- Open Access
Magnetic field direction dependence of the spin liquid state in
Phys. Rev. Research 7, 043237 – Published 1 December, 2025
DOI: https://doi.org/10.1103/wnrc-cjkn
Abstract
Numerous experiments have shown that exhibits a spin liquid (SL) state near 7.1 T when an in-plane magnetic field is applied perpendicular to the Ru-Ru bond directions ([110]), while SL states are rarely observed under fields applied along the Ru-Ru bond ([100]). The underlying mechanism for this pronounced anisotropy remains elusive. In this work, we employ semiclassical spin dynamics simulations to investigate the magnetization evolution of with lattice distortion under varying external magnetic fields. It is found that each [100] direction has its own dominant orientation, and each [110] orientation marks the boundary, where the two alignments compete with each other. The magnetizations are influenced by multiple interactions and perturbations from lattice distortions, leading to the occurrence of the SL state in the [110] direction. These results provide new insights into the field-induced phase transitions in and the microscopic mechanisms underlying the emergence of spin liquid behavior.
Physics Subject Headings (PhySH)
Article Text
Supplemental Material
References (59)
- A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
- S. Kim, B. Yuan, and Y.-J. Kim, and other Kitaev materials, APL Mater. 10, 080903 (2022).
- 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 , Phys. Rev. Lett. 119, 227208 (2017).
- A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
- 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).
- A. Singhania, J. Van Den Brink, and S. Nishimoto, Disorder effects in the Kitaev-Heisenberg model, Phys. Rev. Res. 5, 023009 (2023).
- S. H. Baek, S. H. Do, K. Y. Choi, Y. S. Kwon, A. U. B. Wolter, S. Nishimoto, J. Van Den Brink, and B. Böchner, Evidence for a field-induced quantum spin liquid in , Phys. Rev. Lett. 119, 037201 (2017).
- S. Trebst and C. Hickey, Kitaev materials, Phys. Rep. 950, 1 (2022).
- 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).
- P. A. Maksimov and A. L. Chernyshev, Rethinking , Phys. Rev. Res. 2, 033011 (2020).
- W. Wang, Z. Y. Dong, S. L. Yu, and J. X. Li, Theoretical investigation of magnetic dynamics in , Phys. Rev. B 96, 115103 (2017).
- R. Yadav, N. A. Bogdanov, V. M. Katukuri, S. Nishimoto, J. Van Den Brink, and L. Hozoi, Kitaev exchange and field-induced quantum spin-liquid states in honeycomb , Sci. Rep. 6, 37925 (2016).
- Y. Tian, W. Gao, E. A. Henriksen, J. R. Chelikowsky, and L. Yang, Optically driven magnetic phase transition of monolayer , Nano Lett. 19, 7673 (2019).
- L. Wu et al., Field evolution of magnons in by high-resolution polarized terahertz spectroscopy, Phys. Rev. B 98, 094425 (2018).
- A. Sahasrabudhe et al., High-field quantum disordered state in : Spin flips, bound states, and multiparticle continuum, Phys. Rev. B 101, 140410(R) (2020).
- H. Li, H. K. Zhang, J. Wang, H. Q. Wu, Y. Gao, D. W. Qu, Z. X. Liu, S. S. Gong, and W. Li, Identification of magnetic interactions and high-field quantum spin liquid in , Nat. Commun. 12, 4007 (2021).
- X. Mi et al., Stacking faults in revealed by local electric polarization, Phys. Rev. B 103, 174413 (2021).
- I. S. Villadiego, Pseudoscalar U(1) spin liquids in , Phys. Rev. B 104, 195149 (2021).
- J. Wagner, A. Sahasrabudhe, R. B. Versteeg, L. Wysocki, Z. Wang, V. Tsurkan, A. Loidl, D. I. Khomskii, H. Hedayat, and P. H. M. van Loosdrecht, Magneto-optical study of metamagnetic transitions in the antiferromagnetic phase of , npj Quantum Mater. 7, 28 (2022).
- O. Tanaka, Y. Mizukami, R. Harasawa, K. Hashimoto, K. Hwang, N. Kurita, H. Tanaka, S. Fujimoto, Y. Matsuda, E.-G. Moon, and T. Shibauchi, Thermodynamic evidence for a field-angle-dependent Majorana gap in a Kitaev spin liquid, Nat. Phys. 18, 429 (2022).
- K. Hwang, A. Go, J. H. Seong, T. Shibauchi, and E. G. Moon, Identification of a Kitaev quantum spin liquid by magnetic field angle dependence, Nat. Commun. 13, 323 (2022).
- A. N. Ponomaryov, L. Zviagina, J. Wosnitza, P. Lampen-Kelley, A. Banerjee, J. Q. Yan, C. A. Bridges, D. G. Mandrus, S. E. Nagler, and S. A. Zvyagin, Nature of magnetic excitations in the high-field phase of , Phys. Rev. Lett. 125, 037202 (2020).
- A. N. Ponomaryov et al., Unconventional spin dynamics in the honeycomb-lattice material : High-field electron spin resonance studies, Phys. Rev. B 96, 241107(R) (2017).
- S. Bachus, D. A. S. Kaib, A. Jesche, V. Tsurkan, A. Loidl, S. M. Winter, A. A. Tsirlin, R. Valentí, and P. Gegenwart, Angle-dependent thermodynamics of , Phys. Rev. B 103, 054440 (2021).
- Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, Sixiao Ma, K. Sugii, 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) 559, 227 (2018).
- A. Loidl, P. Lunkenheimer, and V. Tsurkan, On the proximate Kitaev quantum-spin liquid : Thermodynamics, excitations and continua, J. Phys.: Condens. Matter 33, 443004 (2021).
- S. Widmann, V. Tsurkan, D. A. Prishchenko, V. G. Mazurenko, A. A. Tsirlin, and A. Loidl, Thermodynamic evidence of fractionalized excitations in , Phys. Rev. B 99, 094415 (2019).
- K. Imamura et al., Majorana-fermion origin of the planar thermal Hall effect in the Kitaev magnet , Sci. Adv. 10, eadk3539 (2024).
- V. Dantas and E. C. Andrade, Disorder, low-energy excitations, and topology in the Kitaev spin liquid, Phys. Rev. Lett. 129, 037204 (2022).
- D. Wulferding, Y. Choi, S. H. Do, C. H. Lee, P. Lemmens, C. Faugeras, Y. Gallais, and K. Y. Choi, Magnon bound states versus anyonic Majorana excitations in the Kitaev honeycomb magnet , Nat. Commun. 11, 1603 (2020).
- L. J. Sandilands, Y. Tian, K. W. Plumb, Y. J. Kim, and K. S. Burch, Scattering continuum and possible fractionalized excitations in , Phys. Rev. Lett. 114, 147201 (2015).
- H. B. Cao, A. Banerjee, J. Q. Yan, C. A. Bridges, M. D. Lumsden, D. G. Mandrus, D. A. Tennant, B. C. Chakoumakos, and S. E. Nagler, Low-temperature crystal and magnetic structure of , Phys. Rev. B 93, 134423 (2016).
- H. Suzuki et al., Proximate ferromagnetic state in the Kitaev model material , Nat. Commun. 12, 4512 (2021).
- K. Ran et al., Spin-wave excitations evidencing the Kitaev interaction in single crystalline , Phys. Rev. Lett. 118, 107203 (2017).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/wnrc-cjkn for comparative analysis of reported spin-Hamiltonian parameter sets, parameter selection and sensitivity analysis, excitations in a defect-free crystal, lattice distortion parameters, total energy variation, and magnetic increment under lattice distortion, effect of enhanced lattice distortion on SL states, and impact of temperature and parameter variation on the SL states.
- C. Eichstaedt, Y. Zhang, P. Laurell, S. Okamoto, A. G. Eguiluz, and T. Berlijn, Deriving models for the Kitaev spin-liquid candidate material from first principles, Phys Rev B 100, 075110 (2019).
- P. Laurell and S. Okamoto, Dynamical and thermal magnetic properties of the Kitaev spin liquid candidate , npj Quantum Mater 5, 2 (2020).
- D. A. S. Kaib, S. Biswas, K. Riedl, S. M. Winter, and R. Valentí, Magnetoelastic coupling and effects of uniaxial strain in from first principles, Phys Rev B 103, L140402 (2021).
- H. S. Kim, V. Shankar, A. Catuneanu, and H. Y. Kee, Kitaev magnetism in honeycomb with intermediate spin-orbit coupling, Phys. Rev. B 91, 241110(R) (2015).
- J. S. Gordon, A. Catuneanu, E. S. Sørensen, and H.-Y. Kee, Theory of the field-revealed Kitaev spin liquid, Nat. Commun. 10, 2470 (2019).
- R. D. Johnson et al., Monoclinic crystal structure of and the zigzag antiferromagnetic ground state, Phys. Rev. B 92, 235119 (2015).
- H. S. Kim and H. Y. Kee, Crystal structure and magnetism in : An ab initio study, Phys. Rev. B 93, 155143 (2016).
- M. Akram, J. Kapeghian, J. Das, R. Valentí, A. S. Botana, and O. Erten, Theory of moiré magnetism in twisted bilayer , Nano Lett. 24, 890 (2024).
- F. Keffer and C. Kittel, Theory of antiferromagnetic resonance, Phys. Rev. 85, 329 (1952).
- A. Little et al., Antiferromagnetic resonance and terahertz continuum in , Phys. Rev. Lett. 119, 227201 (2017).
- Y. Sizyuk, P. Wölfle, and N. B. Perkins, Selection of direction of the ordered moments in and , Phys. Rev. B 94, 085109 (2016).
- J. A. Sears, Y. Zhao, Z. Xu, J. W. Lynn, and Y. J. Kim, Phase diagram of in an in-plane magnetic field, Phys. Rev. B 95, 180411(R) (2017).
- J. A. Sears, L. E. Chern, S. Kim, P. J. Bereciartua, S. Francoual, Y. B. Kim, and Y. J. Kim, Ferromagnetic Kitaev interaction and the origin of large magnetic anisotropy in , Nat. Phys. 16, 837 (2020).
- C. Balz et al., Finite field regime for a quantum spin liquid in , Phys. Rev. B 100, 060405(R) (2019).
- S. Kim, E. Horsley, J. P. C. Ruff, B. D. Moreno, and Y. J. Kim, Structural transition and magnetic anisotropy in , Phys. Rev. B 109, L140101 (2024).
- V. Kocsis et al., Magnetoelastic coupling anisotropy in the Kitaev material , Phys. Rev. B 105, 094410 (2022).
- B. H. Kim, Field-angle anisotropy of proximate Kitaev systems under an in-plane magnetic field, Phys. Rev. Res. 3, 043032 (2021).
- K. Liu, N. Sadoune, N. Rao, J. Greitemann, and L. Pollet, Revealing the phase diagram of Kitaev materials by machine learning: Cooperation and competition between spin liquids, Phys. Rev. Res. 3, 023016 (2021).
- S. M. Winter, K. Riedl, P. A. Maksimov, A. L. Chernyshev, A. Honecker, and R. Valentí, Breakdown of magnons in a strongly spin-orbital coupled magnet, Nat. Commun. 8, 1152 (2017).
- L. Janssen, E. C. Andrade, and M. Vojta, Magnetization processes of zigzag states on the honeycomb lattice: Identifying spin models for and , Phys. Rev. B 96, 064430 (2017).
- P. Lampen-Kelley, S. Rachel, J. Reuther, J. Q. Yan, A. Banerjee, C. A. Bridges, H. B. Cao, S. E. Nagler, and D. Mandrus, Anisotropic susceptibilities in the honeycomb Kitaev system , Phys. Rev. B 98, 100403(R) (2018).
- S. H. Jang, Y. Kato, and Y. Motome, Vortex creation and control in the Kitaev spin liquid by local bond modulations, Phys. Rev. B 104, 085142 (2021).
- 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).
- H. Xu, Z. Wang, B. Mu, and Y. Liu, Magnetic field direction dependence of the spin liquid state in - open dataset, github (2025), https://github.com/huaxu-dev/rucl3-sl.