  The properties of prototypical examples of one-dimensional fermionic systems
undergoing a sudden quantum quench from a gapless state to a (partially) gapped
state are analyzed. By means of a Generalized Gibbs Ensemble analysis or by
numerical solutions in the interacting cases, we observe an anomalous,
non-monotonic response of steady state correlation functions as a function of
the strength of the mechanism opening the gap. In order to interpret this
result, we calculate the full dynamical evolution of these correlation
functions, which shows a freezing of the propagation of the quench information
(light cone) for large quenches. We argue that this freezing is responsible for
the non-monotonous behaviour of observables. In continuum non-interacting
models, this freezing can be traced back to a Klein-Gordon equation in the
presence of a source term. We conclude by arguing in favour of the robustness
of the phenomenon in the cases of non-sudden quenches and higher
dimensionality.
