  It has been known for decades that thermal conductivity of insulating
crystals becomes proportional to the inverse of temperature when the latter is
comparable to or higher than the Debye temperature. This behavior has been
understood as resulting from Umklapp scattering among phonons. We put under
scrutiny the magnitude of the thermal diffusion constant in this regime and
find that it does not fall below a threshold set by the square of sound
velocity times the Planckian time ($\tau_p=\hbar/k_BT$). The conclusion, based
on scrutinizing the ratio in cubic crystals with high thermal resistivity,
appears to hold even in glasses where Umklapp events are not conceivable.
Explaining this boundary, reminiscent of a recently-noticed limit for charge
transport in metals, is a challenge to theory.
