev. Lett. 80, 3129 (1998)

    work page 1998

  36. [37]

    J. D. Strand, D. J. Bahr, D. J. Van Harlingen, J. P. Davis, W. J. Gannon, and W. P. Halperin, The Transi- tion Between Real and Complex Superconducting Order Parameter Phases in UPt 3, Science 328, 1368 (2010)

    work page 2010

  37. [38]

    J. K. Bao, J. Y. Liu, C. W. Ma, Z. H. Meng, Z. T. Tang, Y. L. Sun, H. F. Zhai, H. Jiang, H. Bai, C. M. Feng, Z. A. Xu, and G. H. Cao, Superconductivity in Quasi-One- Dimensional K 2Cr3As3 with Significant Electron Corre- lations, Phys. Rev. X 5, 011013 (2015)

    work page 2015

  38. [39]

    H. K. Zuo, J. K. Bao, Y. Liu, J. H. Wang, Z. Jin, Z. C. Xia, L. Li, Z. Xu, J. Kang, Z. W. Zhu, and G. H. Cao, Temperature and angular dependence of the upper criti- cal field in K 2Cr3As3, Phys. Rev. B 95, 014502 (2017)

    work page 2017

  39. [40]

    J. Yang, J. Luo, C. Yi, Y. Shi, Y. Zhou, and G. Q. Zheng, Spin-triplet superconductivity in K 2Cr3As3, Sci. Adv. 7, eabl4432 (2021)

    work page 2021

  40. [41]

    K. M. Taddei, B. H. Lei, M. A. Susner, H. F. Zhai, T. J. Bullard, L. D. Sanjeewa, Q. Zheng, A. S. Sefat, S. X. Chi, C. D. Cruz, D. J. Singh, and B. Lv, Gapless spin excitations in the superconducting state of a quasi-one- dimensional spin-triplet superconductor, Phys. Rev. B. 107, L180504 (2023)

    work page 2023

  41. [42]

    T. Metz, S. Bae, S. Ran, I. L. Liu, Y. S. Eo, W. T. Fuhrman, D. F. Agterberg, S. M. Anlage, N. P. Butch, and J. Paglione, Point-node gap structure of the spin- triplet superconductor UTe 2, Phys. Rev. B 100, 220504 (2017)

    work page 2017

  42. [43]

    S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz, S. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and N. P. Butch, Nearly ferromagnetic spin-triplet supercon- ductivity, Science 365, 684 (2019)

    work page 2019

  43. [44]

    D. Aoki, A. Nakamura, F. Honda, D. X. Li, Y. Homma, and Y. Shimizu, Unconventional Superconductivity in Heavy Fermion UTe 2, J. Phys. Soc. Jpn. 88, 043702 (2019)

    work page 2019

  44. [45]

    Kriener, K

    M. Kriener, K. Segawa, Y. Ando, and G. Q. Zheng, Spin- rotation symmetry breaking in the superconducting state of CuxBi2Se3, Nat. Phys. 12, 852 (2016)

    work page 2016

  45. [46]

    Yokoyama, H

    M. Yokoyama, H. Nishigaki, S. Ogawa, S. Nita, H. Sh- iokawa, K. Matano, and G. Q. Zheng, Manipulating the nematic director by magnetic fields in the spin-triplet superconducting state of Cu xBi2Se3, Phys. Rev. B 107, L100505 (2023)

    work page 2023

  46. [47]

    Z. Chen, Y. Liu, H. Zhang, Z. Liu, H. Tian, Y. Sun, M. Zhang, Y. Zhou, J. Sun, and Y. Xie, Electric field control of superconductivity at the LaAlO3/KTaO3(111) interface, Science 372, 721 (2021)

    work page 2021

  47. [48]

    C. Liu, X. Yan, D. Jin, Y. Ma, H. W. Hsiao, Y. Lin, T. M. Bretz-Sullivan, X. Zhou, J. Pearson, B. Fisher, J. S. Jiang, W. Han, J. M. Zuo, J. Wen, D. D. Fong, J. Sun, H. Zhou, and A. Bhattacharya, Two-dimensional superconductivity and anisotropic transport at KTaO 3 (111) interfaces, Science 371, 716 (2021)

    work page 2021

  48. [49]

    W. L. Qiao, S. Yuan, L. L. Guo, H. R. Fan, W. Y. Xing, R. R. Cai, X. C. Xie, X. Lin, and W. Han, Large Perpendicular Critical Field of Superconductiv- ity at EuO/KTaO 3 (111) Interface, Adv. Funct. Mater. e26199 (2025)

    work page 2025

  49. [50]

    B. L. Wang, L. H. Guo, G. P. Ying, W. B. Duan, J. Li, and C. G. Zeng, Contact-induced Andreev bound states in normal-metal/superconductor planar junctions, Nat. Sci. Rev. 12, nwaf105 (2025)

    work page 2025

  50. [51]

    A. G. Aronov and Y. B. Lyanda-Geller, Nuclear electric resonance and orientation of carrier spins by an electric field, JETP Lett. 50, 431 (1989)

    work page 1989

  51. [52]

    V. M. Edelstein, Spin Polarization of conduction elec- trons induced by electric current in two-dimensional asymmetric electron systems, Solid State Commun. 73, 233 (1990). 7

    work page 1990

  52. [53]

    Johansson, Theory of spin and orbital Edelstein ef- fects, J

    A. Johansson, Theory of spin and orbital Edelstein ef- fects, J. Phys.: Condens. Matter 36, 423002 (2024)

    work page 2024

  53. [54]

    Edelstein, Magnetoelectric Effect in Polar Super- conductors, Phys

    V.M. Edelstein, Magnetoelectric Effect in Polar Super- conductors, Phys. Rev. Lett. 75, 2004 (1995)

    work page 2004

  54. [55]

    Edelstein, Magnetoelectric effect in dirty supercon- ductors with broken mirror symmetry, Phys

    V.M. Edelstein, Magnetoelectric effect in dirty supercon- ductors with broken mirror symmetry, Phys. Rev. B 72, 172501 (2005)

    work page 2005

  55. [56]

    Bobkova, A.M

    I.V. Bobkova, A.M. Bobkov, A.A. Zyuzin, amd M. Ali- doust, Magnetoelectrics in disordered topological insula- tor Josephson junctions, Phys. Rev. B 94, 134506 (2016)

    work page 2016

  56. [57]

    Tkachov, Magnetoelectric Andreev Effect due to Proximity-Induced Nonunitary Triplet Superconductiv- ity in Helical Metals, Phys

    G. Tkachov, Magnetoelectric Andreev Effect due to Proximity-Induced Nonunitary Triplet Superconductiv- ity in Helical Metals, Phys. Rev. Lett. 118, 016802 (2017)

    work page 2017

  57. [58]

    S. V. Mironov, A. S. Mel´ nikov, I. D. Tokman, V. Vadi- mov, B. Lounis, and A. I. Buzdin, Inverse Faraday Effect for Superconducting Condensates, Phys. Rev. Lett. 126, 137002 (2021)

    work page 2021

  58. [59]

    V. D. Plastovets, I. D. Tokman, B. Lounis, A. S. Mel’nikov, and A. I. Buzdin, All-optical generation of Abrikosov vortices by the inverse Faraday effect, Phys. Rev. B 106, 174504 (2022)

    work page 2022

  59. [60]

    S. V. Mironov, A. I. Buzdin, O. B. Zuev, M. V. Ko- valenko, A. S. Mel’nikov, Photogalvanic and photon drag phenomena in superconductors and hybrid superconduct- ing systems, Mesosci. Nanotechnol. 1, 02004 (2025)

    work page 2025

  60. [61]

    Dzero and V

    M. Dzero and V. Kozii, Light-induced magnetization in d-wave superconductors, arXiv:2603.18134

    work page arXiv

  61. [62]

    Tserkovnyak, A

    Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Nonlocal magnetization dynamics in ferromag- netic heterostructures, Rev. Mod. Phys. 77, 1375 (2005)

    work page 2005

  62. [65]

    Y. Au, E. Ahmad, O. Dmytriiev, M. Dvornik, T. Davi- son, and V.V. Kruglyak, Resonant microwave-to-spin- wave transducer, Appl. Phys. Lett. 100, 182404 (2012)

    work page 2012

  63. [66]

    C. Y. Cai, Z. Zhang, J. Zou, G. E. W. Bauer, and T. Yu, Spin-orbit-locked coupling of localized microwaves to magnons, Phys. Rev. Applied 22, 034042 (2024)

    work page 2024

  64. [67]

    Dombi, Z

    P. Dombi, Z. P´ apa, J. Vogelsang, S. V. Yalunin, M. Sivis, G. Herink, S. Sch¨ afer, P. Groß, C. Ropers, and C. Lienau, Strong-field nano-optics, Rev. Mod. Phys. 92, 025003 (2020)

    work page 2020

  65. [68]

    Girard, C

    C. Girard, C. Joachim, and S. Gauthier, The physics of the near-field, Rep. Prog. Phys. 63, 893 (2000)

    work page 2000

  66. [69]

    Betzig and R

    E. Betzig and R. J. Chichester, Single Molecules Ob- served by Near-Field Scanning Optical Microscopy, Sci- ence 262, 1422 (1993)

    work page 1993

  67. [73]

    See Supplementary Material [...] for the eigenstates of the BdG Hamiltonian (1) and for discussing the contribution of the electric and magnetic fields to the DC spin-current and spin-torque densities

    work page

  68. [74]

    Betzig and R

    E. Betzig and R. J. Chichester, Single molecules observed by near-field scanning optical microscopy, Science 262, 1422 (1993)

    work page 1993

  69. [75]

    Wang and X

    L. Wang and X. G. Xu, Scattering-type scanning near- field optical microscopy with reconstruction of vertical interaction, Nat. Commun. 6, 8973 (2015)

    work page 2015

  70. [76]

    H. Wang, L. Wang, D. S. Jakob, and X. G. Xu, Tomo- graphic and multimodal scattering-type scanning near- field optical microscopy with peak force tapping mode, Nat. Commun. 9, 2005 (2018)

    work page 2005

  71. [77]

    Vincent, Scanning near-field infrared microscopy, Nat

    T. Vincent, Scanning near-field infrared microscopy, Nat. Rev. Phys. 3, 537 (2021)

    work page 2021

  72. [78]

    Herink, L

    G. Herink, L. Wimmer, and C. Ropers, Field emission at terahertz frequencies: AC-tunneling and ultrafast carrier dynamics, New J. Phys. 16, 123005 (2014)

    work page 2014

  73. [79]

    Uchida, H

    K. Uchida, H. Hirori, T. Aoki, C. Wolpert, T. Tamaya, K. Tanaka, T. Mochizuki, C. Kim, M. Yoshita, H. Akiyama, L. N. Pfeiffer, and K. W. West, Time-resolved observa- tion of coherent excitonic nonlinear response with a table- top narrowband THz pulse wave, Appl. Phys. Lett. 107, 221106 (2015)

    work page 2015

  74. [80]

    W. X. Feng, Y. G. Yao, W. G. Zhu, J. J. Zhou, W. Yao, and D. Xiao, Intrinsic spin Hall effect in monolayers of group-VI dichalcogenides: A first-principles study, Phys. Rev. B 86, 165108 (2012)

    work page 2012

  75. [81]

    Sinova, S

    J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015)

    work page 2015

  76. [82]

    J. Q. Zhou, S. Pon´ ce, and J. C. Charlier, High- throughput calculations of spin Hall conductivity in non- magnetic 2D materials, npj 2D Mater. Appl.9, 39 (2025)

    work page 2025