Abstract
The pure Kitaev honeycomb model harbors a quantum spin liquid in zero magnetic fields, while applying finite magnetic fields induces a topological spin liquid with nonAbelian anyonic excitations. This latter phase has been much sought after in Kitaev candidate materials, such as αRuCl_{3}. 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 magnetic excitations of integer quantum number vs. (ii) Majorana fermionic excitations of possibly nonAbelian nature with a fractional quantum number. To discriminate between these scenarios a detailed investigation of excitations over a wide fieldtemperature phase diagram is essential. Here, we present Raman spectroscopic data revealing lowenergy quasiparticles emerging out of a continuum of fractionalized excitations at intermediate fields, which are contrasted by conventional spinwave excitations. The temperature evolution of these quasiparticles suggests the formation of bound states out of fractionalized excitations.
Introduction
The search for Majorana fermions in solidstate systems has led to the discovery of several promising candidate materials for exchangefrustrated Kitaev quantum spin systems^{1,2,3,4,5,6}. One of the closest realizations of a Kitaev honeycomb lattice is αRuCl_{3}^{7,8}, where the spin Hamiltonian is dominated by Kitaev interaction K. Nevertheless, nonKitaev interactions, such as Heisenberg (J) and offdiagonal symmetric exchange terms (“Γterm”), as well as stacking faults in αRuCl_{3} lead to an antiferromagnetically ordered zigzag ground state below T_{N} ≈ 7 K^{8}. The exact strengths of these interactions have not been pinpointed, yet a general consensus on the minimal model has emerged about a ferromagnetic K ~−6 to −16 meV, as well as Γ ~1 to 7 meV, and J ~−1 to −2 meV^{9,10}. Despite these additional magnetic parameters, the Majorana fermionic quasiparticles are well preserved at high energies and elevated temperatures^{11}. Indeed, many independent and complementary experimental techniques have been used to probe the emergence of itinerant Majorana fermions and localized gauge fluxes from the fractionalization of spin degrees of freedom^{9,12,13}.
A promising route in understanding Kitaev physics might be the suppression of longrange magnetic order in a magnetic field, with the possibility of generating an Ising topological quantum spin liquid. For αRuCl_{3} this field is B_{c} ~6–7 T^{14,15,16}. Higher magnetic fields lead to a trivial spin polarized state. In the intermediate field range of 7–10 T, the magnetic order melts into a quantum disordered phase, in which the halfinteger quantized thermal Hall conductance is reported^{13}. This remarkable finding may be taken as evidence for a fieldinduced topological spin liquid with chiral Majorana edge states and the central charge q = ν/2 (Chern number ν = 1). However, it is less clear whether such a chiral spin liquid state can be stabilized in the presence of a relatively large field and nonKitaev terms in αRuCl_{3}. In the original Kitaev honeycomb model, nonAbelian Majorana excitations are created upon breaking timereversal symmetry, e.g., through applying a magnetic field^{17}. These composite quasiparticles correspond to bound states of localized fluxes and itinerant Majorana fermions^{18}. The composite bound states are of neither fermionic nor bosonic character, but instead they acquire an additional phase in the wavefunction upon interchanging particles, i.e., they follow anyonic statistics^{19}. Although for a Kitaev system bound itinerant Majorana fermions can be formed in the presence of perturbations^{18}, it is unclear how stable a topological spin liquid state is in this case. In particular, an open issue is whether the quantized thermal Hall effect is related to a nonAbelian phase featuring anyonic Majorana excitations.
There exists another scenario in which the intermediate phase is simply a partially polarized phase and smoothly connected to the fully polarized phase. Here, the transition through B_{c} involves conventional multiparticle excitations due to anisotropic interactions^{20}. To resolve these opposing viewpoints, one needs to clarify the characteristics of quasiparticle excitations emergent in the intermediatetohighfield phase. So far, inelastic neutron scattering (INS) ^{21}, THz spectroscopy^{22}, and electron spin resonance (ESR) ^{23} have revealed a significant reconfiguration of the magnetic response through B_{c}, giving tantalizing evidence for a quantum spin liquid state for magnetic fields just below the fully spin polarized state. These experimental techniques dominantly probe ΔS = ±1 excitations. Therefore, complementary experiments sensitive to also singlet (ΔS = 0) excitations are essential for unraveling new aspects of lowenergy properties and for obtaining a complete picture of individual quasiparticles.
Here, we employ Raman spectroscopy capable of sensing single and multiparticle excitations over the sufficiently wide ranges of temperatures T = 2–300 K, fields B = 0–29 T, and energies ℏω = 1–25 meV (8–200 cm^{−1}). At low fields (B < B_{c}) and low temperatures (T < T_{N}) we observe a number of spinwave excitations superimposed onto a continuum of fractionalized excitations. Towards higher fields above 10 T, the magnetic continuum opens progressively a gap and its spectral weight is transferred to welldefined sharp excitations that correspond to onemagnon and multimagnon bound states, marking the crossover to a fieldpolarized phase. In the intermediate phase, a weakly bound state emerges. This bound state is formed via a spectral transfer from the fractionalized continuum through an isosbestic point around 8.75 meV, and does not smoothly connect to the magnon bound states in the highfield phase. Our results suggest that this weakly bound state carries Majorana characteristics^{11,24} and that the intermediatefield phase of αRuCl_{3} hosts a distinct quantum phase.
Results
Field dependence of magnetic excitations
We performed Raman scattering experiments on oriented single crystals to elucidate the fieldevolution of the magnetic excitation spectrum of αRuCl_{3}. A detailed outline of the scattering geometries is given in Supplementary Note 1 and Supplementary Fig. 1, and the full dataset is presented in Supplementary Note 2 and Supplementary Fig. 2. All measurements were carried out with RL circularly polarized light, probing the E_{g} symmetry channel. Fig. 1a–c shows representative raw spectra obtained at increasing magnetic fields aligned along the crystallographic a axis [B//(100)]. We observe two sharp, intense phonon modes at 14.5 and 20.5 meV [marked E_{g}(1) and E_{g}(2)] with a pronounced field dependence, signaling a strong coupling between lattice and spin degrees of freedom (see Supplementary Note 2 and Supplementary Fig. 3 for details). In addition, there are several magnetic excitations with distinct field dependences: at zero field, the magnetic Raman response consists of a broad continuum (C; green shading) and a sharp peak (M1, blue line). The latter M1 peak at 2.5 meV is assigned to onemagnon scattering arising from a spin flip process by strong spin–orbit coupling and enables us to detect a gap of lowlying excitations at the Γpoint as a function of field. The M1 peak energy and its field dependence matches well recent THz magnetooptical data, confirming the ΔS = ±1 scattering process as its origin. The greenshaded continuum C agrees with observations in several previous Raman scattering studies^{12,25,26} and has been identified as Majorana fermionic excitations stemming from a fractionalization of spin degrees of freedom in the Kitaev honeycomb model^{17}. Although we cannot exclude an incoherent multimagnon contribution to the continuum, the thermal evolution of the continuum follows twofermionic statistics. Such exotic behavior is not expected for bosonic spinwave excitations, but rather supports the notion of Majorana fermions^{27}. An analysis of the temperature dependence shows that the twofermionic character increases by about 120% at B_{c} = 6.7 T (when zigzag order is suppressed) compared to B = 0 T (see Supplementary Note 3, and Supplementary Figs. 4 and 5). This can be taken as further evidence for the presence of Majorana fermionic excitations in αRuCl_{3}. As the magnetic field increases above 10 T, C becomes gapped and its spectral width narrows down. This opening of the gap is traced by the dashed curve in Fig. 1d.
Noteworthy is that the gapped continuum has a finite intensity even at high fields B > B_{c}, while several sharp and welldefined excitations emerge additionally with increasing fields. The residual spectral weight of C at sufficiently high fields means that the excitation spectrum in this regime is not solely exhausted by single and multiparticle magnons. Indeed, recent numerical calculations of the Kitaev model under applied fields uncovered a wide Kitaev paramagnetic region, reaching far beyond the critical field at finite temperature^{28}. We therefore ascribe the gapped continuum excitations to fractional quasiparticles pertinent to the Kitaev paramagnetic state. The M1 peak is ubiquitous in all measured fields. The excitation 2M (orange line) is split off from the M1 peak above 12 T, while the higherenergy 3M feature (red line) appears at the lower boundary of the gapped continuum above 10–14 T. In previous experimental fielddependent studies on αRuCl_{3} ranging from INS^{21}, to THz absorption^{22}, to ESR^{23} similar sharp magnetic excitations were reported and interpreted in terms of onemagnon or magnon bound states. In consideration of the narrow spectral form and energy of the corresponding excitations observed in our data, we assign the 2M peak to a twomagnon bound state. The 3M peak could be either the excitation of a threemagnon bound state, or a van Hove singularity of the gapped continuum. Considering the 3M peak sharpens in the highfield regime, the assignment to a threemagnon bound state is more convincing. As the field is lowered to the intermediate phase, the 3M excitation is no longer welldefined. The evolution of all magnetic excitations as a function of fields up to 29 T is depicted in the color contour plots of Fig. 1d–f together with the asmeasured Raman spectra in Fig. 1g–i for field directions along (100), (010), and (110), respectively. A slight anisotropy in magnetic excitations as a function of fielddirection becomes apparent, which is highlighted in Fig. 1j. In particular, the energy and field ranges of 3M are sensitive to the field directions, indicating the presence of nonnegligible inplane anisotropy terms. Our highfield Raman data evidence the coexistence of multimagnon excitations and a gapped continuum that characterizes the quasiparticle landscape of the partially polarized phase. The base temperature is, however, restricted to T ≥ 5 K in this highfield data. As we show below, a richer spectrum emerges at intermediate fields around B_{c} upon further cooling.
Magnetic excitations at intermediate fields
The reported halfinteger thermal Hall conductance^{13} is expected from chiral Majorana states along the edges of the 2D honeycomb layers of αRuCl_{3}. Simultaneously, Majorana bound states with possibly anyonic character emerge in the bulk. A detection of these excitations will provide a clue to the topological nature of the fieldinduced intermediate phase. In the extended Kitaev system, these bound states can occur in different channels^{18}, namely, from either binding itinerant Majorana fermions to localized fluxes (sketched in Fig. 2a), or by binding two itinerant Majorana fermions (sketched in Fig. 2b). With this in mind, we study the intermediate phase in detail by switching to a magnetooptical cryostat setup, enabling us to reach a base temperature of T = 2 K in a field range of B = 0–10 T. In this setup the sample is tilted by an angle of 18^{∘} away from the inplane field geometry, resulting in an additional small but finite outofplane field component. Note that here both magnetic field direction as well as light scattering geometry are slightly different from the highfield setup, which prohibits a strict onetoone comparison of the data. Nonetheless, a good correspondence is found between the highfield B//(110) data at T ≈ 5 K and the magnetooptical data at T = 9 K (see Supplementary Note 1). In Fig. 2c–h we inspect the fielddependence of Raman spectra measured at T = 2 K. Compared to the T = 5 K highfield data shown in Fig. 1a, the T = 2 K data show new sharp M2 and (M3) structures at 5 and 7.5 meV, respectively, in addition to the onemagnon excitation M1 and the fractionalized continuum C. The M2 peak may be tentatively assigned to a twomagnonlike excitation in the singlet sector. Unlike conventional twomagnon scattering, the 2M peak is narrow possibly because a large portion of the twomagnon excitation decays into the continuum. As the field increases, the respective modes evolve in a disparate manner. Initially (at B = 0–4.3 T), the M1 and M2 modes are slightly suppressed, while the continuum C is partially renormalized toward lower energies (Fig. 2c, d). As B_{c} is approached and through 6.7 T, the spectral weight of the continuum is massively redistributed (Fig. 2e, f). A new lowenergy MB mode evolves from the lowfield M2 mode and a shoulder structure (MB’) appears out of the (M3) peak. Apparently, the continuum of Majorana excitations is gapped above 8.1 T (Fig. 2g, h). A recent INS study reported a similarly broad, emerging excitation in the intermediate fieldinduced phase^{21}. It was tentatively discussed as a possible Majorana bound state, but an ultimate assignment was hindered by the lack of a detailed temperature study.
Formation of a Majorana bound state
To analyze the fieldinduced spectral weight redistribution carefully, we replot phononsubtracted Raman data taken at T = 2 K in Fig. 3a. With increasing field a distinct transfer of spectral weight from the midenergy (greenshaded A2) to the lowenergy regime (purpleshaded A1) is observed, with an isosbestic point located around ω_{iso} = 8.75 meV, at which the magnetic Raman response is independent of the external field. The systematic fieldinduced redistribution of spectral weight through this isosbestic point suggests an intimate connection between the continuum and the newly formed excitation, and therefore supports the formation of a lowenergy Majorana bound state (MB) through a confinement of the highenergy broad continuum of Majorana fermionic excitations. We also note that the MB mode appears in the same symmetry channel as the continuum of Majorana fermionic excitations (see Supplementary Note 4 and Supplementary Fig. 6). Consistent with the highfield data presented in Fig. 1, we observe a remaining intensity of the continuum C of Majorana fermions at 9.5 T, i.e., corresponding to the nontrivial quantum phase. A coexistence of massive Majorana fermions that form a broad, gapped continuum together with a subgap MB state is not compatible with the trivial polarized phase that is characterized by the multimagnon bound states. Figure 3b shows the thermal evolution obtained at B = 9.5 T (see Supplementary Note 5, and Supplementary Fig. 7 for full data set). As the temperature increases, the lowenergy mode MB gradually loses in intensity and shifts toward higher energies. Meanwhile, the continuum C slightly gains in intensity. The field and thermal evolution of magnetic modes is visualized in a contour plot in Fig. 3c, based on fits to the asmeasured data. It especially highlights the similar evolution of MB with field and temperature: around 5 meV its spectral weight starts to appear at 4.7 T and grows with increasing field, while it becomes thermally stabilized upon cooling below 12 K at 9.5 T. This suggests that quantum and classical fluctuations play the same role in a confinementdeconfinement transition due to a proximate Kitaev paramagnetic state at elevated temperature^{11}.
Fig. 3d plots the energy of the MB mode as a function of field together with the spinwave excitations observed in the zigzag ordered phase as well as in the high field spinpolarized phase. Here, full squares, empty circles, and empty triangles denote B//(100), B//(010), and B//(110), respectively. We note a smooth transition from M2 to MB through B_{c}. This weak fielddependence suggests that the MB mode corresponds to an excitation in the singlet sector (ΔS_{z} = 0), to which Raman spectroscopy is a natural probe. As the magnon corresponds to a condensation of Majorana fermions, the M2toMB mode evolution may be interpreted in terms of a condensationtoconfinement crossover where the magnon excitations observed at low field gradually evolve into Majorana bound states in the singlet sector. Since the continuum C of deconfined Majorana excitations above B_{c} is massively gapped with an onset energy of 4–6 meV (see Figs. 1d–f and 2g, h), the lowenergy bound state can be created within the gap due to confinement. Unlike the M2toMB mode crossover below B_{c}, the MB mode is not smoothly linked to the 2M bound state for fields above B_{c}. Rather, as the field increases, the 2M mode splits from the M1 mode, and both excitations are observed prominently. This suggests that the highfield phase stabilizes conventional quasiparticles against the fractional excitations. Interestingly, a signature of the MB mode remains absent in our data obtained from the highfield setup, as well as in data from a related recent highfield Raman and THz study^{29}, implying that the parameters temperature, scattering geometry, and magnetic field direction are decisive in stabilizing Majorana bound states. It also suggests that we cannot assert a direct relation between the two excitations M2 (at low fields) and 2M (at high fields). Our data is also contrasted by the rather smooth transition of quasiparticle excitations observed in ESR experiments through 10 T^{23}, due to the different selection rules for quasiparticle excitations in Raman vs. ESR. The discontinuous evolution observed through 10 T in our data admittedly may be expected due to the slightly different experimental conditions between the highfield and the magnetocryostat setups. Yet, the small temperature difference of ≈ 3 K can hardly account for a jump of 1 meV. We also recall that the vanishing thermal Hall conductance around 10 T parallels the disappearance of the MB mode^{13}.
Further support of the MB state interpretation comes from the temperature dependence at two different magnetic fields, 6.7 and 9.5 T (see Fig. 3e and Supplementary Figs. 8 and 9). We see a clear initial increase in the MB mode energy at both fields as the temperature rises. This is contrasted by conventional (magnon) unbound excitations, which continuously soften with increasing temperature. For bound states, however, the thermal energy competes with the binding energy^{30}, until eventually a thermally induced unbinding takes place. We can trace the energy of the MB excitation up to temperatures of 9 K (for B = 6.7 T) and 13 K (for B = 9.5 T). At higher temperatures the MB quasiparticles melt and decay into the higherenergy continuum, and therefore become illdefined. Based on the peakenergy shift observed in the narrow, limited temperature window, we estimate a lower limit of the binding energy of 0.5 meV. Estimating the binding energy from the gap size yields a slightly higher value of about 1 meV (see Supplementary Note 6). The thermal evolution of area A1, summarized in Fig. 3f, highlights the gradual melting of the bound state at 9.5 T (dark red triangles) with increasing temperature, while the conventional magnetic excitations at 0 T (blue squares) abruptly vanish above T_{N}. All these observations are consistent with the picture of a quasibound state of Majorana fermions that is pulled below the gapped fractionalized continuum by residual interactions of nonKitaev origin^{31}.
Discussion
In the Kitaev model, Majorana bound states are created through flux pairs combined with Majorana fermions in a ΔS_{z} = ±1 channel^{17}. However, as flux excitations are largely invisible to the Raman scattering process^{32}, the bound states between the flux and Majorana fermions barely contribute to the magnetic Raman signal. In the presence of additional nonKitaev terms, the creation of bound states from itinerant Majorana fermions is enhanced (see the cartoon in Fig. 2b)^{18}. As Raman scattering probes mainly the ΔS_{z} = 0 channel, we conclude that the MB mode largely consists of the latter Majorana singlet bound states. This interpretation is supported by the smooth crossover from the M2 magnon mode to the bound state MB through B_{c} (see Fig. 3d), as both excitations arise from a spinconserving scattering process. In such a case, αRuCl_{3} as an apt realization of the K − Γ model (see Supplementary Note 6) can host an exotic intermediatefield phase. In relation to this issue, we mention that a numerical study of the \(JK\Gamma \Gamma ^{\prime}\) model shows an extended regime of a chiral spin liquid for the outofplane field. Once the magnetic field is tilted significantly toward the inplane direction, the intermediate topological phase vanishes^{33}. This discrepancy raises the challenging question whether the recently reported fieldinduced phase has a nonAbelian nature and how the inplane intermediate phase transits to the alleged chiral spinliquid phase, if the intermediate phase is of topologically trivial nature. As the Chern number changes its sign as a function of the magnetic field direction, the fieldangular variation of magnetic excitations should be detailed.
Our finding demonstrates that a nontrivial crossover from the zigzag through the intermediate to the highfield phase involves a strong reconfiguration of the fractionalized continuum excitations, calling for future work to shed light on the relation between the observed Majorana bound states in an inplane intermediate field phase of αRuCl_{3} and the nonAbelian phase predicted for outoffield directions.
Methods
Crystal growth
Single crystals of αRuCl_{3} were synthesized by a vacuum sublimation method. A commercial compound of RuCl_{3} (Alfa Aesar) was ground and dried in a quartz tube under vacuum in order to fully dehydrate. The evacuated quartz tube was sealed and placed in a temperature gradient furnace. A powder of RuCl_{3} was heated at 1080 °C for 24 h and then slowly cooled down to 600 °C at a rate of −2 °C h^{−1}. The resulting single crystals have typical sizes of about 5 mm × 5 mm × 0.5 mm, with a shiny black surface. Their thermodynamic and spectroscopic properties have been thoroughly characterized^{9,12,15,16}.
Raman scattering
High magnetic fields up to 29 T were generated using the resistive magnet M10 at the LNCMI Grenoble. The sample was kept at a temperature T ≈ 5–10 K and illuminated with a 515 nm solid state laser (ALS Azur Light Systems) at a laser power P = 0.05 mW and a spot size of 3 μm diameter. Resulting Raman spectra were collected in Voigt geometry for inplane fields, and in Faraday geometry for outofplane geometry, using volume Bragg filters (OptiGrate) in transmission geometry and a 70 cm focal distance Princeton Instruments spectrometer equipped with a liquid N_{2} cooled Pylon CCD camera.
Temperaturedependent Raman scattering experiments in intermediate fields of B = 0–9.5 T were carried out in 90° scattering geometry using a Horiba T64000 triple spectrometer equipped with a Dilor Spectrum One CCD and a Nd:YAG laser emitting at λ = 532 nm (Torus, Laser Quantum). A λ/4plate was used to generate left and rightcircularly polarized light (RL). The laser power was kept to P = 4 mW with a spot diameter of about 100 μm to minimize heating effects. A base temperature of T_{base} = 2 K was achieved by fully immersing the sample in superfluid He. Measurements at elevated temperatures were carried out in He gas atmosphere. From a comparison between Stokes and antiStokes scattering we estimate the laser heating to be of 3 K within the He gas environment. The sample temperatures are corrected accordingly. Inplane magnetic fields were applied via an Oxford Spectromag split coil system (T = 2–300 K, \({B}_{\max }=10\) T).
Data analysis
The midenergy regime of the continuum of Majorana fermionic excitations observed in Raman spectroscopy arises from the simultaneous creation or annihilation of a pair of Majorana fermions. Its temperature dependence can be described by twofermionic statistics, I_{MF} = [1 − f(ϵ_{1})][1 − f(ϵ_{2})]δ(ω − ϵ_{1} − ϵ_{2}); with f(ϵ) = 1∕[\(1+{{\rm{e}}}^{\epsilon /{k}_{{\rm{B}}}T}\)] (see ref. ^{27} for details). Additional terms that stem from deviations of the pure Kitaev model (Γterm, Heisenberg exchange coupling) culminate in an additional bosonic background term, I_{B} = 1∕[\({{\rm{e}}}^{\epsilon /{k}_{{\rm{B}}}T}1\)]. The thermal evolution of the continuum has been fitted to a superposition of both contributions.
Fits to the phonon spectrum were applied using symmetric Lorentzian lineshapes, as well as asymmetric Fano lineshapes^{34} [\(I(\omega )={I}_{0}\frac{{(q+\epsilon )}^{2}}{(1+{\epsilon }^{2})}\), with ϵ = (ω − ω_{0})∕Γ, and Γ = full width at half maximum] in case of a strong coupling between lattice and spin degrees of freedom. The parameter 1∕∣q∣ characterizes the degree of asymmetry and—consequently—gives a measure of the coupling strength.
Data availability
The authors declare that all data supporting the findings of this study are available as plots within the article and its Supplementary Information file. Raw datasets generated during and/or analyzed during the current study are available from the corresponding authors on reasonable request.
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Acknowledgements
We acknowledge important discussions with Natalia Perkins, Yuji Matsuda, and Stephen Nagler. Part of this work was performed at the LNCMI, a member of the European Magnetic Field Laboratory (EMFL). This work was supported by “Niedersächsisches Vorab” through the “Quantum and NanoMetrology (QUANOMET)” initiative within the project NL4, DFGLe96716, and the Excellence Cluster DFGEXC 2123 Quantum Frontiers. The work at CAU was supported by the National Research Foundation (NRF) of Korea (Grant no. 2020R1A2C3012367).
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K.Y.C. together with Y.G. and P.L. conceived and designed the experiments. S.H.D., Y.C., C.H.L. and K.Y.C. synthesized the single crystals. D.W., Y.C., C.F. and Y.G. performed the Raman spectroscopic experiments. Y.C., K.Y.C., D.W., Y.G., C.F. and P.L. analyzed the data. D.W., K.Y.C., Y.G., P.L. and Y.C. participated in the writing of the paper. All authors discussed the results and commented on the paper.
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Wulferding, D., Choi, Y., Do, SH. et al. Magnon bound states versus anyonic Majorana excitations in the Kitaev honeycomb magnet αRuCl_{3}. Nat Commun 11, 1603 (2020). https://doi.org/10.1038/s41467020153701
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DOI: https://doi.org/10.1038/s41467020153701
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Dynamic fingerprint of fractionalized excitations in singlecrystalline Cu3Zn(OH)6FBr
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