Abstract
Quantum spin liquids (QSLs) are topological states of matter exhibiting remarkable properties such as the capacity to protect quantum information from decoherence. Whereas their featureless ground states have precluded their straightforward experimental identification, excited states are more revealing and particularly interesting owing to the emergence of fundamentally new excitations such as Majorana fermions. Ideal probes of these excitations are inelastic neutron scattering experiments. These we report here for a rutheniumbased material, αRuCl_{3}, continuing a major search (so far concentrated on iridium materials) for realizations of the celebrated Kitaev honeycomb topological QSL. Our measurements confirm the requisite strong spin–orbit coupling and lowtemperature magnetic order matching predictions proximate to the QSL. We find stacking faults, inherent to the highly twodimensional nature of the material, resolve an outstanding puzzle. Crucially, dynamical response measurements above interlayer energy scales are naturally accounted for in terms of deconfinement physics expected for QSLs. Comparing these with recent dynamical calculations involving gauge flux excitations and Majorana fermions of the pure Kitaev model, we propose the excitation spectrum of αRuCl_{3} as a prime candidate for fractionalized Kitaev physics.
Main
Exotic physics associated with frustrated quantum magnets is an enduring theme in condensed matter research. The formation of quantum spin liquids (QSLs) in such systems can give rise to topological states of matter with fractional excitations^{1,2,3,4}. Fractionalization describes the counterintuitive phenomenon where an electron breaks apart into welldefined independent quasiparticles. The realization of this physics in real materials is an exciting prospect that may provide a path to a robust quantum computing technology^{5}. Fractional excitations in the form of pairs of S = 1/2 spinons are observed in quasionedimensional (1D) materials containing S = 1/2 Heisenberg antiferromagnetic chains^{6}. Recent evidence for the 2D QSL state, in the form of possible spinon excitations, has been found in quantum antiferromagnets on triangular^{3} and Kagome^{7} lattices. The exactly solvable Kitaev model on the honeycomb lattice^{8} represents a class of 2D QSL that supports two different emergent fractionalized excitations: Majorana fermions and gauge fluxes^{9,10}. The comparatively simple gauge flux can be visualized as a spin–orbit coupled version of a plaquette observable like a resonance energy. The Majorana fermions, by contrast, do not have a straightforward realspace representation because they are not associated with any realspace spin or charge density. At best, an idea of their nature can be gleaned in the strongly anisotropic limit of weakly coupled Ising dimers, where they can be thought of as excitations taking the form of a misaligned nearestneighbour spin pair on top of a ground state consisting of a coherent superposition of satisfied dimers. How to observe such ephemeral entities is one of the central challenges of condensed matter and materials physics today. It has turned out that the signature of the Majorana fermion in the response function measured by means of inelastic neutron scattering is perhaps one of the most direct ways of pinning down the excitation’s existence^{10}. This manuscript reports precisely such a measurement.
The Kitaev model consists of a set of spin1/2 moments {S_{i}} arrayed on a honeycomb lattice. The Kitaev couplings, of strength K in equation (1), are highly anisotropic with a different spin component interacting for each of the three bonds of the honeycomb lattice. In actual materials, a Heisenberg interaction (J) is also generally expected to be present, giving rise to the Heisenberg–Kitaev (H–K) Hamiltonian^{11,12}. where m is the component of the spin directed along the bond connecting spins (i, j). The QSL phase of the pure Kitaev model (J = 0), for both ferromagnetic and antiferromagnetic K, is stable for weak Heisenberg perturbations.
Remarkably, the Hamiltonian (1) has been proposed to accurately describe edgeshared octahedrally coordinated magnetic systems, shown in Fig. 1a, with dominant spin–orbit coupling^{11,12}. The focus so far has centred largely on Ir^{4+} compounds^{13,14,15,16,17,18,19}; however, attempts to measure the dynamical response^{15} by means of inelastic neutron scattering (INS) have met with limited success, owing to the unfavourable magnetic form factor and strong absorption crosssection of the Ir ions. Resonant inelastic Xray scattering (RIXS) has provided important information concerning higherenergy excitations in the iridates^{18}, but cannot provide the meV energy resolution necessary to provide a robust experimental signature of collective fractional excitations that are expected to occur at energy scales of the order of 1–10 meV (ref. 15).
An alternative approach is to explore materials with Ru^{3+} ions^{20}. The realization that the material αRuCl_{3} (refs 20,21,22) also has the requisite honeycomb lattice and strong spin–orbit coupling has stimulated a groundswell of recent investigations^{23,24,25,26,27,28,29}. Although these studies lend support to the material as a potential Kitaev material, conflicting results centring on the lowtemperature magnetic properties have hindered progress. To resolve this, we undertake a comprehensive evaluation of the magnetic and spin–orbit properties of αRuCl_{3}, and further measure the dynamical response, establishing this material as proximate to the widely sought QSL.
We begin by investigating the crystal and magnetic structure of αRuCl_{3}. Samples were synthesized and characterized as described in Methods. The layered structure of the material is shown in Fig. 1a. Figure 1b, c shows the ABCABC stacking arrangement of the layers expected in the trigonal structure (space group P3_{1}12). That the layers are weakly bonded to each other, similar to graphite, is demonstrated by the lattice specific heat (shown for a powder in Fig. 1d). This exhibits a telltale T^{2} behaviour characteristic of highly 2D bonded systems^{30}, rather than the usual T^{3} observed in conventional 3D solids. Because the 2D layers are weakly coupled, the interlayer magnetic exchanges will also be rather weak. In addition, stacking faults are formed easily and significant regions of the sample can crystallize in alternative stacking structures, for example ABAB (ref. 25) (see Supplementary Fig. 2).
Neutron diffraction (see Methods) shows lowtemperature magnetic order. The temperature dependence of the strongest magnetic powder peak, with T_{N} ≍ 14 K, is shown in Fig. 1e. Figure 1f shows the temperature dependence of magnetic peaks in one 22.5 mg single crystal, revealing two ordered phases. The first, which orders below T_{N} ≍ 14 K, is characterized by a wavevector q_{1} = (1/2 0 3/2) (indexed according to the trigonal structure), whereas the other phase (q_{2} = (1/2 0 1)) orders below 8 K (see also Supplementary Fig. 3). These temperatures correspond precisely to anomalies observed in the specific heat and magnetic susceptibility^{25,26,29} (Supplementary Fig. 1). This is readily explained, as the observed L = 3/2 phase corresponds naturally to a stacking order of ABAB type along the caxis, and the L = 1 corresponds to ABCABC stacking. Indeed, the difference in 3D transitions is a residual effect of different interlayer bonding influencing the ordering. Further, a comparison of intensities at (1/2 0 L) with (3/2 0 L)^{16} shows both phases share identical zigzag (ZZ) spin ordering in the honeycomb layers; a phase of the H–K model adjacent to the spin liquid^{11} (see Supplementary Table 1). By calibrating to structural Bragg peaks, the ordered moments are measured to be exceptionally low, with an upper bound of μ = 0.4 ± 0.1μ_{B}. This is at most only 35% of the full moment determined from bulk measurements^{22,25,27}, suggesting strong spin fluctuations consistent with a nearliquidlike quantum state in the material. (See Supplementary Information for more detail.)
Having established the structural and magnetic properties of αRuCl_{3}, we probe the nature of the singleion states to confirm the presence of strong spin–orbit coupling, which is required to generate the Kitaev term K in equation (1). Using INS (see Methods) with E_{i} = 1.5 eV incident neutrons to measure the transition from the Ru^{3+} electronic ground state to its excited state, the spin–orbit coupling λ is extracted. In the octahedral environment shown in Fig. 1, the ground state is a lowspin (J = 1/2) state. The next excited state (J = 3/2) is separated by 3λ/2. Neutrons can activate it by a spinflip process, and the transition is seen in Fig. 2 at 195 ± 11 meV, implying that λ ≍ 130 meV (also see Supplementary Fig. 4 and Supplementary Information). This is close to the expected freeion value (λ_{free} ≍ 150 meV; refs 20,31) and the predictions of recent ab initio calculations^{26}. The J = 3/2 state will be split into two Kramers doublets by small distortions of the octahedron^{32,33}. The resolutionlimited linewidth suggests that such a splitting is relatively small, certainly less than the halfwidth at halfmaximum (HWHM) of 48 meV. In any case, as the higher levels are too energetic to play any role, only the lowest lying doublet needs to be considered. Projecting the interRu^{3+} couplings into this doublet results in Kitaev terms as included in equation (1).
The above results indicate that the H–K Hamiltonian (1) can indeed satisfactorily capture the interactions between Ru^{3+} moments. If this is the case, then given the highly reduced ordered moment and the extended QSL region close to the observed zigzag AFM phase, it is tempting to speculate that signatures of fractionalization characteristic of QSLs will be manifest in the collective magnetic excitations. Figure 3 shows INS data for αRuCl_{3} powder measured using neutrons of E_{i} = 25 meV (more details in Methods). The scattering in the magnetically ordered state is shown in Fig. 3a for T = 5 K. Two distinct features are clearly visible, spanning different energy ranges. The lower among them, M_{1}, is centred near 4 meV and shows a minimum near Q = 0.62 Å^{−1}, which notably corresponds to the M point of the honeycomb lattice, as expected for a quasi2D magnetic system (for 3D behaviour a wavevector Q = 0.81 Å^{−1} is anticipated). The white arrow draws attention to the concave shape of the edge of the scattering, which is expected for magnon excitations in a ZZ ordered state^{15}. This firmly establishes the nature of magnetic order and differentiates it from other potential states, such as a stripy ground state. The second feature is at a higher energy, M_{2}, centred near 6.5 meV.
Both features, M_{1} and M_{2}, correspond to powderaveraged modes which are of magnetic origin, as identified by their wavevector and temperature dependence. The thermal behaviour of these magnetic modes differs significantly from one to the other. Figure 3b shows the scattering at T = 15 K, just above T_{N}. It is seen that M_{1} softens markedly and the intensity shifts towards Q = 0. Conversely, M_{2} is essentially unaffected. ConstantQ cuts through the data are shown in Fig. 3c. The centres are at the positions indicated by the labelled dashed lines in Fig. 3a, b. Comparing cuts (A, B) with (C, D) reinforces the collapse and shift of intensity for M_{1} above T_{N}. Cut B clearly shows two peaks, implying that the density of states sampled by the powder average at T = 5 K has two maxima. The average peak energies determined by fits of the data to Gaussian peaks are given by E_{1} = 4.1(1) meV and E_{2} = 6.5(1) meV. Figure 3d shows constantenergy cuts integrated over the range [2.5, 3.0] meV, near the lower edge of M_{1}. It is seen that, at low temperature, M_{1} is structured with lowenergy features showing up as peaks in cut E. These are centred at Q_{1} = 0.62(3) Å^{−1} and Q_{2} = 1.7(1) Å^{−1}. Above T_{N} this structure disappears, and the broad scattering shifts markedly to lower Q. Fitting the T = 15 K data (cut F) to a Lorentzian with the centre fixed at Q = 0 yields a HWHM of roughly 0.6 Å^{−1}, suggesting that, above T_{N}, spatial correlations of the spin fluctuations are extremely short ranged.
To gain further insight into the magnetic couplings we compare the INS data to the solution of (1) using conventional linear spin wave theory (SWT) for ZZ order^{34,35}. The SWT provides a quasiclassical approximation which works reasonably well when quantum fluctuations are weak. Although strictly speaking it is inapplicable for strongly quantum fluctuating systems, it provides a first starting point for estimating the approximate and relative strengths of the couplings. In the honeycomb lattice appropriate for αRuCl_{3}, SWT predicts four branches, two of which disperse from zero energy at the M point (1/2, 0) to doubly degenerate energies and , respectively, at the Γ point (0,0) (ref. 34). A large density of states in the form of van Hove singularities is expected near ω_{1} and ω_{2}. Figure 4a shows the SWT and Fig. 4b the calculated powderaveraged neutron scattering. Equating ω_{1} and ω_{2} with the peaks E_{1} and E_{2} yields K and J values of (K = 7.0, J = −4.6) meV (shown in Fig. 4) or (K = 8.1, J = −2.9) meV (shown in Supplementary Fig. 5), depending on whether ω_{1} corresponds to E_{1} or E_{2}. These two possibilities lie on either side of the symmetric point K = −2J, where ω_{1} = ω_{2}. The inset of Fig. 4d shows each of these possibilities on the H–K phase diagram^{34}. Either way, the Kitaev term is stronger and antiferromagnetic, whereas the Heisenberg term is ferromagnetic; again consistent with ab initio calculations^{26}.
We note that the M_{1} mode has a gap of at least 1.7 meV near the M point (see Fig. 5a) that is not exhibited in the above SWT calculations. Although such a gapless spectrum is a known artefact of linear SWT for the H–K model^{34}, the experimentally observed gap is too large to be accounted for within systematic 1/S corrections. Extending the Hamiltonian to include further terms can lead to a gap forming within SWT. However, calculations of the SW spectrum (see Supplementary Fig. 5 and Supplementary Information) with additional terms in the Hamiltonian (such as Γ and/or Γ’ terms^{35,36,37,38,39}), when sufficient to generate the observed gap, show features in the powderaveraged scattering that are inconsistent with the observations. Within the SW approximation, a gap can also be generated by adding an additional Isinglike anisotropy, perhaps at the level of 15% of J, which is also equivalent to an anisotropic Kitaev interaction. As discussed below, the resulting SWT is still incompatible with the data.
Although the SWT calculation reproduces many of the features of the observed dynamical response, crucial qualitative disagreements remain. Most importantly, the observed dependence of the M_{2} mode on temperature and energy is incompatible with linear SWT. The constantwavevector cuts shown in Fig. 3c show that M_{2} maintains a totally consistent peak shape and intensity above and below T_{N}. Moreover, for temperatures well above T_{N}, to at least 40 K, the intensity for all measured wavevectors is essentially unchanged, as shown by Fig. 5b, which is a plot of the M_{2} intensity as a function of Q for several temperatures. In fact a welldefined M_{2} peak persists with a similar Q dependence up to at least 70 K, corresponding to T ∼ 5T_{N}. This is in sharp contrast to the typical behaviour of spin waves in conventional magnets, which generally exhibit a pronounced decrease of intensity above the ordering temperature. It should also be noted that in the ordered state the energy width of M_{2} is much broader than the SW calculation over the observed range of Q. Figure 5c shows a constantQ cut around the M_{2} mode (blue triangles). The red line shows the equivalent powderaveraged SWT calculation (Fig. 4b), broadened by the instrumental energy resolution (marked ‘R’) and scaled so that the intensity matches the height of the M_{2} scattering. The lowenergy side of the calculation is affected by the lower mode, and therefore cannot be directly compared with the data; however, it is clear from the highenergy side that there is considerable extra scattering (indicated by the shading) that is not captured by SWT. As discussed in the Supplementary Information, the smooth drop off of intensity on the highenergy side of the M_{2} peak is evidence against the extra width arising from additional features in the spin wave spectrum that can be achieved by adding extra terms to equation (1). Finally, as discussed in the Supplementary Information, for temperatures above T_{N}, the detailed wavevector dependence of the scattering is not what is expected from conventional SWT.
The SWT is a quantization of harmonic excitations from classical order. Moreover, the lowordered moment observed in αRuCl_{3} indicates that linear SWT is inadequate. Indeed, we argue that the behaviour of the observed higherenergy mode M_{2}—which because of its short timescale is least sensitive to 3D couplings—is naturally accounted for through the QSL phase proximate in the H–K phase diagram^{40}.
This QSL viewpoint has the strong quantum limit as its starting point. It can avail itself of the recently computed exact dynamical structure factor of the pure Kitaev model, in which spin excitations fractionalize into static Ising fluxes and propagating Majorana fermions minimally coupled to a Z_{2} gauge field^{10}. Powderaveraged results of the scattering^{10} expected for the isotropic antiferromagnetic Kitaev model are shown in Fig. 5d. Although the QSL is gapless, the structure factor of its excitations shown in Fig. 5d does show a gap. This is due to the fact that a spin flip always excites both quasiparticles—gapless Majorana fermions and a pair of Ising fluxes, the latter with a nonzero excitation gap^{10}. This results in a lowenergy band from 0.125 to 0.5 K, with a peak of intensity near the M point in the Brillouin zone for an antiferromagnetic K. Most interestingly, in addition, a second very broad and nondispersing highenergy band appears, centred at an energy that corresponds approximately to the Kitaev exchange scale, K. (For a similar calculation on the ferromagnetic Kitaev model, and a general discussion, see Supplementary Fig. 6 and Supplementary Information) The intensity of the upper band is strongest at Q = 0, and decreases with increasing Q.
With the Kitaev interaction dominant it is reasonable to expect that αRuCl_{3} is proximate to the QSL phase. The additional nonKitaev interactions lead to longrange order at low temperatures, and strongly affect the lowenergy excitations, which then exhibit spin wave behaviour. Conversely, the highenergy spin fluctuations native to the proximate quantum ground state are more immune, and can persist even in the ordered state. This behaviour is well known in coupled S = 1/2 antiferromagnetic Heisenberg chains^{6}, where at energies large compared to the interchain coupling the spectrum of fractionalized excitations (spinons) of the pure chain dominates the response above and below the magnetic ordering temperature. This leads to a natural interpretation of the M_{2} mode as having the same origin as the upper mode of the Kitaev QSL. The broad width of the M_{2} mode as seen in the measurements can be naturally explained in terms of the fractionalized Majorana fermion excitations. The green line in Fig. 5c shows the calculated powderaveraged QSL scattering, including the effects of instrumental resolution, with the value K = 5.5 meV chosen to match the experimental peak position of M_{2} and the overall height chosen to match the observed scattering. The calculated QSL scattering profile is well matched to the observed additional width of the M_{2} scattering on the highenergy side. This value of K is slightly smaller than that inferred from SWT, but it is very reasonable to expect that the quantum description requires a renormalized parameter. The large energy width is expected for a fractionalized system, because several excitations are excited in a single spinflip process. Moreover, the Q dependence of the intensity of the M_{2} mode (Fig. 5b) strikingly resembles that of the upper band in the pure Kitaev model. The feature is broad in momentum, because the realspace spin correlations of a QSL are short ranged. For convenience, a sidebyside comparison of the Q dependence of the data and the scattering calculated for SWT and a pure Kitaev model is presented in Supplementary Fig. 7.
The fact that M_{2} survives well above T_{N}, even if M_{1} is completely washed out, indicates that the M_{2} mode is not directly connected to the existence of longrange magnetic order. In the strictly 2D Kitaev model there is no true phase transition from the QSL to the hightemperature paramagnet^{41}. However, recent Monte Carlo calculations at finite temperature suggest that highenergy Majorana fermions, thus the M_{2} mode, remain stable up to the highest crossover temperature at an energy scale of K (ref. 42), consistent with the observations reported here.
Taken together, the qualitative features from a complete quantum calculation using a Majorana fermion treatment can successfully provide a broadly consistent account of the inelastic neutron scattering data. This makes αRuCl_{3} a prime candidate for realizing Kitaev and QSL physics. Further support for the presence of Kitaev QSL physics in αRuCl_{3} is seen in recent Raman scattering measurements^{24} which show a broad response similar to that calculated for the pure Kitaev model^{17}, with a value of K = 8 meV, of the same order as that derived here. The Raman continuum also persists to temperatures well above T_{N}. Much more detailed information on the structure of the response functions will require INS in single crystals of both αRuCl_{3} and other relevant compounds, some of which are 3D (refs 43,44). The most instructive measurements on αRuCl_{3} should use single crystals free of the complications induced by stacking faults.
Ideally, a single, fully quantum theoretical treatment should capture the microscopic behaviour across all energy and length scales; however, such a treatment is unavailable for the full Hamiltonian describing the magnetic properties of αRuCl_{3}. Here, we have used the insight that the highenergy shortrange spinliquid physics is well captured by a pure Kitaev model, which permits an analytic treatment, but misses the weak ordering tendency owing to perturbations to the simple model Hamiltonian. These, however, and their concomitant lowenergy spin wave excitations can be approximately captured by SWT. Considering the usual renormalizations inherent in semiclassical descriptions of quantum excitations, these two approximation schemes for different parts of the spectrum can be described by similar microscopic parameters, suggesting that the absence of a full treatment of the complete H–K model is a technical rather than a conceptual issue.
Looking forward, it will also be of great interest to systematically investigate the effects of disorder and doping in these materials^{45}, and there is also the hope of generating a genuinely 2D system by exfoliation techniques.
Methods
Synthesis and bulk measurements.
CommercialRuCl_{3} powder was purified in house to a mixture of αRuCl_{3} and βRuCl_{3}, and converted to 99.9% phasepure αRuCl_{3} by annealing at 500 °C. Single crystals of αRuCl_{3} were grown using vapour transport with TeCl_{4} as the transport agent. The crystals exhibit an anisotropic mosaic for inplane peaks, indicative of stacking faults, as shown in Supplementary Fig. 2. Samples were characterized by standard bulk techniques (see Supplementary Fig. 1). Xray powder diffraction was carried out at room temperature using a Panalytical Empyrean diffractometer employing Cu Kα radiation.
The structure was found to be consistent with the trigonal space group P3_{1}12 (No. 151), with roomtemperature lattice constants a = b = 5.9783(2) Å, c = 17.170(1) Å, with χ^{2} = 13.7 and wRp = 5.16. For C2/m the corresponding fits are worse, with a = 5.982(1), b = 10.3530(7), c = 6.0611(5), β = 109.177(7), with χ^{2} = 16.9, wRp = 6.33. In addition, powder neutron diffraction was carried out at 10 K. For the fit and the lattice constants at T = 10 K refer to Supplementary Fig. 2 and Supplementary Table 2. Magnetic properties were measured with a Quantum Design (QD) Magnetic Property Measurement System in the temperature interval 1.8 K ≤ T ≤ 300 K. Temperaturedependent specific heat data were collected using a 14 T QD Physical Property Measurement System (PPMS) in the temperature range from 1.9 to 200 K. Our measurements of the susceptibility (see Supplementary Fig. 1) are consistent with existing literature^{22,25,27}. The magnetic susceptibility of powders fits a Curie–Weiss law over the range above 150 K, with a temperature intercept of θ ≍ 32 K and a singleion Ru effective moment of 2.2μ_{B}. Magnetic order appears for T ≤ 15 K, leading to a broad specific heat anomaly. The detailed specific heat of singlecrystal specimens is sample dependent, but consistent with other groups^{25,27,29}, and shows the onset of a broad anomaly near 14 K, and a sharper peak near 8 K, possibly with additional structure between those temperatures. This complicated behaviour is a consequence of stacking faults (see main text).
Neutron diffraction.
Neutron diffraction data for structural refinement on a 5.1 g powder sample of αRuCl_{3} were collected at the POWGEN beamline at the Spallation Neutron Source (SNS), at Oak Ridge National Laboratory (ORNL). The sample was loaded in a vanadium sample can under helium, and measured at T ≍ 10 K. Neutron diffraction measurements to characterize the magnetic Bragg peaks in both powder and single crystals were performed at the HB1A Fixed Incident Energy (FIETAX, E_{i} = 14.68 meV) tripleaxis instrument at the HighFlux Isotope Reactor at ORNL. For powder diffraction, 4.7 g of powder was packed into a cylindrical aluminium canister. For singlecrystal diffraction, one ∼0.7 × 1.0 cm^{2}, 22.5 mg crystal was attached to a flat aluminium shim using CytopM glue. It was then sealed with indium into an aluminium canister with helium exchange gas, then aligned and confirmed to be a singledomain sample using neutrons. This was attached to the cold finger of a 4 K closedcycle refrigerator for performing the temperature scans.
Inelastic neutron scattering (INS).
Inelastic neutron scattering of powder αRuCl_{3} was performed using the SEQUOIA chopper spectrometer at the SNS (ref. 46). The sample (5.3 g) was sealed at room temperature in a 5 × 5 × 0.2 cm^{3} flat aluminium sample can using helium exchange gas for thermal contact. This was mounted to the cold finger of a closedcycle helium refrigerator for temperature control. Empty can measurements were performed under the same conditions as the sample measurements. The neutron detector efficiencies were calibrated using vanadium standards, and the neutron counts were normalized to the accumulated incident proton charge. The data presented have the empty can background subtracted, and the uncertainties were calculated assuming Poisson counting statistics with conventional propagation of error calculations. Measurements were made with incident neutron energies E_{i} = 8, 25 and 1,500 meV. The E_{i} = 8 and 25 meV measurements were performed using the fineresolution 100 meV Fermi chopper slit package spinning at 180 Hz and the T_{0} chopper spinning at 30 Hz. The E_{i} = 1,500 meV measurements used the 700 meV coarseresolution Fermi chopper spinning at 600 Hz and the T_{0} chopper spinning at 180 Hz (ref. 47). The E_{i} = 1,500 meV configuration yields a calculated fullwidth at halfmaximum (FWHM) energy resolution of approximately 97 meV at 200 meV energy transfer. The FWHM elastic energy resolution is calculated to be 0.19 and 0.64 meV for the E_{i} = 8 and 25 meV configurations, respectively. Care was taken to minimize the exposure of the sample to air, and after every exposure the sample was pumped for at least 30 min to remove adsorbed moisture. Structural refinements confirmed the purity of the powder sample. Spin wave simulations were performed using SpinW codes^{48} (Version 235) and used the nominal symmetric honeycomb structure for αRuCl_{3} (refs 21,22). The SWT powder average was performed with 3,000 random points distributed over the Brillouin zone. The Ru^{3+} form factor utilized was interpolated using the results of relativistic Dirac–Slater wave functions^{49}.
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Acknowledgements
Research using ORNL’s HFIR and SNS facilities was sponsored by the US Department of Energy, Office of Science, Basic Energy Sciences (BES), Scientific User Facilities Division. A part of the synthesis and the bulk characterization performed at ORNL was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (C.A.B. and J.Q.Y.). The work at University of Tennessee was funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4416 (D.G.M. and L.L.). The work at Dresden was in part supported by DFG grant SFB 1143 (J.K. and R.M.), and by a fellowship within the PostdocProgram of the German Academic Exchange Service (DAAD) (J.K.). D.L.K. is supported by EPSRC Grant No. EP/M007928/1. The collaboration as a whole was supported by the Helmholtz Virtual Institute ‘New States of Matter and their Excitations’ initiative. We thank B. Chakoumakos for overall support in the project, and J. Chalker, J. Rau, S. Toth, G. Khaliullin and F. Ye for valuable discussions. We thank P. Whitfield from the POWGEN beamline and Z. Gai from the CNMS facility for helping with neutron diffraction and magnetic susceptibility measurements.
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Affiliations
Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
 A. Banerjee
 , A. A. Aczel
 , M. B. Stone
 , G. E. Granroth
 , M. D. Lumsden
 & S. E. Nagler
Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
 C. A. Bridges
Material Sciences and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
 J.Q. Yan
 & D. G. Mandrus
Department of Materials Science and Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA
 J.Q. Yan
 & D. G. Mandrus
Department of Physics, University of Tennessee, Knoxville, Tennessee 37996, USA
 L. Li
 & Y. Yiu
Neutron Data Analysis & Visualization Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
 G. E. Granroth
Department of Physics, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, UK
 J. Knolle
 & D. L. Kovrizhin
Max Planck Institute for the Physics of Complex Systems, D01187 Dresden, Germany
 S. Bhattacharjee
 & R. Moessner
International Center for Theoretical Sciences, TIFR, Bangalore 560012, India
 S. Bhattacharjee
Neutron Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
 D. A. Tennant
Bredesen Center, University of Tennessee, Knoxville, Tennessee 37966, USA
 S. E. Nagler
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Contributions
S.E.N., A.B. and D.G.M. conceived the project and the experiment. C.A.B., A.B., L.L., J.Q.Y., Y.Y. and D.G.M. made the sample. J.Q.Y., L.L., A.B. and C.A.B. performed the bulk measurements, A.B., A.A.A., M.B.S., G.E.G., M.D.L. and S.E.N. performed INS measurements, A.B., S.E.N., C.A.B., M.D.L., M.B.S. and D.A.T. analysed the data. Further modelling and interpreting of the neutron scattering data was carried out by A.B., M.D.L., S.E.N., J.K., S.B., D.L.K. and R.M., where A.B., M.D.L., S.B. and S.E.N. performed SWT simulations, and J.K., S.B., D.L.K. and R.M. carried out QSL theory calculations. A.B. and S.E.N. prepared the first draft, and all authors contributed to writing the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to A. Banerjee or S. E. Nagler.
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