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Quantum geometry and linear orbital response in arbitrary SU(2) representation

Rhonald Burgos Atencia

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Phys. Rev. B 113, 075422 – Published 17 February, 2026

DOI: https://doi.org/10.1103/hqsf-v45b

Abstract

We develop a symmetry-based framework to compute band-geometric quantities in multiband systems whose low-energy Hamiltonians realize arbitrary SU(2) representations. Exploiting the existence of a quantization axis, we use the Wigner-Eckart theorem to identify the allowed matrix elements and obtain compact analytic expressions for the quantum geometric tensor, the orbital magnetic moment, and the associated orbital transport coefficients. The formalism applies to multifold fermions as well as to gapped SU(2) models. Its versatility is illustrated through explicit calculations in representative SU(3) and SU(4) settings, where orbital Edelstein and orbital Hall responses emerge naturally from the antisymmetric components of the band geometry. These results establish a direct connection between the algebraic structure of the Hamiltonian and intrinsic orbitronic phenomena.

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References (58)

  1. T. Jungwirth, Q. Niu, and A. H. MacDonald, Anomalous Hall effect in ferromagnetic semiconductors, Phys. Rev. Lett. 88, 207208 (2002).
  2. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  3. N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).
  4. C.-Z. Chang, C.-X. Liu, and A. H. MacDonald, Colloquium: Quantum anomalous Hall effect, Rev. Mod. Phys. 95, 011002 (2023).
  5. Y. Gao, S. A. Yang, and Q. Niu, Field induced positional shift of Bloch electrons and its dynamical implications, Phys. Rev. Lett. 112, 166601 (2014).
  6. I. Sodemann and L. Fu, Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials, Phys. Rev. Lett. 115, 216806 (2015).
  7. C. Ortix, Nonlinear Hall effect with time-reversal symmetry: Theory and material realizations, Adv. Quantum Technol. 4, 2100056 (2021).
  8. Z. Z. Du, C. M. Wang, H.-P. Sun, H.-Z. Lu, and X. C. Xie, Quantum theory of the nonlinear Hall effect, Nat. Commun. 12, 5038 (2021).
  9. Z. Z. Du, H.-Z. Lu, and X. C. Xie, Nonlinear Hall effects, Nat. Rev. Phys. 3, 744 (2021).
  10. M. Suárez-Rodríguez, F. de Juan, I. Souza, M. Gobbi, F. Casanova, and L. E. Hueso, Nonlinear transport in non-centrosymmetric systems, Nat. Mater. 24, 1005 (2025).
  11. Y. Jiang, T. Holder, and B. Yan, Revealing quantum geometry in nonlinear quantum materials, Rep. Prog. Phys. 88, 076502 (2025).
  12. Y. Fang, J. Cano, and S. A. A. Ghorashi, Quantum geometry induced nonlinear transport in altermagnets, Phys. Rev. Lett. 133, 106701 (2024).
  13. G. Sala, M. T. Mercaldo, K. Domi, S. Gariglio, M. Cuoco, C. Ortix, and A. D. Caviglia, The quantum metric of electrons with spin-momentum locking, Science 389, 822 (2025).
  14. D.-F. Shao, S.-H. Zhang, G. Gurung, W. Yang, and E. Y. Tsymbal, Nonlinear anomalous Hall effect for Néel vector detection, Phys. Rev. Lett. 124, 067203 (2020).
  15. Y. Zhao, J. Cao, Z. Zhang, S. Li, Y. Li, F. Ma, and S. A. Yang, Berry curvature dipole and nonlinear Hall effect in two-dimensional Nb2n+1SinTe4n+2, Phys. Rev. B 107, 205124 (2023).
  16. J.-S. You, S. Fang, S.-Y. Xu, E. Kaxiras, and T. Low, Berry curvature dipole current in the transition metal dichalcogenides family, Phys. Rev. B 98, 121109(R) (2018).
  17. C. Wang, Y. Gao, and D. Xiao, Intrinsic nonlinear Hall effect in antiferromagnetic tetragonal CuMnAs, Phys. Rev. Lett. 127, 277201 (2021).
  18. K. Das, S. Lahiri, R. B. Atencia, D. Culcer, and A. Agarwal, Intrinsic nonlinear conductivities induced by the quantum metric, Phys. Rev. B 108, L201405 (2023).
  19. S. Sharma and A. De Sarkar, Intrinsic nonlinear valley Hall physics in PT-symmetric two-dimensional antiferromagnets, Phys. Rev. B 112, 115435 (2025).
  20. Z. Zhang, X.-Z. Li, and W.-Y. He, Orbital magnetization as the origin of the nonlinear Hall effect, Phys. Rev. Res. 7, L042064 (2025).
  21. D. Go, D. Jo, H.-W. Lee, M. Kläui, and Y. Mokrousov, Orbitronics: Orbital currents in solids, Europhys. Lett. 135, 37001 (2021).
  22. R. B. Atencia, A. Agarwal, and D. Culcer, Orbital angular momentum of Bloch electrons: Equilibrium formulation, magneto-electric phenomena, and the orbital Hall effect, Adv. Phys.: X 9, 2371972 (2024).
  23. G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Phys. Rev. B 59, 14915 (1999).
  24. M.-C. Chang and Q. Niu, Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields, J. Phys.: Condens. Matter 20, 193202 (2008).
  25. T. P. Cysne, F. S. M. Guimarães, L. M. Canonico, T. G. Rappoport, and R. B. Muniz, Orbital magnetoelectric effect in zigzag nanoribbons of p-band systems, Phys. Rev. B 104, 165403 (2021).
  26. T. P. Cysne, F. S. M. Guimarães, L. M. Canonico, M. Costa, T. G. Rappoport, and R. B. Muniz, Orbital magnetoelectric effect in nanoribbons of transition metal dichalcogenides, Phys. Rev. B 107, 115402 (2023).
  27. J. M. Lee, M. J. Park, and H.-W. Lee, Orbital Edelstein effect of electronic itinerant orbital motion at edges, Phys. Rev. B 110, 134436 (2024).
  28. W. Gao, L. Liao, H. Isshiki, N. Budai, J. Kim, H.-W. Lee, K.-J. Lee, D. Go, Y. Mokrousov, S. Miwa, and Y. Otani, Nonlocal electrical detection of reciprocal orbital Edelstein effect, Nat. Commun. 16, 6380 (2025).
  29. L. M. Canonico, T. P. Cysne, T. G. Rappoport, and R. B. Muniz, Two-dimensional orbital Hall insulators, Phys. Rev. B 101, 075429 (2020).
  30. L. M. Canonico, T. P. Cysne, A. Molina-Sanchez, R. B. Muniz, and T. G. Rappoport, Orbital Hall insulating phase in transition metal dichalcogenide monolayers, Phys. Rev. B 101, 161409(R) (2020).
  31. S. Bhowal and G. Vignale, Orbital Hall effect as an alternative to valley Hall effect in gapped graphene, Phys. Rev. B 103, 195309 (2021).
  32. T. P. Cysne, M. Costa, L. M. Canonico, M. B. Nardelli, R. B. Muniz, and T. G. Rappoport, Disentangling orbital and valley Hall effects in bilayers of transition metal dichalcogenides, Phys. Rev. Lett. 126, 056601 (2021).
  33. T. P. Cysne, S. Bhowal, G. Vignale, and T. G. Rappoport, Orbital Hall effect in bilayer transition metal dichalcogenides: From the intra-atomic approximation to the Bloch states orbital magnetic moment approach, Phys. Rev. B 105, 195421 (2022).
  34. A. Pezo, D. García Ovalle, and A. Manchon, Orbital Hall effect in crystals: Interatomic versus intra-atomic contributions, Phys. Rev. B 106, 104414 (2022).
  35. O. Busch, I. Mertig, and B. Göbel, Orbital Hall effect and orbital edge states caused by s electrons, Phys. Rev. Res. 5, 043052 (2023).
  36. A. Pezo, D. García Ovalle, and A. Manchon, Orbital Hall physics in two-dimensional Dirac materials, Phys. Rev. B 108, 075427 (2023).
  37. A. L. R. Barbosa, L. M. Canonico, J. H. García, and T. G. Rappoport, Orbital Hall effect and topology on a two-dimensional triangular lattice: From bulk to edge, Phys. Rev. B 110, 085412 (2024).
  38. T. P. Cysne, W. J. M. Kort-Kamp, and T. G. Rappoport, Controlling the orbital Hall effect in gapped bilayer graphene in the terahertz regime, Phys. Rev. Res. 6, 023271 (2024).
  39. H. Liu and D. Culcer, Dominance of extrinsic scattering mechanisms in the orbital Hall effect: Graphene, transition metal dichalcogenides, and topological antiferromagnets, Phys. Rev. Lett. 132, 186302 (2024).
  40. I. Robredo, N. B. M. Schröter, C. Felser, J. Cano, B. Bradlyn, and M. G. Vergniory, Multifold topological semimetals, Europhys. Lett. 147, 46001 (2024).
  41. Z. Ni, B. Xu, M. Á. Sánchez-Martínez, Y. Zhang, K. Manna, C. Bernhard, J. W. F. Venderbos, F. de Juan, C. Felser, A. G. Grushin, and L. Wu, Linear and nonlinear optical responses in the chiral multifold semimetal RhSi, npj Quantum Mater. 5, 96 (2020).
  42. S. Kaushik and J. Cano, Magnetic photocurrents in multifold Weyl fermions, Phys. Rev. B 104, 155149 (2021).
  43. F. Flicker, F. de Juan, B. Bradlyn, T. Morimoto, M. G. Vergniory, and A. G. Grushin, Chiral optical response of multifold fermions, Phys. Rev. B 98, 155145 (2018).
  44. See Supplemental Material at http://link.aps.org/supplemental/10.1103/hqsf-v45b for additional derivations and technical details.
  45. J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  46. J. Yu, B. A. Bernevig, R. Queiroz, E. Rossi, P. Törmä, and B.-J. Yang, Quantum geometry in quantum materials, npj Quantum Mater. 10, 101 (2025).
  47. K. Wen, H.-Y. Xie, A. Auerbach, and B. Uchoa, Thermal and thermoelectric transport in flat bands with nontrivial quantum geometry, Phys. Rev. B 111, 205140 (2025).
  48. S. Zhong, J. E. Moore, and I. Souza, Gyrotropic magnetic effect and the magnetic moment on the Fermi surface, Phys. Rev. Lett. 116, 077201 (2016).
  49. T. Yoda, T. Yokoyama, and S. Murakami, Orbital Edelstein effect as a condensed-matter analog of solenoids, Nano Lett. 18, 916 (2018).
  50. D. Culcer, A. Sekine, and A. H. MacDonald, Interband coherence response to electric fields in crystals: Berry-phase contributions and disorder effects, Phys. Rev. B 96, 035106 (2017).
  51. R. B. Atencia, Q. Niu, and D. Culcer, Semiclassical response of disordered conductors: Extrinsic carrier velocity and spin and field-corrected collision integral, Phys. Rev. Res. 4, 013001 (2022).
  52. D. Green, L. Santos, and C. Chamon, Isolated flat bands and spin-1 conical bands in two-dimensional lattices, Phys. Rev. B 82, 075104 (2010).
  53. L. Ye, M. Kang, J. Liu, F. von Cube, C. R. Wicker, T. Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C. Bell, L. Fu, R. Comin, and J. G. Checkelsky, Massive Dirac fermions in a ferromagnetic kagome metal, Nature (London) 555, 638 (2018).
  54. M. Kang, S. Fang, L. Ye, H. C. Po, J. Denlinger, C. Jozwiak, A. Bostwick, E. Rotenberg, E. Kaxiras, J. G. Checkelsky, and R. Comin, Topological flat bands in frustrated kagome lattice CoSn, Nat. Commun. 11, 4004 (2020).
  55. M. Kang, L. Ye, S. Fang, J.-S. You, A. Levitan, M. Han, J. I. Facio, C. Jozwiak, A. Bostwick, E. Rotenberg, M. K. Chan, R. D. McDonald, D. Graf, K. Kaznatcheev, E. Vescovo, D. C. Bell, E. Kaxiras, J. van den Brink, M. Richter, M. Prasad Ghimire, et al., Dirac fermions and flat bands in the ideal kagome metal FeSn, Nat. Mater. 19, 163 (2020).
  56. Q. Yang, J. Xiao, I. Robredo, M. G. Vergniory, B. Yan, and C. Felser, Monopole-like orbital-momentum locking and the induced orbital transport in topological chiral semimetals, Proc. Natl. Acad. Sci. USA 120, e2305541120 (2023).
  57. O. Pozo and F. de Juan, Computing observables without eigenstates: Applications to Bloch Hamiltonians, Phys. Rev. B 102, 115138 (2020).
  58. A. Graf and F. Piéchon, Berry curvature and quantum metric in n-band systems: An eigenprojector approach, Phys. Rev. B 104, 085114 (2021).

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Phys. Rev. B 113, 075422– Published 17 February, 2026
  • Received 20 November 2025
  • Revised 7 January 2026
  • Accepted 29 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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