  A quantum quench is the simplest protocol to investigate nonequilibrium
many-body quantum dynamics. Previous studies on the entanglement properties of
quenched quantum many-body systems mainly focus on the growth of entanglement
entropy. Several rigorous results and phenomenological guiding principles have
been established, such as the no-faster-than-linear entanglement growth
generated by generic local Hamiltonians and the peculiar logarithmic growth for
many-body localized systems. However, little is known about the dynamical
behavior of the full entanglement spectrum, which is a refined character
closely related to the topological nature of the wave function. Here, we
establish a rigorous and universal result for the entanglement spectra of 1D
SPT systems evolving out of equilibrium. Our result is derived both for
free-fermion SPT systems and interacting ones. For free-fermion systems with AZ
symmetries, we prove that the single-particle entanglement gap after quenches
obeys essentially the same Lieb-Robinson bound as that on the equal-time
correlation, provided that there is no dynamical symmetry breaking. As a
notable byproduct, we obtain a new type of Lieb-Robinson velocity which is
related to the band dispersion with a complex wave number and reaches the
minimum as the maximal (relative) group velocity. Within the framework of
tensor networks, i.e., for SPT MPSs evolved by symmetric and trivial MPUs, we
also identify a Lieb-Robinson bound on the many-body entanglement gap for
general quenched interacting SPT systems. This result suggests high potential
of tensor-network approaches for exploring rigorous results on long-time
quantum dynamics. Influence of partial symmetry breaking, effects of disorder,
and the relaxation property in the long-time limit are also discussed. Our work
establishes a paradigm for exploring rigorous results of SPT systems out of
equilibrium.
