Abstract
The celebrated Kitaev quantum spin liquid (QSL) is the paradigmatic example of a topological magnet with emergent excitations in the form of Majorana Fermions and gauge fluxes. Upon breaking of timereversal symmetry, for example in an external magnetic field, these fractionalized quasiparticles acquire nonAbelian exchange statistics, an important ingredient for topologically protected quantum computing. Consequently, there has been enormous interest in exploring possible material realizations of Kitaev physics and several candidate materials have been put forward, recently including αRuCl_{3}. In the absence of a magnetic field this material orders at a finite temperature and exhibits lowenergy spin wave excitations. However, at moderate energies, the spectrum is unconventional and the response shows evidence for fractional excitations. Here we use timeofflight inelastic neutron scattering to show that the application of a sufficiently large magnetic field in the honeycomb plane suppresses the magnetic order and the spin waves, leaving a gapped continuum spectrum of magnetic excitations. Our comparisons of the scattering to the available calculations for a Kitaev QSL show that they are consistent with the magnetic field induced QSL phase.
Introduction
The exactly soluble Kitaev model consists of S = ½ spins on the honeycomb lattice interacting with anisotropic ising interactions along the three symmetry inequivalent bonds.^{1} The insight that Kitaev physics might be realized in practice^{2} has stimulated investigations of candidate materials, recently including αRuCl_{3.}^{3,4,5,6,7,8,9,10,11,12,13,14,15,16} In all the systems studied to date significant nonKitaev interactions induce magnetic order at low temperature.^{3,17,18} However, inplane magnetic fields of roughly 8 T suppress the longrange magnetic order in αRuCl_{3},^{3,4,5,6,7,8,9,10} raising the intriguing possibility of a fieldinduced quantum spin liquid (QSL)^{11} exhibiting nonAbelian quasiparticle excitations. Here we present inelastic neutron scattering (INS) in αRuCl_{3} in an applied magnetic field.
The insulating magnetic material αRuCl_{3} is comprised of van der Waals coupled honeycomb layers of 4d^{5} Ru^{3+} cations nearly centered in edgesharing RuCl_{6} octahedra. A strong cubic crystal field combined with spinorbit coupling leads to a Kramer’s doublet, nearly perfect J = ½ ground state,^{12,15,16} thus satisfying the conditions necessary for producing Kitaev couplings in the low energy Hamiltonian.^{2} Similar to the widely studied honeycomb^{17} and hyperhoneycomb^{18} Iridates, at low temperatures αRuCl_{3} exhibits smallmoment antiferromagnetic (AFM) zigzag order^{4,14,19,27} with T_{N} ≈ 7 K for crystals with minimal stacking faults. In the zigzag state the magnetic excitation spectrum shows welldefined lowenergy spin waves with minima at the Mpoints (See Supplementary Fig. S1 in Supplementary Materials (SM) for the Brillouin Zone (BZ) definition) as well as a broad continuum that extends to much higher energies centered at the Γ points.^{12,13,14} Above T_{N} the spin waves disappear but the continuum remains, essentially unchanged until high temperatures of the order of 100 K.^{12,13,14} In analogy with the situation for coupled spin ½ AFM Heisenberg chains,^{20} the high energy part of the continuum has been interpreted as a signature of fractionalized excitations.^{12,13,14} The overall features of the INS response resemble those of the Kitaev QSL^{21,22,23} and are consistent with an unusual response seen in Raman scattering,^{22,23,24,25} suggesting that the system is proximate to a QSL state exhibiting magnetic Majorana fermion excitations.^{13,14} It is thus of great interest to investigate the nature of the excitations in the fieldinduced disordered state.^{3,4,5,6,7,8,9,10,11} We show that at a field of 8 T, the spin waves characteristic of the ordered state vanish throughout the BZ. The remaining single dominant feature of the response is a broad continuum centered at the Γ point, signature of fractionalized excitations.^{12,13,14} This provides compelling evidence that a fieldinduced QSL state has been achieved.
Results
To investigate this phenomenon, highquality single crystals of αRuCl_{3} were grown using vaportransport techniques.^{14,19} Figure 1 shows bulk susceptibility and neutron diffraction measurements, demonstrating the suppression of the zigzag order (Fig. 1a) when the field is applied along the ζ = (−1, 2, 0) (trigonal notation, see e.g., refs.^{14,19,26}) or equivalent directions (Fig. 1b). The local maximum in the susceptibility, where dχ/dB = 0, occurs at T_{N} = 7.5 K at 0.1 T, close to the location of the zerofield heat capacity anomaly in singlephase RuCl_{3} crystals with ABC magnetic stacking.^{19} With increasing field, T_{N} shifts to lower temperatures and the transition is not observed beyond B_{C }= 7.3 (3) T.
Magnetic Bragg peaks associated with the zigzag spin order appear below T_{N} at (½, 0, L), (0, ½, L) and (½, −½, L) where L ≠ 3n.^{10,14,19,26,27} Consistent with Sears et al.,^{10} magnetic fields of 2 T along {110}equivalent directions completely suppress the intensity of magnetic Bragg peaks with Q ⊥ B as seen in Fig. 1c. Conversely, peaks with a significant projection of Q along B gain intensity at low fields with increasing fields (Supplementary Fig. S2). This intensity redistribution can signify a reorientation of the ordered moments to lie perpendicular to the field direction or a depopulation of domains with moments aligned along the field direction.^{10} The weighted average intensity of magnetic peaks with L = 1 is roughly flat up to B ≈ 3.5 T and then follows a downward trend. (Supplementary Fig. S3 shows additional details for the peaks at (½, 0, 1) as well as (½, 0, 2)).
Figure 2 shows timeofflight INS from a 740 mg single crystal mounted with the (H, 0, L) scattering plane horizontal, and a vertical magnetic field applied in the ζ = (−1, 2, 0) direction (see Methods section). The scattering at 2 K in zero field (Fig. 2a) shows lowenergy, gapped spin waves with the expected minima at the (±1/2, 0, L) M points. The spin waves also show local minima at the Γ (H = 0) and Y (H = 1) points. (See Supplementary Fig. S4 for constant Qcuts at these locations). The signal near the Γ point, however, is dominated by a broad continuum extending to higher energies, consistent with previous measurements.^{12,13,14}
Figure 2b–e, show the evolution of the spectrum at 2 K in 2 T increments from B = 2 to 8 T. At B = 2 T the spin wave intensity is increased over most of the BZ. This is consistent with expectations for the neutron scattering crosssection. A possible explanation is that at 2 T the ordered moment direction is perpendicular to the applied field, and nearly parallel to (1/2, 0, 1). For simple spin wave models, the neutron scattering crosssection shows diminished intensity of the M point magnetic Bragg peaks with Q ⊥ B and an enhanced scattering intensity for the attendant spin waves at the same wavevectors since the latter represent fluctuations of components perpendicular to the ordered moment. The spin waves persist as the field is increased to 4 T (Fig. 2c), and begin to lose intensity by 6 T (Fig. 2d). At 8 T, above B_{C}, spinwave scattering is totally suppressed and the intensity of the continuum is enhanced (Fig. 2e). The response is an intense, and apparently gapped, column of scattering at the Γ point. Over most of the range, in particular at higher energies, the 8 T spectrum bears a strong resemblance to the zerofield spectrum just above T_{N} (Fig. 2f).^{13,14}
Further details of the infield spectra are presented in Fig. 3 (also Figs. S4S7). ConstantQ cuts at the M point between 0 and 6 T in Fig. 3a and S7 show a spin wave at E = 2.25 ± 0.11 meV, consistent with previous zerofield experimental results.^{14,28} The cuts verify that the spinwave intensity is enhanced at 2 T, gradually reduced by 6 T, and completely absent at 8 T, consistent with the suppression of magnetic order above B_{C}.
Figures 3b and S5 show constantQ cuts at the zone center (Γ point, H = 0). The zerofield cut at 2 K exhibits a peak at E = 2.69 ± 0.11 meV, consistent with a recent observation of the zonecenter magnon by THz spectroscopy.^{29} The scattering is largely unchanged up to 4 T (Supplementary Fig. S5a). Conversely, at 6 T the low energy scattering has “filledin”, suggesting that the energy gap at the Γ point has closed (Fig. 3b, S5a). On further increasing the field to 8 T, the scattering consists of a broad peak centered around 3.4 meV that merges into the higher energy continuum. Extrapolation of the lowenergy Γpoint spectrum to the background intensity level suggests a reopening of a new gap by 8 T (Fig. 3b and S5b). To further elucidate the evolution of the gaps, we show the comparison of the 2 T and 6 T spectra from Fig. 3b, with the 0 T spectrum subtracted, in Fig. 3c. The extra intensity gained at low energies with increasing field to 6 T indicates the closure of the lowenergy spinwave gap by 6 T. In Fig. 3d, the reopening of the spingap at 8 T is emphasized by plotting the 8 T spectra with the 6 T spectrum subtracted, which also clearly shows a shift of the lowenergy spectral weight to higher frequencies (see discussion in Supplementary Materials). While the appearance of a fieldinduced gap in the spectrum of spin excitations above B_{C} has been suggested by nuclear magnetic resonance (NMR)^{9} and inferred from thermal transport^{7} and specific heat^{8,10} measurements, this is a direct observation of the phenomenon.
Figure 3e (also Supplementary Fig. S6) plots the wavevector dependence of the T = 2 K scattering at different fields, integrated over the energy interval^{5,7} meV. As seen previously,^{13,14} the scattering profile is a peak centered at the Γ point. At low temperatures and low fields, the peak width is broader, reflecting contributions associated with enhanced correlations related to the zigzag order.^{14} A similar comparison of the T = 2 K, B = 8 T with the T = 15 K, zerofield scattering in Fig. 3f demonstrates nearquantitative agreement between the two magnetically disordered states. This is true over the available data range for energy transfers above the maximum of the spin wave band.
It has been shown previously^{13,14} that the energy continuum observed at the Γ point in zero field both above and below T_{N} is similar to the T = 0 response function of a Kitaev QSL,^{21,22,23} and therefore may be a signature of fractionalized excitations associated with proximity to a QSL. Combined exact diagonalization and density matrix renormalization group (DMRG) calculations^{11} for extended KitaevHeisenberg Hamiltonians have provided indications for a magnetic fieldinduced transition from zigzag order to a gapped QSL state. Our neutron scattering results are overall consistent with the interpretation of NMR experiments by Baek et al.^{9} The disappearance of spin waves at B_{C}, combined with the appearance of a new gap in the highfield continuum excitation spectrum unconnected with spin waves, provides significant evidence for an interesting fieldinduced QSL in αRuCl_{3}.
Discussion
The removal of magnetic longrange order by the field enables comparing the measured energydependent scattering to a QSLbased theory over a large portion of the bandwidth of the magnetic excitations. The full effective magnetic Hamiltonian describing αRuCl_{3} has not been definitively determined so that the concomitant dynamic response functions remain unknown. In view of this, as a starting point, it is reasonable to compare the results to exact calculations for a Kitaev QSL. Previous such comparisons^{12,14} of the zerofield measurements were restricted to highenergy features with relatively small spectral weight. The expected scattering is broad in energy for both ferromagnetic (FM) and AFM Kitaev QSLs, however in zero field the momentum dependence of the scattering was seen to be similar to the response calculated for an AFM Kitaev QSL at T = 0. Newly available extensions of the QSL based calculations to nonzero T^{23} (see Supplementary Discussion, section D: “Theory of the infield dynamical response of the Kitaev QSL” in SM) provide a theory for the full energy dependence of the scattering intensity at the Γ point. Comparing this theory to the data at 8 T reveals that the constant Q response at the Γpoint is closer to that calculated for a FM Kitaev model in an effective magnetic field (see discussion in SM, and Supplementary Figs. S8 & S9). The latter exhibits the following features that are qualitatively consistent with experiment: For temperatures comparable to, or larger than, the flux gap, the signal near Q = 0 is a broad continuum, exhibiting only a moderately intense peak at the lowerenergy threshold. Secondly, the highfrequency part of the spectrum is resilient as a function of temperature (up to T ~ J_{ K }, the Kitaev constant) or magnetic field (below/above B_{c}). Finally, with field, the lowfrequency response acquires a lowenergy gap with an intensity enhanced at higher energies, similar to the Γ point continuum scattering seen in αRuCl_{3} above B_{c} (Figs. 3b, 3d, S5c and S8). This fits with the idea that the fieldinduced QSL evolves from the zerofield Kitaev QSL as timereversal symmetry breaking opens a gap in the Majorana spectrum and a Majorana flux bound state (broadened by the presence of thermally excited fluxes at nonzero temperature) enhances the lowfrequency response.
Despite possible differences at low energies, the overall similarity between the excitation spectra for T > T_{N} (B = 0 T) and B > B_{C} (T = 2 K) is quite remarkable (see Fig. 2e,f and Fig. 3f), and suggests the possibility of a simple connection between the highenergy excited states in the two regions of parameter space. Generally (see e.g., ref. ^{30}) one would expect Zeeman splitting to drive a softening of the gapped spin wave mode at the zigzag ordering wavevector (Mpoint) as the applied field is increased, eventually driving the system to a phase transition when the gap is closed at the ordering wave vector. Here the lack of any observed splitting or an Mpoint gap softening, coupled with the apparent softening of the gap at the Γ point, is surprising, and suggests the possibility of another fieldinduced transition between the zigzag state and the QSL seen above B_{C}. Figure 3g shows a plot of the intensity and FWHM of the scattering continuum as a function of field. The results suggest an anomaly or discontinuity within the shaded region, in the vicinity of 6 T. Further evidence is provided by the isothermal (T = 2 K) AC susceptibility shown in Fig. 3h. This shows a large anomaly at B_{C}, with a second anomaly near 6 T. Whether or not this indicates a second transition, and thus the presence of an intermediate phase between the ordered magnet and a fieldinduced QSL is the subject of further investigation.
To conclude, we have shown that with a magnetic field applied in the {110} direction, longrange magnetic order in αRuCl_{3} disappears above a threshold field B_{C}. Moreover, above B_{c} the highfrequency spectrum of magnetic excitations is a broad continuum with a response function resembling that expected for a Kitaev QSL. Although the width of the response in energy is similar for AF or FM Kitaev models, when the calculation includes the effects of nonzero temperature and field the distribution of spectral weight in αRuCl_{3} is closer to that expected for a FM model. Our data shows that some aspects of the field dependence of the response functions resemble pure Kitaev model calculations, and above B_{c}, the response at moderate energies is similar to the zero field response above T_{N}. It remains to be answered whether, and how, the fieldinduced gapped QSL observed in αRuCl_{3} is topologically distinct from the Kitaev QSL with or without broken time reversal symmetry [ref. ^{1}, also see discussion in Supplementary Materials, section D]. The next challenge is to identify and explore its most compelling experimental signatures. This also includes more specific identification of the experimental signatures of the possible nonKitaev terms^{8,12,13,31,32,33,34,35} on the field induced QSL, with the aim of understanding whether it is possible to realize topologically protected edge states and quasiparticle excitations with nonAbelian statistics,^{1,36} which have generated much enthusiasm in the context of topological quantum computation.^{36,37,38}
Methods
Synthesis and bulk characterization
Single crystals of αRuCl_{3} were prepared using vaportransport techniques from pure αRuCl_{3} powder as described previously.^{14} Crystals grown by the same method have been extensively characterized via bulk and neutron scattering techniques.^{14,19} All samples measured in the current work exhibit a single magnetic phase at low temperature with a transition temperature Tc ~ 7 K, indicating high crystal quality with minimal stacking faults.^{19} DC magnetization and AC susceptibility in 12 and 17 mg single crystals, respectively, were collected in DC fields of up to 14 T in a Quantum Design Physical Property Measurement System (PPMS). The magnetic field was applied along the reciprocal {110} directions as identified by Laue diffraction as described in Fig. 1b.
Neutron diffraction experiments
Elastic neutron studies in a 5 T verticalfield cryomagnet were performed at Spallation Neutron Source (SNS), Oak Ridge National Laboratory (ORNL), using the CORELLI beamline.^{39} CORELLI is a timeoffight instrument where a pseudostatistical chopper separates the elastic contribution. A 125 mg αRuCl_{3} crystal was mounted on an Al plate and aligned with the (H, 0, L) plane horizontal and B along the ζ = (−1, 2, 0) vertical direction (See Fig. 1b). The crystal was rotated through 170 degrees in 2° steps. Perpendicular coverage of ±8^{o} (limited by the magnet vertical opening) allowed access to the set of magnetic Bragg peaks of the zigzag ordered phase at the Mpoints within the first BZ—at (±½, 0, L) in the (H, 0, L) plane and (±½,\(\mp\)½, L) and (0, ±½, L) out of the (H, 0, L) plane with L = ±1, as well as the outofplane peaks with L = ±2. The data were reduced using Mantid.^{40} Diffraction measurements were also obtained on the same 740 mg crystal measured used in the inelastic study using the HB1A and HB3 triple axis instruments at the High Flux Isotope Reactor (HFIR). For both HFIR measurements the sample was aligned with the (H, 0, L) scattering plane horizontal and with an applied vertical field of up to 5 T. An incident energy E_{i} = 14.7 meV was used. Detailed mesh scans and order parameter measurements confirmed that the magnetic phase transition remains sharp as the field is increased from zero (Supplementary Fig. S3) and the inplane magnetic peaks remain at the commensurate position.
Inelastic scattering experiments
Singlecrystal INS measurements were carried out on a 740 mg αRuCl_{3} crystal in an 8 T verticalfield cryomagnet using the HYSPEC instrument at SNS.^{41} The sample was aligned in the horizontal (H, 0, L) scattering plane, with the magnetic field (B) parallel to the vertical ζ = (−1, 2, 0) direction (See Fig. 1b). An incident energy of E_{i} = 17 meV combined with a Fermi chopper frequency of 240 Hz yielded an experimental energy resolution FWHM = 0.88 ± 0.03 meV based on a Gaussian fit on the elastic line. The zerofield data at 15 K was obtained using a Fermi chopper frequency of 120 Hz, which increases the intensity, but broadens the energy resolution, both by a factor of 2.0, as compared to 240 Hz. The momentum resolution is unaffected. The 15 K data in Fig. 3f is scaled by the factor of 2 to allow a direct comparison of the intensity with the 2 K data taken at 240 Hz. These settings provide reasonable (Q, E) coverage and resolution with which to examine the spinwave spectrum in the (H, 0, 0) direction, however using E_{i} = 17 meV, the higherenergy part of the sample spectrum (i.e., above roughly 6 meV) is limited by kinematic restrictions. Data were collected in 1° steps as the sample was rotated through 200° about the vertical axis for every temperature and field condition. Empty aluminum sample holder measurements were performed under identical conditions as the sample measurements and the resultant scattering has been subtracted from all data presented in the manuscript. Reduction of the raw data was carried out using standard data reduction routines and Python codes available within Mantid software.^{40}
Data availability statement
All neutron data and related Python scripts used for preparing the figures in this manuscript is available at the SNS data servers with permission from the corresponding authors of the manuscript. The bulk susceptibility data is available upon request from PK. All additional numbers used in the plots presented in this manuscript is available from the coauthors upon request.
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Acknowledgements
The authors acknowledge valuable discussions with Christian Batista, Huibo Cao, Matt Stone, Feng Ye, Andrey Podelsnyak. and Matthius Vojta. J.K. and R.M. particularly thank John Chalker and Dmitri Kovrizhin for collaboration on closely related work. A.B. and P.K. thank S. Chi, O. Garlea, N. Helton, J. Werner, R. Moody and M. B. Stone for assistance with the measurement on HB3, CORELLI and HYSPEC and G. Martin for assistance with vectorgraphics. The work at ORNL’s Spallation Neutron Source and the High Flux Isotope Reactor was supported by the United States Department of Energy (USDOE), Office of Science  Basic Energy Sciences (BES), Scientific User Facilities Division, managed by UTBattelle LLC under contract number DEAC0500OR22725. Part of the research was supported by the USDOE, Office of Science  BES, Materials Sciences and Engineering Division (P.K., C.A.B. and JQ.Y.). D.M. and P.K. acknowledge support from the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4416. The work at Dresden was in part supported by DFG grant SFB 1143 (J.K. and R.M.). J.K. is supported by the Marie Curie Programme under EC Grant agreements No.703697.
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Contributions
A.B., D.A.T., P.K., B.C., D.G.M., and S.E.N. conceived the experiments; P.K. made the 740 mg single crystal for HYSPEC data while J.Y., A.B., and C.A.B. made the 125 mg single crystal for CORELLI data; P.K. and J.Q.Y. performed the bulk and AC susceptibility measurements; P.K., A.B., A.A.A., D.P., B.C., C.B., and Y.L. performed the HB1A, HB3A and CORELLI experiments; A.B., B.W., P.K., C.B., M.D.L., D.A.T., and S.E.N. performed the HYSPEC experiments while A.T.S. helped in Mantid reductions; J.K. and R.M. provided theoretical inputs based on QSL calculations; A.B. and P.K. analyzed the data; A.B., P.K., J.K., and S.E.N. produced the first draft; All authors contributed to the finalization of manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Arnab Banerjee or Paula LampenKelley or Stephen E. Nagler.
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