  Motivated by recent experimental progress in the study of quantum systems far
from equilibrium, we investigate the relation between several dynamical
signatures of topology in the coherent time-evolution after a quantum quench.
Specifically, we study the conditions for the appearance of entanglement
spectrum crossings, dynamical quantum phase transitions, and dynamical Chern
numbers. For non-interacting models, we show that in general there is no direct
relation between these three quantities. Instead, we relate the presence of
level crossings in the entanglement spectrum to localized boundary modes that
may not be of topological origin in the conventional sense. Finally, we
investigate how interactions influence the presence of entanglement spectrum
crossings and dynamical quantum phase transitions, by means of time-dependent
density matrix renormalization group simulations.
