  Topological boundary modes can occur at the spatial interface between a
topological and gapped trivial phase and exhibit a wavefunction that
exponentially decays in the gap. Here we argue that this intuition fails for a
temporal boundary between a prequench topological phase that possess
topological boundary eigenstates and a postquench gapped trivial phase that
does not possess any eigenstates in its gap. In particular, we find that
characteristics of states (e.g. probability density) prepared in a
topologically non-trivial system can persist long after it is quenched into a
gapped trivial phase with spatial profiles that appear frozen over long times
postquench. After this near-stationary window, topological boundary mode
profiles decay albeit, slowly in a power-law fashion. This behavior highlights
the unusual features of nonequilibrium protocols enabling quenches to extend
and control localized states of both topological and non-topological origins.
