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 Neel state, we analyze the decay of antiferromagnetic order m(s)(t) and the growth of entanglement entropy S-ent(t) during unitary time evolution. Near the phase transition we find that m(s)(t) decays exponentially to its asymptotic value m(s)(infinity) not equal 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, m(s)(infinity) shows an exponential sensitivity on disorder with a critical exponent nu similar to 0.9. The entanglement entropy in the ergodic phase grows subballistically, S-ent(t) similar to t(alpha), alpha <= 1, with alpha 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.