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Competing magnetic and topological orders in the spin-1 Kitaev-Heisenberg chain with single-ion anisotropy

Sahinur Reja1 and Satoshi Nishimoto2,3

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Phys. Rev. B 114, 084406 – Published 10 August, 2026

DOI: https://doi.org/10.1103/z6xd-4697

Abstract

We investigate the ground-state phase diagram of the spin-1 Kitaev–Heisenberg (KH) chain in the presence of uniaxial single-ion anisotropy Dz using the density-matrix renormalization-group method. The relative strength of the Heisenberg and Kitaev interactions is parametrized by a coupling angle ϕ, defined by J=cosϕ and K=sinϕ. By combining energy-based diagnostics with finite-size extrapolations of order parameters and correlation functions, we obtain a comprehensive phase diagram as a function of ϕ and Dz. The phase diagram contains ferromagnetic and Néel ordered phases, collinear states with four-site spin modulation, magnetically disordered or critical regimes, two Kitaev-dominated spin-liquid-like regions, and a topological Haldane phase near the Heisenberg limit. We find that the Kitaev-dominated disordered regions acquire finite parameter widths in the spin-1 model, whereas the Haldane phase is fragile against Kitaev-type anisotropy, especially for Dz<0. Positive Dz generally suppresses magnetic order and expands nonmagnetic regimes, while negative Dz enhances Ising-like magnetic order. At Dz=0, we also identify an exactly solvable point at tanϕ=−2(ϕ/π≃1.6476), which gives rise to a first-order transition between the Néel-z state and a four-site collinear state. We further contrast these findings with the spin-1/2 KH chain and with the spin-1 honeycomb KH model, highlighting the distinct roles of single-ion anisotropy and dimensionality in Kitaev-type magnets.

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References (60)

  1. L. Balents, Spin liquids in frustrated magnets, Nature (London) 464, 199 (2010).
  2. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
  3. Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017).
  4. J. Knolle and R. Moessner, A field guide to spin liquids, Annu. Rev. Condens. Matter Phys. 10, 451 (2019).
  5. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  6. P. Anderson, Resonating valence bonds: A new kind of insulator? Mater. Res. Bull. 8, 153 (1973).
  7. X. G. Wen, Mean-field theory of spin-liquid states with finite energy gap and topological orders, Phys. Rev. B 44, 2664 (1991).
  8. D. Aasen, R. S. K. Mong, B. M. Hunt, D. Mandrus, and J. Alicea, Electrical probes of the non-Abelian spin liquid in Kitaev materials, Phys. Rev. X 10, 031014 (2020).
  9. S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, Models and materials for generalized Kitaev magnetism, J. Phys.: Condens. Matter 29, 493002 (2017).
  10. S. Trebst and C. Hickey, Kitaev materials, Phys. Rep. 950, 1 (2022).
  11. A. I. Kurbakov, A. N. Korshunov, S. Y. Podchezertsev, M. I. Stratan, G. V. Raganyan, and E. A. Zvereva, Long-range and short-range ordering in 2D honeycomb-lattice magnet Na2Ni2TeO6, J. Alloys Compd. 820, 153354 (2020).
  12. A. M. Samarakoon, Q. Chen, H. Zhou, and V. O. Garlea, Static and dynamic magnetic properties of honeycomb lattice antiferromagnets Na2M2TeO6, M=Co and Ni, Phys. Rev. B 104, 184415 (2021).
  13. A. K. Bera, S. M. Yusuf, L. Keller, F. Yokaichiya, and J. R. Stewart, Magnetism of two-dimensional honeycomb layered Na2Ni2TeO6 driven by intermediate Na-layer crystal structure, Phys. Rev. B 105, 014410 (2022).
  14. E. A. Zvereva, M. I. Stratan, Y. A. Ovchenkov, V. B. Nalbandyan, J.-Y. Lin, E. L. Vavilova, M. F. Iakovleva, M. Abdel-Hafiez, A. V. Silhanek, X.-J. Chen, A. Stroppa, S. Picozzi, H. O. Jeschke, R. Valentí, and A. N. Vasiliev, Zigzag antiferromagnetic quantum ground state in monoclinic honeycomb lattice antimonates A3Ni2SbO6(A=Li,Na), Phys. Rev. B 92, 144401 (2015).
  15. A. I. Kurbakov, A. N. Korshunov, S. Y. Podchezertsev, A. L. Malyshev, M. A. Evstigneeva, F. Damay, J. Park, C. Koo, R. Klingeler, E. A. Zvereva, and V. B. Nalbandyan, Zigzag spin structure in layered honeycomb Li3Ni2SbO6: A combined diffraction and antiferromagnetic resonance study, Phys. Rev. B 96, 024417 (2017).
  16. V. B. Nalbandyan, I. L. Shukaev, Y. V. Popov, A. N. Vasiliev, and T. M. Vasilchikova, Structure and magnetic properties of high-entropy honeycomb-layered A3Q2SbO6 (A = Li or Na; Q = Co1/5Ni1/5Cu1/5Mg1/5Zn1/5), Dalton Trans. 54, 13738 (2025).
  17. Y. Shangguan, S. Bao, Z.-Y. Dong, N. Xi, Y.-P. Gao, Z. Ma, W. Wang, Z. Qi, S. Zhang, Z. Huang, et al., A one-third magnetization plateau phase as evidence for the Kitaev interaction in a honeycomb-lattice antiferromagnet, Nat. Phys. 19, 1883 (2023).
  18. K. M. Taddei, V. O. Garlea, A. M. Samarakoon, L. D. Sanjeewa, J. Xing, T. W. Heitmann, C. dela Cruz, A. S. Sefat, and D. Parker, Zigzag magnetic order and possible Kitaev interactions in the spin-1 honeycomb lattice KNiAsO4, Phys. Rev. Res. 5, 013022 (2023).
  19. A. Koga, H. Tomishige, and J. Nasu, Ground-state and thermodynamic properties of an S = 1 Kitaev model, J. Phys. Soc. Jpn. 87, 063703 (2018).
  20. J. Oitmaa, A. Koga, and R. R. P. Singh, Incipient and well-developed entropy plateaus in spin-S Kitaev models, Phys. Rev. B 98, 214404 (2018).
  21. X.-Y. Dong and D. Sheng, Spin-1 Kitaev-Heisenberg model on a honeycomb lattice, Phys. Rev. B 102, 121102(R) (2020).
  22. Z. Zhu, Z.-Y. Weng, and D. N. Sheng, Magnetic field induced spin liquids in S=1 Kitaev honeycomb model, Phys. Rev. Res. 2, 022047(R) (2020).
  23. C. Hickey, C. Berke, P. P. Stavropoulos, H.-Y. Kee, and S. Trebst, Field-driven gapless spin liquid in the spin-1 Kitaev honeycomb model, Phys. Rev. Res. 2, 023361 (2020).
  24. H.-Y. Lee, N. Kawashima, and Y. B. Kim, Tensor network wave function of S=1 Kitaev spin liquids, Phys. Rev. Res. 2, 033318 (2020).
  25. I. Khait, P. P. Stavropoulos, H.-Y. Kee, and Y. B. Kim, Characterizing spin-one Kitaev quantum spin liquids, Phys. Rev. Res. 3, 013160 (2021).
  26. O. Bradley and R. R. P. Singh, Instabilities of spin-1 Kitaev spin liquid phase in presence of single-ion anisotropies, Phys. Rev. B 105, L060405 (2022).
  27. K. Fukui, Y. Kato, J. Nasu, and Y. Motome, Ground-state phase diagram of spin-S Kitaev-Heisenberg models, Phys. Rev. B 106, 174416 (2022).
  28. R. Pohle, N. Shannon, and Y. Motome, Spin nematics meet spin liquids: Exotic quantum phases in the spin-1 bilinear-biquadratic model with Kitaev interactions, Phys. Rev. B 107, L140403 (2023).
  29. P. M. Cônsoli, L. Janssen, M. Vojta, and E. C. Andrade, Heisenberg-Kitaev model in a magnetic field: 1/S expansion, Phys. Rev. B 102, 155134 (2020).
  30. Y.-H. Chen, J. Genzor, Y. B. Kim, and Y.-J. Kao, Excitation spectrum of spin-1 Kitaev spin liquids, Phys. Rev. B 105, L060403 (2022).
  31. Y. Chen, Y.-C. He, and A. Szasz, Phase diagrams of spin-S Kitaev ladders, Phys. Rev. B 108, 045124 (2023).
  32. T. Mashiko and T. Okubo, Quantum phase transition between spin liquid and spin nematics in spin-1 Kitaev honeycomb model, Phys. Rev. Res. 6, 033110 (2024).
  33. A. Ralko and J. Merino, Chiral bosonic quantum spin liquid in the integer-spin Heisenberg-Kitaev model, Phys. Rev. B 110, 134402 (2024).
  34. A. Singhania, J. van den Brink, and S. Nishimoto, Emergence of vortex state in the S=1 Kitaev-Heisenberg model with single-ion anisotropy, Phys. Rev. Res. 6, 033146 (2024).
  35. Q. Luo, J. Zhao, J. Li, and X. Wang, Chiral spin state and nematic ferromagnet in the spin-1 Kitaev-Γ model, Phys. Rev. B 110, 035121 (2024).
  36. M. Georgiou, I. Rousochatzakis, D. J. J. Farnell, J. Richter, and R. F. Bishop, Spin-S Kitaev-Heisenberg model on the honeycomb lattice: A high-order treatment via the many-body coupled cluster method, Phys. Rev. Res. 6, 033168 (2024).
  37. D. Sasamoto and J. Nasu, Schwinger boson theory for S=1 Kitaev quantum spin liquids, Phys. Rev. Res. 8, 033138 (2026).
  38. J. Chaloupka, G. Jackeli, and G. Khaliullin, Kitaev-Heisenberg model on a honeycomb lattice: Possible exotic phases in iridium oxides A2IrO3, Phys. Rev. Lett. 105, 027204 (2010).
  39. C. E. Agrapidis, J. van den Brink, and S. Nishimoto, Ordered states in the Kitaev-Heisenberg model: From 1D chains to 2D honeycomb, Sci. Rep. 8, 1815 (2018).
  40. W. Yang, A. Nocera, and I. Affleck, Comprehensive study of the phase diagram of the spin-12 Kitaev-Heisenberg-Γ chain, Phys. Rev. Res. 2, 033268 (2020).
  41. W. Yang, C. Xu, S. Ma, A. Nocera, and I. Affleck, Left-left-right-right magnetic order in spin-12 Kitaev-Heisenberg chain, Phys. Rev. B 112, 035104 (2025).
  42. W.-L. You, G. Sun, J. Ren, W. C. Yu, and A. M. Oleś, Quantum phase transitions in the spin-1 Kitaev-Heisenberg chain, Phys. Rev. B 102, 144437 (2020).
  43. W.-L. You, Z. Zhao, J. Ren, G. Sun, L. Li, and A. M. Oleś, Quantum many-body scars in spin-1 Kitaev chains, Phys. Rev. Res. 4, 013103 (2022).
  44. W.-Y. Zhang, Y.-N. Wang, D. Liu, J. Ren, J. Li, N. Wu, A. M. Oleś, and W.-L. You, Quantum many-body scars in spin-1 Kitaev chain with uniaxial single-ion anisotropy, Phys. Rev. B 108, 104411 (2023).
  45. W.-Y. Zhang, Q.-M. Hu, J. Ren, L. Li, and W.-L. You, Exploring quantum criticality and ergodicity-breaking dynamics in spin-1 Kitaev chains via single-ion anisotropies, Phys. Rev. B 110, 134441 (2024).
  46. F. Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model, Phys. Lett. A 93, 464 (1983).
  47. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
  48. Q. Luo, S. Hu, J. Li, J. Zhao, H.-Y. Kee, and X. Wang, Spontaneous dimerization, spin-nematic order, and deconfined quantum critical point in a spin-1 Kitaev chain with tunable single-ion anisotropy, Phys. Rev. B 107, 245131 (2023).
  49. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  50. J. Chaloupka, G. Jackeli, and G. Khaliullin, Zigzag magnetic order in the iridium oxide Na2IrO3, Phys. Rev. Lett. 110, 097204 (2013).
  51. M. Kadosawa, M. Nakamura, Y. Ohta, and S. Nishimoto, Phase diagram of the Kitaev-Heisenberg model using various finite-size clusters, J. Phys. Soc. Jpn. 92, 055001 (2023).
  52. M. den Nijs and K. Rommelse, Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains, Phys. Rev. B 40, 4709 (1989).
  53. T. Kennedy and H. Tasaki, Hidden Z2×Z2 symmetry breaking in Haldane-gap antiferromagnets, Phys. Rev. B 45, 304 (1992).
  54. M. Oshikawa, Hidden Z2*Z2 symmetry in quantum spin chains with arbitrary integer spin, J. Phys.: Condens. Matter 4, 7469 (1992).
  55. S. R. White and D. A. Huse, Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S = 1 Heisenberg chain, Phys. Rev. B 48, 3844 (1993).
  56. W. Chen, K. Hida, and B. C. Sanctuary, Ground-state phase diagram of S=1XXZ chains with uniaxial single-ion-type anisotropy, Phys. Rev. B 67, 104401 (2003).
  57. P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  58. S. Nishimoto, Tomonaga-Luttinger-liquid criticality: Numerical entanglement entropy approach, Phys. Rev. B 84, 195108 (2011).
  59. J. Hauschild and F. Pollmann, Efficient numerical simulations with tensor networks: Tensor Network Python (TeNPy), SciPost Phys. Lect. Notes 5 (2018).
  60. S. Reja and S. Nishimoto, Dataset for “Competing magnetic and topological orders in the spin-1 Kitaev-Heisenberg chain with single-ion anisotropy” [Data set], Zenodo, 2026, https://doi.org/10.5281/zenodo.21470563.

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Phys. Rev. B 114, 084406– Published 10 August, 2026
  • Received 23 December 2025
  • Revised 16 July 2026
  • Accepted 20 July 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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