Sub-Planckian electron diffusion in a model of ultrastrong electron-phonon coupling

The conjecture of the Planckian bound on transport implies a temperature-dependent lower bound on the electron diffusion constant. Motivated by this conjecture, we study a model of electron diffusion in the limit of ultrastrong electron-phonon coupling. In this limit, the phonons can be described by a semiclassical potential energy landscape that fluctuates slowly in time, while electrons are instantaneously equilibrated to lie in local minima of the potential. We study this model for both gapped (optical) and gapless (acoustic) phonon modes. We find diffusive behavior with a diffusion constant that scales as the inverse temperature, as in the naive Planckian bound, but with a small prefactor that can in principle give deeply sub-Planckian diffusion.