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  • Letter
  • Open Access

Hall effect in topologically trivial isolated flat-band systems

Raigo Nagashima1,*, Masao Ogata1,2, and Naoto Tsuji1,3

  • *Contact author: raigo.nagashima@kit.edu
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Phys. Rev. Research 8, L012041 – Published 23 February, 2026

DOI: https://doi.org/10.1103/5s6l-vc71

Abstract

We study the Hall effect in topologically trivial isolated flat-band systems (i.e., flat bands are separated from other bands and have zero Chern number) for a weak magnetic field. In a naive semiclassical picture, the Hall conductivity vanishes when dispersive bands are unoccupied, since there are no mobile carriers. To go beyond the semiclassical picture, we establish a fully quantum mechanical gauge-invariant formula for the Hall conductivity that can be applied to any lattice models. We apply the formula to a general (N+M)-band model with N dispersive bands and M-fold degenerate isolated flat bands, and find that when the dispersive bands are unoccupied, the total conductivity takes a universal form consisting of the energy difference between the dispersive and flat bands, and the non-Abelian quantum geometric tensor of the flat bands, which can be nonzero in systems with vanishing Berry curvature. We numerically confirm the Hall effect for isolated flat-band lattice models on the honeycomb lattice (N=M=1) and two different kagome lattices (N=2, M=1 and N=1, M=2).

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References (70)

  1. E. H. Hall, On a new action of the magnet on electric currents, Am. J. Math. 2, 287 (1879).
  2. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 2004).
  3. G. Grosso and G. P. Parravicini, Solid State Physics (Elsevier, Amsterdam, 2013).
  4. L. Chen, T. Mazaheri, A. Seidel, and X. Tang, The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models, J. Phys. A: Math. Theor. 47, 152001 (2014).
  5. E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett. 62, 1201 (1989).
  6. A. Mielke, Ferromagnetism in the Hubbard model on line graphs and further considerations, J. Phys. A: Math. Gen. 24, 3311 (1991).
  7. H. Tasaki, Ferromagnetism in the Hubbard models with degenerate single-electron ground states, Phys. Rev. Lett. 69, 1608 (1992).
  8. A. Mielke and H. Tasaki, Ferromagnetism in the Hubbard model, Commun. Math. Phys. 158, 341 (1993).
  9. H. Tasaki, Ferromagnetism in Hubbard models, Phys. Rev. Lett. 75, 4678 (1995).
  10. E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum Hall states, Phys. Rev. Lett. 106, 236802 (2011).
  11. K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106, 236803 (2011).
  12. T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett. 106, 236804 (2011).
  13. N. Regnault and B. A. Bernevig, Fractional Chern insulator, Phys. Rev. X 1, 021014 (2011).
  14. E. J. Bergholtz and Z. Liu, Topological flat band models and fractional Chern insulators, Int. J. Mod. Phys. B 27, 1330017 (2013).
  15. A. Zhao and S.-Q. Shen, Quantum anomalous Hall effect in a flat band ferromagnet, Phys. Rev. B 85, 085209 (2012).
  16. Y. Zhou, G. Sethi, H. Liu, Z. Wang, and F. Liu, Excited quantum anomalous and spin Hall effect: Dissociation of flat-bands-enabled excitonic insulator state, Nanotechnology 33, 415001 (2022).
  17. J.-W. Rhim, K. Kim, and B.-J. Yang, Quantum distance and anomalous Landau levels of flat bands, Nature (London) 584, 59 (2020).
  18. T. Misumi and H. Aoki, New class of flat-band models on tetragonal and hexagonal lattices: Gapped versus crossing flat bands, Phys. Rev. B 96, 155137 (2017).
  19. B. Roy and V. Juričić, Unconventional superconductivity in nearly flat bands in twisted bilayer graphene, Phys. Rev. B 99, 121407 (2019).
  20. H. Aoki, Theoretical possibilities for flat band superconductivity, J. Supercond. Nov. Magn. 33, 2341 (2020).
  21. L. H. C. M. Nunes and C. M. Smith, Flat-band superconductivity for tight-binding electrons on a square-octagon lattice, Phys. Rev. B 101, 224514 (2020).
  22. V. Peri, Z.-D. Song, B. A. Bernevig, and S. D. Huber, Fragile topology and flat-band superconductivity in the strong-coupling regime, Phys. Rev. Lett. 126, 027002 (2021).
  23. H. Tian, X. Gao, Y. Zhang, S. Che, T. Xu, P. Cheung, K. Watanabe, T. Taniguchi, M. Randeria, F. Zhang, C. N. Lau, and M. W. Bockrath, Evidence for Dirac flat band superconductivity enabled by quantum geometry, Nature (London) 614, 440 (2023).
  24. T. Mizoguchi and M. Udagawa, Flat-band engineering in tight-binding models: Beyond the nearest-neighbor hopping, Phys. Rev. B 99, 235118 (2019).
  25. N. Regnault, Y. Xu, M.-R. Li, D.-S. Ma, M. Jovanovic, A. Yazdani, S. S. P. Parkin, C. Felser, L. M. Schoop, N. P. Ong, R. J. Cava, L. Elcoro, Z.-D. Song, and B. A. Bernevig, Catalogue of flat-band stoichiometric materials, Nature (London) 603, 824 (2022).
  26. P. M. Neves, J. P. Wakefield, S. Fang, H. Nguyen, L. Ye, and J. G. Checkelsky, Crystal net catalog of model flat band materials, npj Comput. Mater. 10, 39 (2024).
  27. M. Kang, L. Ye, S. Fang, J.-S. You, M. Levitan, Abe Han, J. I. Facio, C. Jozwiak, A. Bostwick, E. Rotenberg, et al., Dirac fermions and flat bands in the ideal kagome metal FeSn, Nat. Mater. 19, 163 (2020).
  28. J. P. Wakefield, M. Kang, P. M. Neves, D. Oh, S. Fang, R. McTigue, S. Y. F. Zhao, T. N. Lamichhane, A. Chen, S. Lee, et al., Three-dimensional flat bands in pyrochlore metal CaNi2, Nature (London) 623, 301 (2023).
  29. E. Suárez Morell, J. D. Correa, P. Vargas, M. Pacheco, and Z. Barticevic, Flat bands in slightly twisted bilayer graphene: Tight-binding calculations, Phys. Rev. B 82, 121407 (2010).
  30. R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108, 12233 (2011).
  31. N. R. Chebrolu, B. L. Chittari, and J. Jung, Flat bands in twisted double bilayer graphene, Phys. Rev. B 99, 235417 (2019).
  32. S. Lisi, X. Lu, T. Benschop, T. A. de Jong, P. Stepanov, J. R. Duran, F. Margot, I. Cucchi, E. Cappelli, A. Hunter, et al., Observation of flat bands in twisted bilayer graphene, Nat. Phys. 17, 189 (2021).
  33. J. Schulenburg, A. Honecker, J. Schnack, J. Richter, and H.-J. Schmidt, Macroscopic magnetization jumps due to independent magnons in frustrated quantum spin lattices, Phys. Rev. Lett. 88, 167207 (2002).
  34. M. E. Zhitomirsky and H. Tsunetsugu, Exact low-temperature behavior of a kagomé antiferromagnet at high fields, Phys. Rev. B 70, 100403 (2004).
  35. S. Tacchi, J. Flores-Farías, D. Petti, F. Brevis, A. Cattoni, G. Scaramuzzi, D. Girardi, D. Cortés-Ortuño, R. A. Gallardo, E. Albisetti, G. Carlotti, and P. Landeros, Experimental observation of flat bands in one-dimensional chiral magnonic crystals, Nano Lett. 23, 6776 (2023).
  36. F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, Bosonic condensation and disorder-induced localization in a flat band, Phys. Rev. Lett. 116, 066402 (2016).
  37. A. A. Daniel Leykam and S. Flach, Artificial flat band systems: From lattice models to experiments, Adv. Phys.: X 3, 1473052 (2018).
  38. T.-X. Ma, Q.-S. Fan, C. Zhang, and Y.-S. Wang, Acoustic flatbands in phononic crystal defect lattices, J. Appl. Phys. 129, (2021).
  39. M. Pan, X. Zhang, Y. Zhou, P. Wang, Q. Bian, H. Liu, X. Wang, X. Li, A. Chen, X. Lei, S. Li, Z. Cheng, Z. Shao, H. Ding, J. Gao, F. Li, and F. Liu, Growth of mesoscale ordered two-dimensional hydrogen-bond organic framework with the observation of flat band, Phys. Rev. Lett. 130, 036203 (2023).
  40. Z. Liu, Z.-F. Wang, J.-W. Mei, Y.-S. Wu, and F. Liu, Flat Chern band in a two-dimensional organometallic framework, Phys. Rev. Lett. 110, 106804 (2013).
  41. M. G. Yamada, T. Soejima, N. Tsuji, D. Hirai, M. Dincă, and H. Aoki, First-principles design of a half-filled flat band of the kagome lattice in two-dimensional metal-organic frameworks, Phys. Rev. B 94, 081102 (2016).
  42. T. Bilitewski and R. Moessner, Disordered flat bands on the kagome lattice, Phys. Rev. B 98, 235109 (2018).
  43. D. L. Bergman, C. Wu, and L. Balents, Band touching from real-space topology in frustrated hopping models, Phys. Rev. B 78, 125104 (2008).
  44. One way to understand this is that as one changes the strength of the magnetic field the magnetic Brillouin zone changes discontinuously, and in the weak-field limit the size of the magnetic Brillouin zone diverges. For example, electrons on a square lattice in a uniform magnetic field show a fractal-like band structure known as the Hofstadter butterfly. Such a fine structure is smeared out if we introduce dissipation on a lattice system.
  45. H. Fukuyama, Theory of Hall effect. II: Bloch electrons, Prog. Theor. Phys. 42, 1284 (1969).
  46. Y. Fuseya, M. Ogata, and H. Fukuyama, Transport properties and diamagnetism of Dirac electrons in bismuth, J. Phys. Soc. Jpn. 84, 012001 (2015).
  47. V. Könye and M. Ogata, Microscopic theory of magnetoconductivity at low magnetic fields in terms of Berry curvature and orbital magnetic moment, Phys. Rev. Res. 3, 033076 (2021).
  48. R. Peierls, Zur theorie des diamagnetismus von leitungselektronen, Z. Phys. 80, 763 (1933).
  49. H. Fukuyama, Theory of orbital magnetism of Bloch electrons: Coulomb interactions, Prog. Theor. Phys. 45, 704 (1971).
  50. M. Ogata and H. Fukuyama, Orbital magnetism of Bloch electrons I. General formula, J. Phys. Soc. Jpn. 84, 124708 (2015).
  51. M. Ogata, Theory of magnetization in Bloch electron systems, J. Phys. Soc. Jpn. 86, 044713 (2017).
  52. M. Ogata, Theory of Hall conductivity and its exact relationship to orbital magnetic susceptibility, J. Phys. Soc. Jpn. 93, 013703 (2024).
  53. J. Mitscherling and T. Holder, Bound on resistivity in flat-band materials due to the quantum metric, Phys. Rev. B 105, 085154 (2022).
  54. B. Mera and J. Mitscherling, Nontrivial quantum geometry of degenerate flat bands, Phys. Rev. B 106, 165133 (2022).
  55. K.-E. Huhtinen and P. Törmä, Conductivity in flat bands from the Kubo-Greenwood formula, Phys. Rev. B 108, 155108 (2023).
  56. Y.-Q. Ma, S. Chen, H. Fan, and W.-M. Liu, Abelian and non-Abelian quantum geometric tensor, Phys. Rev. B 81, 245129 (2010).
  57. H.-T. Ding, C.-X. Zhang, J.-X. Liu, J.-T. Wang, D.-W. Zhang, and S.-L. Zhu, Non-Abelian quantum geometric tensor in degenerate topological semimetals, Phys. Rev. A 109, 043305 (2024).
  58. In principle, our approach can include the spin-orbit coupling. However, our formulation for including the effect of an external magnetic field relies on the Peierls substitution, which introduces a phase arising from the magnetic field when an electron hops from one site to another. When we introduce spin-orbit coupling, the term arising from an electron hopping intertwined with spin degrees of freedom is allowed. Since this hopping term acquires an additional Peierls phase, the weak-field limit will be much more complicated. Including the Zeeman effect should be straightforward.
  59. A. Raoux, F. Piéchon, J.-N. Fuchs, and G. Montambaux, Orbital magnetism in coupled-bands models, Phys. Rev. B 91, 085120 (2015).
  60. K.-T. Chen and P. A. Lee, Unified formalism for calculating polarization, magnetization, and more in a periodic insulator, Phys. Rev. B 84, 205137 (2011).
  61. B. Savoie, On the zero-field orbital magnetic susceptibility of Bloch electrons in graphene-like solids: Some rigorous results, J. Math. Phys. 53, 073302 (2012).
  62. R. Nourafkan and A.-M. S. Tremblay, Hall and Faraday effects in interacting multiband systems with arbitrary band topology and spin-orbit coupling, Phys. Rev. B 98, 165130 (2018).
  63. G. Gómez-Santos and T. Stauber, Measurable lattice effects on the charge and magnetic response in graphene, Phys. Rev. Lett. 106, 045504 (2011).
  64. P. Streda, Theory of quantised Hall conductivity in two dimensions, J. Phys. C 15, L717 (1982).
  65. A. Widom, Thermodynamic derivation of the Hall effect current, Phys. Lett. A 90, 474 (1982).
  66. P. Streda and L. Smrcka, Thermodynamic derivation of the Hall current and the thermopower in quantising magnetic field, J. Phys. C 16, L895 (1983).
  67. We note that we consider the strong topology (independent of the spatial symmetry and characterized by the dimension of the whole system) to define the topologically trivial isolated flat bands. The trace of the non-Abelian Berry curvature vanishes in our setting, since the system preserves time-reversal symmetry (when the external magnetic field is absent) and lacks spin-orbit coupling; hence, we call the system topologically trivial.
  68. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5s6l-vc71 for details of the derivation, model descriptions, plots of the geometric part of the formula, the parameter dependence of the two components that comprise the formula, and the modification of the formula by a small finite dispersion of the flat band for the two-band model.
  69. A. Mielke, Ferromagnetism in the Hubbard model and Hund's rule, Phys. Lett. A 174, 443 (1993).
  70. A. Mielke, Stability of ferromagnetism in Hubbard models with degenerate single-particle ground states, J. Phys. A: Math. Gen. 32, 8411 (1999).

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Phys. Rev. Research 8, L012041– Published 23 February, 2026
  • Received 22 July 2025
  • Accepted 30 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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