On the spreading rate of entanglement in a many-body localized quantum spin chain

Although the many-body localized phase does not allow the transport of local observables, the unbounded logarithmic growth of bipartite entanglement entropy, $S$, has recently been observed (Bardarson et al., Phys. Rev. Lett. {\bf 109}, 017202 (2012)). We aim to elucidate the origin of this logarithmic growth through exact diagonalization methods, analyzing an XXZ spin model with random longitudinal fields. Based on a proposed phenomenology of entanglement spreading (Huse and Oganesyan, arXiv:1305.4915v1), we connect the rate of entanglement spreading with the localization length ($\xi$) of the system and the saturated entanglement entropy per spin ($s_{\infty}$). We find that the time dependence of the entanglement spreading takes the form $S\sim \xi s_{\infty} \log t$.