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 (ξ) 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 ∼ξ×slogt, providing support for the phenomenology of many-body localized systems proposed by Huse and Oganesyan.