  We study the energy absorption in real time of a disordered quantum spin
chain subjected to coherent monochromatic periodic driving. We determine
characteristic fingerprints of the well-known ergodic (Floquet-ETH for slow
driving/weak disorder) and many-body localized (Floquet-MBL for fast
driving/strong disorder) phases. In addition, we identify an intermediate
regime, where the energy density of the system -- unlike the entanglement
entropy a local and bounded observable -- grows logarithmically slowly over a
very large time window.
