Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid

Nature volume 559, pages 227–231 (2018)Cite this article

Subjects

Abstract

The quantum Hall effect in two-dimensional electron gases involves the flow of topologically protected dissipationless charge currents along the edges of a sample. Integer or fractional electrical conductance is associated with edge currents of electrons or quasiparticles with fractional charges, respectively. It has been predicted that quantum Hall phenomena can also be created by edge currents with a fundamentally different origin: the fractionalization of quantum spins. However, such quantization has not yet been observed. Here we report the observation of this type of quantization of the Hall effect in an insulating two-dimensional quantum magnet1, α-RuCl3, with a dominant Kitaev interaction (a bond-dependent Ising-type interaction) on a two-dimensional honeycomb lattice2,3,4,5,6,7. We find that the application of a magnetic field parallel to the sample destroys long-range magnetic order, leading to a field-induced quantum-spin-liquid ground state with substantial entanglement of local spins8,9,10,11,12. In the low-temperature regime of this state, the two-dimensional thermal Hall conductance reaches a quantum plateau as a function of the applied magnetic field and has a quantization value that is exactly half of the two-dimensional thermal Hall conductance of the integer quantum Hall effect. This half-integer quantization of the thermal Hall conductance in a bulk material is a signature of topologically protected chiral edge currents of charge-neutral Majorana fermions (particles that are their own antiparticles), which have half the degrees of freedom of conventional fermions13,14,15,16. These results demonstrate the fractionalization of spins into itinerant Majorana fermions and Z2 fluxes, which is predicted to occur in Kitaev quantum spin liquids1,3. Above a critical magnetic field, the quantization disappears and the thermal Hall conductance goes to zero rapidly, indicating a topological quantum phase transition between the states with and without chiral Majorana edge modes. Emergent Majorana fermions in a quantum magnet are expected to have a great impact on strongly correlated quantum matter, opening up the possibility of topological quantum computing at relatively high temperatures.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Chiral Majorana edge currents and temperature–magnetic field phase diagram of α-RuCl3.
Fig. 2: Longitudinal thermal conductivity in α-RuCl3.
Fig. 3: Half-integer thermal Hall conductance plateau.
Fig. 4: Temperature dependence of the thermal Hall conductance.

References

  1. Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).

    Article  ADS  MathSciNet  MATH  CAS  Google Scholar 

  2. Jackeli, G. & Khaliullin, G. Mott insulators in the strong spin–orbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009).

    Article  ADS  PubMed  CAS  Google Scholar 

  3. Trebst, S. Kitaev materials. Preprint at https://arxiv.org/abs/1701.07056 (2017).

  4. Kim, H.-S., Shankar, V. V., Catuneanu, A. & Kee, H.-Y. Kitaev magnetism in honeycomb RuCl3 with intermediate spin–orbit coupling. Phys. Rev. B 91, 241110 (2015).

    Article  ADS  CAS  Google Scholar 

  5. Banerjee, A. et al. Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016).

    Article  ADS  PubMed  CAS  Google Scholar 

  6. Sandilands, L. J., Tian, Y., Plumb, W., Kim, Y.-J. & Burch, K. S. scattering continuum and possible fractionalized excitations in α-RuCl3. Phys. Rev. Lett. 114, 147201 (2015).

    Article  ADS  PubMed  CAS  Google Scholar 

  7. Nasu, J., Knolle, J., Kovrizhin, D. L., Motome, Y. & Moessner, R. Fermionic response from fractionalization in an insulating two-dimensional magnet. Nat. Phys. 12, 912–915 (2016).

    Article  Google Scholar 

  8. Yadav, R. et al. Kitaev exchange and field-induced quantum spin-liquid states in honeycomb α-RuCl3. Sci. Rep. 6, 37925 (2016).

    Article  ADS  PubMed  PubMed Central  CAS  Google Scholar 

  9. Baek, S.-H. et al. Evidence for a field-induced quantum spin liquid in α-RuCl3. Phys. Rev. Lett. 119, 037201 (2017).

    Article  ADS  PubMed  Google Scholar 

  10. Wolter, A. U. B. et al. Field-induced quantum criticality in the Kitaev system α-RuCl3. Phys. Rev. B 96, 041405 (2017).

    Article  ADS  Google Scholar 

  11. Leahy, I. A. et al. Anomalous thermal conductivity and magnetic torque response in the honeycomb magnet α-RuCl3. Phys. Rev. Lett. 118, 187203 (2017).

    Article  ADS  PubMed  Google Scholar 

  12. Hentrich, R. et al. Unusual phonon heat transport in α-RuCl3: strong spin–phonon scattering and field-induced spin gap. Phys. Rev. Lett. 120, 117204 (2018).

    Article  ADS  PubMed  Google Scholar 

  13. Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000).

    Article  ADS  CAS  Google Scholar 

  14. Sumiyoshi, H. & Fujimoto, S. Quantum thermal hall effect in a time-reversal-symmetry- broken topological superconductor in two dimensions: approach from bulk calculations. J. Phys. Soc. Jpn. 82, 023602 (2013).

    Article  ADS  CAS  Google Scholar 

  15. Nomura, K., Ryu, S., Furusaki, A. & Nagaosa, N. Cross-correlated responses of topological superconductors and superfluids. Phys. Rev. Lett. 108, 026802 (2012).

    Article  ADS  PubMed  CAS  Google Scholar 

  16. Nasu, J., Yoshitake, J. & Motome, Y. Thermal transport in the Kitaev model. Phys. Rev. Lett. 119, 127204 (2017).

    Article  ADS  PubMed  Google Scholar 

  17. Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science 336, 1003–1007 (2012).

    Article  ADS  PubMed  CAS  Google Scholar 

  18. Nadj-Perge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014).

    Article  ADS  PubMed  CAS  Google Scholar 

  19. Das, A. et al. Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions. Nat. Phys. 8, 887–895 (2012).

    Article  CAS  Google Scholar 

  20. He, Q. L. et al. Chiral Majorana fermion modes in a quantum anomalous Hall insulator- superconductor structure. Science 357, 294–299 (2017).

    Article  ADS  MathSciNet  PubMed  CAS  Google Scholar 

  21. Johnson, R. D. et al. Monoclinic crystal structure of α-RuCl3 and the zigzag antiferromagnetic ground state. Phys. Rev. B 92, 235119 (2015).

    Article  ADS  CAS  Google Scholar 

  22. Kasahara, Y. et al. Unusual thermal Hall effect in a Kitaev spin liquid candidate α-RuCl3. Phys. Rev. Lett. 120, 217205 (2018).

    Article  ADS  PubMed  CAS  Google Scholar 

  23. Majumder, M., Schmidt, M., Rosner, H., Tsirlin, A. A., Yasuoka, H. & Baenitz, M. Anisotropic Ru3+ 4d 5 magnetism in the α-RuCl3 honeycomb system: susceptibility, specific heat, and zero-field NMR. Phys. Rev. B 91, 180401 (2015).

    Article  ADS  CAS  Google Scholar 

  24. Chaloupka, L. & Khaliullin, G. Magnetic anisotropy in the Kitaev model systems Na2IrO3 and RuCl3. Phys. Rev. B 94, 064435 (2016).

    Article  ADS  CAS  Google Scholar 

  25. Janša N. et al. Observation of two types of fractional excitation in the Kitaev honeycomb magnet. Nat. Phys. https://doi.org/10.1038/s41567-018-0129-5 (2018).

  26. Banerjee, A. et al. Excitations in the field-induced quantum spin liquid state of α-RuCl3. npj Quantum Mater. 3, 8 (2018).

  27. Banerjee, M. et al. Observed quantization of anyonic heat flow. Nature 545, 75–79 (2017).

    Article  ADS  PubMed  CAS  Google Scholar 

  28. Hirobe, D., Sato, M., Shiomi, Y., Tanaka, H. & Saitoh, E. Magnetic thermal conductivity far above the Néel temperatures in the Kitaev-magnet candidate α-RuCl3. Phys. Rev. B 95, 241112 (2017).

    Article  ADS  Google Scholar 

  29. Yu, Y. J. et al. Ultralow-temperature thermal conductivity of the Kitaev honeycomb magnet α-RuCl3 across the field-induced phase transition. Phys. Rev. Lett. 120, 067202 (2018).

    Article  ADS  PubMed  CAS  Google Scholar 

  30. Gohlke, M., Wachtel, G., Yamaji, Y., Pollmann, F. & Kim, Y. B. Signatures of quantum spin liquid in Kitaev-like frustrated magnets. Phys. Rev. B 97, 075126 (2018).

    Article  ADS  Google Scholar 

  31. Winter, S. M., Li, Y., Jeschke, H. O. & Valenti, R. Challenges in design of Kitaev materials: magnetic interactions from competing energy scales. Phys. Rev. B 93, 214431 (2016).

    Article  ADS  CAS  Google Scholar 

  32. Jiang, H.-C., Gu, Z.-C., Qi, X.-L. & Trebst, S. Possible proximity of the Mott insulating iridate Na2IrO3 to a topological phase: Phase diagram of the Heisenberg-Kitaev model in a magnetic field. Phys. Rev. B 83, 245104 (2011).

    Article  ADS  CAS  Google Scholar 

  33. Kubota, Y., Tanaka, H., Ono, T., Narumi, Y. & Kindo, K. Successive magnetic phase transition in α-RuCl3: XY-like frustrated magnet on the honeycomb lattice. Phys. Rev. B 91, 094422 (2015).

    Article  ADS  CAS  Google Scholar 

  34. Watanabe, D. et al. Emergence of nontrivial magnetic excitations in a spin liquid state of kagomé volborthite. Proc. Natl Acad. Sci. USA 113, 8653–8657 (2016).

    Article  ADS  PubMed  CAS  Google Scholar 

  35. Taylor, O. J., Carrington, A. & Schlueter, J. A. Specific-heat measurements of the gap structure of the organic superconductor κ-(ET)2Cu[N(CN)2]Br and κ-(ET)2Cu(NCS)2. Phys. Rev. Lett. 99, 057001 (2007).

    Article  ADS  PubMed  CAS  Google Scholar 

  36. Han, J. H. & Lee, H. Spin chirality and Hall-like transport phenomena of spin excitations. J. Phys. Soc. Jpn 86, 011007 (2017).

    Article  ADS  Google Scholar 

  37. Sugii, K. et al. Thermal Hall effect in a phonon-glass Ba3CuSb2O9. Phys. Rev. Lett. 118, 145902 (2017).

    Article  ADS  PubMed  CAS  Google Scholar 

Download references

Acknowledgements

We thank S. Fujimoto, H. Ishizuka, N. Kawakami, H.-Y. Kee, Y. B. Kim, E.-G. Moon, N. P. Ong, M. Shimozawa, M. Udagawa and M. Yamashita for useful discussions. We thank N. Abe, Y. Tokunaga and T. Arima for support in X-ray diffraction measurements. This work was supported by Grants-in-Aid for Scientific Research (KAKENHI) (numbers 25220710, 15H02014, 15H02106, 15H05457, 15K13533, 15K17692, 16H02206, 16H00987, 16K05414, 17H01142 and 18H04223) and Grants-in-Aid for Scientific Research on innovative areas “Topological Materials Science” (number JP15H05852) from Japan Society for the Promotion of Science (JSPS).

Reviewer information

Nature thanks K.-Y. Choi, K. Shtengel and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Author information

Authors and Affiliations

  1. Department of Physics, Kyoto University, Kyoto, Japan

    Y. Kasahara, T. Ohnishi, Sixiao Ma & Y. Matsuda

  2. Department of Advanced Materials Science, University of Tokyo, Chiba, Japan

    Y. Mizukami, O. Tanaka & T. Shibauchi

  3. Institute for Solid State Physics, University of Tokyo, Chiba, Japan

    K. Sugii

  4. Department of Physics, Tokyo Institute of Technology, Tokyo, Japan

    N. Kurita, H. Tanaka & J. Nasu

  5. Department of Applied Physics, University of Tokyo, Tokyo, Japan

    Y. Motome

Authors
  1. Y. Kasahara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Ohnishi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Y. Mizukami
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. O. Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Sixiao Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. K. Sugii
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. N. Kurita
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. H. Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. J. Nasu
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Y. Motome
    View author publications

    You can also search for this author in PubMed Google Scholar

  11. T. Shibauchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  12. Y. Matsuda
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Y.K. and Y. Matsuda conceived and designed the study. Y.K., T.O. and S.M. performed the thermal transport measurements. Y. Mizukami, O.T. and K.S. performed the specific heat measurements. N.K. and H.T. synthesized the high-quality single crystalline samples. Y.K., T.O., J.N., Y. Motome, T.S. and Y. Matsuda discussed the results. Y.K., J.N., Y. Motome, T.S. and Y. Matsuda prepared the manuscript.

Corresponding author

Correspondence to Y. Matsuda.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Temperature dependence of the longitudinal thermal conductivity.

a, b, κxx in a field tilted at θ = 60° (a) and 45° (b), plotted as a function of temperature (see inset of Fig. 2a). Arrows indicate the onset temperature of the AFM order TN.

Extended Data Fig. 2 Field dependence of the longitudinal thermal conductivity.

a, b, κxx in field tilted at θ = 60° (a) and 45° (b), plotted as a function of the parallel field component H∥ (see inset of Fig. 2a). Arrows indicate the minimum of κxx, which is attributed to the onset field of the AFM order.

Extended Data Fig. 3 Phase diagram of α-RuCl3 for H ∥ a and H ∥ b.

a, b, Temperature dependence of the specific heat, C, divided by T for H ∥ a (a) and H ∥ b (b). Arrows indicate the Néel temperature TN. c, Field dependence of TN for H ∥ a and H ∥ b, determined by the specific heat measurements. TN, determined from the thermal conductivity and magnetic susceptibility26, is also shown. The critical field for H ∥ a is slightly lower than that for H ∥ b, but both phase diagrams are very similar.

Extended Data Fig. 4 Field dependence of thermal Hall conductivity.

a, b, Thermal Hall conductivity, κxy/T, in a field tilted at θ = 60° (a) and 45° (b), plotted as a function of H⊥ (see inset of Fig. 2a). The top axes show the parallel field component, H∥. The right scales represent the 2D thermal Hall conductance, \({\kappa }_{xy}^{{\rm{2D}}}/T\), in units of \(({\rm{\pi }}/6)({k}_{{\rm{B}}}^{2}/\hbar )\). Violet dashed lines represent the half-integer thermal Hall conductance, \({\kappa }_{xy}^{{\rm{2D}}}/\left[T\left({\rm{\pi }}/6\right)\left({k}_{{\rm{B}}}^{2}/\hbar \right)\right]=1/2\). Error bars represent one standard deviation.

Extended Data Fig. 5 Sample dependence of κxy.

a, κxy/T measured in a different crystal (sample 2) for θ = 60° (see inset of Fig. 2a) at 4.3 K, plotted as a function of H⊥. The right scales represent the 2D thermal Hall conductance, \({\kappa }_{xy}^{{\rm{2D}}}/T\), in units of \(({\rm{\pi }}/6)({k}_{{\rm{B}}}^{2}/\hbar )\). The half-integer thermal Hall conductance plateau is observed at 4.5 T < μ0H⊥ < 5.0 T. The field where the overshoot behaviour from the quantization value is observed is slightly higher than that of sample 1, but the field where κxy/T vanishes (μ0H∥ ≈ 9.3 T) is close to that of sample 1. b, κxy/T of sample 2 in a field tilted at θ = 60°, plotted as a function of H⊥ at 11 K. Error bars represent one standard deviation.

Extended Data Fig. 6 Field dependence of thermal Hall conductivity in tilted fields at high temperatures.

a–d, Thermal Hall conductivity, κxy/T, in a field tilted at θ = 60° (a, b) and 45° (c, d), plotted as a function of H⊥, (see inset of Fig. 2a). The right scales represent the 2D thermal Hall conductance, \({\kappa }_{xy}^{{\rm{2D}}}/T\), in units of \(({\rm{\pi }}/6)({k}_{{\rm{B}}}^{2}/\hbar )\). Violet dashed lines represent the half-integer thermal Hall conductance, \({\kappa }_{xy}^{{\rm{2D}}}/\left[T\left({\rm{\pi }}/6\right)\left({k}_{{\rm{B}}}^{2}/\hbar \right)\right]=1/2\). Error bars represent one standard deviation.

Extended Data Fig. 7 Specific heat above \({{\boldsymbol{H}}}_{{\boldsymbol{\parallel }}}^{{\boldsymbol{\ast }}}\).

a, b, Temperature dependence of C/T for θ = 60° (a; H is tilted within the a–c plane) and 90° (b). c, C/T at 0.47 K plotted as a function of H∥ for θ = 60° and 90°. C(H)/T exhibits a dip-like anomaly for θ = 60° and a kink for θ = 90° at μ0H∥ ≈ 9.2 T (dashed line). This field almost coincides with the characteristic field at which κxy/T vanishes.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kasahara, Y., Ohnishi, T., Mizukami, Y. et al. Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid. Nature 559, 227–231 (2018). https://doi.org/10.1038/s41586-018-0274-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-018-0274-0

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing