  We investigate the role of topology in the space-time scaling limit of
quantum quench dynamics, where both time and system size tend to infinity at a
constant ratio. There, while the standard topological characterization relying
on local unitary transformations becomes ill defined, we show how a different
dynamical notion of topology naturally arises through a dynamical winding
number encoding the linear response of the Berry phase to a magnetic flux.
Specifically, we find that the presence of a locally invisible constant
magnetic flux is revealed by a dynamical staircase behavior of the Berry phase,
whose topologically quantized plateaus characterize the space-time scaling
limit of a quenched Rice-Mele model. These jumps in the Berry phase are also
shown to be related to the interband elements of the DC current operator. We
outline possible experimental platforms for observing the predicted phenomena
in finite systems.

Quantum Gases;Mesoscale and Nanoscale Physics;Quantum Physics