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We study dynamics of entanglement in the thermalization process of anisolated quantum many-body system. We propose a simple setup for measuring thepropagation speed of entanglement entropy (EE) in numerical simulations andapply it to the integrable/non-integrable spin models in 1D - the transverseIsing (TI) model, the chaotic Ising (CI) model, and the extended chaotic Ising(ECI) model. We find that two distinct time-scales $t^\ast$ and $t_{\rm{diff}}$ arise in the dynamics of EE in the thermalization process: the formerrepresents the time-scale for the saturation of EE and the latter characterizesspreading of EE over the entire system. Evaluating the propagation speed ofentanglement from $t_{\rm diff}$, we find that entanglement propagatesballistically with a constant velocity irrespective of the integrability of themodel. The propagation speed of entanglement is found to coincide with themaximum group velocity of quasi-particle excitations in the TI model. We alsoevaluate the propagation speed of entanglement by mutual information and findthe characteristic time-scale $t_{\mathrm{MI}}$. We show that the propagationspeeds of entanglement evaluated by $t_{\mathrm{MI}}$ and $t_{\rm {diff}}$agree well. We discuss the condition for thermalization based on the numericalresults and propose that scrambling of the entire system has to take placebefore saturation of EE for thermalization.

Entanglement propagation in thermalization of an isolated quantum system