  We investigate many-body dynamics in a one-dimensional interacting
periodically driven system, based on a partially filled version of Thouless's
topologically quantized adiabatic pump. The corresponding single-particle
Floquet bands are chiral, with the Floquet spectrum realizing nontrivial cycles
around the quasienergy Brillouin zone. For generic filling, with either bosons
or fermions, the system is gapless and is expected to rapidly absorb energy
from the driving field. We identify parameter regimes where scattering between
Floquet bands of opposite chirality is exponentially suppressed, opening a long
time window in which the system prethermalizes to an infinite-temperature state
restricted to a single Floquet band. In this quasi-steady state, the
time-averaged current takes a universal value determined solely by the density
of particles and the topological winding number of the populated Floquet band.
This remarkable behavior may be readily studied experimentally in recently
developed cold atom systems.
