  We determine the phase diagram and the momentum distribution for a
one-dimensional Bose gas with repulsive short range interactions in the
presence of a two-color lattice potential, with incommensurate ratio among the
respective wave lengths, by using a combined numerical (DMRG) and analytical
(bosonization) analysis. The system displays a delocalized (superfluid) phase
at small values of the intensity of the secondary lattice V2 and a localized
(Bose glass-like) phase at larger intensity V2. We analyze the localization
transition as a function of the height V2 beyond the known limits of free and
hard-core bosons. We find that weak repulsive interactions unfavor the
localized phase i. e. they increase the critical value of V2 at which
localization occurs. In the case of integer filling of the primary lattice, the
phase diagram at fixed density displays, in addition to a transition from a
superfluid to a Bose glass phase, a transition to a Mott-insulating state for
not too large V2 and large repulsion. We also analyze the emergence of a
Bose-glass phase by looking at the evolution of the Mott-insulator lobes when
increasing V2. The Mott lobes shrink and disappear above a critical value of
V2. Finally, we characterize the superfluid phase by the momentum distribution,
and show that it displays a power-law decay at small momenta typical of
Luttinger liquids, with an exponent depending on the combined effect of the
interactions and of the secondary lattice. In addition, we observe two side
peaks which are due to the diffraction of the Bose gas by the second lattice.
This latter feature could be observed in current experiments as characteristics
of pseudo-random Bose systems.

Other Condensed Matter;Statistical Mechanics