Microscopic dynamics in the Holstein model at high-temperature: Absence of diffusion

A single itinerant electron in the Holstein model is known to have a large conductivity at low temperatures as the electron is scattered by dilute thermally-excited phonons. At high temperatures, some theories suggest the dynamics are described by simple Boltzmann diffusion since there are many thermally excited phonons. Here, we investigate the dynamics of an itinerant particle of the Holstein model at high temperature. Our results suggest that the microscopic dynamics is not diffusion in some parameter regimes. In one dimension, the particle travels in a constant direction over time, which suggests an infinite conductivity. In higher dimensions, the particle follows a closed trajectory, a type of localization, and the trajectories are fractal.