  Measurements of thermal diffusivity in several insulators have been shown to
reach a Planckian bound on thermal transport that can be thought of as the
limit of validity of semiclassical phonon scattering. Beyond this regime, the
heat transport must be understood in terms of incoherent motion of the atoms
under strongly anharmonic interactions. In this work, we propose a model for
heat transport in a strongly anharmonic system where the thermal diffusivity
can be lower than the Planckian thermal diffusivity bound. Similar to the
materials which exhibit thermal diffusivity close to this bound, our scenario
involves complex unit cell with incoherent intra-cell dynamics. We derive a
general formalism to compute thermal conductivity in such cases with anharmonic
intra-cell dynamics coupled to nearly harmonic inter-cell coupling. Through
direct numerical simulation of the non-linear unit cell motion, we explicitly
show that our model allows sub-Planckian thermal diffusivity. We find that the
propagator of the acoustic phonons becomes incoherent throughout most of the
Brillouin zone in this limit. We expect these features to apply to more
realistic models of complex insulators showing sub-Planckian thermal
diffusivity, suggesting a multi-species generalization of the thermal
diffusivity bound that is similar to the viscosity bound in fluids.
