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We address the issue of the validity of linear response theory for a closedquantum system subject to a periodic external driving. Linear response theory(LRT) predicts energy absorption at frequencies of the external driving wherethe 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 alattice subject to an extensive perturbation, this approximation should beexpected to be valid only up to a time $t^*$ depending on the strength of thedriving, beyond which the true coherent Schr\"odinger evolution departs fromthe linear response prediction and the system stops absorbing energy form thedriving. We exemplify this phenomenon in detail with the example of a quantumIsing chain subject to a time-periodic modulation of the transverse field, bycomparing an exact Floquet analysis with the standard results of LRT. In thiscontext, we also show that if the perturbation is just local, the system isexpected in the thermodynamic limit to keep absorbing energy, and LRT works atall times. We finally argue more generally the validity of the scenariopresented for closed quantum many-body lattice systems with a bound on theenergy-per-site spectrum, discussing the experimental relevance of our findingsin the context of cold atoms in optical lattices and ultra-fast spectroscopyexperiments.

Linear response as a singular limit for a periodically driven closed quantum system