  We consider a one-dimensional interacting spinless fermion model, which
displays the well-known Luttinger liquid (LL) to charge density wave (CDW)
transition as a function of the ratio between the strength of the interaction,
$U$, and the hopping, $J$. We subject this system to a spatially uniform drive
which is ramped up over a finite time interval and becomes time-periodic in the
long time limit. We show that by using a density matrix renormalization group
(DMRG) approach formulated for infinite system sizes, we can access the
large-time limit even when the drive induces finite heating. When both the
initial and long-time states are in the gapless (LL) phase, the final state has
power law correlations for all ramp speeds. However, when the initial and final
state are gapped (CDW phase), we find a pseudothermal state with an effective
temperature that depends on the ramp rate, both for the Magnus regime in which
the drive frequency is very large compared to other scales in the system and in
the opposite limit where the drive frequency is less than the gap. Remarkably,
quantum defects (instantons) appear when the drive tunes the system through the
quantum critical point, in a realization of the Kibble-Zurek mechanism.
