  In a recent numerical study by Haldar et al. (Phys. Rev. X 11, 031062) it was
shown that signatures of proximate quantum critical points can be observed at
early and intermediate times after certain quantum quenches. Said work focused
mainly on the case of the axial next-nearest neighbour Ising (ANNNI) model.
Here we construct a simple time-dependent mean-field theory that allows us to
obtain a quantitatively accurate description of these quenches at short times,
which for reasons we explain remains a fair approximation at late times (with
some caveats). Our approach provides a simple framework for understanding the
reported numerical results as well as fundamental limitations on detecting
quantum critical points through quench dynamics. We moreover explain the origin
of the peculiar oscillatory behaviour seen in various observables as arising
from the formation of a long-lived bound state.
