## 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 fermions7. Although they are charge-neutral, itinerant Majorana fermions carry heat and are therefore expected to contribute to thermal transport8. Applying a magnetic field gaps the bulk Majorana bands, leading to a topologically protected chiral edge current9. This edge state carries a thermal Hall conductivity per layer divided by temperature, namely, kXY2D/T = $$\frac{{{\uppi}^2k_{\mathrm{B}}}^{2}}{{6h}}$$, where kB and h are the Boltzmann and Planck constants, one-half 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 insulators10 with 5d4 Ir4+ and 4d4 Ru3+ having strong spin–orbit coupling on a honeycomb lattice11,12,13. Here α-RuCl3 is a prime candidate that shows antiferromagnetic zigzag order below the antiferromagnetic ordering temperature TN ≈ 7.5 K. This magnetic order is suppressed with an in-plane critical magnetic field of HC2 ≈ 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 half-quantized value kHQ/T (corresponding to kXY2D/T per atomic plane) was reported in the quantum spin liquid region with an in-plane 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 half-quantized thermal Hall effect should be demonstrated over a reasonably wide range of magnetic fields and temperatures. Several studies report a plateau-like region in a field at around 5 K with a magnitude close to kHQ/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 plateau-onset 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 study22 reported a broad kXY/T dome with a height appreciably smaller than kHQ/T, which was interpreted as not supporting the existence of a half-quantized 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 HC2), 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 field-induced quantum spin liquid, and more specifically, the possible topological edge state.

Here we present comprehensive measurements of kXY, kXX 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 α-RuCl3 single crystal, which reveals that kXY/T stays close to kHQ/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 half-quantized plateau. Below ~10 T, no signature of a plateau is observed and kXY/T is suppressed to zero on lowering the temperature below 6.5 K, which appears to correlate with the multiple high-field (H|| > HC2) anomalies present in both kXX and dM/dH.

### Magnetic-field-induced anomalies in thermal conductivity kXX

The temperature dependence of kXX reproduces the data from previous reports2 very well and displays a sharp minimum at TN = 7.5 K, which moves to lower temperatures as H|| increases and eventually fades out above HC2 = 7.1 T, tracing the extent of the antiferromagnetically ordered region (Fig. 1a,e). As a function of H||, the magnitude of kXX 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 kXX(H||): the broad minimum splits into two at HC1 = 6.05 T and HC2 = 7.10 T. These fields coincide with two sharp peaks in dM/dH (Fig. 1c), which correspond to two reported transitions: from zigzag-ordered phase I to intermediate-ordered phase II and from phase II to the high-field paramagnetic phase23,24,25,26,27,28. The suppression of kXX at the magnetic-phase transitions is consistent with the interpretation that kXX is dominated by phonons that scatter off magnetic excitations29.

At higher in-plane magnetic fields and below 5 K, further anomalies can be seen in kXX, namely, a broad maximum at HP ≈ 10 T flanked by broad minima at HD1 ≈ 8.8 T and HD2 ≈ 10.7 T (Fig. 1b). Features corresponding to HD1 and HP 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 high-field anomalies in kXX arise from crossovers or very weak phase transitions.

In analogy to the minima in kXX at HC1 and HC2, increased phonon scattering by soft magnetic excitations can explain the minima at HD1 and HD2. However, the anomalies at HD1 and HD2 appear broader and weaker than those at the well-defined phase transitions at HC1 and HC2 (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 HD1 and HD2 may not be well-defined phase transitions like HC1 and HC2 but highly probable crossovers with soft but finite energy excitations or alternatively very weak transitions having only a limited number of low-energy excitations involved.

Recent reports of the magnetocaloric effect23 and the magnetic Grüneisen parameter25 at temperatures above 1 K also showed anomalies near HD1 and HD2, whereas the specific heat24 at 0.67 K reported a symmetry-breaking phase transition near 10 T, a conclusion that is contested by several other measurements23,25,27,28. In the course of preparing this manuscript, we noticed another work22 in which broadly similar field-dependent structures are reported in kXX that are ascribed to quantum oscillations from underlying quasiparticles. That scenario, however, does not explain the disappearance of features in the low-temperature limit instead of the expected Lifshitz–Kosevitch behaviour (Extended Data Fig. 2) or the alignment of prominent minima with the well-established magnetic-phase transitions at HC1 and HC2.

### Thermal Hall conductivity close to the half-quantized value

A strong kXY/T signal is resolved above HC2 = 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 kXX and kXY (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, kXY/T is larger than the half-quantized value kHQ/T = 0.87 mW K–2 m–1 at high temperatures and at high magnetic fields (above T = 6.5 K and HC2 = 7.1 T) and decreases to kHQ/T and below by lowering T and H||, giving rise to a region with kXY/T ≈ kHQ/T on the H||T plane. A colour plot of kXY/T across the whole phase diagram (Fig. 4) combines data from all the H and T sweeps, which maps out the region where kXY/T is within ±20% of kHQ/T (white region). We see an L-shaped white region of kXY/T ≈ kHQ/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 kXY/T ≈ kHQ/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 elsewhere2, 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 kXX 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 sample-mounting geometries (Extended Data Fig. 3) and different sequences of data acquisition (H and T sweeps). Recent studies3,4 indeed report the region of kXY/T ≈ kHQ/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 kXY/T ≈ kHQ/T on the H||T plane have a physical implication and support the presence of the topologically protected half-quantized plateau? The enhancement of kXY/T could have different origins, for instance, through (non-quantized) topological magnon5,6 or phonon30 transport. In such scenarios, a crossing may give rise to an accidental kXY/T (H||, T) ≈ kHQ/T line on the H||T plane. In contrast, a quantized, topologically protected kXY/T(H||, T) ≈ kHQ/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 kXY/T (H||, T) ≈ kHQ/T above 10 T represented by the white triangular plane in Fig. 4 argues against an accidental crossing scenario and hints at a half-quantized plateau and hence a Majorana edge state in α-RuCl3. The corresponding T-sweep data (Fig. 2) shows that for fields greater than 10.3 T, a kink-like singularity in the T dependence of kXY/T appears at ~6.5 K (Fig. 2, red arrows). Notably, at the kink, kXY/T is always ~kHQ/T, implying that the magnitude of kHQ/T has special importance in the system. On lowering T, the kink is followed by an almost T-independent kXY/T ≈ kHQ/T region, reminiscent of an incipient half-quantized T plateau and then by a rapid decrease to zero. The width of the T-plateau-like region gradually expands on increasing the field from 10.3 T, giving rise to the white triangle in Fig. 4.

The narrow L-shaped white region below H|| ≈ 10 T (Fig. 4), on the other hand, represents a ‘line’ of kXY/T (H||, T) ≈ kHQ/T, which is clearly demonstrated by the T and H sweeps in Figs. 2 and 3. Along the vertical white region in Fig. 4, kXY/T is smoothly suppressed on cooling at 7.6 T (Fig. 2) and continuously decreases through kHQ/T with decreasing H|| in the H sweeps (Fig. 3). Along the horizontal white region at around 6.5 K, kXY/T shows a plateau in the H sweep at 6.5 K (Fig. 3), but continuously decreases through kHQ/T in the T sweeps up to ~10 T. It is not obvious why kXY/T crosses kHQ/T at a constant temperature of 6.5 K. Nevertheless, because of the line character of the region of kXY/T (H||, T) ≈ kHQ/T, the white region (Fig. 4) below H|| ≈ 10 T is probably due to accidental crossing.

### Low-temperature suppression of kXY/T

What is the origin of the rapid suppression of kXY/T to zero at low temperatures below 12 T (Fig. 4, red-coloured 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 HC2 and the two high-field anomalies at HD1 and HD2 observed in kXX and dM/dH. The maximum at ~10 T reflects the singularly weak T dependence of kXY/T at H|| = 10.3 T (Fig. 2) and coincides with HP. These correlations suggest that the high-field magnetic crossovers/transitions are related to the suppression of kXY/T at low temperatures.

If a half-quantized thermal Hall plateau arises from a chiral Majorana edge mode8, it was theoretically shown that the coupling between a phonon bath and the edge mode31,32 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 low-temperature suppression of kXY/T. From the estimation of the phonon mean free path (lph) from kXX as a function of H|| and T, however, we may exclude the phonon–edge mode decoupling scenario as the origin of low-temperature suppression of kXY/T below kHQ/T from 7 to 12 T.

kXX ﻿approaches a T3 power law at the highest fields and lowest temperatures (Fig. 1f), consistent with phonon transport limited by a temperature-independent scattering length. The calculated lph 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 kXY/T coincide with the minima in kXX, that is, regions with the shortest lph, argues against the decoupling scenario. The low-temperature enhancement of kXY/T to kHQ/T on increasing the magnetic field is accompanied by an increase in lph, 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 half-quantized plateau is only observed in samples with high (phonon) thermal conductivity4.

Recently, the scenario of purely phonon-driven thermal Hall was proposed based on the discovery of a large thermal Hall effect in non-magnetic SrTiO3 and the correlation between the T dependence of kXX and kXY in RuCl3 (ref. 30). In addition, in the magnetic insulator Ba3CuSb2O9, a thermal Hall angle (tan(θH) ≈ 10−4), similar to that of RuCl3, was claimed to arise from the phonon thermal Hall effect alone33. The observed suppression of kXY/T at low temperatures and its correlation with kXX are in line with a phonon-driven scenario. Both show an overall increase on increasing the field, with dip-like suppressions at HD1 and HD2. Although a phonon-only scenario could explain these correlations between kXX and kXY, it is not clear how it would naturally explain the observed plateau-like behaviour of kXY/T close to kHQ/T.

The identification of possible topological phase transitions into and out of the half-quantized state still remains an open question23,24,25,27,28. For the former, unveiling the nature of high-field anomalies may be the key question to be addressed. The high-field transition out of the half-quantized state, if it exists, must be at a field higher than 13 T. A measurement of kXY/T to much higher fields is, therefore, highly desired. Alternative mechanisms leading to an enhancement of kXY/T, such as a topological magnon-driven5,6 or phonon-driven30 Hall effect, have been proposed. Our results do not exclude those scenarios but pose them a challenge to explain the plateau-like behaviour close to half-quantization.

## Methods

### Samples

Large, thin (~2.5 mm × 1.3 mm × 17 µm), high-quality single crystals of α-RuCl3 from the same growth batches as those reported in ref. 2 were measured. The sample under investigation for thermal Hall measurements has a confirmed half-quantized 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 steady-state three-thermometer setup. For the temperature range of 1.8–9.0 K, Cernox CX-1050 chip thermometers (Lake Shore) were used in a 4He 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 field-calibrated 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 kXY/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 (4He) or 90° from the c axis (dilution refrigerator), with the in-plane field component in the a axis. The amplitude of kXX, as well as the measured transition fields (HC1, HC2, HD1 and HD2), 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 kXX were measured, indicating that sample deterioration over time is negligible. The measurement of kXX 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 custom-built 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.