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arxiv: 2605.16830 · v1 · pith:AWRWW33Cnew · submitted 2026-05-16 · ❄️ cond-mat.other · cond-mat.mtrl-sci

Relativistic theory for coupled orbital and spin angular momentum dynamics in magnetic systems

Subhadip Santra , Ritwik Mondal , Marco Berritta , Peter M. Oppeneer This is my paper

Pith reviewed 2026-05-19 19:25 UTC · model grok-4.3

classification ❄️ cond-mat.other cond-mat.mtrl-sci
keywords relativistic angular momentum dynamicsspin and orbital momentsHeisenberg exchange approximationDirac-Kohn-Sham Hamiltonianelectromagnetic fields in magnetismconservation of total angular momentumcoupled spin-orbit dynamicsultrafast magnetic processes
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The pith

Under the atomistic Heisenberg approximation, total angular momentum J equals S plus L stays conserved in magnetic systems even when electromagnetic fields are applied.

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

The paper constructs a relativistic theory for the joint dynamics of spin and orbital angular momenta by beginning with the Dirac-Kohn-Sham Hamiltonian in an electromagnetic field and performing a unitary transformation to reach an extended Pauli form. It first demonstrates that total J is conserved in the absence of a spin-polarized exchange field but loses conservation once an electromagnetic field such as a laser pulse is introduced. When the atomistic Heisenberg approximation is imposed on the exchange interaction for systems possessing local spin and orbital moments, the coupled equations of motion that follow preserve the sum J while allowing the separate spin and orbital parts to vary. This framework keeps the dynamics consistent with the original relativistic Dirac starting point.

Core claim

Considering magnetic systems with atomic spin and orbital momenta, and making the atomistic Heisenberg approximation for the exchange interaction, the total angular momentum J = S + L remains conserved even in the presence of an electromagnetic field, while the individual atomic spin and orbital angular momenta are not conserved.

What carries the argument

The atomistic Heisenberg approximation for the exchange interaction applied within the coupled equations of motion obtained from the semirelativistic Hamiltonian.

Load-bearing premise

The atomistic Heisenberg approximation for the exchange interaction is sufficient to preserve total angular momentum conservation in the presence of electromagnetic fields.

What would settle it

A calculation or measurement in which the sum of spin and orbital angular momenta changes over time for a Heisenberg-model magnetic system driven by a laser pulse or THz field would falsify the conservation result.

read the original abstract

We develop a complete relativistic theory to describe the dynamics of electronic angular momentum including both spin (S) and orbital (L) contributions in magnetic systems. We start with the relativistic Dirac-Kohn-Sham Hamiltonian under the influence of an electromagnetic field and apply a unitary transformation to formulate the extended Pauli Hamiltonian. Using the transformed semirelativistic Hamiltonian, we derive the angular momentum dynamics for the orbital and spin angular momenta. Thereby, we formulate the coupled dynamics of orbital and spin moments consistent with the relativistic Dirac framework. Considering especially the conservation of the total angular momentum, J = S +L, we show first that J is conserved in the absence of a spin-polarized Kohn-Sham exchange field, but is no longer conserved under the application of an electromagnetic field, e.g., laser pulse, THz field, etc. Second, considering magnetic systems with atomic spin and orbital momenta, we derive the coupled equations of motion of angular momenta dynamics whilst making the atomistic Heisenberg approximation for the exchange interaction. Our results suggest that, under these assumptions, the total angular momentum remains conserved, even with electromagnetic field, but atomic spin and orbital angular momenta individually are not conserved.

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 / 2 minor

Summary. The manuscript develops a relativistic theory for the coupled dynamics of orbital (L) and spin (S) angular momenta in magnetic systems subject to electromagnetic fields. It begins with the Dirac-Kohn-Sham Hamiltonian including an external EM field, applies a unitary transformation to obtain the extended Pauli Hamiltonian, and derives the equations of motion for the angular momenta. The authors first establish that total angular momentum J = S + L is conserved without a spin-polarized Kohn-Sham exchange field but is not conserved in the presence of an EM field. They then consider atomistic magnetic systems, invoke the Heisenberg approximation for the exchange interaction, and conclude that total J remains conserved even with EM fields (e.g., laser or THz pulses), while the individual atomic spin and orbital momenta are not conserved.

Significance. If the central conservation result holds, the work supplies a systematic relativistic framework for modeling angular-momentum dynamics in ultrafast magnetism, linking the Dirac equation to atomistic spin dynamics. A clear strength is the derivation from the transformed semirelativistic Hamiltonian rather than phenomenological insertion of torques. The recovery of total-J conservation under the Heisenberg approximation, if demonstrated rigorously, would be useful for interpreting experiments on laser-driven spin-orbit effects and orbital angular-momentum transfer.

major comments (1)
  1. [derived equations section] § on derived equations under Heisenberg approximation: the manuscript must show explicitly that the sum over atomic sites i of all electromagnetic-field contributions (orbital Zeeman, spin-orbit, and diamagnetic terms in the extended Pauli Hamiltonian) to d(S_i + L_i)/dt vanishes identically, independent of the specific values of the exchange parameters J_ij. The Heisenberg form cancels internal exchange torques by construction, but the field-induced torques require an explicit cancellation proof, particularly for spatially inhomogeneous fields.
minor comments (2)
  1. [Hamiltonian transformation section] The notation for the unitary transformation and the resulting extended Pauli Hamiltonian should include an explicit listing of all retained terms up to the order considered, to facilitate verification of the subsequent torque derivations.
  2. [results section] A brief comparison table or paragraph contrasting the conservation properties with and without the Heisenberg approximation would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its potential significance in providing a relativistic framework for angular-momentum dynamics. We have addressed the single major comment in detail below.

read point-by-point responses

  1. Referee: [derived equations section] § on derived equations under Heisenberg approximation: the manuscript must show explicitly that the sum over atomic sites i of all electromagnetic-field contributions (orbital Zeeman, spin-orbit, and diamagnetic terms in the extended Pauli Hamiltonian) to d(S_i + L_i)/dt vanishes identically, independent of the specific values of the exchange parameters J_ij. The Heisenberg form cancels internal exchange torques by construction, but the field-induced torques require an explicit cancellation proof, particularly for spatially inhomogeneous fields.

    Authors: We agree that an explicit demonstration of the cancellation is necessary for rigor. Our derivation starts from the extended Pauli Hamiltonian obtained via unitary transformation of the Dirac-Kohn-Sham equation in an external electromagnetic field. Under the atomistic Heisenberg approximation, the exchange terms cancel internally by construction when summed over sites. For the electromagnetic contributions, the orbital Zeeman, spin-orbit, and diamagnetic terms act on the total electronic angular momentum in a manner that preserves the overall conservation law when summed, because they derive from the minimal-coupling interaction with the vector potential and scalar potential in the underlying relativistic Hamiltonian. However, we acknowledge that the current manuscript presents this cancellation at the level of the equations of motion without expanding every term site-by-site. In the revised manuscript we will add a dedicated appendix that explicitly computes the sum over i of each field-induced torque on (S_i + L_i), retaining the full spatial dependence of the fields to cover inhomogeneous cases such as focused laser pulses. This calculation will be shown to vanish identically and independently of the values of J_ij. revision: yes

Circularity Check

0 steps flagged

Derivation from Dirac-Kohn-Sham Hamiltonian and Heisenberg approximation is self-contained with no circular reductions

full rationale

The paper begins with the relativistic Dirac-Kohn-Sham Hamiltonian in an electromagnetic field, performs a unitary transformation to the extended Pauli Hamiltonian, and derives the equations of motion for orbital and spin angular momenta. Conservation of total J = S + L is shown first without spin-polarized exchange and then recovered under the explicit atomistic Heisenberg approximation for exchange interactions. These steps are presented as direct consequences of the transformed Hamiltonian and the stated approximation rather than being imposed by definition, fitted to prior results, or justified solely via self-citation chains. No load-bearing self-citations, ansatzes smuggled through citations, or renamings of known results are required for the central claims. The derivation remains independent of the target conservation statements and is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on the standard relativistic Dirac equation, the Kohn-Sham framework, and the Heisenberg approximation; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Dirac-Kohn-Sham Hamiltonian correctly describes electrons in the presence of electromagnetic fields and exchange interactions.
    Invoked at the start of the derivation in the abstract.
  • standard math A unitary transformation exists that yields a valid extended Pauli Hamiltonian while preserving the essential relativistic physics.
    Used to move from Dirac to semirelativistic form.

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Reference graph

Works this paper leans on

118 extracted references · 118 canonical work pages

  1. [1]

    traditional

    Orbital angular momentum dynamics The dynamics of the orbital angular momentum is gov- erned by the Heisenberg equation dL dt = 1 iℏ [L,H 0 +H int].(7) Its explicit form is obtained by making use of Eqs. (4) and (7). The calculation is simplified by the fact that the ki- netic energy commutes with the orbital angular momen- tumL. Approximating the crystal...

    work page

  2. [2]

    traditional

    Spin angular momentum dynamics The Heisenberg equation for the spin dynamics in ab- sence of external fields using the unperturbed Hamilto- 5 nian [viz.Eq. (4)] is calculated as dS dt 0 =− 1 2m2c2 drV r S×L.(10) This represents the transfer of angular momentum from spin to orbital degrees of freedom, cf. Eq. (8), keeping the total angular momentumJ=L+Scon...

    work page

  3. [3]

    Total angular momentum dynamics The total magnetic moment is given byM= (L+ gsS)(µB/ℏ) whereg s is the sping-factor. The value of gs for a Dirac point particle is usually considered to have a value of 2, however, quantum electrodynamics predicts the value to be slightly larger than 2 because of vacuum fluctuations and polarizations [94]. The total angular...

    work page

  4. [4]

    (2), specifically in the terms Hxc =−µ B σ·B xc eff + iµB 4m2c2 (p×B xc)·(p−eA)

    General magnetic exchange field The effects of the exchange interaction and its relativis- tic corrections are incorporated into the Hamiltonian in Eq. (2), specifically in the terms Hxc =−µ B σ·B xc eff + iµB 4m2c2 (p×B xc)·(p−eA). (17) In the first, Zeeman-like term, the effective exchange field is given by Bxc eff =B xc +B xc corr,(18) whereB xc repres...

    work page

  5. [5]

    The orbital angular moment is hence taken as a local, atom-centered quantity (and not as an extended itinerant quantity [96])

    Atomic Heisenberg exchange interaction We consider a magnetic system and make an atomistic approximation. The orbital angular moment is hence taken as a local, atom-centered quantity (and not as an extended itinerant quantity [96]). Exchange interactions have successfully been modeled by the most general bi- linear form of magnetic exchange, given by [93]...

    work page

  6. [6]

    Spatial dependence of the Heisenberg exchange interaction In magnetic materials, the exchange interaction is a function of the distance between the neighboring atomic sites. There have been several attempts to compute the exchange integral from first-principles calculations, either using a local spin-density functional theory ap- proach based on multiple ...

    work page arXiv 2022

  7. [7]

    Beaurepaire, J.-C

    E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Ultrafast spin dynamics in ferromagnetic nickel, Phys. Rev. Lett.76, 4250 (1996)

    work page 1996

  8. [8]

    Kirilyuk, A

    A. Kirilyuk, A. V. Kimel, and T. Rasing, Ultrafast op- tical manipulation of magnetic order, Rev. Mod. Phys. 82, 2731 (2010)

    work page 2010

  9. [9]

    Hohlfeld, E

    J. Hohlfeld, E. Matthias, R. Knorren, and K. H. Ben- nemann, Nonequilibrium magnetization dynamics of nickel, Phys. Rev. Lett.78, 4861 (1997)

    work page 1997

  10. [10]

    Scholl, L

    A. Scholl, L. Baumgarten, R. Jacquemin, and W. Eber- hardt, Ultrafast spin dynamics of ferromagnetic thin films observed by fs spin-resolved two-photon photoe- mission, Phys. Rev. Lett.79, 5146 (1997)

    work page 1997

  11. [11]

    Kampfrath, R

    T. Kampfrath, R. G. Ulbrich, F. Leuenberger, M. M¨ unzenberg, B. Sass, and W. Felsch, Ultrafast magneto-optical response of iron thin films, Phys. Rev. B65, 104429 (2002)

    work page 2002

  12. [12]

    St¨ ohr and H

    J. St¨ ohr and H. C. Siegmann, Ultrafast magnetiza- tion dynamics, inMagnetism: From Fundamentals to Nanoscale Dynamics(Springer-Verlag, Berlin Heidel- berg, 2006)

    work page 2006

  13. [13]

    Bigot, M

    J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Coherent ul- trafast magnetism induced by femtosecond laser pulses, Nat. Phys.5, 515 (2009)

    work page 2009

  14. [14]

    Carley, K

    R. Carley, K. D¨ obrich, B. Frietsch, C. Gahl, M. Te- ichmann, O. Schwarzkopf, P. Wernet, and M. Weinelt, Femtosecond laser excitation drives ferromagnetic gadolinium out of magnetic equilibrium, Phys. Rev. Lett.109, 057401 (2012)

    work page 2012

  15. [15]

    C. E. Graves, A. H. Reid, T. Wang, B. Wu, S. de Jong, K. Vahaplar, I. Radu, D. P. Bernstein, M. Messer- schmidt, L. M¨ uller, R. Coffee, M. Bionta, S. W. Epp, R. Hartmann, N. Kimmel, G. Hauser, A. Hartmann, P. Holl, H. Gorke, J. H. Mentink, A. Tsukamoto, A. Fognini, J. J. Turner, W. F. Schlotter, D. Rolles, H. Soltau, L. Str¨ uder, Y. Acremann, A. V. Kime...

    work page 2013

  16. [16]

    Rudolf, C

    D. Rudolf, C. La-O-Vorakiat, M. Battiato, R. Adam, J. M. Shaw, E. Turgut, P. Maldonado, S. Mathias, P. Grychtol, H. T. Nembach, T. J. Silva, M. Aeschli- mann, H. C. Kapteyn, M. M. Murnane, C. M. Schnei- der, and P. M. Oppeneer, Ultrafast magnetization en- hancement in metallic multilayers driven by superdiffu- sive spin current, Nat. Comm.3, 1037 (2012)

    work page 2012

  17. [17]

    Eschenlohr, M

    A. Eschenlohr, M. Battiato, P. Maldonado, N. Pontius, T. Kachel, K. Holldack, R. Mitzner, A. F¨ ohlisch, P. M. Oppeneer, and C. Stamm, Ultrafast spin transport as key to femtosecond demagnetization, Nat. Mater.12, 332 (2013)

    work page 2013

  18. [18]

    I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. D¨ urr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, and A. V. Kimel, Transient ferromagnetic- like state mediating ultrafast reversal of antiferromag- netically coupled spins, Nature472, 205 (2011)

    work page 2011

  19. [19]

    G. P. Zhang and W. H¨ ubner, Laser-induced ultrafast de- magnetization in ferromagnetic metals, Phys. Rev. Lett. 85, 3025 (2000)

    work page 2000

  20. [20]

    Carpene, E

    E. Carpene, E. Mancini, C. Dallera, M. Brenna, E. Pup- pin, and S. De Silvestri, Dynamics of electron-magnon interaction and ultrafast demagnetization in thin iron films, Phys. Rev. B78, 174422 (2008)

    work page 2008

  21. [21]

    Battiato, K

    M. Battiato, K. Carva, and P. M. Oppeneer, Superdiffu- sive spin transport as a mechanism of ultrafast demag- netization, Phys. Rev. Lett.105, 027203 (2010)

    work page 2010

  22. [22]

    Battiato, K

    M. Battiato, K. Carva, and P. M. Oppeneer, Theory of laser-induced ultrafast superdiffusive spin transport 12 in layered heterostructures, Phys. Rev. B86, 024404 (2012)

    work page 2012

  23. [23]

    Carva, M

    K. Carva, M. Battiato, and P. M. Oppeneer, Ab initio investigation of the elliott-yafet electron-phonon mech- anism in laser-induced ultrafast demagnetization, Phys. Rev. Lett.107, 207201 (2011)

    work page 2011

  24. [24]

    Carva, M

    K. Carva, M. Battiato, D. Legut, and P. M. Oppe- neer,Ab initiotheory of electron-phonon mediated ul- trafast spin relaxation of laser-excited hot electrons in transition-metal ferromagnets, Phys. Rev. B87, 184425 (2013)

    work page 2013

  25. [25]

    P. M. Oppeneer and A. Liebsch, Ultrafast demagnetiza- tion in ni: theory of magneto-optics for non-equilibrium electron distributions, J. Phys.: Condens. Matter16, 5519 (2004)

    work page 2004

  26. [26]

    Krieger, J

    K. Krieger, J. K. Dewhurst, P. Elliott, S. Sharma, and E. K. U. Gross, Laser-induced demagnetization at ul- trashort time scales: Predictions of TDDFT, J. Chem. Theory Comput.11, 4870 (2015)

    work page 2015

  27. [27]

    Koopmans, G

    B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. F¨ ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann, Explaining the paradoxical diversity of ultrafast laser-induced demagnetization, Nat. Mater. 9, 259 (2010)

    work page 2010

  28. [28]

    Hofherr, P

    M. Hofherr, P. Maldonado, O. Schmitt, M. Berritta, U. Bierbrauer, S. Sadashivaiah, A. J. Schellekens, B. Koopmans, D. Steil, M. Cinchetti, B. Stadtm¨ uller, P. M. Oppeneer, S. Mathias, and M. Aeschlimann, Speed and efficiency of femtosecond spin current in- jection into a nonmagnetic material, Phys. Rev. B96, 100403 (2017)

    work page 2017

  29. [29]

    Griepe and U

    T. Griepe and U. Atxitia, Evidence of electron-phonon mediated spin flip as driving mechanism for ultrafast magnetization dynamics in 3d ferromagnets, Phys. Rev. B107, L100407 (2023)

    work page 2023

  30. [30]

    S. R. Tauchert, M. Volkov, D. Ehberger, D. Kazenwadel, M. Evers, H. Lange, A. Donges, A. Book, W. Kreuz- paintner, U. Nowak, and P. Baum, Polarized phonons carry angular momentum in ultrafast demagnetization, Nature602, 73 (2022)

    work page 2022

  31. [31]

    Weißenhofer and P

    M. Weißenhofer and P. M. Oppeneer, Ultrafast de- magnetization through femtosecond generation of non- thermal magnons, Adv. Phys. Res.4, 2300103 (2025)

    work page 2025

  32. [32]

    Boeglin, E

    C. Boeglin, E. Beaurepaire, V. Halt´ e, V. L´ opez-Flores, C. Stamm, N. Pontius, H. A. D¨ urr, and J.-Y. Bigot, Distinguishing the ultrafast dynamics of spin and orbital moments in solids, Nature465, 458 (2010)

    work page 2010

  33. [33]

    F¨ ahnle, M

    M. F¨ ahnle, M. Haag, and C. Illg, Is the angular mo- mentum of a ferromagnetic sample after exposure to a fs laser pulse conserved?, J. Magn. Magn. Mater.347, 45 (2013)

    work page 2013

  34. [34]

    Dornes, Y

    C. Dornes, Y. Acremann, M. Savoini, M. Kubli, M. J. Neugebauer, E. Abreu, L. Huber, G. Lantz, C. A. F. Vaz, H. Lemke, E. M. Bothschafter, M. Porer, V. Es- posito, L. Rettig, M. Buzzi, A. Alberca, Y. W. Windsor, P. Beaud, U. Staub, D. Zhu, S. Song, M. Glownia, and S. L. Johnson, The ultrafast Einstein–de Haas effect, Nature565, 209 (2019)

    work page 2019

  35. [35]

    Beaurepaire, G

    E. Beaurepaire, G. M. Turner, S. M. Harrel, M. C. Beard, J.-Y. Bigot, and C. A. Schmuttenmaer, Coher- ent terahertz emission from ferromagnetic films excited by femtosecond laser pulses, Appl. Phys. Lett.84, 3465 (2004)

    work page 2004

  36. [36]

    T. J. Huisman, R. V. Mikhaylovskiy, A. Tsukamoto, Th. Rasing, and A. V. Kimel, Simultaneous measure- ments of terahertz emission and magneto-optical kerr effect for resolving ultrafast laser-induced demagnetiza- tion dynamics, Phys. Rev. B92, 104419 (2015)

    work page 2015

  37. [37]

    B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Orbi- tronics: The intrinsic orbital current inp-doped silicon, Phys. Rev. Lett.95, 066601 (2005)

    work page 2005

  38. [38]

    Tanaka, H

    T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hi- rashima, K. Yamada, and J. Inoue, Intrinsic spin Hall effect and orbital Hall effect in 4dand 5dtransition met- als, Phys. Rev. B77, 165117 (2008)

    work page 2008

  39. [39]

    D. Go, D. Jo, C. Kim, and H.-W. Lee, Intrinsic Spin and Orbital Hall Effects from Orbital Texture, Phys. Rev. Lett.121, 086602 (2018)

    work page 2018

  40. [40]

    Salemi, M

    L. Salemi, M. Berritta, A. K. Nandy, and P. M. Op- peneer, Orbitally dominated Rashba-Edelstein effect in noncentrosymmetric antiferromagnets, Nature Com- mun.10, 5381 (2019)

    work page 2019

  41. [41]

    D. Go, F. Freimuth, J.-P. Hanke, F. Xue, O. Gomonay, K.-J. Lee, S. Bl¨ ugel, P. M. Haney, H.-W. Lee, and Y. Mokrousov, Theory of current-induced angular mo- mentum transfer dynamics in spin-orbit coupled sys- tems, Phys. Rev. Res.2, 033401 (2020)

    work page 2020

  42. [42]

    Go and H.-W

    D. Go and H.-W. Lee, Orbital torque: Torque gener- ation by orbital current injection, Phys. Rev. Res.2, 013177 (2020)

    work page 2020

  43. [43]

    Salemi and P

    L. Salemi and P. M. Oppeneer, First-principles theory of intrinsic spin and orbital hall and nernst effects in metal- lic monoatomic crystals, Phys. Rev. Mater.6, 095001 (2022)

    work page 2022

  44. [44]

    D. Jo, D. Go, G.-M. Choi, and H.-W. Lee, Spintronics meets orbitronics: Emergence of orbital angular mo- mentum in solids, npj Spintronics2, 19 (2024)

    work page 2024

  45. [45]

    D. Lee, D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun, D. Jo, S. Lee, G. Go, J. H. Oh, K.-J. Kim, B.-G. Park, B.-C. Min, H. C. Koo, H.-W. Lee, O. Lee, and K.-J. Lee, Orbital torque in magnetic bilayers, Nature Commun. 12, 6710 (2021)

    work page 2021

  46. [46]

    X. Ning, A. Pezo, K.-W. Kim, W. Zhao, K.-J. Lee, and A. Manchon, Orbital diffusion, polarization, and swap- ping in centrosymmetric metals, Phys. Rev. Lett.134, 026303 (2025)

    work page 2025

  47. [47]

    D. Go, D. Jo, K.-W. Kim, S. Lee, M.-G. Kang, B.-G. Park, S. Bl¨ ugel, H.-W. Lee, and Y. Mokrousov, Long- range orbital torque by momentum-space hotspots, Phys. Rev. Lett.130, 246701 (2023)

    work page 2023

  48. [48]

    Berritta, R

    M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer, Ab initio theory of coherent laser-induced magnetiza- tion in metals, Phys. Rev. Lett.117, 137203 (2016)

    work page 2016

  49. [49]

    Y.-G. Choi, D. Jo, K.-H. Ko, D. Go, K.-H. Kim, H. G. Park, C. Kim, B.-C. Min, G.-M. Choi, and H.-W. Lee, Observation of the orbital Hall effect in a light metal Ti, Nature619, 52 (2023)

    work page 2023

  50. [50]

    Lyalin, S

    I. Lyalin, S. Alikhah, M. Berritta, P. M. Oppeneer, and R. K. Kawakami, Magneto-optical detection of the or- bital hall effect in chromium, Phys. Rev. Lett.131, 156702 (2023)

    work page 2023

  51. [51]

    P. Wang, F. Chen, Y. Yang, S. Hu, Y. Li, W. Wang, D. Zhang, and Y. Jiang, Orbitronics: Mechanisms, Ma- terials and Devices, Adv. Electr. Mater.11, 2400554 (2025)

    work page 2025

  52. [52]

    S. Ding, A. Ross, D. Go, L. Baldrati, Z. Ren, F. Freimuth, S. Becker, F. Kammerbauer, J. Yang, G. Jakob, Y. Mokrousov, and M. Kl¨ aui, Harnessing 13 orbital-to-spin conversion of interfacial orbital currents for efficient spin-orbit torques, Phys. Rev. Lett.125, 177201 (2020)

    work page 2020

  53. [53]

    S. A. Nikolaev, M. Chshiev, F. Ibrahim, S. Krishnia, N. Sebe, J.-M. George, V. Cros, H. Jaffr` es, and A. Fert, Large Chiral Orbital Texture and Orbital Edelstein Ef- fect in Co/Al Heterostructure, Nano Lett.24, 13465 (2024)

    work page 2024

  54. [54]

    L. D. Landau and E. M. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion8, 101 (1935)

    work page 1935

  55. [55]

    T. L. Gilbert and J. M. Kelly, Anomalous rotational damping in ferromagnetic sheets, inAmerican Institute of Electrical Engineers(New York, October 1955) pp. 253–263

    work page 1955

  56. [56]

    Kazantseva, U

    N. Kazantseva, U. Nowak, R. W. Chantrell, J. Hohlfeld, and A. Rebei, Slow recovery of the magnetisation after a sub-picosecond heat pulse, Europhys. Lett.81, 27004 (2007)

    work page 2007

  57. [57]

    Vahaplar, A

    K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, D. Hinzke, U. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, Ultrafast path for optical magnetization reversal via a strongly nonequi- librium state, Phys. Rev. Lett.103, 117201 (2009)

    work page 2009

  58. [58]

    Skubic, J

    B. Skubic, J. Hellsvik, L. Nordstr¨ om, and O. Eriksson, A method for atomistic spin dynamics simulations: im- plementation and examples, J. Phys.: Condens. Matter 20, 315203 (2008)

    work page 2008

  59. [59]

    Ciornei, J

    M.-C. Ciornei, J. M. Rub´ ı, and J.-E. Wegrowe, Mag- netization dynamics in the inertial regime: Nutation predicted at short time scales, Phys. Rev. B83, 020410 (2011)

    work page 2011

  60. [60]

    Wegrowe and M.-C

    J.-E. Wegrowe and M.-C. Ciornei, Magnetization dy- namics, gyromagnetic relation, and inertial effects, Am. J. Phys.80, 607 (2012)

    work page 2012

  61. [61]

    Mondal, M

    R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppe- neer, Relativistic theory of magnetic inertia in ultrafast spin dynamics, Phys. Rev. B96, 024425 (2017)

    work page 2017

  62. [62]

    Mondal, S

    R. Mondal, S. Großenbach, L. R´ ozsa, and U. Nowak, Nutation in antiferromagnetic resonance, Phys. Rev. B 103, 104404 (2021)

    work page 2021

  63. [63]

    Mondal and P

    R. Mondal and P. M. Oppeneer, Influence of intersub- lattice coupling on the terahertz nutation spin dynamics in antiferromagnets, Phys. Rev. B104, 104405 (2021)

    work page 2021

  64. [64]

    Mondal, L

    R. Mondal, L. R´ ozsa, M. Farle, P. M. Oppeneer, U. Nowak, and M. Cherkasskii, Inertial effects in ultra- fast spin dynamic, J. Magn. Magn. Mater.579, 170830 (2023)

    work page 2023

  65. [65]

    Jensen and A

    J. Jensen and A. R. Mackintosh,Rare Earth Magnetism (Clarendon Press, Oxford, 1991)

    work page 1991

  66. [66]

    Frietsch, A

    B. Frietsch, A. Donges, R. Carley, M. Teichmann, J. Bowlan, K. D¨ obrich, K. Carva, D. Legut, P. M. Oppe- neer, U. Nowak, and M. Weinelt, The role of ultrafast magnon generation in the magnetization dynamics of rare-earth metals, Science Advances6, eabb1601 (2020)

    work page 2020

  67. [67]

    Satoh, R

    T. Satoh, R. Iida, T. Higuchi, Y. Fujii, A. Koreeda, H. Ueda, T. Shimura, K. Kuroda, V. I. Butrim, and B. A. Ivanov, Excitation of coupled spin–orbit dynam- ics in cobalt oxide by femtosecond laser pulses, Nature Commun.8, 638 (2017)

    work page 2017

  68. [68]

    Stamm, N

    C. Stamm, N. Pontius, T. Kachel, M. Wietstruk, and H. A. D¨ urr, Femtosecond x-ray absorption spectroscopy of spin and orbital angular momentum in photoexcited Ni films during ultrafast demagnetization, Phys. Rev. B 81, 104425 (2010)

    work page 2010

  69. [69]

    T¨ ows and G

    W. T¨ ows and G. M. Pastor, Many-body theory of ultra- fast demagnetization and angular momentum transfer in ferromagnetic transition metals, Phys. Rev. Lett.115, 217204 (2015)

    work page 2015

  70. [70]

    T¨ ows, G

    W. T¨ ows, G. Stegmann, and G. M. Pastor, Spin and orbital symmetry breakings central to the laser-induced ultrafast demagnetization of transition metals, Symme- try15(2023)

    work page 2023

  71. [71]

    Simoni and S

    J. Simoni and S. Sanvito, Conservation of angular mo- mentum in ultrafast spin dynamics, Phys. Rev. B105, 104437 (2022)

    work page 2022

  72. [72]

    Hayashi, D

    H. Hayashi, D. Jo, D. Go, T. Gao, S. Haku, Y. Mokrousov, H.-W. Lee, and K. Ando, Observation of long-range orbital transport and giant orbital torque, Communications Physics6, 32 (2023)

    work page 2023

  73. [73]

    Mondal, M

    R. Mondal, M. Berritta, and P. M. Oppeneer, Relativis- tic theory of spin relaxation mechanisms in the Landau- Lifshitz-Gilbert equation of spin dynamics, Phys. Rev. B94, 144419 (2016)

    work page 2016

  74. [74]

    Mondal, M

    R. Mondal, M. Berritta, and P. M. Oppeneer, Gener- alisation of Gilbert damping and magnetic inertia pa- rameter as a series of higher-order relativistic terms, J. Phys.: Condens. Matter30, 265801 (2018)

    work page 2018

  75. [75]

    Mondal, M

    R. Mondal, M. Berritta, and P. M. Oppeneer, Unified theory of magnetization dynamics with relativistic and nonrelativistic spin torques, Phys. Rev. B98, 214429 (2018)

    work page 2018

  76. [76]

    Mondal and P

    R. Mondal and P. M. Oppeneer, Dynamics of the rel- ativistic electron spin in an electromagnetic field, J. Phys.: Condens. Matter32, 455802 (2020)

    work page 2020

  77. [77]

    Cr´ epieux and P

    A. Cr´ epieux and P. Bruno, Relativistic corrections in magnetic systems, Phys. Rev. B64, 094434 (2001)

    work page 2001

  78. [78]

    A. H. MacDonald and S. H. Vosko, A relativistic density functional formalism, J. Phys. C: Solid State Phys.12, 2977 (1979)

    work page 1979

  79. [79]

    Eschrig and V

    H. Eschrig and V. D. P. Servedio, Relativistic density functional approach to open shells, J. Comput. Chem. 20, 23 (1999)

    work page 1999

  80. [80]

    Mondal, M

    R. Mondal, M. Berritta, K. Carva, and P. M. Oppeneer, Ab initioinvestigation of light-induced relativistic spin- flip effects in magneto-optics, Phys. Rev. B91, 174415 (2015)

    work page 2015

Showing first 80 references.