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Nature of the topological transition of the Kitaev model in [111] magnetic field

S. Thiagarajan1, C. Watson1, T. Yzeiri1,2, H. Hu3, B. Uchoa4, and F. Krüger1,2,*

  • *Contact author: f.kruger@ucl.ac.uk
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Phys. Rev. B 113, 085108 – Published 5 February, 2026

DOI: https://doi.org/10.1103/jjj6-cx8l

Abstract

We investigate the nature of the topological phase transition of the antiferromagnetic Kitaev model on a honeycomb lattice in the presence of a magnetic field along the [111] direction. The field opens a topological gap in the Majorana fermion spectrum and leads to a sequence of topological phase transitions before the field-polarized state is reached. At mean-field level, the gap first closes at the three M points in the Brillouin zone, where the Majorana fermions form Dirac cones, resulting in a change of Chern number by three. An odd number of Dirac fermions in infrared is unusual and requires Berry curvature compensation in ultraviolet, which occurs via topological, ringlike hybridization gaps with higher-energy bands. We perform a renormalization-group analysis of the topological phase transition at the three M points within the Yukawa theory, allowing for intravalley and intervalley fluctuations of the spin-liquid bond operators. We find that the latter lead to a breaking of Lorentz invariance and hence a different universality compared with the standard Ising Gross-Neveu-Yukawa class.

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References (43)

  1. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  2. G. Jackeli and G. Khaliullin, Mott insulators in the strong spin-orbit coupling limit: From Heisenberg to a quantum compass and Kitaev models, Phys. Rev. Lett. 102, 017205 (2009).
  3. T. Takayama, A. Kato, R. Dinnebier, J. Nuss, H. Kono, L. S. I. Veiga, G. Fabbris, D. Haskel, and H. Takagi, Hyperhoneycomb iridate β−Li2IrO3 as a platform for Kitaev magnetism, Phys. Rev. Lett. 114, 077202 (2015).
  4. S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I. Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P. Gegenwart, K. R. Choi, et al., Spin waves and revised crystal structure of honeycomb iridate Na2IrO3, Phys. Rev. Lett. 108, 127204 (2012).
  5. S. Hwan Chun, J.-W. Kim, J. Kim, H. Zheng, C. C. Stoumpos, C. D. Malliakas, J. F. Mitchell, K. Mehlawat, Y. Singh, Y. Choi, et al., Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na2IrO3, Nat. Phys. 11, 462 (2015).
  6. P. Warzanowski, N. Borgwardt, K. Hopfer, J. Attig, T. C. Koethe, P. Becker, V. Tsurkan, A. Loidl, M. Hermanns, P. H. M. van Loosdrecht, et al., Multiple spin-orbit excitons and the electronic structure of α−RuCl3, Phys. Rev. Res. 2, 042007(R) (2020).
  7. K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, α−RuCl3: A spin-orbit assisted Mott insulator on a honeycomb lattice, Phys. Rev. B 90, 041112(R) (2014).
  8. X.-G. Zhou, H. Li, Y. H. Matsuda, A. Matsuo, W. Li, N. Kurita, G. Su, K. Kindo, and H. Tanaka, Possible intermediate quantum spin liquid phase in α-RuCl3 under high magnetic fields up to 100 T, Nat. Commun. 14, 5613 (2023).
  9. H. Suzuki, H. Liu, J. Bertinshaw, K. Ueda, H. Kim, S. Laha, D. Weber, Z. Yang, L. Wang, H. Takahashi, et al., Proximate ferromagnetic state in the Kitaev model material α-RuCl3, Nat. Commun. 12, 4512 (2021).
  10. X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. Kim, H. Gretarsson, Y. Singh, P. Gegenwart, and J. P. Hill, Long-range magnetic ordering in Na2IrO3, Phys. Rev. B 83, 220403(R) (2011).
  11. F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. Fernandez-Baca, R. Custelcean, T. F. Qi, O. B. Korneta, and G. Cao, Direct evidence of a zigzag spin-chain structure in the honeycomb lattice: A neutron and x-ray diffraction investigation of single-crystal Na2IrO3, Phys. Rev. B 85, 180403(R) (2012).
  12. J. A. Sears, Y. Zhao, Z. Xu, J. W. Lynn, and Y.-J. Kim, Phase diagram of α−RuCl3 in an in-plane magnetic field, Phys. Rev. B 95, 180411(R) (2017).
  13. A. U. B. Wolter, L. T. Corredor, L. Janssen, K. Nenkov, S. Schönecker, S.-H. Do, K.-Y. Choi, R. Albrecht, J. Hunger, T. Doert, et al., Field-induced quantum criticality in the Kitaev system α−RuCl3, Phys. Rev. B 96, 041405(R) (2017).
  14. Y.-F. Jiang, T. P. Devereaux, and H.-C. Jiang, Field-induced quantum spin liquid in the Kitaev-Heisenberg model and its relation to α−RuCl3, Phys. Rev. B 100, 165123 (2019).
  15. S. Pradhan, N. D. Patel, and N. Trivedi, Two-magnon bound states in the Kitaev model in a [111] field, Phys. Rev. B 101, 180401(R) (2020).
  16. C. Hickey and S. Trebst, Emergence of a field-driven U(1) spin liquid in the Kitaev honeycomb model, Nat. Commun. 10, 530 (2019).
  17. F. Yılmaz, A. P. Kampf, and S. K. Yip, Phase diagrams of Kitaev models for arbitrary magnetic field orientations, Phys. Rev. Res. 4, 043024 (2022).
  18. A. Ralko and J. Merino, Novel chiral quantum spin liquids in Kitaev magnets, Phys. Rev. Lett. 124, 217203 (2020).
  19. S.-S. Zhang, G. B. Halász, and C. D. Batista, Theory of the Kitaev model in a [111] magnetic field, Nat. Commun. 13, 399 (2022).
  20. M. Gohlke, R. Moessner, and F. Pollmann, Dynamical and topological properties of the Kitaev model in a [111] magnetic field, Phys. Rev. B 98, 014418 (2018).
  21. K. B. Yogendra, T. Das, and G. Baskaran, Emergent glassiness in the disorder-free Kitaev model: Density matrix renormalization group study on a one-dimensional ladder setting, Phys. Rev. B 108, 165118 (2023).
  22. H. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (I). Proof by homotopy theory, Nucl. Phys. B 185, 20 (1981).
  23. H. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (II). Intuitive topological proof, Nucl. Phys. B 193, 173 (1981).
  24. H. Nielsen and M. Ninomiya, A no-go theorem for regularizing chiral fermions, Phys. Lett. B 105, 219 (1981).
  25. H. Suzuki, A no-go theorem for the Majorana fermion on a lattice, Prog. Theor. Phys. 112, 855 (2004).
  26. S. D. Drell, M. Weinstein, and S. Yankielowicz, Strong-coupling field theories. II. Fermions and gauge fields on a lattice, Phys. Rev. D 14, 1627 (1976).
  27. T. C. Lang and A. M. Läuchli, Quantum Monte Carlo simulation of the chiral Heisenberg Gross-Neveu-Yukawa phase transition with a single Dirac cone, Phys. Rev. Lett. 123, 137602 (2019).
  28. Y. D. Liao, X. Y. Xu, Z. Y. Meng, and Y. Qi, Caution on Gross-Neveu criticality with a single Dirac cone: Violation of locality and its consequence of unexpected finite-temperature transition, Phys. Rev. B 108, 195112 (2023).
  29. B. H. Wellegehausen, D. Schmidt, and A. Wipf, Critical flavor number of the Thirring model in three dimensions, Phys. Rev. D 96, 094504 (2017).
  30. Z. Wang, F. Assaad, and M. Ulybyshev, Validity of SLAC fermions for the (1+1)-dimensional helical Luttinger liquid, Phys. Rev. B 108, 045105 (2023).
  31. K. G. Wilson, Confinement of quarks, Phys. Rev. D 10, 2445 (1974).
  32. P. H. Ginsparg and K. G. Wilson, A remnant of chiral symmetry on the lattice, Phys. Rev. D 25, 2649 (1982).
  33. D. B. Kaplan, A method for simulating chiral fermions on the lattice, Phys. Lett. B 288, 342 (1992).
  34. R. Narayanan and H. Neuberger, A construction of lattice chiral gauge theories, Nucl. Phys. B 443, 305 (1995).
  35. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006).
  36. X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors, Phys. Rev. B 74, 085308 (2006).
  37. H. Hu and F. Krüger, Nature of topological phase transition of Kitaev quantum spin liquids, Phys. Rev. Lett. 133, 146603 (2024).
  38. H. Isobe, B.-J. Yang, A. Chubukov, J. Schmalian, and N. Nagaosa, Emergent non-Fermi-liquid at the quantum critical point of a topological phase transition in two dimensions, Phys. Rev. Lett. 116, 076803 (2016).
  39. M. D. Uryszek, F. Krüger, and E. Christou, Fermionic criticality of anisotropic nodal point semimetals away from the upper critical dimension: Exact exponents to leading order in 1Nf, Phys. Rev. Res. 2, 043265 (2020).
  40. A. N. Vasil'ev, S. É. Derkachev, N. A. Kivel’, and A. S. Stepanenko, The 1/n expansion in the Gross-Neveu model: Conformal bootstrap calculation of the index η in order 1/n3, Theor. Math. Phys. 94, 127 (1993).
  41. J. Gracey, Calculation of the exponent η to O(1/N2) in the O(N) Gross Neveu model, Int. J. Mod. Phys. A 06, 395 (1991).
  42. J. Gracey, Anomalous mass dimension at O(1/N2) in the O(N) Gross-Neveu model, Phys. Lett. B 297, 293 (1992).
  43. J. Gracey, Computation of critical exponent η at O(1/N3) in the four-fermion model in arbitrary dimension, Int. J. Mod. Phys. A 09, 727 (1994).

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Phys. Rev. B 113, 085108– Published 5 February, 2026
  • Received 16 September 2025
  • Revised 14 January 2026
  • Accepted 20 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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