[Formules analytiques pour l’effet Hall anormal dans les matériaux magnétiques itinérants]
We provide analytical formulas to compute all the contributions to the intrinsic Hall conductivity in the presence of Kondo-coupled spins in any configuration and for any spin orbit coupling, and thereby clarify the origin of what is sometimes called the “topological anomalous Hall effect”. We also identify the relation between a momentum space quantity, the momentum space Berry curvature (which is in direct correspondence with the Hall conductivity — a global observable), and unit cell properties such as hopping parameters and spin configuration. More precisely, we find that the Berry curvature involves the scalar spin chirality on elementary unit cell triangles, $\chi _{ijk}=\vec{S}_i\cdot (\vec{S}_j\times \vec{S}_k)$, but also contains scalar triple products of other quantities (such as hopping parameters with spin-orbit coupling $\vec{t}_{ij}$), $\vec{t}_{ij}\cdot (\vec{t}_{jk}\times \vec{t}_{ki})$, $\vec{t}_{jk}\cdot (\vec{S}_{i}\times \vec{S}_{j}), \dots $, and their dot products, $\vec{S}_i\cdot \vec{S}_j$, $\vec{t}_{ij}\cdot \vec{t}_{jk}$, $\vec{t}_{ij}\cdot \vec{S}_k, \dots $ The relative size of the different contributions depends on the strength of the Kondo coupling and our formula captures all regimes. We apply our method to the case of three-sublattice systems, and prove very generally that in the absence of spin-orbit coupling, the Berry curvature of a three-magnetic-sublattice triangular itinerant magnet identically vanishes. The derivation is technical but we emphasize that the results can be very easily applied.
Nous dérivons des formules analytiques pour calculer toutes les contributions à la conductivité intrinsèque de Hall en présence de spins dans une configuration quelconque, couplés via Kondo aux électrons, et en présence de couplage spin-orbite, et ce faisant, nous clarifions l’origine de ce qui est parfois appelé “effet Hall anormal topologique”. Nous identifions également la relation entre une quantité de l’espace des phases, la courbure de Berry dans l’espace des phases (qui est en correspondance directe avec la conductivité de Hall — une observable globale), et les propriétés locales de la cellule unitaire telles les paramètres de saut et la configuration des spins. Plus précisément, nous trouvons que la courbure de Berry contient la chiralité scalaire des spins, $\chi _{ijk}=\vec{S}_i\cdot (\vec{S}_j\times \vec{S}_k)$, mais aussi le triple produit scalaire d’autres quantités (par exemple celui des paramètres de saut avec couplage spin-orbite $\vec{t}_{ij}$), $\vec{t}_{ij}\cdot (\vec{t}_{jk}\times \vec{t}_{ki})$, $\vec{t}_{jk}\cdot (\vec{S}_{i}\times \vec{S}_{j}), \dots $, et leurs produits scalaires, $\vec{S}_i\cdot \vec{S}_j$, $\vec{t}_{ij}\cdot \vec{t}_{jk}$, $\vec{t}_{ij}\cdot \vec{S}_k, \dots $ L’amplitude relative des différentes contributions dépend de la taille du couplage de Kondo et notre formule s’applique dans tous les régimes. Nous appliquons notre méthode au cas de systèmes à trois sous-réseaux et prouvons de manière très générale qu’en l’absence de couplage spin-orbite, la courbure de Berry d’un matériau triangulaire magnétique à trois sous-réseaux est identiquement nulle. La dérivation est technique mais nous insistons sur le fait que les résultats peuvent être appliqués très facilement.
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Mots-clés : Effet Hall anormal, courbure de Berry, chiralité de spins
Lucile Savary 1 , 2 , 3
CC-BY 4.0
@article{CRPHYS_2026__27_G1_161_0,
author = {Lucile Savary},
title = {Analytic formulas for the anomalous {Hall} effect in itinerant magnets},
journal = {Comptes Rendus. Physique},
pages = {161--181},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {27},
doi = {10.5802/crphys.272},
language = {en},
}
Lucile Savary. Analytic formulas for the anomalous Hall effect in itinerant magnets. Comptes Rendus. Physique, Volume 27 (2026), pp. 161-181. doi: 10.5802/crphys.272
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