Entanglement spreading in a many-body localized system

Motivated by the findings of logarithmic spreading of entanglement in a many-body localized system, we more closely examine the spreading of entanglement in the fully many-body localized phase, where all many-body eigenstates are localized. Performing full diagonalizations of an XXZ spin model with random longitudinal fields, we identify two factors contributing to the spreading rate: the localization length ($\xi$), which depends on the disorder strength, and the final value of entanglement per spin ($s_\infty$), which primarily depends on the initial state. We find that the entanglement entropy grows with time as $\sim \xi \times s_\infty \log t$, providing support for the phenomenology of many-body localized systems recently proposed by Huse and Oganesyan [arXiv:1305.4915v1].