Giant Phonon Anomalies in the Proximate Kitaev Quantum Spin Liquid $\alpha$-RuCl$_3$

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. Detection of these excitations remains challenging, because of their fractional quantum numbers and non-locality. 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 uncover this effect in $\alpha$-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. This fractional excitation induced phonon anomalies uncover the key ingredient of the quantum thermal Hall effect in $\alpha$-RuCl3 and demonstrates a proof-of-principle method to detect fractional excitations in topological quantum materials.

Here J ! # ( = , , ) is the bond-dependent coupling parameter, and < , > stands for nearestneighbor pairs of spins at one of the X, Y, or Z bonds. The two characteristic energy scales are shown in Fig. 1d [26][27][28][29] . crossed at q=0.75), but then reappears at P2, which is higher in energy near the G2 point.

Results
Interestingly, we find that the spectral enhancement is different between the symmetry related points q=0. 45 Supplementary 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 . 6 We note, however, a quantitative understanding of the energy and momentum dependent optical phonon enhancement may require theoretical calculations beyond the pure Kitaev model.
We then turn to the transverse acoustic phonon near G1. Figure 4a and b show the temperaturedependence of ′′( , ) 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, ( , ), 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. Figure 4e and 4f show the relative peak shift 0 ( )/ 0 (300 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 a-RuCl3 where the thermal Hall conductivity, kxy, 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.

Discussions
The discovery of temperature and energy dependent phonon softening provides important where the softening effect is expected to be significantly suppressed for ( ) ≫ 0.065 ! . 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 → 0 29 . This enhances Majorana-phonon coupling that yields a renormalization of the phonon dispersion below TK 28,29 .
To justify this conjecture, we extract the acoustic phonon velocity ;<~1 6 meV•Å (ℏ = 1), which is based on the room-temperature phonon dispersion shown in Fig. 2. In the isotropic limit 16  The observed acoustic phonon softening below 2 meV demonstrate a small energy scale in a-RuCl3 that strongly renormalize the acoustic phonon spectrum and hence may be responsible for the quantized thermal Hall effect.
Finally, we discuss the possibility of a magnon-phonon coupling. Below TN, a gapped magnon excitation between 2~7 meV was observed in a-RuCl3 by previous neutron studies 19,21,22,37 .
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 w =1 meV (see q3 in Fig. 4e and Supplementary Fig. 5), which is well below the magnon gap. Therefore, a magnonphonon coupling is unlikely giving rise to the observed acoustic phonon softening. However, the 8 magnon-phonon coupling may indeed present in a-RuCl3. As we show in Fig. 2, the P2 phonon energy is the same as the magnon energy near the M point 19,21,38 . Interestingly, the P2 phonon intensity at the M point is enhanced at 10 K~TN, supporting magnon-phonon coupling 46  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.
Spectral Intensity