Abstract
A key feature of quantum spin liquids is the predicted formation of fractionalized excitations. They are expected to produce changes in the physical response, providing a way to observe the quantum spin liquid state^{1}. In the honeycomb magnet αRuCl_{3}, a quantum spin liquid has been proposed to explain the behaviour observed on applying an inplane magnetic field H_{}. Previous work reported that the thermal Hall conductivity took on a halfinteger quantized value and suggested this as a signature of a fractionalized Majorana edge mode predicted to exist in Kitaev quantum spin liquids^{2}. However, the temperature and magneticfield range of the halfquantized signal^{2,3,4} and its association with Majorana edge modes are still under debate^{5,6}. Here we present a comprehensive study of the thermal Hall conductivity in αRuCl_{3} showing that approximately halfinteger quantization exists in an extended region of the phase diagram, particularly across a plateaulike parameter regime for H_{} exceeding 10 T and temperature below 6.5 K. At lower fields, the thermal Hall conductivity exhibits correlations with complex anomalies in the longitudinal thermal conductivity and magnetization, and is suppressed by cooling to low temperatures. Our results can be explained by the existence of a topological state in magnetic fields above 10 T.
Main
A prime candidate for experimentally accessible quantum spin liquids is the Kitaev model, which has an exactly solvable ground state with itinerant and localized Majorana fermions^{7}. Although they are chargeneutral, itinerant Majorana fermions carry heat and are therefore expected to contribute to thermal transport^{8}. Applying a magnetic field gaps the bulk Majorana bands, leading to a topologically protected chiral edge current^{9}. This edge state carries a thermal Hall conductivity per layer divided by temperature, namely, k_{XY}^{2D}/T = \(\frac{{{\uppi}^2k_{\mathrm{B}}}^{2}}{{6h}}\), where k_{B} and h are the Boltzmann and Planck constants, onehalf that of an equivalent electronic edge state in the quantum Hall effect due to the fractionalized nature of the Majorana fermion.
The search for material candidates of the Kitaev model has focused on Mott insulators^{10} with 5d^{4} Ir^{4+} and 4d^{4} Ru^{3+} having strong spin–orbit coupling on a honeycomb lattice^{11,12,13}. Here αRuCl_{3} is a prime candidate that shows antiferromagnetic zigzag order below the antiferromagnetic ordering temperature T_{N} ≈ 7.5 K. This magnetic order is suppressed with an inplane critical magnetic field of H_{C2} ≈ 7 T (refs. ^{2,14,15,16,17,18,19,20,21}), where the magnetic moment is not yet fully saturated, revealing a region of the phase diagram where a quantum spin liquid may arise. A thermal Hall effect with a magnitude close to the halfquantized value k_{HQ}/T (corresponding to k_{XY}^{2D}/T per atomic plane) was reported in the quantum spin liquid region with an inplane magnetic field H_{} along the a axis (perpendicular to the Ru–Ru bond direction)^{2,3,4}, which was discussed as a signature of the Majorana edge state expected for the Kitaev quantum spin liquid.
To establish the presence of an edge state, however, the robustness of the halfquantized thermal Hall effect should be demonstrated over a reasonably wide range of magnetic fields and temperatures. Several studies report a plateaulike region in a field at around 5 K with a magnitude close to k_{HQ}/T (refs. ^{2,3,4}). These studies, however, are limited down to ~3.5 K and the plateau behaviour as a function of T is not as clear as it is in the field. The plateauonset fields differ between studies, ranging from H_{} = 7.8 T (ref. ^{2}) to 9.9 T (ref. ^{3}). In preparing this manuscript, we noticed that another study^{22} reported a broad k_{XY}/T dome with a height appreciably smaller than k_{HQ}/T, which was interpreted as not supporting the existence of a halfquantized plateau. Recently, anomalies in the magnetocaloric effect, specific heat and magnetic Grüneisen parameter were reported at around H_{} = 10 T (refs. ^{23,24,25}) in the spin liquid region (above H_{C2}), suggesting that the phase diagram may be even more complex. The question of the relationship between these anomalies and the thermal Hall signature may be of fundamental importance to understand the nature of the possible fieldinduced quantum spin liquid, and more specifically, the possible topological edge state.
Here we present comprehensive measurements of k_{XY}, k_{XX} and the magnetic susceptibility (dM/dH) in a T range from 150 mK to 9 K and H_{} up to 13 T (H_{} along the a axis) on an αRuCl_{3} single crystal, which reveals that k_{XY}/T stays close to k_{HQ}/T over a reasonably wide range of both temperature and magnetic field (below T = 6.5 K and from H_{} ≈ 10 T up to at least 13 T), hinting at the presence of a protected halfquantized plateau. Below ~10 T, no signature of a plateau is observed and k_{XY}/T is suppressed to zero on lowering the temperature below 6.5 K, which appears to correlate with the multiple highfield (H_{} > H_{C2}) anomalies present in both k_{XX} and dM/dH.
Magneticfieldinduced anomalies in thermal conductivity k _{XX}
The temperature dependence of k_{XX} reproduces the data from previous reports^{2} very well and displays a sharp minimum at T_{N} = 7.5 K, which moves to lower temperatures as H_{} increases and eventually fades out above H_{C2} = 7.1 T, tracing the extent of the antiferromagnetically ordered region (Fig. 1a,e). As a function of H_{}, the magnitude of k_{XX} first decreases to a broad minimum at around 7 T and then rapidly increases (Fig. 1b). The data below 2 K reveal additional features in k_{XX}(H_{}): the broad minimum splits into two at H_{C1} = 6.05 T and H_{C2} = 7.10 T. These fields coincide with two sharp peaks in dM/dH (Fig. 1c), which correspond to two reported transitions: from zigzagordered phase I to intermediateordered phase II and from phase II to the highfield paramagnetic phase^{23,24,25,26,27,28}. The suppression of k_{XX} at the magneticphase transitions is consistent with the interpretation that k_{XX} is dominated by phonons that scatter off magnetic excitations^{29}.
At higher inplane magnetic fields and below 5 K, further anomalies can be seen in k_{XX}, namely, a broad maximum at H_{P} ≈ 10 T flanked by broad minima at H_{D1} ≈ 8.8 T and H_{D2} ≈ 10.7 T (Fig. 1b). Features corresponding to H_{D1} and H_{P} can be clearly identified as a broad shoulder and a sudden drop in dM/dH, respectively (Fig. 1c and Extended Data Fig. 1). We argue that these highfield anomalies in k_{XX} arise from crossovers or very weak phase transitions.
In analogy to the minima in k_{XX} at H_{C1} and H_{C2}, increased phonon scattering by soft magnetic excitations can explain the minima at H_{D1} and H_{D2}. However, the anomalies at H_{D1} and H_{D2} appear broader and weaker than those at the welldefined phase transitions at H_{C1} and H_{C2} (Fig. 1b). They are enhanced on cooling to 500 mK but subsequently suppressed below 500 mK, indicative of a reduction in phonon scatterers. This suggests that H_{D1} and H_{D2} may not be welldefined phase transitions like H_{C1} and H_{C2} but highly probable crossovers with soft but finite energy excitations or alternatively very weak transitions having only a limited number of lowenergy excitations involved.
Recent reports of the magnetocaloric effect^{23} and the magnetic Grüneisen parameter^{25} at temperatures above 1 K also showed anomalies near H_{D1} and H_{D2}, whereas the specific heat^{24} at 0.67 K reported a symmetrybreaking phase transition near 10 T, a conclusion that is contested by several other measurements^{23,25,27,28}. In the course of preparing this manuscript, we noticed another work^{22} in which broadly similar fielddependent structures are reported in k_{XX} that are ascribed to quantum oscillations from underlying quasiparticles. That scenario, however, does not explain the disappearance of features in the lowtemperature limit instead of the expected Lifshitz–Kosevitch behaviour (Extended Data Fig. 2) or the alignment of prominent minima with the wellestablished magneticphase transitions at H_{C1} and H_{C2}.
Thermal Hall conductivity close to the halfquantized value
A strong k_{XY}/T signal is resolved above H_{C2} = 7.1 T, as measured by sweeping the temperature at a fixed field (Fig. 2, T sweeps) and a magnetic field at a fixed temperature (Fig. 3, H sweeps). Great care was taken to avoid systematic errors by verifying the linear power dependence of the signal and by avoiding field hysteretic effects that may mix k_{XX} and k_{XY} (Extended Data Figs. 3 and 4 and Supplementary Discussion). Data from H and T sweeps are indeed consistent with each other within the given error bars, which arise from random noise in thermometry (Fig. 2, black points, and Fig. 3, thick lines). Repeated measurements were performed over multiple months to eliminate the possibility of dependence on contact geometry or sample aging and produce a fully consistent dataset.
As shown in Figs. 2 and 3, k_{XY}/T is larger than the halfquantized value k_{HQ}/T = 0.87 mW K^{–2} m^{–1} at high temperatures and at high magnetic fields (above T = 6.5 K and H_{C2} = 7.1 T) and decreases to k_{HQ}/T and below by lowering T and H_{}, giving rise to a region with k_{XY}/T ≈ k_{HQ}/T on the H_{}–T plane. A colour plot of k_{XY}/T across the whole phase diagram (Fig. 4) combines data from all the H and T sweeps, which maps out the region where k_{XY}/T is within ±20% of k_{HQ}/T (white region). We see an Lshaped white region of k_{XY}/T ≈ k_{HQ}/T, which first runs vertically from T = 9.0 K to T ≈ 6.5 K at around H_{} = 7.6 T and then continues horizontally from H_{} ≈ 7.6 T up to at least H_{} ≈ 13.2 T at around 6.5 K. The observation of an almost H_{}independent k_{XY}/T ≈ k_{HQ}/T was also reported in previous studies at a similar temperature. In our data, it starts from a similar field of 7.6 T but extends higher to 13.2 T. As we measured a sample that was also studied elsewhere^{2}, one may argue that the difference arises from a change in the sample quality and/or an experimental error such as dependence on contact geometry or measurement sequence. Sample degradation is unlikely because of the perfect agreement of k_{XX} with that in the original report and reproducibility across multiple measurements (Extended Data Figs. 5 and 6). We also note that our results are reproducible with different samplemounting geometries (Extended Data Fig. 3) and different sequences of data acquisition (H and T sweeps). Recent studies^{3,4} indeed report the region of k_{XY}/T ≈ k_{HQ}/T at higher fields than the original study, although the onset field is also higher than the original study and present study. Sample dependence is beyond the scope of this study but should be carefully checked in the future.
The most intriguing feature in Fig. 4 is the notable broadening of the white region at low temperatures with an increasing field above ~10 T, giving rise to a large, triangular white area that extends down to at least 2 K at 13 T. Does the observation of a region with k_{XY}/T ≈ k_{HQ}/T on the H_{}–T plane have a physical implication and support the presence of the topologically protected halfquantized plateau? The enhancement of k_{XY}/T could have different origins, for instance, through (nonquantized) topological magnon^{5,6} or phonon^{30} transport. In such scenarios, a crossing may give rise to an accidental k_{XY}/T (H_{}, T) ≈ k_{HQ}/T line on the H_{}–T plane. In contrast, a quantized, topologically protected k_{XY}/T(H_{}, T) ≈ k_{HQ}/T plateau should exhibit insensitivity to changes in both magnetic field and temperature across and in the extended area in H_{} and T.
The extended nature of k_{XY}/T (H_{}, T) ≈ k_{HQ}/T above 10 T represented by the white triangular plane in Fig. 4 argues against an accidental crossing scenario and hints at a halfquantized plateau and hence a Majorana edge state in αRuCl_{3}. The corresponding Tsweep data (Fig. 2) shows that for fields greater than 10.3 T, a kinklike singularity in the T dependence of k_{XY}/T appears at ~6.5 K (Fig. 2, red arrows). Notably, at the kink, k_{XY}/T is always ~k_{HQ}/T, implying that the magnitude of k_{HQ}/T has special importance in the system. On lowering T, the kink is followed by an almost Tindependent k_{XY}/T ≈ k_{HQ}/T region, reminiscent of an incipient halfquantized T plateau and then by a rapid decrease to zero. The width of the Tplateaulike region gradually expands on increasing the field from 10.3 T, giving rise to the white triangle in Fig. 4.
The narrow Lshaped white region below H_{} ≈ 10 T (Fig. 4), on the other hand, represents a ‘line’ of k_{XY}/T (H_{}, T) ≈ k_{HQ}/T, which is clearly demonstrated by the T and H sweeps in Figs. 2 and 3. Along the vertical white region in Fig. 4, k_{XY}/T is smoothly suppressed on cooling at 7.6 T (Fig. 2) and continuously decreases through k_{HQ}/T with decreasing H_{} in the H sweeps (Fig. 3). Along the horizontal white region at around 6.5 K, k_{XY}/T shows a plateau in the H sweep at 6.5 K (Fig. 3), but continuously decreases through k_{HQ}/T in the T sweeps up to ~10 T. It is not obvious why k_{XY}/T crosses k_{HQ}/T at a constant temperature of 6.5 K. Nevertheless, because of the line character of the region of k_{XY}/T (H_{}, T) ≈ k_{HQ}/T, the white region (Fig. 4) below H_{} ≈ 10 T is probably due to accidental crossing.
Lowtemperature suppression of k _{XY}/T
What is the origin of the rapid suppression of k_{XY}/T to zero at low temperatures below 12 T (Fig. 4, redcoloured region)? Within this red region, weak structures are observed: three vertical streaks of minima (deeper red) at around H_{} = 7.0, 9.5 and 11.0 T and two vertical streaks of maxima (brighter red) at around H_{} = 8.5 and 10.0 T branching out from the white region at T = 6.5 K. As shown in Figs. 1d and 4, the fields of the three minima apparently coincide with the critical field H_{C2} and the two highfield anomalies at H_{D1} and H_{D2} observed in k_{XX} and dM/dH. The maximum at ~10 T reflects the singularly weak T dependence of k_{XY}/T at H_{} = 10.3 T (Fig. 2) and coincides with H_{P}. These correlations suggest that the highfield magnetic crossovers/transitions are related to the suppression of k_{XY}/T at low temperatures.
If a halfquantized thermal Hall plateau arises from a chiral Majorana edge mode^{8}, it was theoretically shown that the coupling between a phonon bath and the edge mode^{31,32} is necessary to observe the plateau in the presence of dominant phonon conductivity. Phonons must be in the diffuse rather than the ballistic scattering regime. If phonons experience a crossover from the diffuse to the ballistic regimes on cooling or under a magnetic field, the thermal Hall signal from the Majorana edge mode should vanish due to the decoupling of phonons and edge modes, which might account for the lowtemperature suppression of k_{XY}/T. From the estimation of the phonon mean free path (l_{ph}) from k_{XX} as a function of H_{} and T, however, we may exclude the phonon–edge mode decoupling scenario as the origin of lowtemperature suppression of k_{XY}/T below k_{HQ}/T from 7 to 12 T.
k_{XX} approaches a T^{3} power law at the highest fields and lowest temperatures (Fig. 1f), consistent with phonon transport limited by a temperatureindependent scattering length. The calculated l_{ph} reaches ~50 µm at the lowest temperatures, which is somewhat smaller than the sample width of ~1 mm (Fig. 1f and Extended Data Fig. 7), meaning that phonon transport at low temperatures is still marginally in the diffuse regime. Furthermore, the fact that the regions of suppressed k_{XY}/T coincide with the minima in k_{XX}, that is, regions with the shortest l_{ph}, argues against the decoupling scenario. The lowtemperature enhancement of k_{XY}/T to k_{HQ}/T on increasing the magnetic field is accompanied by an increase in l_{ph}, the opposite of what would be expected from phonon–edge mode decoupling. We also note that the decoupling scenario appears to be contrary to a report in which the halfquantized plateau is only observed in samples with high (phonon) thermal conductivity^{4}.
Recently, the scenario of purely phonondriven thermal Hall was proposed based on the discovery of a large thermal Hall effect in nonmagnetic SrTiO_{3} and the correlation between the T dependence of k_{XX} and k_{XY} in RuCl_{3} (ref. ^{30}). In addition, in the magnetic insulator Ba_{3}CuSb_{2}O_{9}, a thermal Hall angle (tan(θ_{H}) ≈ 10^{−4}), similar to that of RuCl_{3}, was claimed to arise from the phonon thermal Hall effect alone^{33}. The observed suppression of k_{XY}/T at low temperatures and its correlation with k_{XX} are in line with a phonondriven scenario. Both show an overall increase on increasing the field, with diplike suppressions at H_{D1} and H_{D2}. Although a phonononly scenario could explain these correlations between k_{XX} and k_{XY}, it is not clear how it would naturally explain the observed plateaulike behaviour of k_{XY}/T close to k_{HQ}/T.
The identification of possible topological phase transitions into and out of the halfquantized state still remains an open question^{23,24,25,27,28}. For the former, unveiling the nature of highfield anomalies may be the key question to be addressed. The highfield transition out of the halfquantized state, if it exists, must be at a field higher than 13 T. A measurement of k_{XY}/T to much higher fields is, therefore, highly desired. Alternative mechanisms leading to an enhancement of k_{XY}/T, such as a topological magnondriven^{5,6} or phonondriven^{30} Hall effect, have been proposed. Our results do not exclude those scenarios but pose them a challenge to explain the plateaulike behaviour close to halfquantization.
Methods
Samples
Large, thin (~2.5 mm × 1.3 mm × 17 µm), highquality single crystals of αRuCl_{3} from the same growth batches as those reported in ref. ^{2} were measured. The sample under investigation for thermal Hall measurements has a confirmed halfquantized Hall plateau in that report, where it was labelled as ‘sample 2’. A second sample with a similarly high thermal conductivity was used for magnetic susceptibility measurements.
Thermal conductivity was measured using a steadystate threethermometer setup. For the temperature range of 1.8–9.0 K, Cernox CX1050 chip thermometers (Lake Shore) were used in a ^{4}He cryostat. For the temperature range of 150 mK to 3 K, ruthenium oxide chip thermometers were used in a dilution fridge. In each case, thermometers were calibrated in situ against a fieldcalibrated reference thermometer. A 10 kΩ NiCr resistive heater was used to apply heater power. The sample was free standing with its ‘cold’ edge mounted with Apiezon N grease onto a LiF single crystal, which was attached with silver paint onto a copper mount and cooled by the cryostat. Several copper mounts machined at different angles were used to allow for measurements in fixed tilted fields.
Geometrical error affects the measurement due to the finite contact sizes and uncertainty in determining the exact crystal dimensions. We estimate the resulting systematic uncertainty in k_{XY}/T to be around 10%, which is in addition to the random error due to thermometry noise (error bars shown in Figs. 2 and 3).
We adopt the usual convention that the crystal a axis is perpendicular to Ru–Ru bonds, the b axis lies along the Ru–Ru bonds and the c axis is perpendicular to the honeycomb plane. The magnetic field was applied at either 70° from the c axis (^{4}He) or 90° from the c axis (dilution refrigerator), with the inplane field component in the a axis. The amplitude of k_{XX}, as well as the measured transition fields (H_{C1}, H_{C2}, H_{D1} and H_{D2}), were confirmed to reproduce at the same values of H_{} for these two field angles (Extended Data Fig. 6); therefore, the two datasets are presented as one. The field angle was determined under an optical microscope with 1° precision. Heat was always applied along the a axis.
Over the course of these measurements, the crystal was thermally cycled between room temperature and <4.2 K for 14 times; each time, identical values of k_{XX} were measured, indicating that sample deterioration over time is negligible. The measurement of k_{XX} is additionally in very close agreement with that previously measured in Kyoto (Extended Data Fig. 5).
Magnetization measurements above 2 K were performed using a Physical Property Measurement System vibrating sample magnetometry option (Quantum Design), and below 1 K using a custombuilt Faraday force magnetometer on a dilution refrigerator (Extended Data Fig. 1). For all the magnetization data, the applied field was parallel to the crystal a axis.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.
References
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
Kasahara, Y. et al. Majorana quantization and halfinteger thermal quantum Hall effect in a Kitaev spin liquid. Nature 559, 227–231 (2018).
Yokoi, T. et al. Halfinteger quantized anomalous thermal Hall effect in the Kitaev material candidate αRuCl_{3}. Science 373, 568–572 (2021).
Yamashita, M., Kurita, N. & Tanaka, H. Sample dependence of the halfinteger quantized thermal Hall effect in a Kitaev candidate αRuCl_{3}. Phys. Rev. B 102, 220404 (2020).
McClarty, P. A. et al. Topological magnons in Kitaev magnets at high fields. Phys. Rev. B 98, 060404 (2018).
Zhang, E. Z., Chern, L. E. & Kim, Y. B. Topological magnons for thermal Hall transport in frustrated magnets with bonddependent interactions. Phys. Rev. B 103, 174402 (2021).
Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).
Nasu, J., Yoshitake, J. & Motome, Y. Thermal transport in the Kitaev model. Phys. Rev. Lett. 119, 127204 (2017).
Motome, Y. & Nasu, J. Hunting Majorana fermions in Kitaev magnets. J. Phys. Soc. Jpn 89, 012002 (2020).
Jackeli, G. & Khaliullin, G. Mott insulators in the strong spinorbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009).
Trebst, S. & Hickey, C. Kitaev materials. Phys. Rep. 950, 1–37 (2022).
Winter, S. M., Li, Y., Jeschke, H. O. & Valentí, R. Challenges in design of Kitaev materials: magnetic interactions from competing energy scales. Phys. Rev. B 93, 214431 (2016).
Takagi, H., Takayama, T., Jackeli, G., Khaliullin, G. & Nagler, S. E. Concept and realization of Kitaev quantum spin liquids. Nat. Rev. Phys. 1, 264–280 (2019).
Baek, S. H. et al. Evidence for a fieldinduced quantum spin liquid in αRuCl_{3}. Phys. Rev. Lett. 119, 037201 (2017).
Wang, Z. et al. Magnetic excitations and continuum of a possibly fieldinduced quantum spin liquid in αRuCl_{3}. Phys. Rev. Lett. 119, 227202 (2017).
Zheng, J. et al. Gapless spin excitations in the fieldinduced quantum spin liquid phase of αRuCl_{3}. Phys. Rev. Lett. 119, 227208 (2017).
Banerjee, A. et al. Excitations in the fieldinduced quantum spin liquid state of αRuCl_{3}. npj Quant. Mater. 3, 8 (2018).
Johnson, R. D. et al. Monoclinic crystal structure of αRuCl_{3} and the zigzag antiferromagnetic ground state. Phys. Rev. 92, 235119 (2015).
Kubota, Y., Tanaka, H., Ono, T., Narumi, Y. & Kindo, K. Successive magnetic phase transitions in αRuCl_{3}: XYlike frustrated magnet on the honeycomb lattice. Phys. Rev. B 91, 094422 (2015).
Yadav, R. et al. Kitaev exchange and fieldinduced quantum spinliquid states in honeycomb αRuCl_{3}. Sci. Rep. 6, 37925 (2016).
Sears, J. A. et al. Magnetic order in αRuCl_{3}: a honeycomblattice quantum magnet with strong spinorbit coupling. Phys. Rev. B 91, 144420 (2015).
Czajka, P. et al. Oscillations of the thermal conductivity observed in the spinliquid state of αRuCl_{3}. Nat. Phys. 17, 915–919 (2021).
Balz, C. et al. Finite field regime for a quantum spin liquid in αRuCl_{3}. Phys. Rev. B 100, 60405 (2019).
Tanaka, O. et al. Thermodynamic evidence for fieldangle dependent Majorana gap in a Kitaev spin liquid. Preprint at https://arxiv.org/abs/2007.06757v1 (2020).
Bachus, S. et al. Thermodynamic perspective on fieldinduced behavior of αRuCl_{3}. Phys. Rev. Lett. 125, 97203 (2020).
Balz, C. et al. Fieldinduced intermediate ordered phase and anisotropic interlayer interactions in αRuCl_{3}. Phys. Rev. B 103, 174417 (2021).
Ponomaryov, A. N. et al. Nature of magnetic excitations in the highfield phase of αRuCl_{3}. Phys. Rev. Lett. 125, 37202 (2020).
Schönemann, R. et al. Thermal and magnetoelastic properties of αRuCl_{3} in the fieldinduced lowtemperature states. Phys. Rev. B 102, 214432 (2020).
Hentrich, R. et al. Unusual phonon heat transport in αRuCl_{3}: strong spinphonon scattering and fieldinduced spin gap. Phys. Rev. Lett. 120, 117204 (2018).
Li, X., Fauqué, B., Zhu, Z. & Behnia, K. Phonon thermal Hall effect in strontium titanate. Phys. Rev. Lett. 124, 105901 (2020).
VinklerAviv, Y. & Rosch, A. Approximately quantized thermal Hall effect of chiral liquids coupled to phonons. Phys. Rev. X 8, 31032 (2018).
Ye, M., Halász, G. B., Savary, L. & Balents, L. Quantization of the thermal Hall conductivity at small Hall angles. Phys. Rev. Lett. 121, 147201 (2018).
Sugii, K. et al. Thermal Hall effect in a phononglass Ba_{3}CuSb_{2}O_{9}. Phys. Rev. Lett. 118, 145902 (2017).
Acknowledgements
We thank Y. Kasahara and Y. Matsuda for insightful discussions and M. Dueller and K. Pflaum for technical assistance. The work done in Germany has been supported in part by the Alexander von Humboldt Foundation. H. Tanaka and N.K. have been supported by JSPS KAKENHI via grant nos. JP17H01142 and JP19K03711, respectively. H. Takagi has been supported in part by JSPS KAKENHI via grant no. JP17H01140.
Funding
Open access funding provided by Max Planck Society.
Author information
Authors and Affiliations
Contributions
J.A.N.B., R.R.C., Y.M. and H. Takagi conceived the research. N.K. and H. Tanaka synthesized the single crystals. J.A.N.B. and R.R.C. designed and performed the thermal conductivity experiments. Y.M. designed and performed the magnetic susceptibility experiments. J.A.N.B., R.R.C., Y.M. and H. Takagi analysed the data and participated in the writing of the paper. All the authors contributed to the manuscript preparation.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Physics thanks David Mandrus and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data
Extended Data Fig. 1 Magnetization isotherms and phase transitions.
Field dependences of dM/dH which were used to construct the phase diagram in Fig. 1a (main text). a, isotherms of dM/dH, with every curve offset by 0.003 emu/mol for clarity. Phase transitions out of the antiferromagnetic phases H_{C1}, H_{C2} are identified by peaks in dM/dH, whereas H_{P} coincides with a drop in dM/dH. b, isotherms of d^{2}M/dH^{2}, where the feature at H_{P} is identified as a local minimum. H_{P} shifts only minimally in field upon heating, and is no longer resolved at 10 K. Curves are offset by 8×10^{−4 }emu/mol T for clarity.
Extended Data Fig. 2 Temperature dependence of highfield features in k_{XX}.
The amplitude of highfield features in k_{XX} is expressed as (k_{XX}  k_{bg})/k_{bg}, where k_{bg} is a smooth background fit which passes through the points where k_{XX} has greatest field derivative (as discussed in Ref. ^{22}). a, field dependence (k_{XX}  k_{bg})/k_{bg} for temperatures between 70 mK (dark blue) and 2.5 K (dark red). b, temperature dependence of (k_{XX}  k_{bg})/k_{bg} at the local minimum at ~8.6 T and the local maximum at ~9.7 T, indicated by arrows in panel a. The amplitude at constant field (dashed lines), and the amplitudes at the fields of maximum amplitude (solid lines and markers) differ slightly due to a temperaturedependent phase shift. The amplitudes initially increase upon cooling but peak around 1 K and then rapidly collapse to zero down to the lowest measured temperature, in marked contrast to the LifshitzKosevitch behavior expected for conventional quantum oscillations.
Extended Data Fig. 3 Excluding error arising from Hall contact offset.
a,b, photographs of sample 2 with wires attached in different configurations 1 and 2. The latter configuration was the one used for the measurements shown in the main text. Purple dashed line are guides to the eye to help identify the contact offset. c, measured Hall contact offset expressed as ΔT_{Y}/ ΔT_{X} (%) at 0 T, showing the expected inverse offsets for the two configurations. In the absence of an offset, ΔT_{Y}/ ΔT_{X} would be zero (dashed line). d, isotherms of k_{XY}/T at 3.0 K for both contact configurations, displaying the same features at H_{D1} and H_{D2}. e, isotherms of k_{XY}/T for configuration 1 at multiple temperatures, demonstrating the reproducibility of all the main features observed with configuration 2 (main text, Fig. 3).
Extended Data Fig. 4 Excluding power dependence in k_{XY}.
Repeated measurements at different levels of heater power demonstrate that the measured thermal conductivity and thermal Hall conductivity are independent of the applied heater power. a, Hall temperature gradient normalized by heater power (ΔT_{Y}/P) for two independent isotherms at 3 K, one of which was performed at half the standard heater power (red markers, ΔT_{X} /T ≈ 5%). The two curves overlap as expected. b, the temperature dependence of k_{XY}/T at 8.64 T is compared for two T sweeps, one of which was performed at half power (green markers, ΔT_{X} /T ≈ 5%), together with fullpower H sweeps. All curves overlap within error. c, temperature dependence of k_{XY}/T is compared for two T sweeps at 13.2 T, one of which was performed at double the usual power (ΔT_{X} /T ≈ 20%), together with normalpower H sweeps. The data overlap within the (substantially fieldenhanced) error bars.
Extended Data Fig. 5 Reproducibility of thermal conductivity measurements.
Comparison of three independent measurements of the thermal conductivity (k_{XX}) of sample 2. Red circles: data measured in Kyoto, Solid blue circles: data taken in a dilution refrigerator in Stuttgart, open blue circles: data taken in a ^{4}He flow cryotstat in Stuttgart. Each measurement had slightly modified contact placements.
Extended Data Fig. 6 Temperature and magnetic fieldangle dependence of the thermal conductivity above 2.0 K.
a, field dependence of k_{XX} up to 12 T for the temperature range 2.0 K – 8.6 K. The sharp features in the field dependence of k_{XX} seen at low temperature rapidly disappear upon heating above 3.0 K. At high temperatures, the field dependence displays a single, broad minimum around 7 T. At 8.6 K, which is above the ordering temperature T_{N} = 7.5 K, the field dependence is dramatically weakened, although a high field upturn persists at this temperature. b, comparison of the field dependence of k_{XX} at 2.0 K and 2.5 K, at field angles of 70° (markers) and 90° (lines) with respect to the caxis. The nearperfect overlap of the data sets demonstrates the insensitivity to magnetic field angle.
Extended Data Fig. 7 Calculated phonon mean free path across the H_{},T phase diagram.
Contour plot of the calculated phonon mean free path l_{ph} = 3 k_{XX} / C v, where C is the phonon specific heat per unit volume and v is the average phonon velocity. We assume a magnetic fieldindependent low temperature phonon specific heat with magnitude C_{ph}/T^{3} = 1.22 mJ mol^{−1}K^{−4} as reported in Ref. ^{24}. The phonon velocity is estimated using the Debye relation \(v = \left( {\frac{{2\pi ^2k_B^4}}{{5\beta \hbar ^3}}} \right)^{\frac{1}{3}} \approx 1700m/s\) where β = C_{ph}/T^{3} as before. The grey lines trace the antiferromagnetic phase transitions, for reference.
Supplementary information
Supplementary Information
Supplementary Discussion and Fig. 1.
Source data
Source Data Fig. 1
Statistical source data.
Source Data Fig. 2
Statistical source data.
Source Data Fig. 3
Statistical source data.
Source Data Fig. 4
Statistical source data.
Source Data Extended Data Fig. 1
Statistical source data.
Source Data Extended Data Fig. 2
Statistical source data.
Source Data Extended Data Fig. 3
Statistical source data.
Source Data Extended Data Fig. 4
Statistical source data.
Source Data Extended Data Fig. 5
Statistical source data.
Source Data Extended Data Fig. 6
Statistical source data.
Source Data Extended Data Fig. 7
Statistical source data.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bruin, J.A.N., Claus, R.R., Matsumoto, Y. et al. Robustness of the thermal Hall effect close to halfquantization in αRuCl_{3}. Nat. Phys. 18, 401–405 (2022). https://doi.org/10.1038/s4156702101501y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s4156702101501y
This article is cited by

Weakcoupling to strongcoupling quantum criticality crossover in a Kitaev quantum spin liquid αRuCl3
npj Quantum Materials (2023)

Planar thermal Hall effect of topological bosons in the Kitaev magnet αRuCl3
Nature Materials (2023)

The phonon thermal Hall angle in black phosphorus
Nature Communications (2023)

An exact chiral amorphous spin liquid
Nature Communications (2023)

Thermal Hall conductivity of αRuCl3
Nature Materials (2023)