Superfluid - Bose-Glass Transition in Weakly Disordered Commensurate One-Dimensional System

We study the effect of commensurability (integer filling factor) on the superfluid (SF) - Bose-glass (BG) transition in a one-dimensional disordered system in the limit of weak disorder, when the effect is most pronounced and, on the other hand, may be traced via the renormalization-group analysis. The equation for the SF-BG phase boundary demonstrating the effect of disorder-stimulated superfluidity implies that the strength of disorder sufficient to restore superfluidity from Mott insulator (MI) is much larger than that enough to turn MI into BG. Thus we provide an explicit proof of the fact that at arbitrarily small disorder the SF and MI phases are always separated by BG.