Mean-field phase diagram of the one-dimensional Bose gas in a disorder potential

We study the quantum phase transition of the 1D weakly interacting Bose gas in the presence of disorder. We characterize the phase transition as a function of disorder and interaction strengths, by inspecting the long-range behavior of the one-body density matrix as well as the drop in the superfluid fraction. We focus on the properties of the low-energy Bogoliubov excitations that drive the phase transition and find that the transition to the insulator state is marked by a diverging density of states and a localization length that diverges as a power-law with power 1. We draw the phase diagram and we observe that the boundary between the superfluid and the insulator phase is characterized by two different algebraic relations. These can be explained analytically by considering the limiting cases of zero and infinite disorder correlation length.