Mott-glass phase induced by long-range correlated disorder in a one-dimensional Bose gas

We determine the phase diagram of a one-dimensional Bose gas in the presence of disorder with short- and long-range correlations, the latter decaying with distance as 1/|x|1+σ. When σ<0, the Berezinskii-Kosterlitz-Thouless transition between the superfluid and the localized phase is driven by the long-range correlations and the Luttinger parameter K takes the critical value Kc(σ)=3/2-σ/2. The localized phase is a Bose glass for σ>σc=3-π²/3 -0.289868, and a Mott glass -- characterized by a vanishing compressibility and a gapless conductivity -- when σ<σc. Our conclusions, based on the nonperturbative functional renormalization group and perturbative renormalization group, are confirmed by the study of the case σ=-1, corresponding to a perfectly correlated disorder in space, where the model is exactly solvable in the semiclassical limit K→ 0⁺.