Abstract
Geometrical constraints to the electronic degrees of freedom within condensedmatter systems often give rise to topological quantum states of matter such as fractional quantum Hall states, topological insulators, and Weyl semimetals^{1,2,3}. In magnetism, theoretical studies predict an entangled magnetic quantum state with topological ordering and fractionalized spin excitations, the quantum spin liquid^{4}. In particular, the socalled Kitaev spin model^{5}, consisting of a network of spins on a honeycomb lattice, is predicted to host Majorana fermions as its excitations. By means of a combination of specific heat measurements and inelastic neutron scattering experiments, we demonstrate the emergence of Majorana fermions in single crystals of αRuCl_{3}, an experimental realization of the Kitaev spin lattice. The specific heat data unveils a twostage release of magnetic entropy that is characteristic of localized and itinerant Majorana fermions. The neutron scattering results corroborate this picture by revealing quasielastic excitations at low energies around the Brillouin zone centre and an hourglasslike magnetic continuum at high energies. Our results confirm the presence of Majorana fermions in the Kitaev quantum spin liquid and provide an opportunity to build a unified conceptual framework for investigating fractionalized excitations in condensed matter^{1,6,7,8}.
Main
Quantum spin liquids (QSLs) are an unconventional electronic phase of matter characterized by an absence of magnetic longrange order down to zero temperature. They are typically predicted to occur in geometrically frustrated magnets such as triangular, kagome, and pyrochlore lattices^{4}, and typically display a macroscopic degeneracy that stabilizes a topologically ordered ground state. The Kitaev QSL state arises as an exact solution of the ideal twodimensional (2D) honeycomb lattice with bonddirectional Isingtype interactions (H = J_{K}^{γ}S_{i}^{γ}S_{j}^{γ}; γ = x, y, z) on the three distinct links (Fig. 1a) by expressing the spin excitations in terms of noninteracting Majorana fermions^{5,9}. The elementary excitations of a Kitaev QSL are localized and itinerant Majorana fermions^{5}, which are associates with static Z_{2}fluxes and propagating quasiparticles (Fig. 1b). These two types of excitation have ramifications for the observable physics and potential technological applications of QSL in quantum computers^{10,11,12,13,14}.
As candidates for realizing a QSL, honeycomb iridates A_{2}IrO_{3} (A = Li, Na) with a spin–orbit coupled J_{eff} = 1/2 Ir^{4+} (5d^{5}) state^{15} have been intensively studied. This is due to the orbital state forming the three orthogonal bonds required for the bonddirectional exchange interactions in the geometry^{16}. The iridates, however, cannot avoid monoclinic distortions with anisotropic Ir–Ir bonds disturbing the exchange frustration away from the ideal values, and their magnetism is apparently dominated by antiferromagnetic (AFM) ordering^{17,18}.
A promising candidate for the Kitaev model system is the van der Waals ruthenate αRuCl_{3} with J_{eff} = 1/2 Ru^{3+} (4d^{5}) ions^{19,20}. There is a growing body of evidence that αRuCl_{3} hosts predominantly Isinglike Kitaev interactions and that the ground state could be proximate to the QSL state^{21,22}. Most crystallographic studies reported the presence of the monoclinic distortions^{23,24}, resulting in considerable contribution of Heisenberg and asymmetric exchange interactions^{25,26}. However, these distortions are probably due to stacking faults of the RuCl_{3} layers, and even lead to multiple magnetic transitions^{24}. Recently, significant advances in the synthesis of highquality αRuCl_{3} crystals have been achieved. These crystals are almost free from stacking faults and have a rhombohedral () phase, while preserving the Isingtype AFM state below 6.5 K due to nonvanishing interlayer couplings^{27}. Importantly, this highsymmetry structure renders isotropic Kitaev interactions (J_{K} = J_{K}^{x} = J_{K}^{y} = J_{K}^{z}) with a 94° Ru–Cl–Ru bond angle maximizing the Kitaev interaction, and the Heisenberg contribution becomes minimal^{26}. Furthermore, recent methodological progress in the quantum Monte Carlo (QMC) method and cluster dynamic meanfield theory (CDMFT) for thermally excited quantum states provides a route to identify Majorana fermions emerging from the QSL ground state^{12,13,14}. It is predicted that thermally fluctuating quantum spins are successively fractionalized into itinerant and localized Majorana fermions at crossover temperatures T_{L} (lowT) and T_{H} (highT), respectively. At very lowtemperature (T < T_{L}), Z_{2}fluxes are mostly frozen in the topologically ordered zerotemperature QSL state and the thermal energy excites only lowenergy itinerant Majorana fermions (see Fig. 1b). On increasing the temperature across T_{L}, the fluxes fluctuate to activate localized Majorana fermions (Kitaev paramagnet). Upon further heating, itinerant Majorana fermions are additionally activated and the spin–spin correlation fades out across T_{H}. Finally, the system ends in a conventional paramagnetic phase well above T_{H}.
Figure 2 displays the thermodynamic signatures in the magnetic susceptibility χ(T), magnetic specific heat C_{M} and entropy S_{M} for fractionalized spin excitations. The static χ(T) of αRuCl_{3} deviates from the Curie–Weiss curve below 140 K, indicating the onset of shortrange spin correlations (Fig. 2a). The anomalies in χ(T) and C_{M} at T_{N} = 6.5 K represent the onset of zigzagtype AFM order (Fig. 2a, b). C_{M} is obtained by subtracting the lattice contribution from the total specific heat (C_{P}) as described in the Supplementary Information. Besides the sharp anomaly at T_{N}, C_{M} exhibits two broad maxima, one near T_{N} and the other around T_{H} ≈ 100 K, although the lowT maximum feature is obscured by the AFM anomaly. As predicted in theory^{12,13}, the high and lowT structures can be ascribed to the thermal excitations of itinerant and localized Majorana fermions, respectively. It is worth noting that C_{M} follows a linear Tdependence in the intermediate range T_{N} < T < T_{H}, reflecting metalliclike behaviour of itinerant Majorana fermions (inset of Fig. 2b).
Rather firm evidence is provided by the twostage release of the entropy gain S_{M}(T) = ∫ C_{M}/TdT (Fig. 2c). The obtained S_{M} at T = 200 K is 5.13 J mol^{−1} K^{−1}, which corresponds to about 90 % of the ideal value Rln2 (R: ideal gas constant) of the spin1/2 system. Upon cooling, nearly half of the entropy is released stepwise with the plateaulike behaviour at 0.46Rln2, signifying two maxima of C_{M}. Indeed, S_{M}(T) above T_{N} agrees well with the simulated sum (red line) of two phenomenological Schottkylike functions with about an equal weight (ρ_{H} = 0.92, ρ_{L} = 1.08, T_{H} ≃ 101 K, and T_{L} ≃ 22 K), which involve itinerant and localized Majorana fermions in the QMC simulation (see Supplementary Information). Considering the predicted temperature ratio T_{L}/T_{H} ≈ 0.03 in the isotropic Kitaev model, T_{L} would be somewhat lower than T_{N} if the AFM order were absent. S_{M} involving AFM order below T_{N} was estimated to be 1.09 J mol^{−1} K^{−1}, about 20% of the total entropy Rln2 (40% of 1/2Rln2) (ref. 27), indicating that the entropy held by the AFM order is partially released and roughly 3/5ths of the frozen Z_{2}flux is maintained just above T_{N}.
The microscopic and dynamic properties of the Majorana fermions can be visualized by the thermally fractionalized spin excitations obtained from the INS measurements. Figure 3a shows the neutron scattering function S_{tot}(Q, ω) as a function of momentum transfer Q and energy transfer ω measured at T = 10 K above T_{N} along the X–K–Γ–M–Y direction. S_{tot}(Q, ω) at sufficiently low T can be approximated as the magnetic scattering function S_{mag}(Q, ω) although weak phonon features are still observable as marked with black stars in the figure (see Supplementary Information). S_{tot}(Q, ω) displays an hourglass shape spectrum centred at the Γpointextending to about 20 meV with strong lowenergy excitations around the Γpoint and highenergy Yshaped excitations. Similar features are reproduced in the simulated spectra of the isotropic Kitaev model with a FM Kitaev interaction J_{K} = −16.5 meV by using the CDMFT + continuoustime QMC method^{14} (see Fig. 3b). It is worth noting that the spectral centre would move to the Mpoint for an AFM J_{K} (>0) (ref. 14). The lowenergy feature represents the quasielastic responses associated with the flux excitations, and the Yshaped Qω dependence in the highenergy region reflects the dispersive itinerant Majorana fermions extending to ω ∼ J_{K} (refs 11,14). Both features are also clearly observable in the constantenergy cuts S_{tot}(Q), which also agree well with the theoretical calculations (Fig. 3c). According to the simulation, the excitation energy of the itinerant MF at the K and Mpoints corresponds to Kitaev J_{K}. S_{tot}(Q) data (Fig. 3d) are again compared with the simulated values (Fig. 3e) in 2D reciprocal space (Fig. 3f). The overall features are well reproduced by the simulations, except the hexagramshaped Qdependence of the lowenergy S_{tot}(Q) (ω ≲ 6 meV), indicating that the key character of the Majorana fermions is rather robust. The hexagramshaped Qdependence is considered to be induced by the second nearestneighbour Kitaev interactions^{28} and/or symmetric anisotropy exchange interactions^{29,30} involving direct Ru–Ru electron hopping, both of which are not considered in the pure Kitaev model. These interactions are weak, but become important at low energies and temperatures.
Figure 4a, b presents the thermal evolution of the experimental and simulated S_{mag}(Q, ω) (see Methods), respectively. At T = 16 K, the hourglass shape spectrum is maintained with minor reduction in the overall intensity. Upon heating up to T_{H} ∼ 100 K (Kitaev paramagnetic phase), the lowenergy intensity involving localized Majorana fermions is significantly reduced while the highenergy intensity from itinerant Majorana fermions is almost maintained, although the dichotomic feature becomes smeared with increasing thermal fluctuations. Further heating across T_{H} causes the highenergy intensity to begin to decrease considerably. Well above T_{H}(T = 240 K), S_{mag}(Q, ω) exhibits only a featureless low background as in conventional paramagnets. The evolution of localized and itinerant Majorana fermions with temperature are visualized in the temperature–energy contour plots of S_{mag} around the Γpoint, as presented in Fig. 4c (experiment) and 4d (simulation). The lowenergy excitations below ω ≈ 4 meV appear at T ≲ T_{H} while the highenergy excitations extend out to ω ∼ J_{K}. This is also evident from the S_{mag}(Γ, ω) plots in Fig. 4e, which are consistent with the simulations.
The quantitative agreement between the experiment and the simulation is also excellent in the INS intensities for the low and highenergy excitations in an overall temperature range, as shown in Fig. 4f, g, presenting the temperature dependences of the corresponding integrations ∫ S_{mag}(Γ, ω)dω. Meanwhile, one also notices that the experiment deviates somewhat from the simulation below ∼50 K only in the integration involving the lowenergy excitations (Fig. 4f). This is probably due to the presence of the additional perturbing magnetic interactions in the real system, whose influence might be apparent in the lowenergy scale to be detrimental to the lowenergy flux excitations at low temperature. Those perturbing interactions contribute the hexagramshaped Qdependence in the lowenergy S_{mag}(Q) (see Fig. 3d), which becomes isotropic above ∼50 K, as expected in the Kitaev model (see Supplementary Information).
Tracing the magnetic entropy and evolution of the spin excitations as a function of temperature, energy, and momentum, we provide strong evidence for thermal fractionalization to Majorana fermions of spin excitations. αRuCl_{3} is well described in the ferromagnetic Kitaev model and is proximate to the Kitaev QSL. The key features of the thermal fractionalization predicted in the pure Kitaev model are reproduced well in the thermodynamic and spectroscopic results, although AFM order is developed below T_{N} = 6.5 K and additional perturbing magnetic interactions deteriorate QSL behaviour, especially in the lowenergy scale. When the temperature is higher than the energy scale related to the perturbing magnetic interactions, the two distinct Majorana fermions predicted in the Kitaev honeycomb model are unveiled. This finding lays a cornerstone for an indepth understanding of emergent Majorana quasiparticles in condensed matter, and also possibly for future implementation in quantum computations.
Methods
Crystal growth.
Highquality single crystals of αRuCl_{3} and their isostructural counterpart ScCl_{3} were grown by a vacuum sublimation method. A commercial RuCl_{3} (ScCl_{3}) powder (Alfa Aesar) was thoroughly ground, and dehydrated in a quartz ampoule for a day. The ampoule was sealed in vacuum and placed in a temperature gradient furnace. The temperature of the RuCl_{3} (ScCl_{3}) powder is set at 1,080 °C (900 °C). After dwelling for 5 h, the furnace is cooled to 650 °C (600 °C) at a rate of −2 °C per hour. We obtained αRuCl_{3} (ScCl_{3}) crystals black coloured (transparent) with shiny surfaces. Electrondispersive Xray measurements confirmed the stoichiometry of the Ru(Sc):Cl = 1:3 ratio for the crystals.
Magnetic susceptibility and specific heat measurement.
Magnetic susceptibility measurements were performed using a commercial superconducting quantum interference device (SQUID) (Quantum Design, model: MPMS5XL). A single domain crystal (3 × 3 × 1 mm^{3}, 20 mg) was chosen for the measurements under an external magnetic field parallel to the abplane. Specific heat C_{P} was measured by using a conventional calorimeter of the Quantum Design Physical Property Measurement System (model: PPMS DynaCool) in a temperature range of T = 1.8–300 K. The magnetic specific heat C_{M} of αRuCl_{3} was determined by subtracting the lattice contribution, which is supposed to be equivalent to the specific heat of the isostructural nonmagnetic ScCl_{3} with mass scaling (see Supplementary Information).
Inelastic neutron scattering.
Inelastic neutron scattering data were collected by using the timeofflight spectrometers MERLIN (high intensity) and LET (high resolution) at the ISIS Spallation Neutron Source, the Rutherford Appleton Laboratory in the United Kingdom. Total 46 pieces (∼1.35 g) of αRuCl_{3} single crystals for MERLIN, and 153 pieces (∼5.1 g) for LET were prepared, and coaligned with crystallographic caxis surface normal on aluminium plates, resulting in a mosaic within 3° (Supplementary Fig. 1). The samples were mounted in a liquid helium cryostat for temperature control ranging from 1.5 K to 270 K. Due to the highly twodimensional structure of αRuCl_{3}, magnetic correlations between honeycomb layers are extremely weak and insensitive. Therefore, crystals are aligned with the caxis parallel to the incident neutron beam, so that the area detector measures the energy spectrum over the 2D qspace of the hkplane. To observe the intensity at the Γpoint (LET measurement), we rotated the crystal by 30 degrees to the incident beam direction, so that it filled the blank region of the beam mask.
Data were obtained with the incident neutron energy set to E_{i} = 5.66, 10, 22 (LET), and 31 meV (MERLIN). With incoherent neutron scattering intensity measured from a vanadium standard sample, all data were normalized and converted to the value of the neutron scattering function S_{tot}(Q, ω), which is proportional to the differential neutron crosssection (d^{2}σ)/(dΩdE) and the ratio of the incident to the scattered neutron wavevector k_{i}/k_{f} (ref. 31),
Since S_{tot}(Q, ω) contains both the nuclear and magnetic scattering contributions, the magnetic scattering function S_{mag}(Q, ω)_{T} at temperature T in Fig. 4 is extracted from S_{tot}(Q, ω)_{T} after subtraction of the scaled {\text{S}}_{tot}{(Q,\omega )}_{{T}_{0}=290\text{K}} with the Bose factor correction n(T)/n({T}_{0})=(1{e}^{\hslash \omega /{k}_{\text{B}}{T}_{0}})/(1{e}^{\hslash \omega /{k}_{\text{B}}T}), which represents the approximate phonon contribution in the experiment.
All of the data processes, including Bose factor correction and projection of the scattering function along appropriate directions, were performed using the HORACE software, which is published by ISIS^{32}.
Calculation of the magnetic scattering function.
The calculation of S_{mag}(Q, ω)_{T} is performed by using the CDMFT + continuoustime QMC method as described in ref. 15. The Bose factor correction of the equation (2) is also applied to the simulation results for quantitative comparison with the experimental results in Fig. 4. All calculated results include the magnetic form factor of the Ru^{3+} ion, which is obtained by using the density functional theory method considering solidstate effects in αRuCl_{3}, as described in the Supplementary Information.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work is supported by the National Research Foundation (NRF) through the Ministry of Science, ICP & Future Planning (MSIP) (NRF2016K1A4A4A01922028). S.J. acknowledges support from the NRF grant (NRF2017R1D1A1B03034432). S.H.D. and K.Y.C. are supported by the Korea Research Foundation (KRF) grant (No. 20090076079) funded by the Korea government (MEST). J.Y., J.N. and Y.M. acknowledges GrantinAid for Scientific Research under Grant No. JP15K13533, JP16K17747, and JP16H02206. Y.S.K. is supported from the NRF grant (NRF2015M2B2A9028507). K.K. acknowledges partial support from NRF grant (NRF2016R1D1A1B02008461).
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K.Y.C. and S.J. conceived and designed the experiments. S.H.D., Y.S.K. and K.Y.C. synthesized single crystals. S.H.D., S.Y.P., T.H.J. and K.Y.C. measured and analysed the magnetic susceptibility and specific heat. S.H.D., S.Y.P. and S.J. performed inelastic neutron scattering measurements with the support from D.T.A. and D.J.V. S.H.D., S.Y.P. and S.J. analysed neutron data. J.Y., J.N. and Y.M. carried out the CDMFT + CTQMC calculations. K.K. calculated magnetic form factors. S.H.D., J.H.P., K.Y.C. and S.J. participated in writing of the manuscript. All authors discussed the results and commented on the manuscript. The project was led by S.J. and K.Y.C. under the supervision of J.H.P.
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Do, SH., Park, SY., Yoshitake, J. et al. Majorana fermions in the Kitaev quantum spin system αRuCl_{3}. Nature Phys 13, 1079–1084 (2017). https://doi.org/10.1038/nphys4264
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DOI: https://doi.org/10.1038/nphys4264
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