- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review B

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Electric field effects in the one-dimensional spin-12K1J1Γ1Γ1′K2J2 model with ferromagnetic Kitaev coupling

Wang Yang1, Helin Wang1, and Chao Xu2,3

PDF

Phys. Rev. B 113, 134444 – Published 29 April, 2026

DOI: https://doi.org/10.1103/wrk3-c6bb

Abstract

We perform a systematic study on the effects of electric fields in the Luttinger liquid phase of the one-dimensional spin-1/2 K1J1Γ1Γ1′K2J2 model in the region of ferromagnetic nearest-neighboring Kitaev coupling. We find that while electric fields along (1,1,1) direction maintain the Luttinger liquid behavior, fields along other directions drive the system to a dimerized state. An estimation is made on how effective a (1,1,1) field is for tuning the Luttinger parameter in real materials. Our work is useful for understanding the effects of electric fields in one-dimensional generalized Kitaev spin models, and provides a starting point for exploring the electric-field-related physics in two dimensions based on a quasi-one-dimensional approach.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (92)

  1. 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).
  2. 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).
  3. Y. Singh and P. Gegenwart, Antiferromagnetic mott insulating state in single crystals of the honeycomb lattice material Na2IrO3, Phys. Rev. B 82, 064412 (2010).
  4. C. C. Price and N. B. Perkins, Critical properties of the Kitaev-Heisenberg Model, Phys. Rev. Lett. 109, 187201 (2012).
  5. Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Relevance of the Heisenberg-Kitaev model for the honeycomb lattice iridates A2IrO3, Phys. Rev. Lett. 108, 127203 (2012).
  6. 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).
  7. H.-S. Kim, V. S. V., A. Catuneanu, H.-Y. Kee, Kitaev magnetism in honeycomb RuCl3 with intermediate spin-orbit coupling, Phys. Rev. B 91, 241110(R) (2015).
  8. S. M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, Challenges in design of Kitaev materials: Magnetic interactions from competing energy scales, Phys. Rev. B 93, 214431 (2016).
  9. S.-H. Baek, S.-H. Do, K.-Y. Choi, Y. S. Kwon, A. U. B. Wolter, S. Nishimoto, J. van den Brink, B. Büchner, Evidence for a field-induced quantum spin liquid in α−RuCl3, Phys. Rev. Lett. 119, 037201 (2017).
  10. I. A. Leahy, C. A. Pocs, P. E. Siegfried, D. Graf, S. H. Do, K. Y. Choi, B. Normand, and M. Lee, Anomalous thermal conductivity and magnetic torque response in the honeycomb magnet α−RuCl3, Phys. Rev. Lett. 118, 187203 (2017).
  11. 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).
  12. A. U. B. Wolter, L. T. Corredor, L. Janssen, K. Nenkov, S. Schonecker, S. H. Do, K. Y. Choi, R. Albrecht, J. Hunger, T. Doert, M. Vojta, and B. Buchner, Field-induced quantum criticality in the Kitaev system α−RuCl3, Phys. Rev. B 96, 041405(R) (2017).
  13. J. Zheng, K. Ran, T. Li, J. Wang, P. Wang, B. Liu, Z.-X. Liu, B. Normand, J. Wen, and W. Yu, Gapless spin excitations in the field-induced quantum spin liquid phase of α−RuCl3, Phys. Rev. Lett. 119, 227208 (2017).
  14. I. Rousochatzakis and N. B. Perkins, Classical spin liquid instability driven by off-diagonal exchange in strong spin-orbit magnets, Phys. Rev. Lett. 118, 147204 (2017).
  15. Y. Kasahara, T. Ohnishi, N. Kurita, H. Tanaka, J. Nasu, Y. Motome, T. Shibauchi, and Y. Matsuda, Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid, Nature (London) (London) 559, 227 (2018).
  16. J. G. Rau, E. K. H. Lee, and H. Y. Kee, Generic spin model for the honeycomb iridates beyond the Kitaev limit, Phys. Rev. Lett. 112, 077204 (2014).
  17. K. Ran, J. Wang, W. Wang, Z.-Y. Dong, X. Ren, S. Bao, S. Li, Z. Ma, Y. Gan, Y. Zhang, J. T. Park, G. Deng, S. Danilkin, S.-L. Yu, J.-X. Li, and J. Wen, Spin-wave excitations evidencing the Kitaev interaction in single crystalline α−RuCl3, Phys. Rev. Lett. 118, 107203 (2017).
  18. W. Wang, Z.-Y. Dong, S.-L. Yu, and J.-X. Li, Theoretical investigation of magnetic dynamics in α−RuCl3, Phys. Rev. B 96, 115103 (2017).
  19. A. Catuneanu, Y. Yamaji, G. Wachtel, Y. B. Kim, and H.-Y. Kee, Path to stable quantum spin liquids in spin-orbit coupled correlated materials, npj Quantum Mater. 3, 23 (2018).
  20. M. Gohlke, G. Wachtel, Y. Yamaji, F. Pollmann, and Y. B. Kim, Quantum spin liquid signatures in Kitaev-like frustrated magnets, Phys. Rev. B 97, 075126 (2018).
  21. 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).
  22. J. Chaloupka, G. Jackeli, and G. Khaliullin, Zigzag magnetic order in the iridium oxide Na2IrO3, Phys. Rev. Lett. 110, 097204 (2013).
  23. R. D. Johnson, S. C. Williams, A. A. Haghighirad, J. Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O. Jeschke, R. Valentí, and R. Coldea, Monoclinic crystal structure of α−RuCl3 and the zigzag antiferromagnetic ground state, Phys. Rev. B 92, 235119 (2015).
  24. Motome, R. Sano, S. H. Jang, Y. Sugita, and Y. Kato, Materials design of Kitaev spin liquids beyond the Jackeli–Khaliullin mechanism, J. Phys.: Condens. Matter 32, 404001 (2020).
  25. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  26. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  27. M. Hermanns, I. Kimchi, and J. Knolle, Physics of the Kitaev model: Fractionalization, dynamic correlations, and material connections, Annu. Rev. Condens. Matter Phys. 9, 17 (2018).
  28. H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and realization of Kitaev quantum spin liquids, Nat. Rev. Phys. 1, 264 (2019).
  29. S. Trebst and C. Hickey, Kitaev materials, Phys. Rep. 950, 1 (2022).
  30. Y. Matsuda, T. Shibauchi, and H.-Y. Kee, Kitaev quantum spin liquids, Rev. Mod. Phys. 97, 045003 (2025).
  31. T. Yokoi, S. Ma, Y. Kasahara, S. Kasahara, T. Shibauchi, N. Kurita, H. Tanaka, J. Nasu, Y. Motome, C. Hickey, S. Trebst, and Y. Matsuda, Half-integer quantized anomalous thermal Hall effect in the Kitaev material α−RuCl3, Science 373, 568 (2021).
  32. J. A. N. Bruin, R. R. K. Singh, M. S. Griffin, Y. Kasahara, N. Kurita, H. Tanaka, and Y. Matsuda, Robustness of the thermal Hall effect close to half-quantization in α−RuCl3, Nat. Phys. 18, 401 (2022).
  33. P. Czajka, T. Gao, M. McGuire, D. G. Mandrus, P. Lampen-Kelley, J.-Q. Yan, S. E. Nagler, and N. P. Butch, Oscillations of the thermal conductivity in the spin-liquid state of α−RuCl3, Nat. Phys. 17, 915 (2021).
  34. H. Liu, J. Chaloupka, and G. Khaliullin, Exchange interactions in d5 Kitaev materials: From Na2IrO3 to α−RuCl3, Phys. Rev. B 105, 214411 (2022).
  35. F. D. M. Haldane, Effective harmonic-fluid approach to low-energy properties of one-dimensional quantum fluids, J. Phys. C: Solid State Phys. 14, 2585 (1981).
  36. F. D. M. Haldane, Luttinger liquid theory of one-dimensional quantum fluids: I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas, Phys. Rev. Lett. 47, 1840 (1981).
  37. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241, 333 (1984).
  38. V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess–Zumino model in two dimensions, Nucl. Phys. B 247, 83 (1984).
  39. I. Affleck, Critical behaviour of two-dimensional systems with continuous symmetries, Phys. Rev. Lett. 55, 1355 (1985).
  40. I. Affleck, Field theory methods and quantum critical phenomena, in Fields, Strings and Critical Phenomena, Proceedings of the Les Houches Summer School (1988), edited by E. Brézin and J. Zinn-Justin (North-Holland, Amsterdam, 1990), pp. 563–640.
  41. I. Affleck, Conformal field theory approach to the Kondo effect, Acta Phys. Polon. B 26, 1869 (1995).
  42. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  43. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  44. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  45. E. Sela, H.-C. Jiang, M. H. Gerlach, and S. Trebst, Order-by-disorder and spin-orbital liquids in a distorted Heisenberg-Kitaev model, Phys. Rev. B 90, 035113 (2014).
  46. C. E. Agrapidis, J. van den Brink, S. Nishimoto, Ordered states in the Kitaev-Heisenberg model: From 1D chains to 2D honeycomb, Sci. Rep. 8, 1815 (2018).
  47. C. E. Agrapidis, J. van den Brink, S. Nishimoto, Ground state and low-energy excitations of the Kitaev-Heisenberg two-leg ladder, Phys. Rev. B 99, 224418 (2019).
  48. A. Catuneanu, E. S. Sørensen, and H.-Y. Kee, Nonlocal string order parameter in the S=1/2 Kitaev-Heisenberg ladder, Phys. Rev. B 99, 195112 (2019).
  49. W. Yang, A. Nocera, T. Tummuru, H.-Y. Kee, and I. Affleck, Phase diagram of the spin-1/2 Kitaev-gamma chain and emergent SU(2) symmetry, Phys. Rev. Lett. 124, 147205 (2020).
  50. W. Yang, A. Nocera, and I. Affleck, Comprehensive study of the phase diagram of the spin-1/2 Kitaev-Heisenberg-Gamma chain, Phys. Rev. Res. 2, 033268 (2020).
  51. W. Yang, A. Nocera, and I. Affleck, Spin wave theory of a one-dimensional generalized Kitaev model, Phys. Rev. B 102, 134419 (2020).
  52. W. Yang, A. Nocera, E. S. Sørensen, H.-Y. Kee, and I. Affleck, Classical spin order near the antiferromagnetic Kitaev point in the spin-1/2 Kitaev-Gamma chain, Phys. Rev. B 103, 054437 (2021).
  53. W. Yang, A. Nocera, P. Herringer, R. Raussendorf, and I. Affleck, Symmetry analysis of bond-alternating Kitaev spin chains and ladders, Phys. Rev. B 105, 094432 (2022).
  54. W. Yang, C. Xu, S. Xu, A. Nocera, and I. Affleck, Nonsymmorphic Luttinger liquids in generalized antiferromagnetic Kitaev spin-1/2 chains, Phys. Rev. B 109, L180403 (2024).
  55. W. Yang, C. Xu, A. Nocera, and I. Affleck, Origin of nonsymmorphic bosonization formulas in generalized antiferromagnetic Kitaev spin-1/2 chains from a renormalization-group perspective, Phys. Rev. B 106, 064425 (2022).
  56. W. Yang, A. Nocera, C. Xu, S. Ma, A. Adhikary, and I. Affleck, Emergent SU(2)1 conformal symmetry in the spin-1/2 Kitaev-Gamma chain with a Dzyaloshinskii-Moriya interaction, Phys. Rev. B 111, 174414 (2025).
  57. W. Yang, C. Xu, S. Ma, A. Nocera, and I. Affleck, Left-left-right-right magnetic order in spin-1/2 Kitaev-Heisenberg chain, Phys. Rev. B 112, 035104 (2025).
  58. W. Yang, A. Nocera, C. Xu, and I. Affleck, Nonsymmorphic spin-space cubic groups and SU(2)1 conformal invariance in one-dimensional spin-1/2 models, SciPost Phys. 17, 097 (2024).
  59. Q. Luo, J. Zhao, X. Wang, and H.-Y. Kee, Unveiling the phase diagram of a bond-alternating spin-12K−Γ chain, Phys. Rev. B 103, 144423 (2021).
  60. Q. Luo, S. Hu, and H.-Y. Kee, Unusual excitations and double-peak specific heat in a bond-alternating spin-1 chain, Phys. Rev. Res. 3, 033048 (2021).
  61. E. S. Sørensen, A. Catuneanu, J. S. Gordon, and H.-Y. Kee, Heart of entanglement: Chiral, nematic, and incommensurate phases in the Kitaev-Gamma ladder in a field, Phys. Rev. X 11, 011013 (2021).
  62. Z.-A. Liu, T.-C. Yi, J.-H. Sun, Y.-L. Dong, and W.-L. You, Lifshitz phase transitions in a one-dimensional Gamma model, Phys. Rev. E 102, 032127 (2020).
  63. Z.-A. Liu, Y.-L. Dong, N. Wu, Y. Wang, and W.-L. You, Quantum criticality and correlations in the Ising-Gamma chain, Physica A 579, 126122 (2021).
  64. Z. Zhao, T.-C. Yi, M. Xue, and W.-L. You, Characterizing quantum criticality and steered coherence in the XY-Gamma chain, Phys. Rev. A 105, 063306 (2022).
  65. S. Kheiri, H. Cheraghi, S. Mahdavifar, and N. Sedlmayr, Information propagation in one-dimensional XY-Γ chains, Phys. Rev. B 109, 134303 (2024).
  66. S. Mahdavifar and D. C. Liu, Anisotropic spin-1/2 XXZ chains with uniform gamma interaction, Sci. Rep. 14, 30024 (2024).
  67. M. Abbasi, S. Mahdavifar, and M. Motamedifar, Spin-1/2 XX chains with modulated gamma interaction, SciPost Phys. Core 8, 001 (2025).
  68. X. Jin, D. C. Liu, Y. Shi, A. M. Chen, and Y. H. Su, Quantum phase transitions in the spin-1/2 XXZ model with staggered γ interaction, Phys. Rev. E 111, 014110 (2025).
  69. M. Abbasi, S. Mahdavifar, and M. Motamedifar, Hybrid critical line in the spin-1/2 XX chain with gamma interaction under transverse fields, Sci. Rep. 15, 20469 (2025).
  70. M. Abbasi, S. Mahdavifar, and M. Motamedifar, Concurrenceless and QDless line in the ground state phase diagram of the spin-1/2 XX chain with uniform gamma interaction, Results Phys. 76, 108362 (2025).
  71. H. Saito and C. Hotta, Exact matrix product states at the quantum Lifshitz tricritical point in a spin-1/2 zigzag-chain antiferromagnet with anisotropic Γ-term, Phys. Rev. Lett. 132, 166701 (2024).
  72. A. Ali, S. Elghaayda, S. Al-Kuwari, M. I. Hussain, M. T. Rahim, H. Kuniyil, C. Seida, A. El Allati, and M. Mansour, Super-extensive scaling of ergotropy in a spin-1/2 XY-Γ chain, arXiv:2411.14074.
  73. E. S. Sørensen, J. Gordon, J. Riddell, T. Wang, and H.-Y. Kee, Field-induced chiral soliton phase in the Kitaev spin chain, Phys. Rev. Res. 5, L012027 (2023).
  74. E. S. Sørensen, J. Riddell, and H.-Y. Kee, Islands of chiral solitons in integer-spin Kitaev chains, Phys. Rev. Res. 5, 013210 (2023).
  75. P. Laurell, G. Alvarez, and E. Dagotto, Spin dynamics of the generalized quantum spin compass chain, Phys. Rev. B 107, 104414 (2023).
  76. C. M. Morris, N. Desai, J. Viirok, D. Hüvonen, U. Nagel, T. Rõõm, J. W. Krizan, R. J. Cava, T. M. McQueen, S. M. Koohpayeh, R. K. Kaul, and N. P. Armitage, Duality and domain wall dynamics in a twisted Kitaev chain, Nat. Phys. 17, 832 (2021).
  77. D. Churchill and H.-Y. Kee, Transforming from Kitaev to disguised Ising chain: Application to CoNb2O6, Phys. Rev. Lett. 133, 056703 (2024).
  78. V. Subrahmanyam, Block entropy for Kitaev-type spin chains in a transverse field, Phys. Rev. A 88, 032315 (2013).
  79. M. Bhullar, H. Xu, and H. Y. Kee, Field-induced ordered phases in anisotropic spin-1/2 Kitaev chains, Phys. Rev. B 111, 104439 (2025).
  80. S. Birnkammer, J. Knolle, and M. Knap, Signatures of domain-wall confinement in Raman spectroscopy of Ising spin chains, Phys. Rev. B 110, 134408 (2024).
  81. A. Metavitsiadis and W. Brenig, Flux mobility delocalization in the Kitaev spin ladder, Phys. Rev. B 103, 195102 (2021).
  82. W.-L. You, Y. Wang, T.-C. Yi, C. Zhang, and A. M. Oles, Quantum coherence in a compass chain under an alternating transverse magnetic field, Phys. Rev. B 97, 224420 (2018).
  83. N. Wu and W.-L. You, Exact zero modes in a quantum compass chain under transverse fields, Phys. Rev. B 100, 085130 (2019).
  84. S. C. Furuya and M. Sato, Electric-field control of magnetic anisotropies: Applications to Kitaev spin liquids and topological spin textures, Phys. Rev. Res. 6, 013228 (2024).
  85. N. Laflorencie, E. S. Sørensen, M.-S. Chang, and I. Affleck, Boundary effects in the critical scaling of entanglement entropy in 1D systems, Phys. Rev. Lett. 96, 100603 (2006).
  86. P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009).
  87. I. Garate and I. Affleck, Interplay between symmetric exchange anisotropy, uniform Dzyaloshinskii-Moriya interaction, and magnetic fields in the phase diagram of quantum magnets and superconductors, Phys. Rev. B 81, 144419 (2010).
  88. S. Gangadharaiah, J. Sun, and O. A. Starykh, Spin-orbital effects in magnetized quantum wires and spin chains, Phys. Rev. B 78, 054436 (2008).
  89. A. P. Schnyder, O. A. Starykh, and L. Balents, Spatially anisotropic Heisenberg kagome antiferromagnet, Phys. Rev. B 78, 174420 (2008).
  90. R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veenstra, J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker, J. N. Hancock, D. van der Marel, I. S. Elfimov, and A. Damascelli, Na2IrO3 as a novel relativistic mott insulator with a 340-meV Gap, Phys. Rev. Lett. 109, 266406 (2012).
  91. W. Yang, A. Nocera, C. Xu, H.-Y. Kee, and I. Affleck, Counter-rotating spiral, zigzag, and 120∘ orders from coupled-chain analysis of Kitaev-Gamma-Heisenberg model, and relations to honeycomb iridates, arXiv:2207.02188.
  92. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).

Outline

Information

Phys. Rev. B 113, 134444– Published 29 April, 2026
  • Received 15 January 2026
  • Revised 19 March 2026
  • Accepted 6 April 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation