  We consider the time-dependent transverse field Ising chain with
time-periodic perturbations. Without perturbations, this model is one of the
famous models that obeys the scaling in the adiabatic limit predicted by the
quantum Kibble-Zurek mechanism (QKZM). However, it is known that when
oscillations are added to the system, the non-perturbative contribution becomes
larger and the scaling may break down even if the perturbation is small.
Therefore, we analytically analyze the density of defects in the model and
discuss how much the oscillations affect the scaling. As a result, although the
non-perturbative contribution does not become zero in the adiabatic limit, the
scaling does not change from the prediction of the QKZM. This indicates that
the QKZM is robust to the perturbations.
