  Electron-phonon interactions play a key role in many branches of solid-state
physics. Here, our focus is on the transport properties of one-dimensional
systems, and we apply efficient real-time matrix-product state methods to
compute the optical and thermal conductivities of Holstein chains at finite
temperatures and filling. We validate our approach by comparison with
analytical results applicable to single polarons valid in the small polaron
limit. Our work provides a systematic study of contributions to the thermal
conductivity at finite frequencies and elucidates differences in the spectrum
compared to the optical conductivity, covering both the Luttinger-liquid and
charge-density-wave regimes. Finally, we demonstrate that our approach is
capable of extracting the DC conductivities as well. Beyond this first
application, several future extensions seem feasible, such as, the inclusion of
dispersive phonons, different types of local electron-phonon coupling, and a
systematic study of drag effects in this electron-phonon coupled system.
