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 $1/{\left|x\right|}^{1+\sigma }$. When $\sigma <0$, 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 $K_c\left(\sigma \right)=3/2-\sigma /2$. The localized phase is a Bose glass for $\sigma >{\sigma }_c=3-{\pi }^2/3\simeq -0.289868$, and a Mott glass—characterized by a vanishing compressibility and a gapless conductivity—when $\sigma <{\sigma }_c$. Our conclusions, based on the nonperturbative functional renormalization group and perturbative renormalization group, are confirmed by the study of the case $\sigma =-1$, corresponding to a perfectly correlated disorder in space, where the model is exactly solvable in the semiclassical limit $K\to 0^{+}$.

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• Bose gases
• 1-dimensional systems
• Disordered systems
• Quantum fluids & solids
• Functional renormalization group