  We address the issue of the validity of linear response theory for a closed
quantum system subject to a periodic external driving. Linear response theory
(LRT) predicts energy absorption at frequencies of the external driving where
the imaginary part of the appropriate response function is different from zero.
Here we show that, for a fairly general non-linear many-body system on a
lattice subject to an extensive perturbation, this approximation should be
expected to be valid only up to a time $t^*$ depending on the strength of the
driving, beyond which the true coherent Schr\"odinger evolution departs from
the linear response prediction and the system stops absorbing energy form the
driving. We exemplify this phenomenon in detail with the example of a quantum
Ising chain subject to a time-periodic modulation of the transverse field, by
comparing an exact Floquet analysis with the standard results of LRT. In this
context, we also show that if the perturbation is just local, the system is
expected in the thermodynamic limit to keep absorbing energy, and LRT works at
all times. We finally argue more generally the validity of the scenario
presented for closed quantum many-body lattice systems with a bound on the
energy-per-site spectrum, discussing the experimental relevance of our findings
in the context of cold atoms in optical lattices and ultra-fast spectroscopy
experiments.
