"The superfluid-insulator phase diagram of 1-D weakly interacting bosons in a
disorder potential"
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A one-dimensional system of non-interacting particles in presence of disorder is
always in an insulating state at zero temperature. Interactions can induce a
quantum phase transition to a superfluid state, characterized by quasi
long-range order. This Bose-glass to superfluid transition has been the subject
of intense theoretical studies and of several recent experiments carried out on
ultracold atomic clouds.

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I will present a theoretical study of the weakly interacting Bose gas at zero
temperature in a disorder potential with finite spatial correlation length. To
leading order in the interaction strength, the boundary between the superfluid
and insulating phases can be determined from a symmetry-breaking approach: the
extended Bogolyubov model. This approach accounts correctly for diverging low
energy phase fluctuations - that occur in a low-dimensional system and are
ultimately responsible for the suppression of superfluidity. In this context,
the phase diagram on the interaction-disorder plane (U,D) can be characterized
by inspecting the long-range behaviour of the one-body density matrix as well as
the drop in superfluid fraction. It turns out in particular that the phase
boundary between the two phases follows two different power laws on the phase
diagram in the white-noise and Thomas-Fermi limits respectively. This feature is
peculiar of a spatially correlated disorder and is expected to govern the
behaviour of disordered ultracold atomic clouds in current experiments.

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The in-situ density profile is perhaps the feature of an ultracold atomic cloud
that can be most easily measured in an experiment. I will show that a direct
link exists between the fragmentation of the density profile and the occurrence
of the phase transition. This link is given by the probability distribution of
the gas density. In particular, the appearance of a superfluid fraction
coincides with a vanishing probability distribution in the limit of zero
density, namely with the disappearance of fragmentation. This analysis sets the
intuitive relation between fragmentation and insulating behaviour into a
rigorous framework, and opens the way to the experimental detection of the phase
transition.