Entanglement spreading in a many-body localized system

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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 (ξ), which depends on the disorder strength, and the final value of entanglement per spin (s∞), which primarily depends on the initial state. We find that the entanglement entropy grows with time as ξ× s∞ t, providing support for the phenomenology of many-body localized systems recently proposed by Huse and Oganesyan [arXiv:1305.4915v1].