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

## 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.

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## 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

### 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

• K. Sugii
4. #### Department of Physics, Tokyo Institute of Technology, Tokyo, Japan

• N. Kurita
• , H. Tanaka
•  & J. Nasu

• Y. Motome

### 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.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Y. Matsuda.

## Extended data figures and tables

1. ### 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.

2. ### 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.

3. ### 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.

4. ### 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, $κ x y 2D /T$, in units of $( π / 6 ) ( k B 2 / ħ )$. Violet dashed lines represent the half-integer thermal Hall conductance, $κ x y 2D / T π / 6 k B 2 / ħ =1/2$. Error bars represent one standard deviation.

5. ### 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, $κ x y 2D /T$, in units of $( π / 6 ) ( k B 2 / ħ )$. 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.

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

ad, 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, $κ x y 2D /T$, in units of $( π / 6 ) ( k B 2 / ħ )$. Violet dashed lines represent the half-integer thermal Hall conductance, $κ x y 2D / T π / 6 k B 2 / ħ =1/2$. Error bars represent one standard deviation.

7. ### Extended Data Fig. 7 Specific heat above $H ∥ ∗$.

a, b, Temperature dependence of C/T for θ = 60° (a; H is tilted within the ac 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.