  Motivated by recent bounds for charge diffusion in critical matter, we
investigate the question: What sets the scale for charge diffusion in a
scale-invariant system? To make our statements precise, we analyze the
diffusion pole in an exactly solvable model for a Mott transition in the
presence of a long-range interaction term. To achieve scale invariance, we
limit our discussion to the flat-band regime. We find in this limit that the
diffusion pole which would normally obtain at finite energy is pushed to zero
energy resulting in a vanishing of the diffusion constant. This occurs even in
the presence of interactions in certain limits, indicating the robustness of
this result to the inclusion of a scale in the problem. Consequently,
scale-invariance precludes any reasonable definition of the diffusion constant.
Nonetheless, we do find that a scale can be defined, all be it, irrelevant to
diffusion, which is the product of the squared band velocity and the density of
states.
