Magnon bound states vs. anyonic Majorana excitations in the Kitaev honeycomb magnet α -RuCl 3
Dirk Wulferding, Youngsu Choi, Seung-Hwan Do, Chan Hyeon Lee, Peter Lemmens, Clement Faugeras, Yann Gallais, Kwang-Yong Choi
MMagnon bound states vs. anyonic Majorana excitations in the Kitaev honeycombmagnet α -RuCl Dirk Wulferding,
1, 2, ∗ Youngsu Choi, † Seung-Hwan Do, Chan Hyeon Lee, Peter Lemmens,
1, 2
Cl´ement Faugeras, Yann Gallais, and Kwang-Yong Choi ‡ Institute for Condensed Matter Physics, TU Braunschweig, D-38106 Braunschweig, Germany Laboratory for Emerging Nanometrology (LENA),TU Braunschweig, D-38106 Braunschweig, Germany Department of Physics, Chung-Ang University, Seoul 156-756, Republic of Korea LNCMI, CNRS, EMFL, Univ. Grenoble Alpes, 38000 Grenoble, France Laboratoire Matriaux et Phnomnes Quantiques (UMR 7162 CNRS),Universit Paris Diderot - Paris 7, 75205 Paris cedex 13, France (Dated: October 3, 2019)The pure Kitaev honeycomb model harbors a quantum spin liquid in zero magnetic fields, whileapplying finite magnetic fields induces a topological spin liquid with non-Abelian anyonic excitations.This latter phase has been much sought after in Kitaev candidate materials, such as α -RuCl .Currently, two competing scenarios exist for the intermediate field phase of this compound ( B =7 −
10 T), based on experimental as well as theoretical results: (i) conventional multiparticle magneticexcitations of integer quantum number vs. (ii) Majorana fermionic excitations of possibly non-Abelian nature with a fractional quantum number. To discriminate between these scenarios adetailed investigation of excitations over a wide field-temperature phase diagram is essential. Here wepresent Raman spectroscopic data revealing low-energy quasiparticles emerging out of a continuumof fractionalized excitations at intermediate fields, which are contrasted by conventional spin-waveexcitations. The temperature evolution of these quasiparticles suggests the formation of boundstates out of fractionalized excitations.
I. INTRODUCTION
The search for Majorana fermions in solid-state sys-tems has led to the discovery of several promising can-didate materials for exchange-frustrated Kitaev quan-tum spin systems . One of the closest realizationsof a Kitaev honeycomb lattice is α -RuCl , where thespin Hamiltonian is dominated by Kitaev interaction K .Nevertheless, non-Kitaev interactions, such as Heisen-berg ( J ) and off-diagonal symmetric exchange terms (“Γ-term”), as well as stacking faults in α -RuCl lead to anantiferromagnetically ordered zigzag ground state below T N ≈ . The exact strengths of these interactionshave not been pinpointed, yet a general consensus onthe minimal model has emerged about a ferromagnetic K ∼ -6 – -16 meV, as well as Γ ∼ J ∼ -1 – -2 meV . Despite these additional magnetic pa-rameters, the Majorana fermionic quasiparticles are wellpreserved at high energies and elevated temperatures .Indeed, many independent and complementary experi-mental techniques have been used to probe the emer-gence of itinerant Majorana fermions and localized gaugefluxes from the fractionalization of spin degrees of free-dom .A promising route in understanding Kitaev physicsmight be the suppression of long-range magnetic orderin a magnetic field, with the possibility of generating anIsing topological quantum spin liquid. For α -RuCl thisfield is B c ∼ − . Higher magnetic fields leadto a trivial spin polarized state. In the intermediate fieldrange of 7-10 T, the magnetic order melts into a quan-tum disordered phase, in which the half-integer quantized thermal Hall conductance is reported . This remark-able finding may be taken as evidence for a field-inducedtopological spin liquid with chiral Majorana edge statesand the central charge q = ν/ ν = 1).However, it is less clear whether such a chiral spin liquidstate can be stabilized in the presence of a relatively largefield and non-Kitaev terms in α -RuCl . In the originalKitaev honeycomb model, non-Abelian Majorana excita-tions are created upon breaking time-reversal symmetry,e.g., through applying a magnetic field . These compos-ite quasiparticles correspond to bound states of localizedfluxes and itinerant Majorana fermions . The compositebound states are of neither fermionic nor bosonic char-acter, but instead they acquire an additional phase inthe wavefunction upon interchanging particles, i.e., theyfollow anyonic statistics . Although for a Kitaev sys-tem bound itinerant Majorana fermions are possible inthe presence of perturbations , it is unclear how stablea topological spin liquid state is in this case. In particu-lar, an open issue is whether the quantized thermal Halleffect is related to a non-Abelian phase featuring anyonicMajorana excitations.There exists another scenario in which the interme-diate phase is simply a partially polarized phase andsmoothly connected to the fully polarized phase. Herethe transition through B c involves conventional multi-particle excitations due to anisotropic interactions . Toresolve these opposing scenarios, one needs to clarifythe nature of quasiparticle excitations emergent in theintermediate-to-high-field phase. So far, neutron scatter-ing , THz spectroscopy , and electron spin resonance have revealed a significant reconfiguration of the mag- a r X i v : . [ c ond - m a t . s t r- e l ] O c t netic response through B c . These methods generallyprobe ∆ S = ± S = 0) excitationsare essential for unraveling new aspects of low-energyproperties and for obtaining a complete picture of in-dividual quasiparticles. In this work, we employ Ramanspectroscopy capable of sensing single- and multiparticleexcitations over the sufficiently wide ranges of temper-atures T = 2 −
300 K, fields B = 0 −
29 T, and ener-gies (cid:126) ω = 1 −
25 meV (8 −
200 cm − ). At low fields( B < B c ) and low temperatures ( T < T N ) we observea number of spin wave excitations superimposed onto acontinuum of fractionalized excitations. Towards higherfields above 10 T, the magnetic continuum opens pro-gressively a gap and its spectral weight is transferredto well-defined sharp excitations that correspond to one-magnon and magnon bound states, marking the crossoverto a field-polarized phase. In the intermediate phase, aweakly bound state emerges. This bound state is formedvia a spectral transfer from the fractionalized contin-uum through an isosbestic point around 8.75 meV, anddoes not smoothly connect to the magnon bound statesin the high-field phase. Our results suggest that thisweakly bound state carries Majorana characteristics and that the intermediate-field phase of α -RuCl hosts anovel quantum phase. II. RESULTS
We performed Raman scattering experiments on ori-ented single crystals to elucidate the field-evolution ofthe magnetic excitation spectrum of α -RuCl (see Sup-plementary Information, section S1 for a detailed out-line of the scattering geometries and section S2 for thefull dataset). All measurements were carried out with RL circularly polarized light, probing the E g symme-try channel. Fig. 1a shows representative raw spectraobtained at increasing fields aligned along the crystal-lographic a axis [B // (100)]. Besides two sharp, in-tense phonon modes at 14.5 meV and 20.5 meV [marked E g (1) and E g (2)], we observe several magnetic excita-tions with a pronounced field dependence: At zero field,the magnetic Raman response consists of a broad con-tinuum (C; green shading) and a sharp peak (M1, blueline). The latter M1 excitation at 2.5 meV is assignedto one-magnon scattering arising from a spin flip processby strong spin-orbit coupling and enables us to detect agap of low-lying excitations at the Γ-point as a functionof field. The M1 mode energy and its field dependencematches well recent THz magneto-optical data, confirm-ing the ∆ S = ± and hasbeen identified as Majorana fermionic excitations stem-ming from a fractionalization of spin degrees of freedomin the Kitaev honeycomb model . Although we cannotexclude an incoherent multimagnon contribution to the continuum, the thermal evolution of the continuum fol-lows two-fermionic statistics. Such exotic behavior is notexpected for bosonic spin wave excitations, but rathersupports the notion of Majorana fermions . As detailedin Supplementary Information, section S3, a more promi-nent two-fermionic character is present at B c = 6 . B = 0 T.This can be taken as further evidence for the presenceof Majorana fermionic excitations in α -RuCl . As themagnetic field increases above 10 T, C becomes gappedand its spectral width narrows down. This leads to abuild-up of spectral weight towards the edge of the gap(solid line in Fig. 1b-d).Noteworthy is that the gapped continuum has a finiteintensity even at high fields B > B c , while several sharpand well-defined excitations emerge additionally with in-creasing fields. The residual spectral weight of C at suffi-ciently high fields means that the excitation spectrum inthis regime is not solely comprised of single- and multi-particle magnons. Indeed, recent numerical calculationsof the Kitaev model under applied fields uncovered a wideKitaev paramagnetic region, reaching far beyond the crit-ical field at finite temperature . We therefore ascribethe gapped continuum excitations to fractional quasipar-ticles pertinent to the Kitaev paramagnetic state. TheM1 peak is ubiquitous in all measured fields. The ex-citation M2’ (orange line; the notation “prime” is usedto differentiate distinct modes in the high field regime B > B c ) is split off from the M1 peak above 12 T, whilethe higher-energy M3’ excitation (red line) appears at thelower boundary of the gapped continuum above 10-14 T.In previous experimental field-dependent studies on α -RuCl ranging from inelastic neutron scattering (INS) ,to THz absorption , to ESR similar sharp magneticexcitations were reported and interpreted in terms of one-magnon or magnon bound states. In consideration of thenarrow spectral form and energy of the corresponding ex-citations observed in our data, we assign the M2’ peakto a two-magnon bound state and the M3’ peak to ei-ther a multimagnon excitation or a van Hove singularityof the gapped continuum. The evolution of all magneticexcitations as a function of fields up to 29 T is depictedin the color contour plots of Figs. 1b, 1c, and 1d to-gether with the as-measured Raman spectra in Figs. 1e,1f, and 1g for field directions along (100), (010), and(110), respectively. A slight anisotropy in magnetic exci-tations as a function of field-direction becomes apparent,which is highlighted in Fig. 1h. In particular, the energyand field ranges of M3’ are sensitive to the in-field direc-tions, indicating the presence of non-negligible in-planeanisotropy terms. Our high-field Raman data evidencethe existence of multimagnon excitations and a gappedcontinuum that characterizes the spin dynamics of thepartially polarized phase. The base temperature is how-ever restricted to T ≥ B c upon further cooling.The reported half-integer thermal Hall conductance FIG. 1.
Field evolution of magnetic excitations in α -RuCl through field-induced phases. a , As-measured Ramanspectra at T ≈ H//a , α -RuCl passes successively from a zigzag antiferromagnetic- through an intermediate- to afield-polarized-phase with increasing magnetic fields. The color shading denotes the broad continuum (C) on top of well-definedsharp peaks (M1, M2’, M3’) and phonon modes ( E g (1) and E g (2)). b , c , and d , Color contour plots of the Raman scatteringintensity evolution with magnetic fields aligned along (100), (010), and (110), respectively. e - g , Respective sets of raw Ramandata. h , Field-dependence of the sharp low-energy magnetic excitations compared for different field directions.FIG. 2. Magnetic excitations at intermediate magnetic fields. a , The creation of a bound state from itinerant fermionsbound to localized fluxes and b from binding only itinerant Majorana fermions. c , Evolution of Raman data obtained at T = 2 K (open circles) with increasing fields from 0 T - 9.5 T. The shaded regions denote the decomposition of the magneticexcitations into well-defined peaks and a continuum of excitations. The solid line is a sum of all excitations. The M1 (blue)and M2 (purple shading) modes at low fields of 0 - 4.3 T correspond to spin wave excitations. The excitation MB (dark red)above the critical field of 6.7 T is assigned to a Majorana bound state. is expected from chiral Majorana states along the edgesof the 2D honeycomb layers of α -RuCl . Simultaneously,anyonic excitations emerge in the bulk. A detection ofthese anyonic Majorana bound states will provide an ulti-mate confirmation of the field-induced intermediate non-Abelian phase. In the extended Kitaev system, thesebound states can occur in different channels , namely,from either binding itinerant Majorana fermions to lo-calized fluxes (sketched in Fig. 2a), or by binding twoitinerant Majorana fermions (sketched in Fig. 2b). Withthis in mind, we study the intermediate phase in detail byswitching to a magneto-optical cryostat setup, enablingus to reach a base temperature of T = 2 K in a field rangeof B = 0 −
10 T. In this setup the sample is tilted by anangle of 18 ◦ away from the in-plane field geometry, re-sulting in an additional small but finite out-of-plane fieldcomponent. Note that here both magnetic field directionas well as light scattering geometry are slightly differ-ent from the high-field setup, which prohibits a strictone-to-one comparison of the data. Nonetheless, a goodcorrespondence is found between the high-field B//(110)data at T ≈ T = 9K (see Suppl. Information, section S1). In Fig. 2c weinspect the field-dependence of Raman spectra measuredat T = 2 K. Compared to the T = 5 K high-field datashown in Fig. 1a, the T = 2 K data show a new sharpstructure at 5 meV (M2) in addition to the one-magnonexcitation (M1) and the fractionalized continuum (C). Asthe field increases, the respective modes evolve in a dis-parate manner. Initially (at B = 0 − . B c is approached and through 6.7 T, the spectral weightof the continuum is massively redistributed. A new low-energy mode (MB) evolves from the low-field M2 modewith a shoulder structure (M3) and the continuum of Ma-jorana excitations is gapped above 8.1 T. A recent INSstudy reported a similarly broad, emerging excitation inthe intermediate field-induced phase . It was tentativelydiscussed as a possible Majorana bound state, but an ul-timate assignment was hindered by the lack of a detailedtemperature study.To analyze the field-induced spectral weight redistri-bution carefully, we replot phonon-subtracted Ramandata taken at T = 2 K in Fig. 3a. With increas-ing field a distinct transfer of spectral weight from themid-energy (green-shaded A2) to the low-energy regime(purple-shaded A1) is observed, with an isosbestic pointlocated around ω iso = 8 .
75 meV, at which the magneticRaman response is independent of the external field. Thesystematic field-induced redistribution of spectral weightthrough this isosbestic point suggests an intimate con-nection between the continuum and the newly formedexcitation, and therefore supports the formation of a low-energy Majorana bound state (MB) through a confine-ment of the high-energy broad continuum of Majoranafermionic excitations. Consistent with the high-field datapresented in Fig. 1, we observe a remaining intensity of the continuum C of Majorana fermions at 9.5 T, i.e., cor-responding to the non-trivial quantum phase. A coexis-tence of massive Majorana fermions that form a broad,gapped continuum together with a sub-gap MB state isnot compatible with the trivial polarized phase that ischaracterized by the multimagnon bound states. Figure3b shows the thermal evolution obtained at B = 9 . .Fig. 3d plots the energy of the MB mode as a func-tion of field together with the spin-wave excitations ob-served in the zigzag ordered phase as well as in the highfield spin-polarized phase. We note a smooth transitionfrom M2 to MB through B c . This weak field-dependencesuggests that the MB mode corresponds to an excita-tion in the singlet sector (∆ S z = 0), to which Ramanspectroscopy is a natural probe. As the magnon corre-sponds to a condensation of Majorana fermions, the M2-to-MB mode evolution may be interpreted in terms of acondensation-to-confinement crossover where the multi-magnon excitations observed at low field gradually evolveinto Majorana bound states. Since the continuum C ofdeconfined Majorana excitations above B c is massivelygapped with an onset energy of 4-6 meV (see Figs. 1b-1d, Fig. 2c), the low-energy bound state can be naturallycreated within the gap due to confinement. Unlike theM2-to-MB mode crossover below B c , the MB mode isnot smoothly linked to the M2’ bound state for fieldsabove B c . Rather, as the field increases, the M2’ modesplits from the M1 mode, and both excitations are ob-served prominently, while a signature of the MB moderemains absent in the data obtained from the high-fieldsetup. This suggests that the MB mode is of differentnature than the excitations M1 and M2’, and that theparameters temperature, scattering geometry, and mag-netic field direction play a crucial role in the creation andobservation of Majorana bound states. Our data is alsocontrasted by the rather smooth transition of quasipar-ticle excitations observed in ESR experiments through10 T , due to the different selection rules for quasipar-ticle excitations in Raman vs. ESR. The discontinuousevolution observed through 10 T in our data admittedlymay be expected in our data due to the slightly differ-ent experimental conditions between the high-field andthe magneto-cryostat setups, the small temperature dif- FIG. 3.
Spectral weight redistribution and formation of a bound state through B c . a , Raman spectra obtained at T = 2 K with increasing magnetic fields. b , Raman spectra at B = 9 . c , Contour plot of the magnetic Raman scatteringintensity as a function of temperature and field. d , Field-dependence of excitations around the intermediate phase (spheres)together with spin-wave excitations at low and high magnetic fields. e , Thermally induced binding-unbinding crossover at B = 6 . f , Thermal melting of the low-temperature magnetic modes at 0 T, 6.7T, and 9.5 T. Standard deviations in e-f are indicated by error bars. ference of ≈ .Further support of the MB state interpretation comesfrom the temperature dependence at two different mag-netic fields, 6.7 T and 9.5 T (Fig. 3e). We see a clearinitial increase in the MB mode energy at both fieldsas the temperature rises. This is contrasted by con-ventional (magnon) unbound excitations, which contin-uously soften with increasing temperature. For boundstates, however, the thermal energy competes with thebinding energy , until eventually an unbinding takesplace. Therefore, a sudden drop in mode energy occursat around 15 K (at B = 9 . ≈ . T ≈
10 K. In contrast, the shoulder of built-up Majo-rana fermionic excitations gives no clear sign of binding.The thermal evolution of area A1, summarized in Fig. 3f, highlights the gradual melting of the bound stateat 9.5 T (dark red triangles) with increasing tempera-ture, while the conventional magnetic excitations at 0T (blue squares) abruptly vanish above T N . All theseobservations are consistent with the picture of a quasi-bound state of Majorana fermions that is pulled belowthe gapped fractionalized continuum by residual interac-tions of non-Kitaev origin . III. DISCUSSION
In the Kitaev model, Majorana bound states arecreated through flux pairs combined with Majoranafermions in a ∆ S z = ± . However, as fluxexcitations are largely invisible to the Raman scatteringprocess , the bound states between the flux and Ma-jorana fermions barely contribute to the magnetic Ra-man signal. In the presence of additional non-Kitaevterms, the creation of bound states from itinerant Majo-rana fermions is enhanced (see the cartoon in Fig. 2b) .As Raman scattering probes mainly the ∆ S z = 0 chan-nel, we conclude that the MB mode largely consists ofthe latter Majorana singlet bound states. This interpre-tation is supported by the smooth crossover from themultimagnon M2 mode to the bound state MB through B c (see Figure 3d). In such a case, α -RuCl as an apt re-alization of the K − Γ model (see Suppl. Information, sec-tion S5) can host an exotic intermediate-field phase. Inrelation to this issue, we mention that a numerical studyof the J − K − Γ − Γ (cid:48) model shows an extended regimeof a chiral spin liquid for the out-of-plane field. Oncethe magnetic field is tilted significantly towards the in-plane direction, the intermediate topological phase van-ishes . This discrepancy raises the challenging questionwhether the recently reported field-induced phase has anon-Abelian nature and how the in-plane intermediatephase transits to the alleged chiral spin-liquid phase, ifthe intermediate phase is of topologically trivial nature.Our finding demonstrates that a non-trivial crossoverfrom the zigzag through the intermediate to the high-field phase involves a strong reconfiguration of the frac-tionalized continuum excitations, calling for future workto shed light on the relation between the observed Majo-rana bound states in an in-plane intermediate field phaseof α -RuCl and the non-Abelian phase predicted for out-of-field directions. IV. METHODS
Crystal growth
Single crystals of α -RuCl weregrown via a vacuum sublimation method, as describedelsewhere . For Raman experiments, samples with di-mensions of about 5 × × . were selected fromthe same batch whose thermodynamic and spectroscopicproperties have been thoroughly characterized . Raman scattering
High magnetic fields up to 29 Twere generated using the resistive magnet M10 at theLNCMI Grenoble. The sample was kept at a temper-ature T ≈ −
10 K and illuminated with a 515 nmsolid state laser (ALS Azur Light Systems) at a laserpower P = 0 .
05 mW and a spot size of 3 µ m diame-ter. Resulting Raman spectra were collected in Voigt ge-ometry for in-plane fields, and in Faraday geometry forout-of-plane geometry, using volume Bragg filters (Opti-Grate) in transmission geometry and a 70 cm focal dis-tance Princeton Instruments spectrometer equipped witha liquid N cooled Pylon CCD camera.Temperature-dependent Raman scattering experi-ments in intermediate fields of B = 0 − . ◦ scattering geometry using a HoribaT64000 triple spectrometer equipped with a Dilor Spec-trum One CCD and a Nd:YAG laser emitting at λ = 532nm (Torus, Laser Quantum). A λ/ RL ).The laser power was kept to P = 4 mW with a spot di-ameter of about 100 µ m to minimize heating effects. Abase temperature of T base = 2 K was achieved by fullyimmersing the sample in superfluid He. Measurementsat elevated temperatures were carried out in He gas at-mosphere. From a comparison between Stokes- and anti-Stokes scattering we estimate the laser heating to be of 3K within the He gas environment. The sample tempera-tures are corrected accordingly. In-plane magnetic fieldswere applied via an Oxford Spectromag split coil system( T = 2 K – 300 K, B max = 10 T). Data analysis
The mid-energy regime of the contin-uum of Majorana fermionic excitations observed in Ra-man spectroscopy arises from the simultaneous creationor annihilation of a pair of Majorana fermions. Its tem-perature dependence can be described by two-fermionicstatistics, I MF = [1 − f ( (cid:15) )][1 − f ( (cid:15) )] δ ( ω − (cid:15) − (cid:15) ); with f ( (cid:15) ) = 1 / [1+e (cid:15)/k B T ] (see for details). Additional termsthat stem from deviations of the pure Kitaev model (Γ-term, Heisenberg exchange coupling) culminate in an ad-ditional bosonic background term, I B = 1 / [e (cid:15)/k B T − [ I ( ω ) = I q + (cid:15) ) (1+ (cid:15) ) , with (cid:15) = ( ω − ω ) / Γ, andΓ = full width at half maximum] in case of a strongcoupling between lattice- and spin degrees of freedom.The parameter 1 / | q | characterizes the degree of asymme-try and - consequently - gives a measure of the couplingstrength. ACKNOWLEDGMENTS
We acknowledge important discussions with NataliaPerkins. Part of this work was performed at the LNCMI,a member of the European Magnetic Field Laboratory(EMFL). This work was supported by “Nieders¨achsischesVorab” through the “Quantum- and Nano-Metrology(QUANOMET)” initiative within the project NL-4,DFG-Le967-16, and the Excellence Cluster DFG-EXC2123 Quantum Frontiers. The work at CAU was sup-ported by the National Research Foundation (NRF) ofKorea (Grant No. 2017R1A2B3012642). ∗ Contributed equally to this work.; Corresponding author:[email protected] † Contributed equally to this work. ‡ Corresponding author: [email protected] Jackeli, G. & Khaliullin, G., Phys. Rev. Lett. , 017205(2009). Singh, Y. et al. , Phys. Rev. Lett. , 127203 (2012). Modic, K. A. et al. , Nat. Commun. , 4203 (2014). Chun, S. H. et al. , Nat. Phys. , 462466 (2015). Abramchuk, M. et al. , J. Am. Chem. Soc. , 1537115376(2017). Kitagawa, K. et al. , Nature , 341-345 (2018). Plumb, K. W., Clancy, J. P., Sandilands, L. J. & Shankar,V. V., Phys. Rev. B , 041112(R) (2014). Sears, J. A. et al. , Phys. Rev. B , 144420 (2015). Do, S.-H. et al. , Nat. Phys. , 1079-1084 (2017). Suzuki, T. & Suga, S.-I., Phys. Rev. B , 134424 (2018). Rousochatzakis, I., Kourtis, S., Knolle, J., Moessner, R. &Perkins, N. B., Phys. Rev. B , 045117 (2019). Glamazda, A., Lemmens, P., Do, S.-H., Kwon, Y. S. &Choi, K.-Y., Phys. Rev. B , 174429 (2017). Kasahara, Y. et al. , Nature , 227-231 (2018). Sears, J. A., Zhao, Y., Xu, Z., Lynn, J. W. & Kim, Y.-J.,Phys. Rev. B , 180411(R) (2017). Wolter, A. U. B. et al. , Phys. Rev. B , 041405(R) (2017). Baek, S.-H. et al. , Phys. Rev. Lett. , 037201 (2017). Kitaev, A., Annals of Physics , 2111 (2006). Th´eveniaut, H. & Vojta, M., Phys. Rev. B , 054401(2017). Stern, A., Nature , 187 (2010). Winter, S. M., Riedl, K., Kaib, D., Coldea, R. & Valenti,R., Phys. Rev. Lett. , 077203 (2018). Banerjee, A. et al. , npj Quantum Materials , 8 (2018). Wang, Z. et al. , Phys. Rev. Lett. , 227202 (2017). Ponomaryov, A. N. et al. , Phys. Rev. B , 241107(R)(2017). Takikawa, D. & Fujimoto, S., Phys. Rev. B , 224409(2019). Sandilands, L. J., Tian, Y., Plumb, K. W., Kim, Y.-J. &Burch, K. S., Phys. Rev. Lett. , 147201 (2015). Glamazda, A., Lemmens, P., Do, S.-H., Choi, Y. S. & Choi,K.-Y., Nat. Commun. , 12286 (2016). Nasu, J., Knolle, J., Kovrizhin, D. L., Motome, Y. &Moessner, R., Nat. Phys. , 912 (2016). Yoshitake, J., Nasu, J., Kato, Y. & Motome, Y., Preprintat https://arxiv.org/abs/1907.07299 (2019). Choi, K.-Y. et al. , Phys. Rev. Lett. , 117204 (2013). Knolle, J., Kovrizhin, D. L., Chalker, J. T. & Moessner,R., Phys. Rev. B , 115127 (2015). Knolle, J., Chern, G.-W., Kovrizhin, D. L., Moessner, R.& Perkins, N. B., Phys. Rev. Lett. , 187201 (2014). Gordon, J. S., Catuneanu, A., Sørensen, E. S. & Kee, H.-Y., Nat. Commun. , 2470 (2019). Fano, U., Phys. Rev. , 1866 (1961).
V. SUPPLEMENTARY INFORMATIONVI. S1. SCATTERING GEOMETRIES
In-plane and out-of-plane Raman scattering experi-ments at the high magnetic field lab in Grenoble havebeen carried out in Voigt- and in Faraday geometry, re-spectively, as sketched in Figs. S1a and S1b. At present,the high-field Raman setup is not equipped with vari-able temperature inserts (VTIs). For a temperature scan,a magneto-optical cryostat with built-in VTI was used.These experiments were performed in 90 ◦ scattering ge-ometry (Fig. S1c) in applied magnetic fields up to 10T. To achieve large enough in-plane field components, the sample was irradiated under a grazing angle of 18 ◦ .A maximum field of 10 T hence corresponds to an in-plane component of 9.5 T. In the main text and in theSupplementary all applied fields have been corrected ac-cordingly. All Raman experiments were performed us-ing circular polarized light: incident right-circular polar-ized laser light ( R ) was created via a λ /4 waveplate, andbackscattered light was passed through a left-circular po-larized analyzer ( L ). This RL configuration allows us toprobe the E g symmetry channel, which carries the mag-netic Raman contribution in the pure Kitaev model .In Fig. S1d we compare two spectra obtained from themagneto-optical setup and from the high-field facility.Despite the difference in temperature and scattering ge-ometry, we find a nearly one-to-one correspondence. VII. S2. HIGH FIELD DATA
In the left panel of Fig. S2 high-field Raman spectrataken at T ≥ T N are presented for magnetic fields appliedalong various crystallographic directions. In the rightpanel the corresponding color contour maps are given.While in-plane magnetic fields along the (100), (010), and(110) directions yield phenomenologically similar behav-ior concerning the intensity and spectral distribution ofspectral weight, the energy scales of the emerging mag-netic excitations are slightly different. This is owed tothe finite in-plane anisotropy of α -RuCl .A substantial spin-phonon-coupling is observed, as the E g (1) mode gets heavily damped with increasing fieldthrough 10 T, while the E g (2) mode is gradually sup-pressed. This is owed to the spectral redistribution of C:With increasing magnetic fields the continuum of frac-tionalized excitations is confined to higher energies as agap opens and increases in size. In contrast, out-of-planemagnetic fields (along the 001-direction) do not affect theRaman data up to at least 29 T. This is consistent withthe anisotropy of the extended Kitaev model. VIII. S3. DECONFINED MAGNETICEXCITATIONS AT THE CRITICAL FIELD
In order to study the competition of quantum andthermal fluctuations and to analyze the statistics of ex-citation spectrum C in the intermediate field phase withsuppressed LRO we perform experiments as a functionof temperature at a fixed magnetic field B = B c . Thisenables us to distinguish conventional bosonic multipar-ticle excitations from fractionalized Majorana excitationsas quantum fluctuations deconfine the bosonic multipar-ticle excitations . In Figs. S3a and S3b we plotRaman spectra obtained at zero fields and around thequantum critical point at B c = 6 . B = 0 (blue diamonds) and B = B c (red FIG. 4.
S1: Sketch of the scattering geometries and data comparison. a , Raman scattering in Voigt geometry. b , Raman scattering in Faraday geometry. c , 90 ◦ scattering geometry with the sample tilted by 18 ◦ from the in-plane fielddirection for the magneto-cryostat measurements. d , Comparison of Raman data obtained in the magneto-optical cryostat(black line; scattering geometry c) and in the high-field facility (red line, scattering geometry a). circles). This mid-energy regime of the continuum ofMajorana fermionic excitations observed in Raman spec-troscopy arises from the simultaneous creation or anni-hilation of a pair of Majorana fermions. Its temperaturedependence can be described by two-fermionic statis-tics, I MF = [1 − f ( (cid:15) )][1 − f ( (cid:15) )] δ ( ω − (cid:15) − (cid:15) ); with f ( (cid:15) ) = 1 / [1+e (cid:15)/k B T ] (see for details). Additional termsthat stem from deviations of the pure Kitaev model (Γ-term, Heisenberg exchange coupling) culminate in an ad-ditional bosonic background term, I B = 1 / [e (cid:15)/k B T − E g (1) phonon overlapping withC is more pronounced at B c , indicative of a stronger cou-pling between lattice and Majorana fermionic excitations.Our observations suggest that the Majorana-related exci-tations at the quantum critical field B c become more pro-nounced than the B = 0 T excitation in spite of the de-velopment of the low-energy Majorana bound state MB.This suggests that the field-induced phase is thus closerto a spin liquid than the zero-field phase. IX. S4. TEMPERATURE- AND FIELDDEPENDENCE
In Fig. S4 we present the full dataset of temperature-and field-dependent as-measured Raman data. The up-per panel (a-f) presents the decomposition into variousmagnetic excitations at base temperature and with in-creasing magnetic field (see also main text). In Figs.S4g-l the thermal evolution at a given field is shown. For
B < B c = 6 . T N . For B > B c , the bound state (MB) is gradually suppressedwith temperature, while a finite remnant spectral weightabout 6.25 meV is observed even at T = 23 K. This sug-gests that the nature of low-energy excitations is alteredthrough B c . X. S5. K − Γ MODEL: THEORY VSEXPERIMENT
Recent theoretical modeling of the Raman response ina Kitaev magnet has been extended to include a Γ-term.The key impact of the considerable Γ-term is to shift the
FIG. 5.
S2: Raman data obtained over a wide field range in various field directions.
Left panel: As-measuredRaman spectra for magnetic fields applied along the (100), (110), (010), and (001) directions. The spectra have been shiftedvertically for clarity. Right panel: The respective color contour plots extending up to E = 25 meV. FIG. 6.
S3: Temperature dependence and analysis of the quantum statistics. a , Raman spectra obtained at B = 0T in RL polarization over a wide temperature range 2 K – 290 K. b , Raman spectra obtained around B c = 6 . RL polarization over a wide temperature range 2 K – 290 K. c , Intensity of the broad continuum C integrated over a frequencyrange of 8.75 – 15 meV as a function of temperature at B = 0 T (blue diamonds) and B = 6 . S4: Field- and temperature dependence of Raman spectra. a - f , Field dependence of Raman spectra at T = 2K and in circular RL polarization. With increasing external fields through the critical regime B c ≈ . B c . g - l , Thermal evolution of Raman spectra at the givenapplied magnetic fields. spectral weight of itinerant Majorana fermionic excita-tions to a lower energy . More specifically, the Γ-termon the one hand generates strong confinement (by open-ing a massive excitation gap), and on the other hand,rearranges the spectral weight towards lower energies, offering abundant low-energy states. Consequently, the K − Γ model is in favor of stabilizing low-energy boundstates. At zero fields, the K − Γ model gives a reasonabledescription of our T = 2 K data (see Fig. S5) as themagnon modes contribute only small spectral weight. In1 FIG. 8.
S5: K − Γ Model, Experiment vs. Theory.
As-measured temperature- and field-dependent Raman data togetherwith calculations of the spectral weight based on the K − Γ model (solid red line). The parameters to fit the data were chosenas K = − .
43 meV and Γ = 1 .
48 meV, and the calculations were performed for small but finite temperatures. contrast, at B c , the itinerant Majarana fermions are con-fined to form the bound state at low temperatures. As aconsequence, the K − Γ model reproduces the Majoranacontinuum excitation at around the temperature wherethe Majorana fermions start to bind ( T = 13 −
18 K).The comparison between our data and the K − Γ modeltheory strongly supports the notion that the K − Γ modelcan capture a key feature of the α -RuCl magnetism pos-sibly because the residual perturbations are weaker thanthe K and the Γ terms, and can be effectively quenchedin applied magnetic fields and at slightly elevated tem-perature (see Fig. S5).Based on calculations of the dynamical spin response itwas shown that with increasing strength of the anisotropyΓ-term a second mode can occur as a consequence of an inequality in gap size ∆ x and ∆ z . This might accountfor the rather large linewidth of MB (see, e.g., Fig. S4e).From an analysis of the T dependence of the peak en-ergy (see Fig. 3e, main text), we estimate the bindingenergy as E B = ω MB − ∆, where ω MB = 5 meV and ∆are the energy of MB and the excitation gap of the con-tinuum C, respectively. With a binding-unbinding tran-sition temperature around 15 K we estimate E B to bearound 1.3 meV, which places the gap energy around∆ = 6 .
25 meV. This estimation is in very good agree-ment with the fit of the continuum in Fig. 2c, maintext (at B = 9 ..