Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology

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 general result for the entanglement spectra of one-dimensional symmetry-protected topological (SPT) systems evolving out of equilibrium. We prove that the single-particle entanglement gap in particle-hole symmetric topological insulators after quenches obeys essentially the same Lieb-Robinson bound as that on the equal-time correlation. 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 matrix-product states evolved by symmetric and trivial matrix-product unitaries, we identify a Lieb-Robinson bound on the many-body entanglement gap. This result suggests high potential of tensor-network approaches for exploring rigorous results on long-time quantum dynamics. The effects of disorder and the relaxation property in the long-time limit are discussed. Our work establishes a paradigm for exploring rigorous results of SPT systems out of equilibrium.