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The recent report of a half-quantized thermal Hall effect in the Kitaevmaterial $\alpha$-RuCl$_3$ has sparked a strong debate on whether it isgenerated by Majorana fermion edge currents or whether other more conventionalmechanisms involving magnons or phonons are at its origin. A more directevidence for Majorana fermions which could be expected to arise from acontribution to the longitudinal heat conductivity $\kappa_{xx}$ at$T\rightarrow0$ is elusive due to a very complex magnetic field dependence of$\kappa_{xx}$. Here, we report very low temperature (below 1~K) thermalconductivity ($\kappa$) of another candidate Kitaev material,Na$_2$Co$_2$TeO$_6$. The application of a magnetic field along differentprincipal axes of the crystal reveals a strong directional-dependentmagnetic-field ($\bf B$) impact on $\kappa$. We show that no evidence formobile quasiparticles except phonons can be concluded at any field from 0~T tothe field polarized state. In particular, severely scattered phonon transportis observed across the $B-T$ phase diagram, which is attributed to prominentmagnetic fluctuations. Cascades of phase transitions are uncovered for all $\bfB$ directions by probing the strength of magnetic fluctuations via a preciserecord of $\kappa$($B$). Our results thus rule out recent proposals foritinerant magnetic excitations in Na$_2$Co$_2$TeO$_6$, and emphasise theimportance of discriminating true spin liquid transport properties fromscattered phonons in candidate materials.

Phonon thermal transport shaped by strong spin-phonon scattering in a Kitaev material Na$_2$Co$_2$TeO$_6$