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 $1/|x|^{1+\sigma}$. 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 $K_c(\sigma)=3/2-\sigma/2$. 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 $K\to 0^+$.