Pith. sign in

REVIEW 3 major objections 5 minor 3 cited by

Transport theory and spin-transfer physics in d-wave altermagnets

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper derives the altermagnetic spin-transfer torque $\tau = \eta_{\rm FL}(j^e_x\partial_y + j^e_y\partial_x)n + \eta_{\rm DL} s\, n\times(j^e_x\partial_y + j^e_y\partial_x)n$ and its reciprocal spin pumping from a microscopic model.

desk verdict A serious microscopic derivation of transverse spin-transfer torques in d-wave altermagnets; the novelty is real but narrower than claimed, and the advertised stress-test red flag on Eq. (19) does not hold up. read the letter →

arxiv 2412.13763 v1 pith:I5BWLQ3Q submitted 2024-12-18 cond-mat.mes-hall

Ricardo Zarzuela , Rodrigo Jaeschke-Ubiergo , Olena Gomonay , Libor Šmejkal , Jairo Sinova This is my paper · ORCID
classification cond-mat.mes-hall
keywords altermagnetismd-wavealtermagnetspin-transfertorquespinpumpingspin-splittereffectslave-bosont-Jmodeldomain-wallmotionmesoscaletransporttheory
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

Using a slave-boson formulation of the $t$--$J$ model, this paper derives a mesoscale transport theory for itinerant carriers in a d-wave altermagnet, a compensated magnet whose opposite-spin sublattices are linked by a fourfold rotation and whose band structure shows momentum-dependent d-wave spin splitting. Its central result is the altermagnetic spin-transfer torque $\tau^{\rm AM}_{m,\mathrm{ST}} = \eta_{\rm FL}(j^e_x\partial_y+j^e_y\partial_x)n + \eta_{\rm DL} s\, n\times(j^e_x\partial_y+j^e_y\partial_x)n$, whose crossed combination of current and spatial derivative exists only for d-wave altermagnets. The same crossed structure appears in the reciprocal spin-pumping current. If this is right, charge currents flowing transverse to a domain wall's normal can move the wall, and the spin-splitter effect follows from an off-diagonal spin-polarized mass tensor. The paper also shows that strain renormalizes the spin splitting and hence tunes these effects.

What carries the argument

The load-bearing object is the effective Euclidean Lagrangian (3), obtained from a spin-rotation-invariant slave-boson mean-field expansion of the single-band $t$--$J$ model on a rutile lattice. Its decisive ingredient is the spin-polarized mass tensor $(M^s)^{-1}$, whose off-diagonal elements $\propto n/m^s_{xy}$ parametrize the transverse intertwining of charge and spin currents. Together with the emergent gauge fields $A_x = g^{xy}_4\, n\,\partial_x n + g^{\rm AM}_4\, m\,\partial_y n$ and $A_y = g^{xy}_4\, n\,\partial_y n + g^{\rm AM}_4\, m\,\partial_x n$, the energy term $E_{\rm ST} = -\frac{4m_\parallel}{\hbar^2}(J_x\cdot A_x + J_y\cdot A_y)$ generates the spin-transfer torque through the Poisson-bracket dynamics of the N\'eel order and the macroscopic magnetization.

What would settle it

Measure the current-direction dependence of domain-wall motion in a rutile d-wave altermagnet such as RuO$_2$. For a wall whose N\'eel order varies only along $x$, Eq. (20) predicts a torque proportional to $j^e_y\,\partial_x n$, so the wall should move when the current flows along $y$ (transverse to the wall normal) and stay put for current along $x$; a conventional antiferromagnet shows the opposite pattern. Observing this 90-degree switching rule, or finding its absence, would settle whether the central claim is correct.

Watch

Extended reading notes

Core claim

The paper's central claim is that the long-wavelength theory of itinerant carriers in a d-wave altermagnet contains a spin-polarized diffusive term with off-diagonal mass-tensor elements proportional to the N\'eel order, and that this term couples charge currents to spatial gradients of the order parameter in a crossed way. Combining the spin-splitter relation $J_\kappa = \frac{m_\parallel}{m^s_{xy}} \frac{\hbar}{2e}\, n\, \sigma^x_{\kappa\sigma} j^e_\sigma$ with the gauge-field coupling in the energy functional yields the spin-transfer torque of Eq. (20), $\tau^{\rm AM}_{m,\mathrm{ST}} = \eta_{\rm FL}(j^e_x\partial_y + j^e_y\partial_x)n + \eta_{\rm DL} s\, n\times(j^e_x\partial_y + j^e_y\partial_x)n$, with $\eta_{\rm FL}$ and $\eta_{\rm DL}$ proportional to the d-wave spin splitting. The Onsager-reciprocal spin-pumping current has the same crossed structure. The same framework produces the spin-splitter effect as a real-space off-diagonal spin conductivity and predicts that elastic strain modifies these effects by renormalizing the spin-splitting parameter $t-t'$.

Load-bearing premise

The load-bearing premise is the adiabatic slave-boson mean-field assumption that the vector boson fields stay rigidly pinned to the localized sublattice spin densities, i.e. the $J_{sd}\to\infty$ limit; if that pinning fails, the effective Lagrangian (3) and every torque and pumping formula derived from it inherit the error.

Editorial extensions

If this is right

  • The spin-splitter effect follows from the off-diagonal spin-polarized mass tensor: an electric field generates a spin current polarized along the N\'eel order and flowing transversely, with no spin-orbit coupling needed.
  • Altermagnetic spin-transfer torques contain a fieldlike term linear in the N\'eel order, allowed because the sublattice symmetry is broken, plus a dissipative term transverse to it; both vanish when altermagnetism is switched off.
  • Spin pumping is Onsager reciprocal to the torque and carries the same crossed $j^e_x\partial_y + j^e_y\partial_x$ structure, so a moving texture pumps spin currents that have no counterpart in bipartite antiferromagnets.
  • Domain walls and skyrmions experience a spin-transfer force with a distinct angular dependence; in particular, the reactive force on a skyrmion is isotropic in the current direction, unlike the transverse dependence in ferrimagnets and magnetoelectric antiferromagnets.
  • Elastic strain renormalizes the spin-splitting parameter $t-t'$: shear strain can enhance the altermagnetic transport effects when the unstrained hoppings have the same sign, and merely renormalizes them when the signs are opposite.

Reading between the lines

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

  • If Eq. (20) is confirmed, the 90-degree switching rule for current-driven domain-wall motion becomes a practical diagnostic for d-wave altermagnetism, and devices could steer textures by rotating the current direction instead of changing its magnitude.
  • Because the crossed derivative combination is the real-space fingerprint of d-wave symmetry, g- and i-wave altermagnets would plausibly show higher-order derivative combinations of the same type; deriving them would give a testable symmetry hierarchy, though this goes beyond the paper.
  • Strain tuning suggests that placing an altermagnet on a piezoelectric substrate could modulate the spin-splitter efficiency and the spin-transfer torque in situ; this is an extrapolation from the paper's strain analysis, not a claim it makes.
  • Since the derivation is presented at the $\Gamma$-valley and the $Z$-valley flips the sign of $z$-derivative terms, quasi-two-dimensional devices that average over valleys could partially cancel or enhance the predicted torque; the paper notes the valley-independence only for in-plane physics.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript develops an effective long-wavelength theory for itinerant carriers in d-wave altermagnets, starting from a slave-boson representation of a t-J model on a rutile lattice. The central results are a spin-polarized mass tensor with off-diagonal elements proportional to the Néel order, which yields the spin-splitter effect, and spin-transfer torques and spin-pumping currents exhibiting a transverse derivative structure, τ ∝ (j_x^e ∂_y + j_y^e ∂_x) n. The paper also discusses consequences for current-driven domain-wall motion and the effect of elastic strain on the transport coefficients.

Significance. If the central results hold, the paper identifies a new, symmetry-required spin-transfer torque in d-wave altermagnets that couples charge currents to magnetization gradients in a transverse, anisotropic combination not present in conventional antiferromagnets. The derivation is self-contained within a microscopic model and produces falsifiable predictions, such as transverse-current-driven domain-wall motion in the same geometry where the torque vanishes in bipartite antiferromagnets. The main limitations are the load-bearing mean-field slave-boson ansatz, the partly asserted passage from Eq. (18) to Eq. (19), and the phenomenological status of the dissipative torque. Nevertheless, the qualitative form of Eq. (20) is a valuable contribution to altermagnetic spintronics.

major comments (3)
  1. [IV.A, Eqs. (18)–(19)] The reduction from Eq. (18) to Eq. (19) is not shown, and the cancellations involved are nontrivial. After substituting the spin-splitter relation J_κ = (m_∥/m_{sxy})(ℏ/2e) n σ_x j^e and using n·m=0 with uniform j^e, the g_4^AM terms in Eq. (18) cancel pairwise, and the surviving g_4^xy contribution emerges only after a partial cancellation among the third, fourth, and fifth terms of that equation. The authors should present this simplification explicitly, because the microscopic expression for η_FL is a central result. In addition, the factor s^2 in Eq. (19) is unexplained: it does not appear in the spin-splitter relation (15) or in the energy functional (16), and its presence changes the spin-density scaling of η_FL. Note that, as printed, Eq. (19) contains g_4^xy (not g_4^AM), so it is linear in the spin splitting once 1/m_{sxy} is accounted for, consistent with the abstract; however, the derivation must be made transparent and the s^2 factor either derived or corrected.
  2. [IV.A, Eq. (20)] The dissipative (antidamping-like) torque is introduced by appealing to "the usual phenomenological arguments" rather than by deriving it from the effective Lagrangian. Since the central physical prediction in Sec. VI — transverse-current-driven domain-wall motion — relies specifically on the dissipative component (see the discussion of Fig. 2), the paper should either derive η_DL within the same microscopic framework or explicitly state that Eq. (20) is a phenomenological ansatz with η_DL as an independent parameter. As written, the abstract's claim to "elucidate the spin-transfer response" overstates the status of the dissipative term.
  3. [IV.B, Eq. (24)] The derivation of the spin-pumping currents from Onsager reciprocity is under-specified. In particular, the thermodynamic force f_m ≈ −n ∂_t n/s^2 stated after Eq. (22) does not follow directly from the LLG equations (22); the standard relation in the exchange-dominated limit involves f_m ∝ n × ∂_t n up to factors of the spin density and susceptibility. Please provide the derivation and clarify the definitions of s and f_m, since the pumped currents in Eq. (24) depend sensitively on these factors. The qualitative reciprocal relation between Eqs. (20) and (24) is plausible, but the quantitative coefficients need support.
minor comments (5)
  1. [Eqs. (19)–(20) and Appendix A] The symbol s is used both for the sublattice spin magnitude (Appendix A) and for the saturation spin density in the dissipative torque (Eq. (20)); please use distinct notation to avoid ambiguity.
  2. [After Eq. (19)] The sentence stating that terms "quadratic in the spin splitting ∝ t′−t (via g_4^AM and 1/m_{sxy})" are disregarded should be rephrased: it is the combination g_4^AM / m_{sxy} that is quadratic, not the factors individually.
  3. [Sec. II, footnote 31] The neglect of valley-off-diagonal terms is justified by an energy-cost argument; a quantitative estimate, even an order-of-magnitude one, would make the long-wavelength expansion more convincing.
  4. [Sec. V, Eq. (27)] The strain dependence of t−t′ is a useful and clear result; however, the discussion after Eq. (27) could be shortened and the conditions for shear-strain enhancement stated more precisely (e.g., relative magnitude of |t0−t′0| vs. |t0+t′0|).
  5. [References] There are several typographical errors and formatting inconsistencies in the reference list, including "Pys." in Ref. 36 and inconsistent spelling of "Jaeschke-Ubiergo" in Refs. 25 and 34; please harmonize the bibliography.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the spin-transfer torque and spin-pumping expressions are derived from the microscopic t-J model through the slave-boson Lagrangian, not fitted or assumed.

full rationale

The central results, Eq. (19)-(20) and Eq. (24), are obtained by a closed derivation chain: the slave-boson mean-field treatment in Appendix A1 produces the effective Lagrangian (3) with coupling constants expressed in terms of microscopic tight-binding parameters (A21); the constitutive equations (9) and (11) yield the spin-splitter relation (12)-(15); and the reactive torque follows from Poisson brackets of E_ST (16)-(18) using functional derivatives in Appendix B. No parameter is fitted to the target torque or pumping, and no external benchmark is required. The transverse derivative combination {j_x^e, ∂_y} plus {j_y^e, ∂_x} emerges from the off-diagonal mass tensor 1/m_s_xy and the g4^xy terms, which are derived rather than imposed. Self-citations are present (e.g., Ref. 35 for the slave-boson ansatz), but the ansatz is restated and the relevant formulas are re-derived in the appendices; they are not used as an unexamined uniqueness theorem. The reviewer's algebraic concern about g4^AM versus g4^xy in Eq. (19) is a correctness issue, not a circularity: even if the printed coefficient is wrong, the derivation chain is not circular. Accordingly, no circular step is identified.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central transport theory rests on the t-J model, a slave-boson mean-field ansatz with adiabatic spinon pinning, a second-order long-wavelength truncation, and the exchange-dominated limit. These are stated in the paper but not independently proven. No experimental fitting is involved; the only hand-inserted quantities are the phenomenological torque coefficients eta_FL and eta_DL.

free parameters (2)
  • eta_FL (reactive spin-transfer coefficient) = not assigned; implicitly defined via Eq. (19) in terms of m_parallel, m_sxy, g_xy4, and s
    Eq. (20) introduces eta_FL as a phenomenological constant proportional to the spin splitting, and Eq. (19) fixes it in terms of microscopic constants, but no numeric value is given.
  • eta_DL (dissipative spin-transfer coefficient) = not determined; asserted by transversality argument
    The dissipative torque is not derived from the microscopic model. The paper states that it must be transversal to the fieldlike torque, which is a symmetry assumption rather than a calculated result.
assumptions (6)
  • domain assumption The single-band t-J model faithfully describes the itinerant electron physics in strongly correlated systems.
    Used as the microscopic starting point in Section II, Eq. (1). The entire derivation inherits this modeling assumption.
  • ad hoc to paper Slave-boson mean-field ansatz: vector slave-boson fields pin to the localized sublattice spin densities, and the spinon adjusts adiabatically to the magnetic background.
    Introduced in Section II and Appendix A1, Eqs. (A4)-(A5), with the J_sd to infinity analogy. This is the structural premise that generates the effective Lagrangian.
  • ad hoc to paper Long-wavelength expansion truncated at second order in derivatives and order parameters, with valley-off-diagonal terms neglected.
    Section II around Eq. (2) and Appendix A2. This truncation selects which terms survive in Eq. (3) and all later transport equations.
  • domain assumption Exchange-dominated regime with no spin-orbit interactions or Dzyaloshinskii-Moriya coupling.
    Stated in Section VI. The torque expressions and domain wall dynamics neglect relativistic corrections that can matter in real altermagnets.
  • domain assumption Saddle-point or mean-field treatment of the Fermi field, with fluctuations entering only as spatial dependence of coupling constants.
    Justified in Section VI. The transport equations are derived from saddle-point equations, not from a full quantum treatment of fluctuations.
  • standard math Onsager reciprocity and the Landau-Lifshitz-Gilbert form with Rayleigh dissipation.
    Used in Section IV.B, Eqs. (22)-(23), to obtain the spin-pumping current from the spin-transfer torque. This is a standard linear-response assumption.
invented entities (1)
  • Emergent non-Abelian gauge field A_k with components A_x, A_y, A_z
    purpose: Encodes the coupling between itinerant spin currents and spatial gradients of the magnetic order; generates spin-transfer torque, spin pumping, and topological Hall response.
    Defined in Eq. (4) as a function of gradients of n and m. It is an effective field derived from the order parameter, not an independent degree of freedom with an external falsifiable handle.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Transport theory and spin-transfer physics in d-wave altermagnets." pith.science (2026). https://pith.science/paper/I5BWLQ3Q

@misc{pith2026241213763,
  author       = {Pith},
  title        = {Pith review of: Transport theory and spin-transfer physics in d-wave altermagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5BWLQ3Q}},
  note         = {Machine review of arXiv:2412.13763}
}
abstract

We develop a mesoscale transport theory for the charge and spin degrees of freedom of itinerant carriers in a $d$-wave altermagnet. Our effective Lagrangian description is built upon the slave-boson formulation of the microscopic $t-J$ model. We obtain a spin-polarized diffusive contribution to the effective Hamiltonian, with no counterpart in conventional antiferromagnetism and parametrized by the spin splitting, that is responsible for the so-called spin-splitter effect in $d$-wave altermagnets. We also elucidate the spin-transfer response of the itinerant fluid as well as the spin pumping into the altermagnet, which show previously unidentified combinations of the charge current and spatial partial derivatives (namely, {$j_{x}^{e}$,$\partial_{y}$} and {$j_{y}^{e}$,$\partial_{x}$}). The emergent spin-transfer physics in $d$-wave altermagnets opens up new possibilities for the dynamics of spin textures, such as the domain-wall motion driven by transverse charge currents. We also consider the effect of elastic distortions in the aforementioned transport properties.

Figures

Figures reproduced from arXiv: 2412.13763 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Rutile unit cell and tight-binding constants of the model. Turquoise and fuchsia spheres correspond to magnetic [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Bloch-type domain-wall spin configuration extending along the crystallographic [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

  1. Quantum transport theory for unconventional magnets: Interplay of altermagnetism and p-wave magnetism with superconductivity

    cond-mat.supr-con 2024-12 conditional novelty 7.0 of 10

    A symmetry-based quantum transport theory for unconventional magnets, unified with superconductivity, yields testable predictions for spin-polarized currents, proximity-induced magnetization, and spin-galvanic effects.

  2. Electrical Control of the Exchange Bias Effect at Model Ferromagnet-Altermagnet Junctions

    cond-mat.mtrl-sci 2025-01 conditional novelty 6.0 of 10

    Ferromagnet/altermagnet junctions are predicted to show exchange bias, including on compensated surfaces, with strength tunable by interface coupling and electric field.

  3. Spin-Transfer Torque in Altermagnets with Magnetic Textures

    cond-mat.mes-hall 2024-12 accept novelty 6.0 of 10

    Spin-splitter torques in textured altermagnets are predicted to make domain walls precess and skyrmions move sideways with d-wave anisotropy.

Reference graph

Works this paper leans on

47 extracted references · 42 canonical work pages · cited by 3 Pith papers

  1. [1]

    S mejkal, J

    L. S mejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12 , 031042 (2022)

    work page 2022

  2. [2]

    S mejkal, J

    L. S mejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12 , 040501 (2022)

    work page 2022

  3. [3]

    Mazin, and the PRX Editors, Phys

    I. Mazin, and the PRX Editors, Phys. Rev. X 12 , 040002 (2022)

    work page 2022

  4. [4]

    L Bai, W. Feng, S. Liu, L. S mejkal, Y. Mokrousov and Y. Yao, Adv. Funct. Mater. 2409327 (2024)

    work page 2024

  5. [5]

    Jungwirth, R

    T. Jungwirth, R. Fernandes, J. Sinova and L. S Smejkal, arXiv:2409.10034 (2024)

    arXiv 2024

  6. [6]

    S mejkal, R

    L. S mejkal, R. Gonz\' a lez-Hern\' a ndez, T. Jungwirth, and J. Sinova, Sci. Adv. 6 , eaaz8809 (2020)

    work page 2020

  7. [7]

    Mazin, K

    I. Mazin, K. Koepernik, M.D. Johannes, R. Gonz \'a lez-Hern \'a ndez and L. S mejkal, Proc. Natl. Acad. Sci. U.S.A. 118 , e2108924118 (2021)

    work page 2021

  8. [8]

    Z. Feng, X. Zhou, L. S mejkal, L. Wu, Z. Zhu, H. Guo, R. Gonz\' a lez-Hern\' a ndez, X. Wang, H. Yan, P. Qin, X. Zhang, H. Wu, H. Chen, Z. Meng, L. Liu, Z. Xia, J. Sinova, T. Jungwirth, and Z. Liu, Nat. Electron. 5 735 (2022)

    work page 2022

Show all 47 references
  1. [9]

    Reichlova, R

    H. Reichlova, R. Lopes Seeger, R. González-Hernández, I. Kounta, R. Schlitz, D. Kriegner, P. Ritzinger, M. Lammel, M. Leivisk\" a , A.B. Hellenes, K. Olejník, V. Pet r i c ek, P. Dole z al, L. Horak, E. Schmoranzerova, A. Badura, S. Bertaina, A. Thomas, V. Baltz, L. Michez, J....

    2024

  2. [10]

    M. Naka, S. Hayami, H. Kusunose. Y. Yanagi, Y. Motome, and H. Seo, Nat. Comms. 10 , 4305 (2019)

    2019

  3. [11]

    S mejkal, A

    L. S mejkal, A. Marmodoro, K.-H. Ahn, R. González-Hern\' a ndez, I. Turek, S. Mankovsky, H. Ebert, S.W. D’Souza, O. S ipr, J. Sinova and T. Jungwirth, Phys. Rev. Lett. 131 , 256703 (2023)

    2023

  4. [12]

    Krempask\' y , L

    J. Krempask\' y , L. S mejkal, S.W. D’Souza, M. Hajlaoui, G. Springholz, K. Uhl\' i r ov\' a , F. Alarab, P. C. Constantinou, V. Strocov, D. Usanov, W. R. Pudelko, R. Gonz\' a lez-Hern\' a ndez, A. Birk Hellenes, Z. Jansa, H. Reichlov\' a , Z. S ob\' a n , R.D. Gonzalez Betanc...

    2024

  5. [13]

    Reimers, L

    S. Reimers, L. Odenbreit, L. S mejkal, V.N. Strocov, P. Constantinou, A.B. Hellenes, R. Jaeschke Ubiergo, W.H. Campos, V.K. Bharadwaj, A. Chakraborty, T. Denneulin, W. Shi, R.E. Dunin-Borkowski, S. Das, M. Kläui, J. Sinova and M. Jourdan, Nat. Comms. 15 , 2116 (2024)

    2024

  6. [14]

    S. Lee, S. Lee, S. Jung, J. Jung, D. Kim, Y. Lee, B. Seok, J. Kim, B.G. Park, L. S mejkal, C.-J. Kang and C. Kim, Phys. Rev. Lett. 132 , 036702 (2024)

    2024

  7. [15]

    Osumi, S

    T. Osumi, S. Souma, T. Aoyama, K. Yamauchi, A. Honma, K. Nakayama, T. Takahashi, K. Ohgushi and T. Sato, Phys. Rev. B 109 , 115102 (2024)

    2024

  8. [16]

    G. Yang, Z. Li, S. Yang, J. Li, H. Zheng, W. Zhu, S. Cao, W. Zhao, J. Zhang, M. Ye, Y. Song, L.-H. Hu, L. Yang, M. Shi, H. Yuan, Y. Zhang, Y. Xu and Y. Liu, arXiv:2405.12575 (2024)

    2024 arXiv

  9. [17]

    J. Ding, Z. Jiang, X. Chen, Z. Tao, Z. Liu, T. Li, J. Liu, J. Sun, J. Cheng, J. Liu, Y. Yang, R. Zhang, L. Deng, W. Jing, Y. Huang, Y. Shi, M. Ye, S. Qiao, Y. Wang, Y. Guo, D. Feng, and D. Shen, Phys. Rev. Lett. 133 , 206401 (2024)

    2024

  10. [18]

    Zeng, M.-Y

    M. Zeng, M.-Y. Zhu, Y.-P. Zhu, X.-R. Liu, X.-M. Ma, Y.-J. Hao, P. Liu, G. Qu, Y. Yang, Z. Jiang, K. Yamagami, M. Arita, X. Zhang, T.-H. Shao, Y. Dai, K. Shimada, Z. Liu, M. Ye, Y. Huang, Q. Liu and Chang Liu, Adv. Sci. 2406529 (2024)

    2024

  11. [19]

    C. Li, M. Hu, Z. Li, Y. Wang, W. Chen, B. Thiagarajan, M. Leandersson, C. Polley, T. Kim, H. Liu, C. Fulga, M.G. Vergniory, O. Janson, O. Tjernberg and J. van den Brink, arXiv:2405.14777 (2024)

    2024 arXiv

  12. [20]

    Fedchenko, J

    O. Fedchenko, J. Minar, A. Akashdeep, S.W. D'Souza, D. Vasilyev, O. Tkach, L. Odenbreit, Q.L. Nguyen, D. Kutnyakhov, N. Wind, L. Wenthaus, M. Scholz, K. Rossnagel, M. Hoesch, M. Aeschlimann, B. Stadtmueller, M. Klaeui, G. Schoenhense, G. Jakob, T. Jungwirth, L. Smejkal, J. Sin...

    2024

  13. [21]

    Gonz\' a lez-Hern\' a ndez, L

    R. Gonz\' a lez-Hern\' a ndez, L. S mejkal, K. V\' y born\' y , Y. Yahagi, J. Sinova, T. Jungwirth, and J. Z elezn\' y , Phys. Rev. Lett. 126 , 127701 (2021)

    2021

  14. [22]

    Bose, N.J

    A. Bose, N.J. Schreiber, R. Jain, D. Shao, H.P. Nair, J. Sun, X.S. Zhang, D.A. Muller, E.Y. Tsymbal, D.G. Schlom, and D.C. Ralph, Nat. Electron. 5 , 267 (2022)

    2022

  15. [23]

    S mejkal, A.B

    L. S mejkal, A.B. Hellenes, R. Gonz \'a lez-Hern \'a ndez, J. Sinova and T. Jungwirth, Phys. Rev. X 12 , 011028 (2022)

    2022

  16. [24]

    Shao, S.-H

    D.-F. Shao, S.-H. Zhang, M. Li, C.-B. Eom and E.Y. Tsymbal, Nat. Comms. 12 , 7061 (2021)

    2021

  17. [25]

    S mejkal, A

    L. S mejkal, A. MacDonald, J. Sinova, S. Nakatsuji, and T. Jungwirth, Nat. Rev. Mater. 7 , 482 (2022)

    2022

  18. [26]

    H. Bai, L. Han, X.Y. Feng, Y.J. Zhou, R.X. Su, Q. Wang, L.Y. Liao, W.X. Zhu, X.Z. Chen, F. Pan, X.L. Fan, and C. Song, Phys. Rev. Lett. 128 , 197202 (2022)

    2022

  19. [27]

    Karube, T

    S. Karube, T. Tanaka, D. Sugawara, N. Kadoguchi, M. Kohda, and J. Nitta, Phys. Rev. Lett. 129 , 137201 (2022)

    2022

  20. [28]

    Gonzalez Betancourt, J

    R.D. Gonzalez Betancourt, J. Zub\' a c , R.J. Gonzalez-Hernandez, K. Geishendorf, Z. S ob\' a , G. Springholz, K. V\' y born\' y , K. Olejn\' i k, L. S mejkal, J. Sinova, T. Jungwirth, S.T.B. Goennenwein, A. Thomas, H. Reichlov\' a , J. Z elezn\' y , and D. Kriegner, Phys. Rev...

    2023

  21. [29]

    Gomonay, V.P

    O. Gomonay, V.P. Kravchuk, R. Jaeschke-Ubiergo, K.V. Yershov, T. Jungwirth, L. S mejkal, J. van den Brink and J. Sinova, npj Spintronics 2 , 35 (2024)

    2024

  22. [30]

    Z. Jin, Z. Zeng, Y. Cao and P. Yan, arXiv:2407.03959 (2024)

    2024 arXiv

  23. [31]

    Kotliar and A

    G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. 57 , 1362 (1986)

    1986

  24. [32]

    T. Li, P. W\" o lfe and P. J. Hirshfeld, Phys. Rev. B 40 , 6817 (1989)

    1989

  25. [33]

    Fr\' e sard, M

    R. Fr\' e sard, M. Dzierawa and P. W\" o lfe, Europhys. Lett. 15 , 325 (1991)

    1991

  26. [34]

    Fr\' e sard and P

    R. Fr\' e sard and P. W\" o lfe, J. Condens. Matter 4 , 3625 (1992)

    1992

  27. [35]

    In other words, these off-diagonal terms cost more energy due to the presence of phases of the form e^ i( k _ - k _ ) r 0 , and therefore are neglected in the low-energy long-wavelength limit

  28. [36]

    Grimmer-ACSA1993

    which is the case of the rutile crystal structure, see Ref. Grimmer-ACSA1993

  29. [37]

    Grimmer, Acta Crystallogr

    H. Grimmer, Acta Crystallogr. Sect. A 49 , 763 (1993)

    1993

  30. [38]

    Gomonay, V.P

    O. Gomonay, V.P. Kravchuk, R. Jaeschke-Ubiergo, K.V. Yershov, T. Jungwirth, L. S mejkal, J. van den Brink, and J. Sinova. npj Spintronics 2 , 35 (2024)

    2024

  31. [39]

    Zarzuela and J

    R. Zarzuela and J. Sinova, Phys. Rev. B 105 , 024423 (2022)

    2022

  32. [40]

    B. I. Shraiman and E. D. Siggia, Pys. Rev. Lett. 61 , 467 (1988)

    1988

  33. [41]

    Jin, Y.-Q

    P.-Q. Jin, Y.-Q. Li and F.-C. Zhang, J. Phys. A: Math. Gen. 39 , 7115 (2006)

    2006

  34. [42]

    Baltz, A

    V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono and Y. Tserkovnyak. Rev. Mod. Phys. 90 , 015005 (2018)

    2018

  35. [43]

    Shraiman and E.D

    B.I. Shraiman and E.D. Siggia, Phys. Rev. B 42 , 2485 (1990)

    1990

  36. [44]

    Kim, K.-J

    S.K. Kim, K.-J. Lee and Y. Tserkovnyak, Phys. Rev. B 95 , 140404(R) (2017)

    2017

  37. [45]

    Zarzuela, S.K

    R. Zarzuela, S.K. Kim and Y. Tserkovnyak, Phys. Rev. B 97 , 014418 (2018)

    2018

  38. [46]

    Kokkeler, I.V

    T. Kokkeler, I.V. Tokatly and F.S. Bergeret, arXiv:2412.10236 (2024)

    2024 arXiv

  39. [47]

    Vakili, E

    H. Vakili, E. Schwartz and A.A. Kovalev, arXiv:2412.11274 (2024)

    2024 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.