Introduction

In correlated quantum materials, the nature of electronic interactions and their ground state topology is intimately linked to the geometry of the underlying lattice1,2,3,4,5,6. The low-energy excitations arising from pure electronic degrees of freedom inevitably interact with the crystal lattice, leaving behind their fingerprints in the phonon spectrum. Hitherto, the interactions of phonons with “conventional” quasiparticles of either Bose–Einstein or Fermi–Dirac statistics, such as magnons in magnets7, phasons and amplitudons in density waves8,9,10 and Bogoliubons in superconductors11, have been explored extensively. In contrast, the coupling between phonons and fractional excitations, including spinons in one-dimensional magnets1,2,3,12,13,14,15, and Majorana fermions (MFs) and Z2 gauge fluxes that are thought to exist in the Kitaev quantum spin liquids (QSL)16,17,18,19,20,21,22,23,24,25, have remained elusive. The discovery of such fractional-excitation-phonon coupling (FPC) is of fundamental importance, as they carry key information of the intertwined quantum state12,13,14,15. In particular, the coupling of phonons to the itinerant MFs has been predicted to play a pivotal role in the realization of the field-induced quantum thermal Hall effect in α-RuCl326,27,28,29, which is a signature of quantum entanglement in Kitaev-QSLs30,31,32.

Numerous studies have shown that the low-temperature phase of α-RuCl3 is a promising Kitaev-QSL candidate17,18,19,20,21,22,23,24,30,31,32. As displayed in Fig. 1a, the edge-sharing Ru–Cl octahedra form an effective spin-1/2 honeycomb network. The destructive quantum-interference 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 plane33 (Fig. 1b). Figure 1c schematically depicts the phase diagram of α-RuCl3. At zero magnetic field, the low-energy excitations in the paramagnetic phase are primarily determined by the Kitaev term3,4,17,18,19,20,21,22,23,24,30,31,32

$$H=\mathop{\sum}\limits_{\gamma ,\, <\, i,j \,> \, }{J}_{K}^{\,\gamma }{S}_{i}^{\gamma }{S}_{j}^{\gamma }$$
(1)
Fig. 1: Schematics of a Kitaev-QSL and the phase diagram of α-RuCl3.
figure 1

a Structure motif of α-RuCl3 based on a honeycomb lattice of edge-sharing Ru–Cl octahedra. The red, green, and blue bonds represent three orthogonal Kitaev interactions \({J}_{K}^{\,\gamma }\). In the pure Kitaev model [Eq. (1)], the low-energy excitations fractionalize into itinerant MFs (red arrows) and anyonic Z2 flux (blue and yellow hexagons) W = ±1 represents the Z2 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-RuCl3 on a logarithmic-scale: below TK ~ JK ~8 meV (the yellow area) the thermal Hall conductivity, κxy, becomes finite, indicating a proximate Kitaev-QSL with MF and Z2 gauge flux. In the green area (T < TN =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 excitation18 (yellow area) that is broadly peaked around JK and the Z2 gauge flux excitations (blue area) near 0.065 JK.

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Here \({J}_{K}^{\,\gamma }\,\)(\(\gamma =X,Y,Z\)) 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 (\({J}_{K}^{\,\gamma }={J}_{K}\)). Below the Kitaev temperature scale \({T}_{k} \sim {J}_{K}\), the low-energy excitations of Eq. (1) start to fractionalize into itinerant MFs and fluctuating Z2 gauge fluxes34. The former features a continuum that peaks broadly near JK, while the latter is a local excitation with an energy around 0.065 JK16,20,25,34. Below TN = 7 K, non-Kitaev interactions such as remnant Heisenberg magnetic exchange couplings, stabilize zigzag antiferromagnetic order that is suppressed under magnetic field35,36,37,38,39. Above B ~7 T, a quantized thermal Hall conductivity (red region in Fig.1c) is observed, indicating strongly an entangled topological phase30,31,32. 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 FPCs26,27,28,29. Here we report experimental signature of the FPC in α-RuCl3 by uncovering two-types of phonon anomalies at zero magnetic field: a 35% enhancement of the phonon spectral weight near the Kitaev energy JK, and a giant phonon softening of ~15% below 2 meV26,27,28,29.

Results

Figure 2a shows the imaginary part of the dynamical phonon susceptibility \(\chi {\prime\prime} ({\bf{Q}},\omega )\) along Γ1(6, −3, 0)−M(6, −2.5, 0)−Γ2(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 fluctuation-dissipation theorem via \(\chi {\prime\prime} ({\bf{Q}},\omega )=S({\bf{Q}},\omega )\times (1-{e}^{-\omega /{k}_{B}T})\), where \(S({\bf{Q}},\omega )\) 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 Γ1 to Γ2, reflecting their different Bragg peak intensities that are plotted in Fig. 2c. Two low-energy optical phonons, P1 and P2, are observed at the Brillouin zone center Γ2, corresponding to \({\omega }_{1}=2.7\) and \({\omega }_{2}=7\) meV, which are in good agreement with previous optical and neutron studies19,40. 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-lines41,42.

Fig. 2: Room temperature phonon excitations in α-RuCl3.
figure 2

a Low-energy phonon excitations determined by IXS along the Γ1 (6, −3, 0)–M (6, −2.5, 0)– Γ2 (6, −2, 0) direction. The plot shows the Bose-factor corrected IXS intensity. The extracted peak positions are presented in (b) revealing interlaced optical phonons intercepting with the transverse acoustic phonon branch. The optical phonon energies at Γ2 are denoted by the green (P1) and orange (P2) hexagons, which are consistent with the phonon modes previously found by THz-spectroscopy38. The yellow and blue shaded areas correspond to the two characteristic Kitaev energy scales displayed in Fig. 1d. c IXS spectra at Γ1 and Γ2. The intensity is shown on a logarithmic scale. Note that due to the large intensity difference at Γ1 and Γ2, the acoustic phonon intensity is expected to be extremely weak near Γ2. 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 Γ2 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.

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In α-RuCl3, JK is estimated to be 5–9 meV in the low-temperature phase below 150 K4,17,18,19,20,21,22,23,24,25,30 (more discussions in Supplementary Note 5), which roughly corresponds to the top of the P1–P2 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 point28. To uncover the energy and momentum-dependent coupling between the optical phonons and the suggested MFs, we compare the temperature-dependent \(\chi {\prime\prime} ({\bf{q}},\omega )\) along the M–Γ2 path. A large spectral enhancement can be observed clearly in Fig. 3a–f. Near the M point, the peak intensity of P1 increases dramatically upon cooling from 300 K to 10 K. In contrast, the peak intensity of P2 is unchanged except the 10 K data at the M point. When approaching the Γ2 point (towards larger |q|), the intensity enhancement first decreases near the crossing-point (P1 and P2 crossed at q = 0.75), but then reappears at P2, which is higher in energy near the Γ2 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 \(\omega \sim {J}_{K}\). To quantitatively show the spectral enhancement effect, we extract the temperature-induced difference in the integrated phonon intensity, \(\Delta \chi {\prime\prime} ({\bf{q}},{\omega }_{0})={\int }_{{\omega }_{0}-\infty }^{{\omega }_{0}+\infty }[\chi {\prime\prime} ({\bf{q}},\omega ,10{\rm{K}})-\chi {\prime\prime} ({\bf{q}},\omega ,300{\rm{K}})]d\omega\), and plot \(\Delta \chi {\prime\prime} ({\bf{q}},{\omega }_{0})\) as function of \(\Delta E={\omega }_{0}-{\omega }_{{{\max }}}\) in Fig. 3g. Here \({\omega }_{0}\) denotes the phonon peak position and \({\omega }_{max}=7\) meV is the band-top energy of P1 and P2. Unlike the broad continuum observed in the spin correlation function17,18,19,21,22,23, \(\Delta \chi ({\bf{q}},{\omega }_{0})\) decreases rapidly as \({\omega }_{0}\) moves away from JK (Fig. 3). It also shows strong momentum dependence with the enhancement occuring around the high symmetry points M and Γ2 (see Supplementary Fig. 9). This observation is in qualitative agreement with theoretical calculation that shows energy and momentum dependent Majorana–phonon coupling28 (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 excitations7. We note, however, a quantitative understanding of the energy and momentum-dependent optical phonon enhancement may require theoretical calculations beyond the pure Kitaev model.

Fig. 3: Itinerant MF-phonon coupling near ω ~ JK.
figure 3

a–f Spectra of the interlaced optical phonons at different reduced momentum transfer q. Here, we define q = (0, 0, 0) and (0, 1, 0) as Γ1 and Γ2, respectively, where the M point is at q = (0, 0.5, 0). The labels P1 and P2 denote the two optical phonon branches. Note the relative peak position of P1 and P2 switches at q = (0, 0.75, 0). The temperature dependent \({\chi }^{\prime\prime}({\bf{q}},\omega )\) shows a spectral weight enhancement at \(\omega \sim {J}_{K}\) at low temperature. In b we notice a shoulder on P2 that may come from the acoustic mode. g The difference in the integrated phonon spectral weight, \(\Delta \chi {\prime\prime} ({\bf{q}},{\omega }_{0})\), between 10 and 300 K as a function of\(\,\Delta E={\omega }_{0}-{\omega }_{max}\). Here \({\omega }_{0}\) is the phonon peak position, \({\omega }_{max}\) = 7 meV is the band top of the interlaced optical phonons. The drastic increase of \(\Delta \chi {\prime\prime} ({\bf{q}},{\omega }_{0})\) 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.

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We then turn to the transverse acoustic phonon near Γ1. Figure 4a and b show the temperature-dependence of \(\chi {\prime\prime} ({\bf{q}},\omega )\) at q1 = (0, 0.1, 0) (or Q1 = (6, −2.9, 0)) and q2 = (0, 0.15, 0) (or Q2 = (6, −2.85, 0)), respectively. At q1, the phonon peak position gradually shifts to lower energies. In contrast, it remains nearly unchanged at q2. The softening-effect is confirmed by directly comparing the raw data, \(S({\bf{q}},\omega )\), at 10 and 300 K (Fig. 4c and d). The peak position is softened by about 13% at q1, which corresponds to ~0.3 meV shift in energy. Figures 4e and f show the relative peak shift \({\omega }_{0}(T)/{\omega }_{0}(300\,{\rm{K}})\) at q1 and q2 as function of temperature. We find that the acoustic phonon softening at q1 becomes progressively stronger below 80 K, consistent with the thermal Hall effect in α-RuCl3 where the thermal Hall conductivity, κxy, starts to increase. In Fig. 4e, we further show the phonon softening at q3 = (0, 0.05, 0). The error-bars returned from fittings are larger at q3 as the elastic intensity becomes stronger when approaching the Bragg peak. Interestingly, the relative phonon softening at q3 (~15%) is even larger when compared to q1. This suggests an enhanced renormalization for long wavelength acoustic phonons.

Fig. 4: Giant acoustic phonon softening.
figure 4

a, b Show the temperature-dependent \(\chi {\prime\prime} ({\bf{q}},\omega )\) at q1 = (0, 0.1, 0) in reciprocal lattice units (r.l.u.), ω ~2 meV and q2 = (0, 0.15, 0) r.l.u., ω ~3 meV, respectively. c direct comparison of the IXS raw data, \(S({{\bf{q}}}_{{\bf{1}}},\omega ),\) at T = 10 and 300 K. d shows the same plot as (c) but at q2. There is an apparent phonon softening at q1, while at q2, the effect is negligible. e, f The relative peak shift at q1, q2 and q3. The ~13% phonon softening at q1 (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 systems10. The blue diamonds in e represent the relative peak shifts at q3 = (0,0.05,0) that show even larger softening-effect (~15% at 60 K), whereas q2 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-RuCl3. g, h Schematically show two phonon coupling mechanisms. g The flatband of the Z2 flux mode intercepts the acoustic phonon near ω ~0.065JK. h The nearly identical linear dispersion of the itinerant MF and the acoustic phonon at \({\bf{q}}\to 0\) 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.

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Discusson

The discovery of temperature and energy dependent phonon softening provides important information on the FPC in α-RuCl3. In the pure Kitaev model [Eq. (1)], quantum fractionalization occurs at TK ~ JK ~100 K43, in agreement with our observations. Below TK, the dispersionless gauge flux excitation crosses the linear dispersing acoustic phonon near \(\omega =0.065{J}_{K} \sim 0.5\) meV23 and induces a phonon anomaly near this energy scale (Fig. 4g). The observation of enhanced phonon softening [\(\omega ({{\bf{q}}}_{1})=2\) meV and \(\omega ({{\bf{q}}}_{3})=\) 1 meV as \(\omega \to 0.065{J}_{K}\) is consistent with this picture, where the softening effect is expected to be significantly suppressed for \(\omega ({\bf{q}})\gg 0.065{J}_{K}\). 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 \({\bf{q}}\to 0\)29. This enhances Majorana–phonon coupling that yields a renormalization of the phonon dispersion below TK28,29. To justify this conjecture, we extract the acoustic phonon velocity \({v}_{{\rm{ph}}} \sim\)16 meV Å (\(\hslash =1\)), which is based on the room-temperature phonon dispersion shown in Fig. 2. In the isotropic limit16, the velocity of the itinerant MF is \({v}_{{\rm{MF}}}=\frac{\sqrt{3}}{4}{J}_{K}a\), where the in-plane lattice constant \(a=5.9639\) Å. Comparing \({v}_{{\rm{ph}}}\) and \({v}_{{\rm{MF}}}\) gives \({J}_{K}\) ~ 6.2 meV, comparable to the experimental value. Besides the Z2 gauge flux and MFs, in more realistic models with non-Kitaev interactions44,45, 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-RuCl3. In particular, direct experimental evidence of Z2 gauge flux is not well established yet. The observed acoustic phonon softening below 2 meV demonstrate a small energy scale in α-RuCl3 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 TN, a gapped magnon excitation between 2–7 meV was observed in α-RuCl3 by previous neutron studies19,21,22,37. However, as we show in Figs. 3 and 4, the phonon anomalies onset at TK, which is well above TN. More importantly, evidence of an enhanced phonon softening is observed at ω =1 meV (see q3 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-RuCl3. As we show in Fig. 2, the P2 phonon energy is the same as the magnon energy near the M point19,21,38. Interestingly, the P2 phonon intensity at the M point is enhanced at 10 K~TN, supporting magnon–phonon coupling46. In addition, strong anharmonicity is proposed in the magnon excitation of this material47, which represents the break-down of the spin quasiparticles. Such excitations contain extremely broad features47 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-QSL26,27. 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 candidates1,3,4, such as iridates4,48, 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 2kF, which induces phonon anomalies at q = 2kF12. Both kagome and triangular lattices have been speculated to host such charge neutral Fermi surfaces49,50. More recently, a giant thermal Hall effect has been observed in the cuprate high-Tc superconductors13 with large phonon contributions51. While mechanisms based on chiral spin liquid or topological spinons14,15 have also been proposed, the theoretically predicted κxy is 50% smaller than the experimental value14, suggesting large phonon effect. Our observation of FPC in α-RuCl3 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-RuCl3 crystals were grown by the sublimation of RuCl3 powder sealed in a quartz tube under vacuum52. 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 heat21.

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 Ei = 23.7 keV (l = 0.5226 Å) was focused on the sample with a beam cross section of ∼35 × 15 mm2 (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-RuCl3 were calculated using with density functional perturbation theory (DFPT) and the Vienna Ab initio Simulation Package (VASP). The exchange-correlation 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.