Revealing the origin of this universal THE in crystals requires a comprehensive understanding of the thermal Hall signal in elemental semiconductors Si and Ge. Even though the scenario of ionic vibrations36 may explain the thermal Hall phenomena in SiO2 and MgO, it falls short in explaining the case of elemental Si and Ge. For another elemental crystal BP, which also exhibits large *κ*xy, it has been suggested that the unevenly distributed positive and negative charges in BP could pave the way for coupling between phonons and the magnetic field. However, the high symmetry of Si and Ge crystals poses challenge in forming such uneven charge distributions. Note again, the data of BP also exhibit the same scaling law indicating a common origin with our compounds.

Considering the high quality of our non-magnetic single crystals, especially the high-purity Si and Ge (with purity higher than 4N), extrinsic mechanisms for phonon THE such as coupling to the magnetic environment, impurity scattering, and AFD domains are unlikely. In this context, an intrinsic mechanism, i.e. the direct influence of Lorentz forces on the coupled phonon modes during phonon propagation, should exist for this universal phonon THE. Ceresoli *et al*. conducted calculations on the model of hydrogen molecules and revealed that the magnetic field effect survives for the relative motion of the two atoms, despite perfect screening for center-of-mass motion38. Previous studies have proposed Raman-type interactions between the magnetic field and phonons, leading to an intrinsic THE in non-magnetic band insulators39-41. However, in long-wavelength limit, the coupling of acoustic phonons to the magnetic field occurs through higher order gradient terms, resulting in an intrinsic effect much smaller than the measured values. In particular, the calculation of intrinsic phonon thermal Hall conductivity for Si gave a positive *κ*xy ~ 10-6 W m−1 K−1 at 300 K, which decreases with decreasing temperature39, far away from our experimental result. Therefore, the microscopic mechanism for this universal phonon THE with a scaling law |*κ*xy| ~ *κ*xx2 remains an open question. New theoretical approach and calculations are highly desired to clarify this fundamental physics of phonons in magnetic field.

Since the direct coupling between atom vibrations and the field gives this universal phonon THE, it explains why there exists planar phonon THE. In fact, previously the observation of planar THE in α-RuCl3 (refs. 12,13,42), Na2Co2TeO6 (ref. 23) and cuprates22 was quite puzzling. Now this planar THE is also observed in our STO and SiO2 samples. Furthermore, the planar *κ*xy curves of SiO2 follow the same scaling law (see Supplemental Fig. S8), confirming that the planar THE originates from the same source as the conventional one. In three-dimensional crystals, phonon vibration contains two transverse waves and one longitudinal wave, allowing for the convenient coupling of a magnetic field in any direction to affect phonon vibrations. This ultimately manifests in the THE through coupling between phonon modes in any direction, including the planar configurations.

Starting from this universal phonon THE in crystals, all previous interpretations of THE in magnetic or non-magnetic materials need to be reconsidered. It is a conceptional change. Previously, people were trying hard to find extrinsic mechanisms for phonons to generate the THE, such as impurity scattering or interaction with the magnetic environment, since phonon itself is usually believed not affected by magnetic field. Now one must accept that phonon THE is a universal intrinsic property of any crystal. Bearing this in mind first, then one may consider various factors which will affect this phonon THE, such as magnetic environment, impurity, and doped carriers.

In ferromagnetic insulators such as Lu2V2O7 (ref. 1), Fe2Mo3O8 (ref. 8), VI3 (ref. 3) and CrI3 (ref. 20), the field-dependent *κ*xy tracks the magnetization curve. This means that the spin polarization affects the phonon *κ*xy significantly. We notice that *κ*xy of CrI3 is negative and only changes slightly at *T*c, roughly following the *κ*xx curve20. On the contrary, *κ*xy of Lu2V2O7, Fe2Mo3O8 and VI3 is positive, and changes abruptly at *T*c, not following the *κ*xx curve1,3,8. More works are needed to clarify the effect of spin polarization on the phonon THE.

In the quantum spin-liquid candidate *α*-RuCl3, initially half-integer quantized anomalous THE was reported in both conventional and planar configurations, which was argued as evidence for Majorana fermions5,12. However, this half-integer quantized anomalous THE is not robust, since it cannot be reproduced by some other groups17,42. Based on current work, we know that there must exist an intrinsic phonon THE in *α*-RuCl3, as reported in ref. 17 in a completely conventional configuration. Therefore, the observed field dependence of *κ*xy in *α*-RuCl3 around the critical in-plane field 7.5 T may only reflect the effect of magnetic fluctuations on the phonon THE, thus no need for introducing exotic Majorana fermions.

In cuprates, starting from this universal phonon THE, the observation of large *κ*xy in La2CuO4 (ref. 7), Na2CuO4 and Sr2CuO2Cl2 (ref. 15) is not surprising now. While the energy gap of magnons in these AF insulators is large, short-range magnetic fluctuations may still affect this phonon THE. The main question in cuprates is why the phonon *κ*xy vanishes at high doping14. It is likely that the increasing carriers, as scatterers of phonon, gradually diminish the phonon THE. In another system Sr2Ir1-*x*RhxO4, the intrinsic phonon *κ*xy is detected in the AF parent compound Sr2IrO4 (ref. 16). The huge change in *κ*xy emerging from element substitution on the spin-carrying site shows that the phonon THE is strongly affected by spin-phonon coupling and impurity scattering in this system16. All above mentioned factors will affect the phonon THE, therefore the scaling law |*κ*xy| ~ *κ*xx2 may no longer hold. Indeed, as seen in Supplementary Fig. S9, most magnetic insulators do not obey this scaling law.

In summary, we have experimentally discovered a universal phonon THE in crystals, characterized by a scaling law of |*κ*xy| ~ *κ*xx2, by measuring a series of non-magnetic insulators and intrinsic semiconductors, including STO, SiO2, MgO, MgAl2O4, Si and Ge. It shows that phonons do not require extrinsic factors to generate THE. This is counter-intuitive since phonon itself is usually believed not affected by field. This completely changes the starting point for interpreting the THE results in magnetic or non-magnetic materials. The nature behind this fundamental physics of phonons in magnetic field should relate to the direct coupling of atom vibrations to the magnetic field, which require further theoretical investigations.