On the Planckian bound for heat diffusion in insulators

High temperature thermal transport in insulators has been conjectured to be subject to a Planckian bound on the transport lifetime $\tau \gtrsim \tau_\text{Pl} \equiv \hbar/(k_B T)$, despite phonon dynamics being entirely classical at these temperatures. We argue that this Planckian bound is due to a quantum mechanical bound on the sound velocity: $v_s < v_M$. The `melting velocity' $v_M$ is defined in terms of the melting temperature of the crystal, the interatomic spacing and Planck's constant. We show that for several classes of insulating crystals, both simple and complex, $\tau/\tau_\text{Pl} \approx v_M/v_s$ at high temperatures. The velocity bound therefore implies the Planckian bound.