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We formulate a dynamical real space renormalization group approach to
describe the time evolution of a random spin-1/2 chain, or interacting
fermions, initialized in a state with fixed particle positions. Within
this approach we identify a many-body localized state of the chain as a
dynamical infinite randomness fixed point. Near this fixed point our
method becomes asymptotically exact, allowing analytic calculation of time
dependent quantities. In particular we explain the striking universal
features in the growth of the entanglement seen in recent numerical
simulations: unbounded logarithmic growth delayed by a time inversely
proportional to the interaction strength. The particle number fluctuations
by contrast exhibit much slower growth as log(log(t)) indicating blocked
particle transport. Lack of true thermalization in the long time limit is
attributed to an infinite set of approximate integrals of motion revealed
in the course of the RG flow, which become asymptotically exact
conservation laws at the fixed point. Hence we identify the many-body
localized state with an emergent generalized Gibbs ensemble.
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