Anatomy of the modern theory of orbital magnetism from first-principles: term-by-term analysis in the gauge-covariant formalism

Dongwook Go; Hojun Lee; Hyun-Woo Lee; Insu Baek; Mirco Sastges; Yuriy Mokrousov

REVIEW 2 major objections 5 minor 3 cited by

Anatomy of the modern theory of orbital magnetism from first-principles: term-by-term analysis in the gauge-covariant formalism

T0 review · 2 major / 5 minor · reviewed 2026-07-13 · grok-4.5

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

Pith's one-line read Berry-phase terms, not just atomic orbitals, set how large orbital magnetism can get.

desk verdict Solid computational anatomy of modern orbital magnetism that cleanly shows when ACA is enough and when Berry-phase hybridization terms dominate. read the letter →

arxiv 2603.19875 v3 pith:OJ24434K submitted 2026-03-20 cond-mat.mes-hall cond-mat.mtrl-sci

Hojun Lee , Insu Baek , Mirco Sastges , Yuriy Mokrousov , Hyun-Woo Lee , Dongwook Go This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-sci

keywords orbitalmagnetizationmoderntheoryBerryphaseWannierfunctionsatom-centeredapproximationgaugecovariancetransitionmetalsmetaldichalcogenides

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that the full modern (Berry-phase) theory of orbital magnetization can be split into pieces that track how much comes from localized atomic-like motion versus coherent band hybridization. Using a gauge-covariant Wannier formulation, the authors compute every term for d-transition metals, sp metals, and two-dimensional dichalcogenides. In most 3d magnets the atom-centered muffin-tin approximation already recovers the bulk of the modern-theory value because d electrons stay localized; 5d metals, free-electron-like sp metals, and especially valley materials such as MoS2 show large extra contributions from interband hybridization that the atomic approximation misses completely. The practical message is that orbital magnetism can be engineered far beyond the atomic limit by exploiting Berry-phase geometry in the band structure.

What carries the argument

The J-decomposition of the modern-theory orbital magnetization (M = M^(0) + M^(1) + M^(2)) obtained from the gauge-covariant Wannier objects A, B, C together with the occupation-weighted covariant derivative; it isolates atomic-like Wannier self-rotation from band-hybridization contributions while keeping the sum gauge-invariant.

What would settle it

Recompute the modern-theory terms for the same materials with deliberately delocalized or differently projected Wannier bases and check whether M^(0) still tracks the muffin-tin ACA and whether the total remains unchanged; a large residual interstitial contribution inside M^(0) would break the claimed ACA correspondence.

Watch

Extended reading notes

Core claim

When orbital magnetization is evaluated with the gauge-covariant modern theory and decomposed by powers of the Wannier-to-Hamiltonian gauge connection J, the atom-centered approximation equals the leading (J^0) intracell self-rotation term for localized d electrons and therefore captures most of the total moment, while in sp metals and valley TMDs the higher-order hybridization terms dominate and can exceed the atomic value by factors of several.

Load-bearing premise

The claim that the intracell piece of the lowest-order Wannier term is quantitatively the same as the muffin-tin atom-centered approximation when the Wannier functions are chosen to look atomic.

Editorial extensions

If this is right

For ordinary 3d magnets the simpler atom-centered approximation is already a reliable estimate of the full modern-theory orbital magnetization.
In sp metals and TMDs, orbital moments can be many times larger than the atomic limit once Berry-phase hybridization is included.
Valley materials with direct gaps (e.g., MoS2) offer a route to giant, chemically tunable orbital moments without needing strong atomic spin-orbit coupling.
Effective tight-binding models that keep only the J^2 term systematically miss the dominant atomic contribution in localized systems.
Orbitronic device design can target band geometry rather than only atomic orbital character.

Reading between the lines

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

The same J-decomposition should diagnose when orbital Hall or orbital Edelstein calculations based on atom-centered operators become unreliable.
Materials near avoided crossings or van-Hove singularities are natural places to look for hybridization-enhanced orbital responses far above atomic estimates.
If the occupation-weighted covariant derivative operator can be promoted to a true current operator, nonequilibrium orbital transport formulas could inherit the same gauge consistency.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This is a careful, useful paper. What is new is the systematic J-decomposition (M(0)+M(1)+M(2)) of the Lopez gauge-covariant modern theory across a real materials set, plus an explicit occupation-weighted orbital-moment operator (Eqs. 45–48) that recovers the ground-state totals and lets them do band-resolved analysis. The main claim is well supported: for most d metals ACA already captures >70 % of the modern-theory Mz because d states are localized and M(0) dominates; for sp metals and TMDs the hybridization/Berry terms (especially M(2)) take over and can exceed the atomic limit by large factors (hex-Bi peak ~12 imes ACA, MoS2 valleys ~4 imes). They recover known Fe/Co/Ni numbers, check the Středa relation on MoS2, and supply the gauge- and space-selection proofs in the appendices.

The softest link is exactly the one the reader flagged: the quantitative identification of the intracell self-rotation piece of M(0) with muffin-tin ACA. But the paper’s own App. G already bounds it—for the localized cases that underwrite the >70 % claim the discrepancy is only 3–15 % at EF—so the central trend does not rest on a fragile assumption. For the delocalized cases they never claim the equivalence; they report the opposite. Free parameters (U/J, Zeeman field, MT radii, frozen window) are conventional and tabulated; self-citations are present but the numbers in Table I and Figs. 4–10 are independent first-principles results.

This is for people who actually compute orbital moments or design orbitronic materials. It tells you when you can keep using ACA and when you must go to the full modern theory, and it points to gap hybridizations as a practical enhancement route. Math and data look solid. I would send it to referees without hesitation and would cite the material trends and the operator construction myself.

Referee Report

2 major / 5 minor

Summary. The manuscript delivers a systematic first-principles anatomy of the modern (Berry-phase) theory of orbital magnetization, implemented in the gauge-covariant Wannier formalism of Lopez et al. It decomposes the magnetization into self-rotation versus center-of-mass pieces and, more originally, into a J-power series M^(0)+M^(1)+M^(2) that isolates atomic-like Wannier contributions from coherent interband hybridizations. Across d-transition metals, sp metals, and two TMD monolayers the authors compare these terms with the conventional atom-centered (muffin-tin) approximation, recover known experimental and prior theoretical values for Fe/Co/Ni, and show that ACA accounts for the bulk of the modern-theory result when d electrons are localized, while M^(2) (Berry-phase/hybridization) terms dominate and can exceed the atomic limit by large factors in sp metals and at the valleys of 1H-MoS2.

Significance. If the numerical trends hold, the work supplies a practical, gauge-controlled diagnostic that tells the community when the widely used ACA is quantitatively reliable and when Berry-phase enhancements must be retained. The explicit construction of an occupation-weighted orbital-moment operator (Eqs. 44–48), the gauge- and space-selection proofs (Appendices C–D), the tabulated computational parameters (Table II), and the recovery of experimental orbital moments for the 3d ferromagnets are concrete strengths that make the results reproducible and immediately usable for orbitronics materials screening. The demonstration that valley moments in MoS2 and avoided-crossing peaks in Td-WTe2 far exceed the atomic limit points to a concrete materials-design route beyond atomic-orbital control.

major comments (2)

Sec. I B and the ACA–modern-theory comparisons throughout Sec. III: the introduction correctly notes that ACA results can depend on the muffin-tin radius R_μ and that saturation with increasing R_μ must be verified. No such R_μ-dependence test (or statement that the chosen R_MT values already saturate) appears for the materials in Table I / Figs. 4–8. Because the central claim that ACA captures >70 % of the modern-theory magnetization for most d metals rests on these numbers, a short supplementary check for at least Fe, Ni and W would remove residual doubt about the quantitative percentages.
Sec. II F and Appendix G: the identification of the intracell self-rotation piece of M^(0) with the muffin-tin ACA is the interpretive link that lets the authors call M^(0) “atomic.” Appendix G already shows 3–15 % discrepancies for the localized d cases that underwrite the >70 % claim, which is reassuring. The manuscript should, however, state explicitly in the main text (not only in the appendix) that this quantitative equivalence is claimed only for atomic-like Wannier functions of well-localized d states, and that for sp metals and TMDs the residual interstitial/intercell content inside M^(0) itself is large (as their own hex-Bi and MoS2 data already demonstrate). A single sentence of this form would prevent over-reading of the correspondence.

minor comments (5)

Throughout the text (abstract, Sec. III B–D) compound words such as “dtransition,” “spelectrons,” “delectrons,” “1H-MoS2” appear without spaces or hyphens; these are formatting artifacts that should be cleaned for readability.
Fig. 7 caption and the accompanying discussion of (r/r_WS)^3 cite Ref. [100] but do not list the numerical values used for each element; a short table or explicit numbers would make the localization argument fully self-contained.
Eq. (32) and the numerical implementation introduce a finite η = 0.0259 eV; a one-sentence remark on the sensitivity of M^(2) (especially near avoided crossings in WTe2 and the Bi van-Hove peak) to this broadening would be useful.
Table I header “SQA” is never expanded; “spin-quantization axis” should be written out once.
Author name “Mirco Sastges” appears once; confirm spelling against the institutional record.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: modern-theory formulas and Wannier objects are taken from independent literature; numerical ACA vs. modern comparisons and J-decomposition results are independent first-principles evaluations, not forced by definition or self-citation.

full rationale

The paper adopts the gauge-invariant modern-theory expression (Eq. 11, from Ceresoli et al.) and the gauge-covariant Wannier implementation (Lopez et al. PRB 85, 014435) as established external inputs, then evaluates them term-by-term via independent FLAPW+Wannier90 calculations (Fleur, Orbitrans). The J-decomposition (M = M(0)+M(1)+M(2)) and SR/CM split are analytic rearrangements of those formulas (Sec. II F, Eqs. 52–53), not redefinitions of the target observables. The claimed ACA–modern correspondence is not definitional: ACA is the independent muffin-tin integral (Eq. 2), while M_SR(on) is the intracell piece of M(0) extracted from Wannier ABC matrices; the paper verifies their numerical proximity only for localized d cases (App. G, 3–15 % discrepancy) and reports large deviations elsewhere (hex-Bi ~12 imes, MoS2 valleys ~4 imes). Gauge-invariance proofs (Apps. C–D) and recovery of known benchmarks (Table I vs. prior DFT/experiment) are self-contained. Self-citations (own orbitronics papers, Orbitrans code) supply context or implementation but do not underwrite the formulas or the material-class trends. No fitted parameters are relabeled as predictions, no uniqueness theorems are imported from the authors, and no ansatz is smuggled via self-citation. Central numerical claims therefore stand as independent first-principles results.

Assumptions & free parameters 5 free parameters · 4 assumptions · 2 invented entities

The central comparative claims rest on standard DFT/Wannier machinery plus a handful of computational choices (U/J, artificial Zeeman, MT radii, inner windows) that are conventional but free. The gauge-covariant objects and J-decomposition are taken from Lopez et al.; the new operator is a definitional construct. No exotic particles or forces are invented.

free parameters (5)

Hubbard U and J (DFT+U) = see Appendix I
Material-specific values (Fe: 1.2/1.0 eV, Co: 1.6/0.9 eV, Ni: 1.9/1.2 eV, WTe2: 5.5/0 eV) chosen to improve agreement with experiment or open a gap; they shift absolute magnetization numbers.
artificial spin-Zeeman field = 0.544 eV
Fixed at 0.544 eV for all non-magnetic metals to induce orbital magnetism; magnitude is conventional but arbitrary and affects absolute scale.
muffin-tin radii R_MT = Table II
Species-dependent cut-offs that define the ACA integral; results can depend on the choice if wave-functions spill outside.
inner (frozen) energy window = 5–12 eV above EF
Set 5–12 eV above EF; controls which bands remain invariant under Wannierization and therefore which geometric contributions are retained.
smearing temperature and eta = 300 K / 0.0259 eV
T=300 K and eta=0.0259 eV used for occupation and J-matrix regularization; affect peak heights near degeneracies.

assumptions (4)

domain assumption Modern theory of orbital magnetization (Berry-phase formula of Ceresoli/Thonhauser/Vanderbilt/Resta and Lopez et al.) is the correct total orbital magnetization.
Taken as given throughout Sec. II; all comparisons are relative to this formula.
domain assumption DFT (PBE + optional DFT+U) plus FLAPW basis yields sufficiently accurate ground-state projectors and Wannier functions for the materials studied.
Standard electronic-structure assumption; no beyond-DFT validation is performed.
standard math Zero-temperature projector limit (f_nk = 0 or 1) and the occupation-weighted covariant derivative correctly recover the finite-T trace formulas.
Used to define the orbital-moment operator (Eqs. 44–48); justified in Appendices.
domain assumption Inner window placed above EF guarantees gauge and space invariance of the occupied-state contribution.
Proved under that condition in Appendices C–D; if violated the invariance claims fail.

invented entities (2)

occupation-weighted covariant derivative / orbital-moment operator (Eqs. 44–48)
purpose: Allows band-resolved and potentially non-equilibrium evaluation of modern-theory orbital magnetization while remaining gauge-covariant.
Definitional construct introduced by the authors; recovers the known ground-state formula by construction but has no independent experimental handle yet.
J-decomposition of orbital magnetization into M(0)+M(1)+M(2)
purpose: Separates Wannier-basis (atomic-like) contributions from Hamiltonian-induced hybridization contributions for term-by-term analysis.
Borrowed from earlier Berry-curvature numerics but applied here as a physical diagnostic; not independently measurable.

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Pith. "Pith review of Anatomy of the modern theory of orbital magnetism from first-principles: term-by-term analysis in the gauge-covariant formalism." pith.science (2026). https://pith.science/paper/OJ24434K

@misc{pith2026260319875,
  author       = {Pith},
  title        = {Pith review of: Anatomy of the modern theory of orbital magnetism from first-principles: term-by-term analysis in the gauge-covariant formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJ24434K}},
  note         = {Machine review of arXiv:2603.19875}
}

read the original abstract

We present an in-depth analysis of the orbital magnetism by means of the so-called modern theory based on the Berry phase across distinct classes of materials-d transition metals, sp metals, and transition metal dichalcogenides-highlighting the microscopic nature of band structure characteristics. We adopt a gauge-covariant formulation of the modern theory proposed in [Lopez et al. Phys. Rev. B 85, 014435 (2012)], which enables the calculation of orbital magnetism in a controlled manner in any chosen gauge of Wannier functions and gives the total contribution as a gauge-invariant measurable. This captures consistently the contributions due to the anomalous position, velocity, and orbital angular momentum of Wannier basis, as well as the contributions due to Hamiltonian such that their sum is gauge-invariant. For d transition metals, we find that the atom-centered approximation captures the majority of the total contribution given by modern theory, which we attribute to localized nature of d electrons. However, 5d metals tend to exhibit larger deviation between the two methods than 3d metals do, as 5d electrons are more delocalized than 3d electrons. On the other hand, sp metals exhibit a strong deviation between the two methods, where large kinetic energy of sp electrons is important. Finally, in 1H-MoS2, we find that the valley orbital moment far exceeds the atomic limit of d electrons due to coherent hybridization between valence and conduction bands in direct band gaps. Our work elucidates the interplay of the chemical nature of electronic orbitals and the effect of band structures in a consistent manner and highlights the role of Berry phase in orbital magnetism. The results suggest a promising direction of orbitronics beyond controlling atomic orbitals, in which the orbital magnetism can be greatly enhanced by exploiting Berry phase.

Figures

Figures reproduced from arXiv: 2603.19875 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p029_12.png]

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Forward citations

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Reference graph

Works this paper leans on

133 extracted references · 2 linked inside Pith · cited by 3 Pith papers

[1]

For all three materials, modern theory and ACA show good agreement:M ACA z reproduces both the value ofM z atE F and its overall trend with respect toE

[93]. For all three materials, modern theory and ACA show good agreement:M ACA z reproduces both the value ofM z atE F and its overall trend with respect toE. In partic- ular, the discrepancies betweenM z andM ACA z atE F– defined as|(M z −M ACA z )/Mz|–are 8% in hcp Co (0.2% in fcc Co) and 6% in Ni. The discrepancy is largest in Fe and smallest in Ni, re...

2024
[2]

D” as the “active subspace

Introduction to the gauge-covariant derivative In this section, we revisit the derivation of the gauge- invariant formula in Eq. (11). Since the partial deriva- tive of a state does not, in general, transform consistently (i.e., in a gauge-covariant manner) under a gauge trans- formation, the covariant derivative must be introduced when evaluating physica...
[3]

First, we focus on the quantum geometric tensor ˆFk,αβ =|u nk⟩ ⟨∂αunk|∂βumk⟩ ⟨umk|, 25 one of the building blocks of the orbital magnetiza- tion and Berry curvature

Gauge-covariant objects for orbital magnetization Now, we construct the gauge-covariant objects for or- bital magnetization. First, we focus on the quantum geometric tensor ˆFk,αβ =|u nk⟩ ⟨∂αunk|∂βumk⟩ ⟨umk|, 25 one of the building blocks of the orbital magnetiza- tion and Berry curvature. This is not covariant un- der the gauge transformation and thus ge...
[4]

J. H. V. Vleck,The Theory of Electric and Magnetic Susceptibilities(Oxford at the Clarendon Press, 1932)

1932
[5]

D. Go, D. Jo, H.-W. Lee, M. Kl¨ aui, and Y. Mokrousov, EPL135, 37001 (2021)

2021
[6]

D. Jo, D. Go, G.-M. Choi, and H.-W. Lee, npj Spin- tronics2, 19 (2024)

2024
[7]

P. Wang, F. Chen, Y. Yang, S. Hu, Y. Li, W. Wang, D. Zhang, and Y. Jiang, Adv. Electron. Mater.11, 32 2400554 (2025)

2025
[8]

T. P. Cysne, L. M. Canonico, M. Costa, R. B. Muniz, and T. G. Rappoport, npj Spintronics3, 39 (2025)

2025

Show all 133 references

[9]

R. B. Atencia, A. Agarwal, and D. Culcer, Adv. Phys.: X9, 2371972 (2024)

2024
[10]

Osumi, T

K. Osumi, T. Zhang, and S. Murakami, Commun. Phys. 4, 211 (2021)

2021
[11]

S. R. Park, C. H. Kim, J. Yu, J. H. Han, and C. Kim, Phys. Rev. Lett.107, 156803 (2011)

2011
[12]

Go, J.-P

D. Go, J.-P. Hanke, P. M. Buhl, F. Freimuth, G. Bihlmayer, H.-W. Lee, Y. Mokrousov, and S. Bl¨ ugel, Sci. Rep.7, 46742 (2017)

2017
[13]

T. Yoda, T. Yokoyama, and S. Murakami, Nano Lett. 18, 916 (2018)

2018
[14]

El Hamdi, J.-Y

A. El Hamdi, J.-Y. Chauleau, M. Boselli, C. Thibault, C. Gorini, A. Smogunov, C. Barreteau, S. Gariglio, J.- M. Triscone, and M. Viret, Nat. Phys.96, 1855 (2023)

2023
[15]

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Phys. Rev. Lett.95, 066601 (2005)

2005
[16]

Kontani, T

H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Phys. Rev. Lett.102, 016601 (2009)

2009
[17]

D. Go, D. Jo, C. Kim, and H.-W. Lee, Phys. Rev. Lett. 121, 086602 (2018)

2018
[18]

D. Jo, D. Go, and H.-W. Lee, Phys. Rev. B98, 214405 (2018)

2018
[19]

Salemi and P

L. Salemi and P. M. Oppeneer, Phys. Rev. Mater.6, 095001 (2022)

2022
[20]

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, Nature619, 52 (2023)

2023
[21]

Lyalin, S

I. Lyalin, S. Alikhah, M. Berritta, P. M. Oppeneer, and R. K. Kawakami, Phys. Rev. Lett.131, 156702 (2023)

2023
[22]

H. Liu, J. H. Cullen, D. P. Arovas, and D. Culcer, Phys. Rev. Lett.134, 036304 (2025)

2025
[23]

Liu and D

H. Liu and D. Culcer, Phys. Rev. Lett.132, 186302 (2024)

2024
[24]

B. C. Park, T. Ha, H. Lee, K. I. Sim, I. Baek, S. J. Yun, K. Lee, H.-W. Lee, and Y. H. Lee, Adv. Mater. , e03808 (2025)

2025
[25]

Go and H.-W

D. Go and H.-W. Lee, Phys. Rev. Res.2, 013177 (2020)

2020
[26]

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, Nat. Commun.12, 6710 (2021)

2021
[27]

J. Kim, D. Go, H. Tsai, D. Jo, K. Kondou, H.-W. Lee, and Y. Otani, Phys. Rev. B103, L020407 (2021)

2021
[28]

Sala and P

G. Sala and P. Gambardella, Phys. Rev. Res.4, 033037 (2022)

2022
[29]

Hayashi, D

H. Hayashi, D. Jo, D. Go, T. Gao, S. Haku, Y. Mokrousov, H.-W. Lee, and K. Ando, Commun. Phys.6, 32 (2023)

2023
[30]

D. Go, D. Jo, K.-W. Kim, S. Lee, M.-G. Kang, B.-G. Park, S. Bl¨ ugel, H.-W. Lee, and Y. Mokrousov, Phys. Rev. Lett.130, 246701 (2023)

2023
[31]

Fukunaga, S

R. Fukunaga, S. Haku, H. Hayashi, and K. Ando, Phys. Rev. Res.5, 023054 (2023)

2023
[32]

Ding, M.-G

S. Ding, M.-G. Kang, W. Legrand, and P. Gambardella, Phys. Rev. Lett.132, 236702 (2024)

2024
[33]

Lee and H.-W

H. Lee and H.-W. Lee, Curr. Appl. Phys.67, 60 (2024)

2024
[34]

Moriya, M

H. Moriya, M. Taniguchi, D. Jo, D. Go, N. Soya, H. Hayashi, Y. Mokrousov, H.-W. Lee, and K. Ando, Nano Lett.24, 6459 (2024)

2024
[35]

Chiba, Y

S. Chiba, Y. Marui, H. Ohno, and S. Fukami, Nano Lett. 24, 14028 (2024)

2024
[36]

Hayashi, D

H. Hayashi, D. Go, S. Haku, Y. Mokrousov, and K. Ando, Nat. Electron.7, 646 (2024)

2024
[37]

Han, H.-W

S. Han, H.-W. Ko, J. H. Oh, H.-W. Lee, K.-J. Lee, and K.-W. Kim, Phys. Rev. Lett.134, 036305 (2025)

2025
[39]

S. R. Park, J. Han, C. Kim, Y. Y. Koh, C. Kim, H. Lee, H. J. Choi, J. H. Han, K. D. Lee, N. J. Hur, M. Arita, K. Shimada, H. Namatame, and M. Taniguchi, Phys. Rev. Lett.108, 046805 (2012)

2012
[40]

Hagiwara, Y.-J

K. Hagiwara, Y.-J. Chen, D. Go, X. L. Tan, S. Grytsiuk, K.-H. O. Yang, G.-J. Shu, J. Chien, Y.-H. Shen, X.-L. Huang, I. Cojocariu, V. Feyer, M.-T. Lin, S. Bl¨ ugel, C. M. Schneider, Y. Mokrousov, and C. Tusche, Adv. Mater.37, 2418040 (2025)

2025
[41]

Bhowal and S

S. Bhowal and S. Satpathy, Phys. Rev. B101, 121112 (2020)

2020
[42]

Bhowal and S

S. Bhowal and S. Satpathy, Phys. Rev. B102, 035409 (2020)

2020
[43]

T. P. Cysne, M. Costa, L. M. Canonico, M. B. Nardelli, R. B. Muniz, and T. G. Rappoport, Phys. Rev. Lett. 126, 056601 (2021)

2021
[44]

T. P. Cysne, S. Bhowal, G. Vignale, and T. G. Rap- poport, Phys. Rev. B105, 195421 (2022)

2022
[45]

Costa, B

M. Costa, B. Focassio, L. M. Canonico, T. P. Cysne, G. R. Schleder, R. B. Muniz, A. Fazzio, and T. G. Rap- poport, Phys. Rev. Lett.130, 116204 (2023)

2023
[46]

S. D. Wilson and B. R. Ortiz, Nat. Rev. Mater.9, 420 (2024)

2024
[47]

Resta, Phys

R. Resta, Phys. Rev. Lett.80, 1800 (1998)

1998
[48]

Sharma, S

S. Sharma, S. Pittalis, S. Kurth, S. Shallcross, J. K. Dewhurst, and E. K. U. Gross, Phys. Rev. B76, 100401 (2007)

2007
[49]

Hanke, F

J.-P. Hanke, F. Freimuth, A. K. Nandy, H. Zhang, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B94, 121114 (2016)

2016
[50]

Thonhauser, Int

T. Thonhauser, Int. J. Mod. Phys. B25, 1429 (2011)

2011
[51]

Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge university press, 2018)

D. Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge university press, 2018)

2018
[52]

Aryasetiawan and K

F. Aryasetiawan and K. Karlsson, J. Phys. Chem. Solids 128, 87 (2019)

2019
[53]

R. D. King-Smith and D. Vanderbilt, Phys. Rev. B47, 1651 (1993)

1993
[54]

Vanderbilt and R

D. Vanderbilt and R. D. King-Smith, Phys. Rev. B48, 4442 (1993)

1993
[55]

Resta, D

R. Resta, D. Ceresoli, T. Thonhauser, and D. Vander- bilt, ChemPhysChem6, 1815 (2005)

2005
[56]

Thonhauser, D

T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Phys. Rev. Lett.95, 137205 (2005)

2005
[57]

Ceresoli, T

D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, Phys. Rev. B74, 024408 (2006)

2006
[58]

Ceresoli, U

D. Ceresoli, U. Gerstmann, A. P. Seitsonen, and F. Mauri, Phys. Rev. B81, 060409 (2010)

2010
[59]

Chang and Q

M.-C. Chang and Q. Niu, Phys. Rev. B53, 7010 (1996)

1996
[60]

Sundaram and Q

G. Sundaram and Q. Niu, Phys. Rev. B59, 14915 (1999)

1999
[61]

D. Xiao, J. Shi, and Q. Niu, Phys. Rev. Lett.95, 137204 (2005)

2005
[62]

D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Phys. Rev. Lett. 97, 026603 (2006). 33

2006
[63]

J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys. Rev. Lett.99, 197202 (2007)

2007
[64]

Bianco and R

R. Bianco and R. Resta, Phys. Rev. Lett.110, 087202 (2013)

2013
[65]

K. J. U. Vidarte, H. P. Veiga, J. M. V. P. Lopes, R. Car- dias, A. Ferreira, T. P. Cysne, and T. G. Rappoport, Real-space spectral approach to orbital magnetization (2025), arXiv:2512.01575 [cond-mat.mtrl-sci]

2025
[66]

Marrazzo and R

A. Marrazzo and R. Resta, Phys. Rev. Lett.116, 137201 (2016)

2016
[67]

S.-S. Wang, Y. Yu, J.-H. Guan, Y.-M. Dai, H.-H. Wang, and Y.-Y. Zhang, Phys. Rev. B106, 075136 (2022)

2022
[68]

L. M. Canonico, T. P. Cysne, A. Molina-Sanchez, R. B. Muniz, and T. G. Rappoport, Phys. Rev. B101, 161409 (2020)

2020
[69]

Bhowal and G

S. Bhowal and G. Vignale, Phys. Rev. B103, 195309 (2021)

2021
[70]

A. Pezo, D. Garc´ ıa Ovalle, and A. Manchon, Phys. Rev. B106, 104414 (2022)

2022
[71]

Busch, I

O. Busch, I. Mertig, and B. G¨ obel, Phys. Rev. Res.5, 043052 (2023)

2023
[72]

G¨ obel and I

B. G¨ obel and I. Mertig, Phys. Rev. Lett.133, 146301 (2024)

2024
[73]

M. G. Lopez, D. Vanderbilt, T. Thonhauser, and I. Souza, Phys. Rev. B85, 014435 (2012)

2012
[74]

Go, H.-W

D. Go, H.-W. Lee, P. M. Oppeneer, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B109, 174435 (2024)

2024
[75]

G. Go, D. An, H.-W. Lee, and S. K. Kim, Nano Lett. 24, 5968 (2024)

2024
[76]

J. M. Lee, Phys. Rev. B112, 054441 (2025)

2025
[77]

X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Phys. Rev. B74, 195118 (2006)

2006
[78]

Souza and D

I. Souza and D. Vanderbilt, Phys. Rev. B77, 054438 (2008)

2008
[79]

Resta, Phys

R. Resta, Phys. Rev. Res.2, 023139 (2020)

2020
[80]

Mahfouzi, M

F. Mahfouzi, M. D. Stiles, and P. M. Haney, Phys. Rev. B111, 014415 (2025)

2025
[81]

Marzari and D

N. Marzari and D. Vanderbilt, Phys. Rev. B56, 12847 (1997)

1997
[82]

Souza, N

I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001)

2001
[83]

Marzari, A

N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Rev. Mod. Phys.84, 1419 (2012)

2012
[84]

Marrazzo, S

A. Marrazzo, S. Beck, E. R. Margine, N. Marzari, A. A. Mostofi, J. Qiao, I. Souza, S. S. Tsirkin, J. R. Yates, and G. Pizzi, Rev. Mod. Phys.96, 045008 (2024)

2024
[85]

A. Urru, I. Souza, O. P. Oca˜ na, S. S. Tsirkin, and D. Vanderbilt, Phys. Rev. B112, 045201 (2025)

2025
[86]

J. C. Slater and G. F. Koster, Phys. Rev.94, 1498 (1954)

1954
[87]

X. Liu, S. S. Tsirkin, and I. Souza, Covariant deriva- tives of Berry-type quantities: Application to nonlinear transport (2023), arXiv:2303.10129 [cond-mat.mtrl-sci]. [85]http://www.flapw.de

2023 arXiv
[88]

Wimmer, H

E. Wimmer, H. Krakauer, M. Weinert, and A. J. Free- man, Phys. Rev. B24, 864 (1981)

1981
[89]

J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.77, 3865 (1996)

1996
[90]

H. J. Monkhorst and J. D. Pack, Phys. Rev. B13, 5188 (1976)

1976
[91]

A. B. Shick, A. I. Liechtenstein, and W. E. Pickett, Phys. Rev. B60, 10763 (1999)

1999
[92]

Pizzi, V

G. Pizzi, V. Vitale, R. Arita, S. Bl¨ ugel, F. Freimuth, G. G´ eranton, M. Gibertini, D. Gresch, C. John- son, T. Koretsune, J. Iba˜ nez-Azpiroz, H. Lee, J.- M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Ponc´ e, T. Ponwe...

2020
[93]

Freimuth, Y

F. Freimuth, Y. Mokrousov, D. Wortmann, S. Heinze, and S. Bl¨ ugel, Phys. Rev. B78, 035120 (2008). [92]https://iffgit.fz-juelich.de/dgo/wann_interpol

2008
[94]

A. J. P. Meyer and G. Asch, J. Appl. Phys.32, S330 (1961)

1961
[95]

Strigl, C

F. Strigl, C. Espy, M. B¨ uckle, E. Scheer, and T. Pietsch, Nat. Commun.6, 6172 (2015)

2015
[96]

H. J. Albert and L. R. Rubin, Adv. Chem.98, 1 (1971)

1971
[97]

Simopoulos, E

A. Simopoulos, E. Devlin, A. Kostikas, A. Jankowski, M. Croft, and T. Tsakalakos, Phys. Rev. B54, 9931 (1996)

1996
[98]

Meier, S

F. Meier, S. Lounis, J. Wiebe, L. Zhou, S. Heers, P. Mavropoulos, P. H. Dederichs, S. Bl¨ ugel, and R. Wiesendanger, Phys. Rev. B83, 075407 (2011)

2011
[99]

Wilhelm, P

F. Wilhelm, P. Poulopoulos, G. Ceballos, H. Wende, K. Baberschke, P. Srivastava, D. Benea, H. Ebert, M. Angelakeris, N. K. Flevaris, D. Niarchos, A. Rogalev, and N. B. Brookes, Phys. Rev. Lett.85, 413 (2000)

2000
[100]

W. J. Antel, M. M. Schwickert, T. Lin, W. L. O’Brien, and G. R. Harp, Phys. Rev. B60, 12933 (1999)

1999
[101]

van der Marel and G

D. van der Marel and G. A. Sawatzky, Phys. Rev. B37, 10674 (1988)

1988
[102]

T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, and J. Feng, Nat. Commun. 3, 887 (2012)

2012
[103]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nat. Nanotechnol.7, 490 (2012)

2012
[104]

K. F. Mak, K. He, J. Shan, and T. F. Heinz, Nat. Nan- otechnol.7, 494 (2012)

2012
[105]

J. Lee, Z. Wang, H. Xie, K. F. Mak, and J. Shan, Nat. Mater.16, 887 (2017)

2017
[106]

S. Wu, J. S. Ross, G.-B. Liu, G. Aivazian, A. Jones, Z. Fei, W. Zhu, D. Xiao, W. Yao, D. Cobden, and X. Xu, Nat. Phys.9, 149 (2013)

2013
[107]

Tahir, A

M. Tahir, A. Manchon, and U. Schwingenschl¨ ogl, Phys. Rev. B90, 125438 (2014)

2014
[108]

F. Xue, V. Amin, and P. M. Haney, Phys. Rev. B102, 161103 (2020)

2020
[109]

Sallen, L

G. Sallen, L. Bouet, X. Marie, G. Wang, C. R. Zhu, W. P. Han, Y. Lu, P. H. Tan, T. Amand, B. L. Liu, and B. Urbaszek, Phys. Rev. B86, 081301 (2012)

2012
[110]

Arakawa and K

N. Arakawa and K. Yonemitsu, Phys. Rev. B111, L121110 (2025)

2025
[111]

Streda, J

P. Streda, J. Phys. C: Solid State Phys.15, L717 (1982)

1982
[112]

Xiao, M.-C

D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010)

1959
[113]

Feng, C.-C

W. Feng, C.-C. Liu, G.-B. Liu, J.-J. Zhou, and Y. Yao, Comput. Mater. Sci.112, 428 (2016)

2016
[114]

S.-Y. Xu, Q. Ma, H. Shen, V. Fatemi, S. Wu, T.-R. Chang, G. Chang, A. M. M. Valdivia, C.-K. Chan, Q. D. Gibson, J. Zhou, Z. Liu, K. Watanabe, T. Taniguchi, H. Lin, R. J. Cava, L. Fu, N. Gedik, and P. Jarillo- Herrero, Nat. Phys.14, 900 (2018)

2018
[115]

X. Qian, J. Liu, L. Fu, and J. Li, Science346, 1344 (2014)

2014
[116]

S. Tang, C. Zhang, D. Wong, Z. Pedramrazi, H.-Z. Tsai, C. Jia, B. Moritz, M. Claassen, H. Ryu, S. Kahn, 34 J. Jiang, H. Yan, M. Hashimoto, D. Lu, R. G. Moore, C.-C. Hwang, C. Hwang, Z. Hussain, Y. Chen, M. M. Ugeda, Z. Liu, X. Xie, T. P. Devereaux, M. F. Crommie, S.-K. Mo, and...

2017
[117]

S. Wu, V. Fatemi, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, Sci- ence359, 76 (2018)

2018
[118]

J.-S. You, S. Fang, S.-Y. Xu, E. Kaxiras, and T. Low, Phys. Rev. B98, 121109 (2018)

2018
[119]

Torbatian, D

Z. Torbatian, D. Novko, and R. Asgari, Phys. Rev. Appl. 14, 044014 (2020)

2020
[120]

Vila, C.-H

M. Vila, C.-H. Hsu, J. H. Garcia, L. A. Ben´ ıtez, X. Waintal, S. O. Valenzuela, V. M. Pereira, and S. Roche, Phys. Rev. Res.3, 043230 (2021)

2021
[121]

L. M. Canonico, J. H. Garcia, A. W. Cummings, and S. Roche, Spin polarization driven by itinerant orbital angular momentum in van der waals heterostructures (2025), arXiv:2507.12587 [cond-mat.mtrl-sci]

2025 arXiv
[122]

Nakahara,Geometry, Topology and Physics(Adam Hilger, Bristol and New York, 1990)

M. Nakahara,Geometry, Topology and Physics(Adam Hilger, Bristol and New York, 1990)

1990
[123]

A. Pezo, D. Garc´ ıa Ovalle, and A. Manchon, Phys. Rev. B108, 075427 (2023)

2023
[124]

G¨ obel, L

B. G¨ obel, L. Schimpf, and I. Mertig, Commun. Phys.8, 17 (2025)

2025
[125]

Ghosh and B

S. Ghosh and B. L. Chittari, Sci. Rep.15, 11085 (2025)

2025
[126]

M. A. W. Schoen, J. Lucassen, H. T. Nembach, T. J. Silva, B. Koopmans, C. H. Back, and J. M. Shaw, Phys. Rev. B95, 134410 (2017)

2017
[127]

Kirklin, J

S. Kirklin, J. E. Saal, B. Meredig, A. Thompson, J. W. Doak, M. Aykol, S. R¨ uhl, and C. Wolverton, npj Com- put. Mater.1, 15010 (2015)

2015
[128]

A. Fang, C. Adamo, S. Jia, R. J. Cava, S.-C. Wu, C. Felser, and A. Kapitulnik, Sci. Adv.4, eaaq0330 (2018)

2018
[129]

Cucka and C

P. Cucka and C. S. Barrett, Acta Crystallogr.15, 865 (1962)

1962
[130]

H. He, P. Lu, L. Wu, C. Zhang, Y. Song, P. Guan, and S. Wang, Nanoscale Res. Lett.11, 330 (2016)

2016
[131]

A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Nature527, 495 (2015)

2015
[132]

S ¸a¸ sıo˘ glu, C

E. S ¸a¸ sıo˘ glu, C. Friedrich, and S. Bl¨ ugel, Phys. Rev. B 83, 121101 (2011)

2011
[133]

Tung, H.-R

J.-C. Tung, H.-R. Fuh, and G.-Y. Guo, Phys. Rev. B 86, 024435 (2012)

2012
[134]

I. Yang, S. Y. Savrasov, and G. Kotliar, Phys. Rev. Lett. 87, 216405 (2001)

2001

Pith tools

Reviewed July 13, 2026 · model on record in the stance chip above.

[1] [1]

For all three materials, modern theory and ACA show good agreement:M ACA z reproduces both the value ofM z atE F and its overall trend with respect toE

[93]. For all three materials, modern theory and ACA show good agreement:M ACA z reproduces both the value ofM z atE F and its overall trend with respect toE. In partic- ular, the discrepancies betweenM z andM ACA z atE F– defined as|(M z −M ACA z )/Mz|–are 8% in hcp Co (0.2% in fcc Co) and 6% in Ni. The discrepancy is largest in Fe and smallest in Ni, re...

2024

[2] [2]

D” as the “active subspace

Introduction to the gauge-covariant derivative In this section, we revisit the derivation of the gauge- invariant formula in Eq. (11). Since the partial deriva- tive of a state does not, in general, transform consistently (i.e., in a gauge-covariant manner) under a gauge trans- formation, the covariant derivative must be introduced when evaluating physica...

[3] [3]

First, we focus on the quantum geometric tensor ˆFk,αβ =|u nk⟩ ⟨∂αunk|∂βumk⟩ ⟨umk|, 25 one of the building blocks of the orbital magnetiza- tion and Berry curvature

Gauge-covariant objects for orbital magnetization Now, we construct the gauge-covariant objects for or- bital magnetization. First, we focus on the quantum geometric tensor ˆFk,αβ =|u nk⟩ ⟨∂αunk|∂βumk⟩ ⟨umk|, 25 one of the building blocks of the orbital magnetiza- tion and Berry curvature. This is not covariant un- der the gauge transformation and thus ge...

[4] [4]

J. H. V. Vleck,The Theory of Electric and Magnetic Susceptibilities(Oxford at the Clarendon Press, 1932)

1932

[5] [5]

D. Go, D. Jo, H.-W. Lee, M. Kl¨ aui, and Y. Mokrousov, EPL135, 37001 (2021)

2021

[6] [6]

D. Jo, D. Go, G.-M. Choi, and H.-W. Lee, npj Spin- tronics2, 19 (2024)

2024

[7] [7]

P. Wang, F. Chen, Y. Yang, S. Hu, Y. Li, W. Wang, D. Zhang, and Y. Jiang, Adv. Electron. Mater.11, 32 2400554 (2025)

2025

[8] [8]

T. P. Cysne, L. M. Canonico, M. Costa, R. B. Muniz, and T. G. Rappoport, npj Spintronics3, 39 (2025)

2025