Giant phonon anomalies in the proximate Kitaev quantum spin liquid α-RuCl3

The Kitaev quantum spin liquid epitomizes an entangled topological state, for which two flavors of fractionalized low-energy excitations are predicted: the itinerant Majorana fermion and the Z2 gauge flux. It was proposed recently that fingerprints of fractional excitations are encoded in the phonon spectra of Kitaev quantum spin liquids through a novel fractional-excitation-phonon coupling. Here, we detect anomalous phonon effects in α-RuCl3 using inelastic X-ray scattering with meV resolution. At high temperature, we discover interlaced optical phonons intercepting a transverse acoustic phonon between 3 and 7 meV. Upon decreasing temperature, the optical phonons display a large intensity enhancement near the Kitaev energy, JK~8 meV, that coincides with a giant acoustic phonon softening near the Z2 gauge flux energy scale. These phonon anomalies signify the coupling of phonon and Kitaev magnetic excitations in α-RuCl3 and demonstrates a proof-of-principle method to detect anomalous excitations in topological quantum materials.

Numerous studies have shown that the low-temperature phase of α-RuCl 3 is a promising Kitaev-QSL candidate [17][18][19][20][21][22][23][24][30][31][32] . As displayed in Fig. 1a, the edge-sharing Ru-Cl octahedra form an effective spin-1/2 honeycomb network. The destructive quantuminterference through the close-to-90°Ru-Cl-Ru bonds significantly suppresses the Heisenberg magnetic exchange interaction, yielding a dominant Ising-type interaction (J) perpendicular to the Ru-Cl-Ru plane 33 (Fig. 1b). Figure 1c schematically depicts the phase diagram of α-RuCl 3 . At zero magnetic field, the lowenergy excitations in the paramagnetic phase are primarily determined by the Kitaev term 3,4,[17][18][19][20][21][22][23][24][30][31][32] Here J γ K (γ ¼ X; Y; Z) is the bond-dependent coupling parameter, and <i; j> stands for nearest-neighbor pairs of spins at one of the X, Y, or Z bonds. The two characteristic energy scales are shown in Fig. 1d for the isotropic limit (J γ K ¼ J K ). Below the Kitaev temperature scale T k $ J K , the low-energy excitations of Eq. (1) start to fractionalize into itinerant MFs and fluctuating Z 2 gauge fluxes 34 . The former features a continuum that peaks broadly near J K , while the latter is a local excitation with an energy around 0.065 J K 16,20,25,34 . Below T N = 7 K, non-Kitaev interactions such as remnant Heisenberg magnetic exchange couplings, stabilize zigzag antiferromagnetic order that is suppressed under magnetic field [35][36][37][38][39] . Above B~7 T, a quantized thermal Hall conductivity (red region in Fig.1c) is observed, indicating strongly an entangled topological phase [30][31][32] . However, unlike the quantum Hall effect of electrons, it has been theoretically predicted that the quantum thermal Hall effect can only be approximate and requires strong FPCs [26][27][28][29] . Here we report experimental signature of the FPC in α-RuCl 3 by uncovering two-types of phonon anomalies at zero magnetic field: a 35% enhancement of the phonon spectral weight near the Kitaev energy J K , and a giant phonon softening of~15% below 2 meV [26][27][28][29] . (1)], the low-energy excitations fractionalize into itinerant MFs (red arrows) and anyonic Z 2 flux (blue and yellow hexagons) W = ±1 represents the Z 2 index. b The nearly 90°Ru-Cl-Ru bonds and the moderate spin-orbit-coupling favor an Ising-type magnetic interaction that is perpendicular to the Ru-Cl-Ru plane highlighted in a blue-green color. Lattice vibrations perturbatively modify the magnetic interactions, which induce a coupling between phonons and fractional excitations. c illustrates the phase diagram of α-RuCl 3 on a logarithmic-scale: below T K~JK~8 meV (the yellow area) the thermal Hall conductivity, κ xy , becomes finite, indicating a proximate Kitaev-QSL with MF and Z 2 gauge flux. In the green area (T < T N =7 K), non-Kitaev terms drive the system into zigzag antiferromagnetic order. Under an external magnetic field (B > 7 T) that completely suppresses the magnetic order, the system is driven into a quantum thermal Hall state at finite temperature (the red area). d schematically shows two characteristic Kitaev energy scales in the isotropic limit: the itinerant MF excitation 18 (yellow area) that is broadly peaked around J K and the Z 2 gauge flux excitations (blue area) near 0.065 J K . Figure 2a shows the imaginary part of the dynamical phonon susceptibility χ00ðQ; ωÞ along Γ 1 (6, −3, 0)−M(6, −2.5, 0)−Γ 2 (6, −2, 0) in reciprocal lattice units (r.l.u.) at room temperature (see Supplementary Note 6 for first-principles calculations of phonon dispersion). The dynamical susceptibility is given by the fluctuationdissipation theorem via χ00ðQ; ωÞ ¼ SðQ; ωÞ ð1 À e Àω=k B T Þ, where SðQ; ωÞ is the dynamical phonon structure factor that is directly measured by inelastic x-ray scattering (IXS). The total momentum transfer Q = q + G, is composed of the reduced momentum transfer in the first Brillouin zone q and the reciprocal lattice vector G. The elastic contribution at ω = 0 was subtracted by fitting the IXS raw data in the entire energy window (see Supplementary Note 1 and Note 2). We selectively probe in-plane transverse phonon modes, whose dispersions (open circles and open squares) and sinusoidal fits (dashed curves) are shown in Fig. 2b. As shown in Fig. 2a, the intensity of the transverse acoustic phonon changes significantly from Γ 1 to Γ 2 , reflecting their different Bragg peak intensities that are plotted in Fig. 2c. Two low-energy optical phonons, P 1 and P 2 , are observed at the Brillouin zone center Γ 2 , corresponding to ω 1 ¼ 2:7 and ω 2 ¼ 7 meV, which are in good agreement with previous optical and neutron studies 19,40 . The two optical phonons carry opposite phonon velocities and form an interlaced structure that intercepts the acoustic phonon. An apparent phonon crossing occurs between Γ and M ( Fig. 2d and e), suggesting possible Dirac-cone and topological phononic nodal-lines 41,42 .

Results
In α-RuCl 3 , J K is estimated to be 5-9 meV in the lowtemperature phase below 150 K 4,17-25,30 (more discussions in Supplementary Note 5), which roughly corresponds to the top of the P 1 -P 2 phonon band. Thus, if Majorana-phonon coupling is present, phonon anomalies are expected in the energy range shown in Fig. 2b. Moreover, a recent theoretical study of the pure Kitaev model predicts that the Majorana-phonon coupling is momentum dependent and peaks near the M and K point 28 . To uncover the energy and momentum-dependent coupling between the optical phonons and the suggested MFs, we compare the temperature-dependent χ00ðq; ωÞ along the M-Γ 2 path. A large spectral enhancement can be observed clearly in Fig. 3a-f. Near the M point, the peak intensity of P 1 increases dramatically upon cooling from 300 K to 10 K. In contrast, the peak intensity of P 2 is unchanged except the 10 K data at the M point. When approaching the Γ 2 point (towards larger |q|), the intensity enhancement first decreases near the crossing-point (P 1 and P 2 crossed at q = 0.75), but then reappears at P 2 , which is higher in energy near the Γ 2 point. Interestingly, we find that the spectral enhancement is different between the symmetry related points q = 0.45 and q = 0.55. As we show in Fig. 2a, the transverse acoustic phonon starts to merge with the optical phonon near the M point. Since the acoustic phonon intensity is stronger at q = 0.45, the asymmetric intensity enhancement suggests that the Majorana-phonon coupling is larger on the acoustic mode than the optical mode near ω $ J K . To quantitatively show the spectral enhancement effect, we extract the temperatureinduced difference in the integrated phonon intensity, Δχ00ðq; ω 0 Þ ¼ R ω 0 þ1 ω 0 À1 ½χ00ðq; ω; 10KÞ À χ00ðq; ω; 300KÞdω, and plot Δχ00ðq; ω 0 Þ as function of ΔE ¼ ω 0 À ω max in Fig. 3g. Here ω 0 denotes the phonon peak position and ω max ¼ 7 meV is the band-top energy of P 1 and P 2 . Unlike the broad continuum observed in the spin correlation function [17][18][19][21][22][23] , Δχðq; ω 0 Þ decreases rapidly as ω 0 moves away from J K (Fig. 3). It also shows strong momentum dependence with the enhancement occuring around the high symmetry points M and Γ 2 (see Supplementary The optical phonon energies at Γ 2 are denoted by the green (P 1 ) and orange (P 2 ) hexagons, which are consistent with the phonon modes previously found by THz-spectroscopy 38 . The yellow and blue shaded areas correspond to the two characteristic Kitaev energy scales displayed in Fig. 1d. c IXS spectra at Γ 1 and Γ 2 . The intensity is shown on a logarithmic scale. Note that due to the large intensity difference at Γ 1 and Γ 2 , the acoustic phonon intensity is expected to be extremely weak near Γ 2 . d Constant momentum transfer cuts around the phonon-crossing. The two optical branches cross each other without imposing a hybridization gap. e The extracted phonon peak positions from M to Γ 2 at different temperatures reveal a temperature-independent massless Dirac-cone. The error bars in b, e denote the 2σ returned from the fittings (see Supplementary Note 2). The error bars in d represent one standard deviation assuming Poisson counting statistics. Fig. 9). This observation is in qualitative agreement with theoretical calculation that shows energy and momentum dependent Majorana-phonon coupling 28 (spectrum near the K point with spectral peak at higher energy is shown in Supplementary Fig. 6). The observed phonon enhancement is also consistent with a recent study of frustrated magnetic systems, which predicts large IXS cross-section for magnetic excitations 7 . We note, however, a quantitative understanding of the energy and momentumdependent optical phonon enhancement may require theoretical calculations beyond the pure Kitaev model.
We then turn to the transverse acoustic phonon near Γ 1 . Figure 4a and b show the temperature-dependence of χ00ðq; ωÞ at q 1 = (0, 0.1, 0) (or Q 1 = (6, −2.9, 0)) and q 2 = (0, 0.15, 0) (or Q 2 = (6, −2.85, 0)), respectively. At q 1 , the phonon peak position gradually shifts to lower energies. In contrast, it remains nearly unchanged at q 2 . The softening-effect is confirmed by directly comparing the raw data, Sðq; ωÞ, at 10 and 300 K (Fig. 4c  and d). The peak position is softened by about 13% at q 1 , which corresponds to~0.3 meV shift in energy. Figures 4e and f show the relative peak shift ω 0 ðTÞ=ω 0 ð300 KÞ at q 1 and q 2 as function of temperature. We find that the acoustic phonon softening at q 1 becomes progressively stronger below 80 K, consistent with the thermal Hall effect in α-RuCl 3 where the thermal Hall conductivity, κ xy , starts to increase. In Fig. 4e, we further show the phonon softening at q 3 = (0, 0.05, 0). The error-bars returned from fittings are larger at q 3 as the elastic intensity becomes stronger when approaching the Bragg peak. Interestingly, the relative phonon softening at q 3 (~15%) is even larger when compared to q 1 . This suggests an enhanced renormalization for long wavelength acoustic phonons.

Discusson
The discovery of temperature and energy dependent phonon softening provides important information on the FPC in α-RuCl 3 .
In the pure Kitaev model [Eq. (1)], quantum fractionalization occurs at T K~JK~1 00 K 43 , in agreement with our observations. Below T K , the dispersionless gauge flux excitation crosses the linear dispersing acoustic phonon near ω ¼ 0:065J K $ 0:5 meV 23 and induces a phonon anomaly near this energy scale (Fig. 4g). The observation of enhanced phonon softening [ωðq 1 Þ ¼ 2 meV and ωðq 3 Þ ¼ 1 meV as ω ! 0:065J K is consistent with this picture, where the softening effect is expected to be significantly suppressed for ωðqÞ ) 0:065J K . Figure 4h depicts another scenario that attempts to explain the phonon-softening. Here, the acoustic phonon and the itinerant MFs possess nearly identical linear dispersions at q ! 0 29 . This enhances Majorana-phonon coupling that yields a renormalization of the phonon dispersion below T K 28,29 . To justify this conjecture, we extract the acoustic phonon velocity v ph $16 meV Å (_ ¼ 1), which is based on the room-temperature phonon dispersion shown in Fig. 2. In the isotropic limit 16 where the in-plane lattice constant a ¼ 5:9639 Å. Comparing v ph and v MF gives J K~6 .2 meV, comparable to the experimental value. Besides the Z 2 gauge flux and MFs, in more realistic models with non-Kitaev interactions 44,45 , other fractional excitations may also be consistent with the observed phonon anomalies. It is important to note that the charge and magnetic excitations below 2 meV still remain unresolved in α-RuCl 3 . In particular, direct experimental evidence of Z 2 gauge flux is not well established yet. The observed acoustic phonon softening below 2 meV demonstrate a small energy scale in α-RuCl 3 that strongly renormalizes the acoustic phonon spectrum and hence may be responsible for the quantized thermal Hall effect.
Finally, we discuss the possibility of magnon-phonon coupling. Below T N , a gapped magnon excitation between 2-7 meV was observed in α-RuCl 3 by previous neutron studies 19,21,22,37 . However, as we show in Figs. 3 and 4, the phonon anomalies onset at T K , which is well above T N . More importantly, evidence of an enhanced Here, we define q = (0, 0, 0) and (0, 1, 0) as Γ 1 and Γ 2 , respectively, where the M point is at q = (0, 0.5, 0). The labels P 1 and P 2 denote the two optical phonon branches. Note the relative peak position of P 1 and P 2 switches at q = (0, 0.75, 0). The temperature dependent χ 00 ðq; ωÞ shows a spectral weight enhancement at ω $ J K at low temperature. In b we notice a shoulder on P 2 that may come from the acoustic mode. g The difference in the integrated phonon spectral weight, Δχ00ðq; ω 0 Þ, between 10 and 300 K as a function of ΔE ¼ ω 0 À ω max . Here ω 0 is the phonon peak position, ω max = 7 meV is the band top of the interlaced optical phonons. The drastic increase of Δχ00ðq; ω 0 Þ is fitted to an exponential function (dashed line). The vertical error bars in all panels represent one standard deviation based on Poisson counting statistics. The horizontal error bars in g denote the 2σ returned from the fitting algorithm that extract the spectral peak positions. ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23826-1 phonon softening is observed at ω =1 meV (see q 3 in Fig. 4e and Supplementary Fig. 5), which is well below the magnon gap. Therefore, a magnon-phonon coupling is unlikely giving rise to the observed acoustic phonon softening. However, the magnon-phonon coupling may indeed be present in α-RuCl 3 . As we show in Fig. 2, the P 2 phonon energy is the same as the magnon energy near the M point 19,21,38 . Interestingly, the P 2 phonon intensity at the M point is enhanced at 10 K~T N , supporting magnon-phonon coupling 46 . In addition, strong anharmonicity is proposed in the magnon excitation of this material 47 , which represents the break-down of the spin quasiparticles. Such excitations contain extremely broad features 47 that are contradictory to the well-defined energy scale of the phonon anomalies observed here.
Our discovery of two-types of phonon anomalies, i.e., the spectral enhancement in the optical phonon and the acoustic mode softening, provides experimental signature of FPC in the proximity of Kitaev-QSL 26,27 . Beyond the aforementioned implications, our observation has an even deeper impact on correlated topological quantum states. First of all, our approach can be immediately applied to other Kitaev-QSL candidates 1,3,4 , such as iridates 4,48 , where the inelastic neutron scattering experiments are difficult to perform due to strong neutron absorption of Ir. Moreover, it has been predicted that in U(1) spin liquids the spinon Fermi surface features a large singularity at 2k F , which induces phonon anomalies at q = 2k F 12 . Both kagome and triangular lattices have been speculated to host such charge neutral Fermi surfaces 49,50 . More recently, a giant thermal Hall effect has been observed in the cuprate high-T c superconductors 13 with large phonon contributions 51 . While mechanisms based on chiral spin liquid or topological spinons 14,15 have also been proposed, the theoretically predicted κ xy is 50% smaller than the experimental value 14 , suggesting large phonon effect. Our observation of FPC in α-RuCl 3 validates phonons as a sensitive probe to uncover hidden fractional and non-local excitations, and hence can help resolving key puzzles in correlated and entangled quantum states.

Methods
Sample preparation and characterizations. Millimeter-sized α-RuCl 3 crystals were grown by the sublimation of RuCl 3 powder sealed in a quartz tube under vacuum 52 . The growth was performed in a box furnace. After dwelling at 1060°C for 6 h, the furnace was cooled to 800°C at 4°C/h. Magnetic order was confirmed to occur at 7 K by measuring magnetic properties and specific heat 21 .
Inelastic X-ray scattering. The experiments were conducted at beam line 30-ID-C (HERIX) at the Advanced Photon Source (APS). The highly monochromatic X-ray beam of incident energy E i = 23.7 keV (l = 0.5226 Å) was focused on the sample with a beam cross section of ∼35 × 15 mm 2 (horizontal × vertical). The total energy resolution of the monochromatic X-ray beam and analyzer crystals was ΔE ∼1.3 meV (full width at half maximum). The measurements were performed in transmission geometry. Typical counting times were in the range of 30-120 s per point in the energy scans at constant momentum transfer Q. H, K, L are defined in the trigonal structure with a = b = 5.9639 Å, c = 17.17 Å at the room temperature.
Density functional theory calculations of phonon spectrum. Phonon dispersions for α-RuCl 3 were calculated using with density functional perturbation theory (DFPT) and the Vienna Ab initio Simulation Package (VASP). The exchangecorrelation potential was treated within the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof variety, where the kinetic energy cutoff was set to 400 eV. Integration for the Brillouin zone was done by using a Monkhorst-Pack k-point grids which is equivalent to 8 × 8 × 9.

Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request. There is an apparent phonon softening at q 1 , while at q 2 , the effect is negligible. e, f The relative peak shift at q 1 , q 2 and q 3. The~13% phonon softening at q 1 (red squares in e) corresponds to a~0.3 meV phonon peak shift. This value is as large as some well-known electron-phonon coupled systems 10 . The blue diamonds in e represent the relative peak shifts at q 3 = (0,0.05,0) that show even larger softening-effect (~15% at 60 K), whereas q 2 displays negligible change as shown in (f). This acoustic phonon anomaly, together with the spectral enhancement discussed in Fig. 3, present a full picture of the FPC in α-RuCl 3 . g, h Schematically show two phonon coupling mechanisms. g The flatband of the Z 2 flux mode intercepts the acoustic phonon near ω~0.065J K . h The nearly identical linear dispersion of the itinerant MF and the acoustic phonon at q ! 0 causes a phonon renormalization at low temperature. The error bars in a-d represent one standard deviation assuming Poisson counting statistics. The error bars in e, f denote the 2σ returned from the fitting.