Abstract
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 $(\ensuremath{\xi})$ of the effective interaction between localized pseudospins (l-bits), which depends on the disorder strength, and on the final value of entanglement per spin $({s}_{\ensuremath{\infty}})$, which primarily depends on the initial state. The entanglement entropy thus grows with time as $\ensuremath{\sim}\ensuremath{\xi}\ifmmode\times\else\texttimes\fi{}{s}_{\ensuremath{\infty}}logt$, providing support for the phenomenology of many-body localized systems proposed by Huse and Oganesyan.
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