Entanglement spreading in a many-body localized system

Motivated by the findings of logarithmic spreading of entanglement in a many-body localized system, we identify and untangle two factors contributing to 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 demonstrate a linear dependence of the spreading rate on the decay length () of the effective interaction between localized pseudospins (l-bits), which depends on the disorder strength, and on the final value of entanglement per spin (s_), which primarily depends on the initial state. The entanglement entropy thus grows with time as s_logt, providing support for the phenomenology of many-body localized systems proposed by Huse and Oganesyan.