Many-body localization in infinite chains

We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin-$1/2$ Heisenberg chains with binary disorder. Starting from the N\'eel state, we analyze the decay of antiferromagnetic order mₛ(t) and the growth of entanglement entropy Sₑₙₜ(t) during unitary time evolution. Near the phase transition we find that mₛ(t) decays exponentially to its asymptotic value mₛ(∞)≠0 in the localized phase while the data are consistent with a power-law decay at long times in the ergodic phase. In the localized phase, mₛ(∞) shows an exponential sensitivity on disorder with a critical exponent ν~0.9. The entanglement entropy in the ergodic phase grows subballistically, Sₑₙₜ(t)~t^α, α≤1, with α varying continuously as a function of disorder. Exact diagonalizations for small systems, on the other hand, do not show a clear scaling with system size and attempts to determine the phase boundary from these data seem to overestimate the extent of the ergodic phase.