  An important and incompletely answered question is whether a closed quantum
system of many interacting particles can be localized by disorder. The time
evolution of simple (unentangled) initial states is studied numerically for a
system of interacting spinless fermions in one dimension described by the
random-field XXZ Hamiltonian. Interactions induce a dramatic change in the
propagation of entanglement and a smaller change in the propagation of
particles. For even weak interactions, when the system is thought to be in a
many-body localized phase, entanglement shows neither localized nor diffusive
behavior but grows without limit in an infinite system: interactions act as a
singular perturbation on the localized state with no interactions. The
significance for proposed atomic experiments is that local measurements will
show a large but nonthermal entropy in the many-body localized state. This
entropy develops slowly (approximately logarithmically) over a diverging time
scale as in glassy systems.
