The quasi-periodic Bose-Hubbard model and localization in   one-dimensional cold atomic gases
  We compute the phase diagram of the one-dimensional Bose-Hubbard model with a quasi-periodic potential by means of the density-matrix renormalization group technique. This model describes the physics of cold atoms loaded in an optical lattice in the presence of a superlattice potential whose wave length is incommensurate with the main lattice wave length. After discussing the conditions under which the model can be realized experimentally, the study of the density vs. the chemical potential curves for a non-trapped system unveils the existence of gapped phases at incommensurate densities interpreted as incommensurate charge-density wave phases. Furthermore, a localization transition is known to occur above a critical value of the potential depth V_2 in the case of free and hard-core bosons. We extend these results to soft-core bosons for which the phase diagrams at fixed densities display new features compared with the phase diagrams known for random box distribution disorder. In particular, a direct transition from the superfluid phase to the Mott insulating phase is found at finite V_2. Evidence for reentrances of the superfluid phase upon increasing interactions is presented. We finally comment on different ways to probe the emergent quantum phases and most importantly, the existence of a critical value for the localization transition. The later feature can be investigated by looking at the expansion of the cloud after releasing the trap. 
G. Roux, T. Barthel, I. P. McCulloch, C. Kollath, U. Schollwoeck, T.   Giamarchi