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 $1\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^{1+\sigma }$. 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 $K_c(\sigma )=3\mathrm{/}2-\sigma \mathrm{/}2$. The localized phase is a Bose glass for $\sigma >{\sigma }_c=3-{\pi }^2\mathrm{/}3\simeq -0.289868$, 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 $K\to 0^{+}$.